ct12005-2010

Faculty of Actuaries Institute of Actuaries

EXAMINATION

6 April 2005 (am)

Subject CT1

Financial Mathematics Core Technical

Time allowed: Three hours

INSTRUCTIONS TO THE CANDIDATE

1. Enter all the candidate and examination details as requested on the front of your answer booklet.

2. You must not start writing your answers in the booklet until instructed to do so by the supervisor.

3. Mark allocations are shown in brackets.

4. Attempt all 11 questions, beginning your answer to each question on a separate sheet.

5. Candidates should show calculations where this is appropriate.

Graph paper is not required for this paper.

AT THE END OF THE EXAMINATION

Hand in BOTH your answer booklet, with any additional sheets firmly attached, and this question paper.

In addition to this paper you should have available the 2002 edition of the Formulae and Tables and your own electronic calculator.

Faculty of Actuaries CT1 A2005 Institute of Actuaries

CT1 A2005 2

1 A bond is priced at £95 per £100 nominal, has a coupon rate of 5% per annum payable half-yearly, and has an outstanding term of five years.

An investor holds a short position in a forward contract on £1 million nominal of this bond, with a delivery price of £98 per £100 nominal and maturity in exactly one year, immediately following the coupon payment then due.

The continuously compounded risk-free rates of interest for terms of six months and one year are 4.6% per annum and 5.2% per annum, respectively.

Calculate the value of this forward contract to the investor assuming no arbitrage. [5]

2 An investment fund had a market value of £2.2 million on 31 December 2001 and £4.2 million on 31 December 2004. It had received a net cashflow of £1.44 million on 31 December 2003.

The money weighted rate of return and the time weighted rate of return for the period from 31 December 2001 to 31 December 2004 are equal (to two decimal places).

Calculate the market value of the fund immediately before the net cashflow on 31 December 2003. [7]

3 A computer manufacturer is to develop a new chip to be produced from 1 January 2008 until 31 December 2020. Development begins on 1 January 2006. The cost of development comprises £9 million payable on 1 January 2006 and £12 million payable continuously during 2007.

From 1 January 2008 the chip will be ready for production and it is assumed that income will be received half yearly in arrear at a rate of £5 million per annum.

(i) Calculate the discounted payback period at an effective rate of interest of 9% per annum. [6]

(ii) Without doing any further calculations, explain whether the discounted payback period would be greater than, less than or equal to that given in part (i) if the effective interest rate were substantially greater than 9% per annum.

[2] [Total 8]

CT1 A2005 3 PLEASE TURN OVER

4 The force of interest, ( )t , is a function of time and at any time t (measured in years) is given by

0.07 0.005 for 8( )

0.06 for 8

t tt

t

(i) Calculate the accumulation at time t = 10 of £500 invested at time t = 0. [3]

(ii) Calculate the present value at time t = 0 of a continuous payment stream at the

rate of 0.1£200 te paid from t = 10 to t = 18. [5] [Total 8]

5 A university student receives a 3-year sponsorship grant. The payments under the grant are as follows:

Year 1 £5,000 per annum paid continuously. Year 2 £5,000 per annum paid monthly in advance. Year 3 £5,000 per annum paid half yearly in advance.

Calculate the total present value of these payments at the beginning of the first year using a rate of interest of 8% per annum convertible quarterly. [8]

6 At time t = 0 an investor purchased an annuity-certain which paid her £10,000 per annum annually in arrear for three years. The purchase price paid by the investor was £25,000.

The value of the retail price index at various times was as shown in the table below:

Time t (years): t = 0 t = 1 t = 2 t = 3 Retail price index: 170.7 183.3 191.0 200.9

(i) Calculate, to the nearest 0.1%, the following effective rates of return per annum achieved by the investor from her investment in the annuity:

(a) the real rate of return; and (b) the money rate of return

[7]

(ii) By considering the average rate of inflation over the three-year period, explain the relationship between your answers in (a) and (b) of (i). [2]

[Total 9]

CT1 A2005 4

7 A loan of nominal amount £100,000 is to be issued bearing coupons payable quarterly in arrear at a rate of 5% per annum. Capital is to be redeemed at 103 on a single coupon date between 15 and 20 years after the date of issue, inclusive. The date of redemption is at the option of the borrower.

An investor who is liable to income tax at 20% and capital gains tax of 25% wishes to purchase the entire loan at the date of issue. Calculate the price which the investor should pay to ensure a net effective yield of at least 4% per annum. [9]

8 A small insurance fund has liabilities of £4 million due in 19 years time and £6 million in 21 years time. The manager of the fund has sold the assets previously held and is creating a new portfolio by investing in the zero-coupon bond market. The manager is able to buy zero-coupon bonds for whatever term he requires and has adequate monies at his disposal.

(i) Explain whether it is possible for the manager to immunise the fund against small changes in the rate of interest by purchasing a single zero-coupon bond.

[2]

(ii) In fact, the manager purchases two zero-coupon bonds, one paying £3.43 million in 15 years time and the other paying £7.12 million in 25 years time. The current interest rate is 7% per annum effective.

Investigate whether the insurance fund satisfies the necessary conditions to be immunised against small changes in the rate of interest.

[8] [Total 10]

9 The one-year forward rate of interest at time t = 1 year is 5% per annum effective.

The gross redemption yield of a two-year fixed interest stock issued at time t = 0 which pays coupons of 3% per annum annually in arrear and is redeemed at 102 is 5.5% per annum effective.

The issue price at time t = 0 of a three-year fixed interest stock bearing coupons of 10% per annum payable annually in arrear and redeemed at par is £108.9 per £100 nominal.

(i) Calculate the one-year spot rate per annum effective at time t = 0. [4]

(ii) Calculate the one-year forward rate per annum effective at time t = 2 years. [3]

(iii) Calculate the two-year par yield at time t = 0. [3] [Total 10]


10 (i) In any year, the interest rate per annum effective on monies invested with a given bank has mean value j and standard deviation s and is independent of the interest rates in all previous years.

Let Sn be the accumulated amount after n years of a single investment of 1 at time t = 0.

(a) Show that [ ] = (1 )nnE S j .

(b) Show that 2 2 2Var [ ] = (1 2 ) (1 )n nnS j j s j .

[5]

(ii) The interest rate per annum effective in (i), in any year, is equally likely to be

1 2 1 2or ( )i i i i . No other values are possible.

(a) Derive expressions for j and s2 in terms of i1 and i2.

(b) The accumulated value at time t = 25 years of £1 million invested with the bank at time t = 0 has expected value £5.5 million and standard deviation £0.5 million.

Calculate the values of i1 and i2. [8]

[Total 13]

CT1 A2005 6

11 (i) A loan is repayable over 20 years by level instalments of £1,000 per annum made annually in arrear. Interest is charged at the rate of 5% per annum effective for the first 10 years, increasing to 7% per annum effective for the remaining term.

Show that the amount of the original loan is £12,033.56. (Minor discrepancies due to rounding will not be penalised). [2]

(ii) The following are the details from the loan schedule for year x, i.e. the year running from exact duration x

1 years to exact duration x years.

Instalment paid at the end of the year Loan outstanding at the beginning of the year Interest Capital

Year x £8,790.48 £439.52 £560.48

Determine the value of x. [4]

(iii) At the beginning of year 11, it is agreed that the increase in the rate of interest will not take place, so that the rate remains at 5% per annum effective for the remainder of the loan. The annual instalment will continue to be payable at the same level so that there may be a reduced term and a reduced final instalment.

(a) Calculate by how many years, if any, the repayment schedule is shortened.

(b) Calculate the amount of the reduced final instalment.

(c) Calculate the reduction in the total interest paid during the existence of the loan as a result of the interest rate not increasing.

[7] [Total 13]

END OF PAPER


EXAMINATION

April 2005

Subject CT1


EXAMINERS REPORT

Introduction

The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable.

M Flaherty

Chairman of the Board of Examiners

15 June 2005


Subject CT1 (Financial Mathematics Core Technical) April 2005

Examiners Report

Page 2

1 r T tf S I Ke

where:

t is the present time T is the time of maturity of the forward contract r is the continuously compounded risk-free rate of interest for the interval from t

to T S is the spot price of the security at time t I is the present value, at the risk-free interest rate, of the income generated by the

security during the interval from t to T K is the delivery price of the forward contract f is the value of a long position in the forward contract

Here, working with £100 nominal,

S = 95, K = 98, T

t =1, r = 0.052

0.046 0.5 0.052 12.5 4.81648I e e

0.05295 4.81648 98 2.85071f e

The value of the investor s short position in a forward contract on £1 million is therefore

1,000,00010,000 2.85071

100f

= £28,507

2 MWRR: 2.2 3

1 1.44 1 4.2i i

Estimate 6% , 4.1466i LHS

7% , 4.2359i LHS

4.2 4.14660.06 0.01

4.2359 4.1466i

= 6.60% p.a. to two decimal places

Let F = Fund value before net cashflow on 31 December 2003


Examiners Report

Page 3

Then,

TWRR = 6.60% p.a. means that

3 4.21.066

2.2 1.44

F

F

0.634521.44

F

F

0.63452 0.63452F x 1.44 F

F =£2.5m

3 (i) Work in millions:

1

of liabilities 9 12PV v a

at 9%

9 12 .i

v v

29 12 0.91743 1.044354

= 19.54811

The assets up to 2k years from 1 January 2006 have:

22 22

5 5kk

iPV v a v a

i

5 0.84168 1.022015k

a

4.301048k

a

With 6, 4.301048 4.4859k PV

= 19.2941

The next payment of 2.5 million at k = 6.5 is made at time

8.5 and has present value = 8.52.5 1.2018v


Examiners Report

Page 4

This would make PV of assets (20.5m) > PV of liabilities (19.5m)

Discounted payback period = 8.5 years.

(ii) The income of the development is received later than the costs are incurred. Hence an increase in the rate of interest will reduce the present value of the income more than the present value of the outgo. Hence the DPP will increase.

4 (i) Accumulation =10

0500s ds

e

= 8 10

0 80.07 0.005 0.06

500s ds ds

e

=

81028

0

0.0050.07 0.06

2500

S S S

e

= 0.40 0.12500e

= 841.01

(ii) 018 0.1

10200 .

ts dstPV e e dt

8

0 80.07 0.005 ) 0.0618 0.1

10200 .

ts ds ds

te e

18 0.1 0.40 0.48 0.06

10200 . .t te e e dt

180.08 0.04

10200 te e dt

0.08 180.04

10

200

0.04te

e

0.08 0.72 0.405000 3047.33e e e


Examiners Report

Page 5

5 Present Value = 12 22

1 1 15000 . .a v a v a at %i

where

41 1.02 8.24322% p.a.i i

effective

1

0.0824322 1. .

1.0824322 1.0824322

ia v

Ln

0.9614201

and 11212

121

11.0824322 .

va

i

where

1.0824322 =

1212

121 0.079472512

ii

12

10.9645970a

and 12

2

21

11.0824322 .

va

i

where 1.0824322

22

21 0.08080002

ii

2

1a = 0.9805844

So 25000 0.9614201 0.9645970 0.9805844 13, 447.39PV v v

Examiners Comment: There are other valid methods for obtaining the required answer which also received full credit.


Examiners Report

Page 6

6 (i) (a) Work in t = 0 monetary values

25000 = 10000 2 3170.7 170.7 170.7

183.3 191.0 200.9v v v

where

1

1v

i with i = real rate of return

Try 4% RHS = 24770.94

3% RHS = 25241.25

25241.25 250000.03 0.01

25241.25 24770.94i

= 0.0351 i.e. 3.5%

(b) 25000 = 10000 3

a at %i p.a.

32.5a

From tables, 3

2.5313a at 9%

= 2.4869 at 10%

2.5313 2.50.09 0.01

2.5313 2.4869i

= 0.097

i.e. 9.7% p.a.

(ii) We should find that 1

11

ie

i

where e = average annual rate of inflation over the period.

Hence 1

1 1.0971.06

1.0351

i

i

which implies 6% p.a. inflation over the period


Examiners Report

Page 7

The actual average inflation rate was:

3 200.91 5.6%

170.7e e p.a.

The inflation rate would not be expected to be exactly 6% p.a. since the Retail Price Index is not increasing by a constant amount each year.

7 4

441 1.04 0.039414

4

ii

10.05

1 0.80 0.0388351.03

g t

411i t g

Capital gain on contract

Assume redeemed as late as possible (ie: after 20 years) to obtain minimum yield.

Price of stock, P:

P 4

20100000 0.05 0.80 a

20103000 0.25 103000 at 4%P v

4 2020

20

4000 77250

1 0.25

a vP

v

4000 1.014877 13.5903 77250 0.45639

1 0.25 0.45639

= 102,072.25


Examiners Report

Page 8

8 (i) No, because the spread (convexity) of the liabilities would always be greater than the spread (convexity) of the assets 3rd Redington condition would never be satisfied.

(ii) Conditions required: (a) A LV V

(b) ' '.

A LV V

(c) " "A LV V

where differentiation can be in respect of delta or i. In this solution, it is in respect of delta.

(a) 15 253.43 7.12 @7%AV v v

= 2.5550

19 214 6LV v v

= 2.5551

A LV V (ignoring rounding)

(b) ' 15 253.43 15 7.12 25AV v v

= 51.444

' 19 214 19 6 21LV v v

= 51.445

' 'A LV V (ignoring rounding)

(c) " 2 15 2 253.43 15 7.12 25AV v v

= 1099.627

" 2 19 2 214 19 6 21LV v v

1038.322

" "A LV V

all 3 conditions are satisfied.


Examiners Report

Page 9

Examiners Comment: There are other valid methods for obtaining the required answer which also received full credit.

9 (i) From two year stock information:

Price = 22

3 102 at 5.5%a v

= 3 1.84632 + 102 0.89845

= 97.1811

Therefore, from one-year forward rate information,

1 1 1,1

3 3 10297.1811

1 1 1i i f

where 1i =one-year spot rate

1,1f = one-year forward rate from t = 1

1 1

3 10597.1811

1 1 1.05i i

1

10397.1811

1 i

1 5.9877%i p.a.

(ii) From three-year stock information:

108.9 = 1 1 2,1

10 10 110

1 1 1.05 1 1.05 1ii i i f

where 2,1f =one-year forward rate from t = 2

Hence

2,1

10 10 110108.9

1.059877 1.059877 1.05 1.059877 1.05 1 f

2,1

110108.9 9.4351 8.9858

1.11287 1 f


Examiners Report

Page 10

2,1 9.245%f p.a.

(iii) Let 2%y p.a. be the two-year par yield

100 = 21 1 1,1 1 1,1

1 1 100

1 1 1 1y

i i i f i f

100 = 21 1 100

1.059877 1.059877 1.05 1.059877 1.05y

2100 1.84208 89.8577y

2 5.506%y p.a.

10 (i) (a) Let ti be the (random) rate of interest in year t . Let nS be the

accumulation of a single investment of 1 unit after n years:

1 21 1 1n nE S E i i i

1 21 1 1 as n n tE S E i E i E i i are independent

tE i j

1n

nE S j

(b) 22

1 21 1 1n nE S E i i i

2 2 2

1 21 1 1 nE i E i E i (using independence)

2 2 21 1 2 21 2 1 2 1 2 n nE i i E i i E i i

2 21 2n

j s j


Examiners Report

Page 11

as 22 2 2

i t tE i V i E i s j

22 2Var 1 2 1

n nnS j s j j

(ii) (a) 1 21

Interest 2

E j i i

22 2Interest Interest InterestVar s E E

22 21 2 1 2

1 1

2 2i i i i

= 2 21 2 1 2

1 1.

4 2i i i i

2

1 21

2i i

(b) 25

25 1 5.5E S j

0.0705686j

Var 25 50 22 2

25 1 2 1 0.5S j j s j

252 21 2 0.0705686 0.0705686 s50

1.0705686 0.25

2 0.000377389s

Hence, 22

1 21

0.0003773894

s i i

1 2 0.0388530i i (taking positive root since 1 2i i )

1 2 2 0.07056862i i = 0.1411372

12 0.0388530 0.1411372i

1 0.089995 8.9995%p.a.i


Examiners Report

Page 12

and 2 0.051142 5.1142%p.a.i

11 (i) Loan 5% 10 7%

5%10 101000 a v a

= 1000 7.7217 0.61391 7.0236

= 12033.56

(ii) Note 439.52

0.05 108790.48

x

5% 11 7%5%11 10

8790.48 1000 xx

a v a

1111

18.79048 7.0236

0.05

xx

vv

118.79048 20 20 7.0236 xv

11 11.209520.86384

12.9764xv at 5%

8x

(iii) Let Y = reduced final payment n = new total term of loan

Loan outstanding after 10 years = 7%10

1000 a = £7,023.60

After change is made:

7023.60 = 1011

1000 at 5%nn

a Yv

try n = 20 (i.e., keep to original term)

RHS = 1000 7.1078 0.61391Y

137.15Y

doesn t work

try n = 19


Examiners Report

Page 13

RHS = 1000 6.4632 0.64461Y

869.36Y

Hence:

(a) Term shortened by 1 year

(b) Final instalment = £869.36

(c) Under original terms, total interest paid is:

20 1000 12033.56 7966.44

Under changed terms, total interest paid is:

18 1000 869.36 12033.56 6835.80

difference = £1,130.64

END OF EXAMINERS REPORT


EXAMINATION

7 September 2005 (am)

Subject CT1













Faculty of Actuaries CT1 S2005 Institute of Actuaries

CT1 S2005 2

1 Describe how cashflows are exchanged in an interest rate swap . [2]

2 An investor has earned a money rate of return from a portfolio of bonds in a particular country of 1% per annum effective over a period of ten years. The country has experienced deflation (negative inflation) of 2% per annum effective during the period.

Calculate the real rate of return per annum over the ten years. [2]

3 Calculate the time in days for £1,500 to accumulate to £1,550 at:

(a) a simple rate of interest of 5% per annum (b) a force of interest of 5% per annum

[4]

4 The force of interest (t) at time t is a + bt2 where a and b are constants. An amount of £200 invested at time t = 0 accumulates to £210 at time t = 5 and £230 at time t = 10.

Determine a and b. [5]

5 (i) Calculate the present value of £100 over ten years at the following rates of interest/discount:

(a) a rate of interest of 5% per annum convertible monthly (b) a rate of discount of 5% per annum convertible monthly (c) a force of interest of 5% per annum

[4]

(ii) A 91-day treasury bill is bought for $98.91 and is redeemed at $100. Calculate the annual effective rate of interest obtained from the bill. [3]

[Total 7]

6 (i) State the features of a eurobond. [3]

(ii) An investor purchases a eurobond on the date of issue at a price of £97 per £100 nominal. Coupons are paid annually in arrear. The bond will be redeemed at par twenty years from the issue date. The rate of return from the bond is 5% per annum effective.

(a) Calculate the annual rate of coupon paid by the bond. (b) Calculate the duration of the bond.

[6] [Total 9]

CT1 S2005 3 PLEASE TURN OVER

7 A bank makes a loan to be repaid in instalments annually in arrear. The first instalment is 50, the second 48 and so on with the payments reducing by 2 per annum until the end of the 15th year after which there are no further payments. The rate of interest charged by the lender is 6% per annum effective.

(i) Calculate the amount of the loan. [6]

(ii) Calculate the interest and capital components of the second payment. [3]

(iii) Calculate the amount of capital repaid in the instalment at the end of the fourteenth year. [3]

[Total 12]

8 An insurance company has just written contracts that require it to make payments to policyholders of £1,000,000 in five years time. The total premiums paid by policyholders amounted to £850,000. The insurance company is to invest half the premium income in fixed interest securities that provide a return of 3% per annum effective. The other half of the premium income is to be invested in assets that have an uncertain return. The return from these assets in year t, it, has a mean value of 3.5% per annum effective and a standard deviation of 3% per annum effective. (1 + it) is independently and lognormally distributed.

(i) Deriving all necessary formulae, calculate the mean and standard deviation of the accumulation of the premiums over the five-year period. [9]

(ii) A director of the company suggests that investing all the premiums in the assets with an uncertain return would be preferable because the expected accumulation of the premiums would be greater than the payments due to the policyholders.

Explain why this still may be a more risky investment policy. [2] [Total 11]

CT1 S2005 4

9 (i) Explain what is meant by the expectations theory for the shape of the yield curve. [2]

(ii) Short-term, one-year annual effective interest rates are currently 8%; they are expected to be 7% in one years time, 6% in two years time and 5% in three years time.

(a) Calculate the gross redemption yields (spot rates of interest) from 1-year, 2-year, 3-year and 4-year zero coupon bonds assuming the expectations theory explanation of the yield curve holds.

(b) The price of a coupon paying bond is calculated by discounting individual payments from the bond at the zero-coupon bond yields in (a).

Calculate the gross redemption yield of a bond that is redeemed at par in exactly four years and pays a coupon of 5 per annum annually in arrear.

(c) A two-year forward contract has just been issued on a share with a price of 400p. A dividend of 4p is expected in exactly one year.

Calculate the forward price using the above spot rates of interest, assuming no arbitrage. [12]

[Total 14]

10 An investor purchased a bond with exactly 15 years to redemption. The bond, redeemable at par, has a gross redemption yield of 5% per annum effective. It pays coupons of 4% per annum, half yearly in arrear. The investor pays tax at 25% on the coupons only.

(i) Calculate the price paid for the bond. [3]

(ii) After exactly eight years, immediately after the payment of the coupon then due, this investor sells the bond to another investor who pays income tax at a rate of 25% and capital gains tax at a rate of 40%. The bond is purchased by the second investor to provide a net return of 6% per annum effective.

(a) Calculate the price paid by the second investor.

(b) Calculate, to one decimal place, the annual effective rate of return earned by the first investor during the period for which the bond was held. [10]

[Total 13]

CT1 S2005 5

11 (i) Explain what is meant by the following terms:

(a) equation of value (b) discounted payback period from an investment project

[4]

(ii) An insurance company is considering setting up a branch in a country in which it has previously not operated. The company is aware that access to capital may become difficult in twelve years time. It therefore has two decision criteria. The cashflows from the project must provide an internal rate of return greater than 9% per annum effective and the discounted payback period at a rate of interest of 7% per annum effective must be less than twelve years.

The following cashflows are generated in the development and operation of the branch.

Cash Outflows

Between the present time and the opening of the branch in three years time the insurance company will spend £1.5m per annum on research, development and the marketing of products. This outlay is assumed to be a constant continuous payment stream. The rent on the branch building will be £0.3m per annum paid quarterly in advance for twelve years starting in three years time. Staff costs are assumed to be £1m in the first year, £1.05m in the second year, rising by 5% per annum each year thereafter. Staff costs are assumed to be incurred at the beginning of each year starting in three years time and assumed to be incurred for 12 years.

Cash Inflows

The company expects the sale of products to produce a net income at a rate of £1m per annum for the first three years after the branch opens rising to £1.9m per annum in the next three years and to £2.5m for the following six years. This net income is assumed to be received continuously throughout each year. The company expects to be able to sell the branch operation 15 years from the present time for £8m.

Determine which, if any, of the decision criteria the project fulfils. [17]

[Total 21]

END OF PAPER


EXAMINATION

September 2005

Subject CT1


EXAMINERS REPORT


Subject CT1 (Financial Mathematics Core Technical)

September 2005

Examiners Report

Page 2

As is in some recent diets, the questions requiring descriptions of concepts, definitions or verbal reasoning (such as Q1, Q8(ii) and Q9(i)) tended not to be well answered with candidates producing vague statements which did not demonstrate that they understood the relevant points. It is important that candidates understand the subject well enough to express important topics and issues in their own words as well as in mathematical language. In show that questions or questions where students are asked to derive formulae (such as Q8 part (i)) candidates are required to show detailed steps in deriving the results required in order to obtain full marks.

Please note that differing answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates were not penalised for this. However, candidates were penalised where excessive rounding had been used or where insufficient working had been shown.


September 2005

Examiners Report

Page 3

1 One party agrees to pay to the other a regular series of fixed amounts for a certain term. In exchange the second party agrees to pay a series of variable amounts based on the level of a short term interest rate.

2 If f = the rate of inflation; j = the real rate of return and i = the money rate of return, then j = (i

f)/(1 + f). In this case, f = 2%, i= 1% and therefore j = 3.061%.

3 (a) Let the answer be t days

1,500(1 + 0.05

t/365) = 1,550

t = 243.333 days

(b) Let the answer be t days

1,500e0.05(t/365) = 1,550

0.05 (t/365) = ln (1,550/1500)

t = 239.366 days

4 5 52 31

3 00

210 200exp 200exp 200 5 41.667a bt dt at bt a b

10 102 313 0

0

230 200exp 200exp 200 10 333.333a bt dt at bt a b

ln(1.05) 5 41.667a b

ln(1.15) 10 333.333a b

The second expression less twice the first expression gives:

ln(1.15) 2ln(1.05) 250 0.0001687b b

ln(1.15) 333.333 0.00016870.0083520

10a


September 2005

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Page 4

5 (i) (a) 100 (1 + 0.05/12) 12 10 = £60.716 (b) 100 (1 0.05/12)12 10 = £60.590 (c) 100

e 10 = £60.6531

(ii) 98.91 = 100(1 + i) 91/365

ln(1 + i) = ( 65/91)

ln(98.91/100) = 0.04396

therefore i = 0.04494

6 (i)

Used for medium or long-term borrowing

Unsecured

Regular annual coupon payments

Generally repayable at par

Generally issued by large companies and on behalf of governments

Yields depend on risk and marketability

Generally innovative market designed to attract different types of investor

Issued internationally (normally by a syndicate of banks)

Can be issued in any currency (not necessarily the domestic currency of the borrower)

(ii) (a) 97 = 20ga + 100v20 at 5% per annum effective

20a = 12.4622; v20 = 0.37689 therefore 97 = 12.4622g + 100

0.37689

g = (97 37.689)/12.4622 = 4.75927

(b) Duration = Ct tvt/ Ctv

t where Ct is the amount of the cash flow at time t

20( )Ia = tvt Therefore duration of the eurobond is:

(4.75927 20( )Ia + 100

20v20)/(4.75927 20a + 100v20)

20( )Ia = 110.9506 all other values have been used in (a) above

therefore duration is:

(4.75927

110.9506 + 100

20

0.37689)/(4.75927

12.4622 + 100

0.37689) =1281.8239/97 = 13.2147


September 2005

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Page 5

7 (i) Value of loan = 50v + 48v2 + 46v3 + 44v4 + + 22v15

= 52(v +v2 +v3 + + v14 + v15) 2(v + 2v2 +4v3 + + 28v14+30 v15)

= 5215

a

- 2

15( )Ia

15

( )Ia = 67.2668

15a = 9.7122

Therefore amount of the loan is 52 9.7122 - 2 67.2668 = 370.501

Candidates who derived an appropriate formula for a decreasing annuity directly or who calculated the value of the loan by summing the individual terms received full credit.

(ii) Interest component in first year is 0.06

370.504 = 22.23024; therefore capital component is 50 22.23024 = 27.76976.

Capital remaining after first instalment is 370.504 27.76976 = 342.73424. Interest paid in second instalment is 0.06

342.73424 = 20.56405

Capital in second instalment is 48 20.56405 = 27.43595.

(iii) At the end of the thirteenth year, the capital outstanding is:

24v + 22v2 = 24

0.94340 + 22

0.89000 = 42.2216

The interest due in the fourteenth instalment 0.06 42.2216 = 2.53330

The capital payment is therefore 24 2.53330 = 21.46670


September 2005

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Page 6

8 (i) Let ti be the (random) rate of interest in year t . Let 5S be the accumulation of

a single investment of 1 unit after 5 years:

5

51

5

1

1

1

tt

tt

E S E i

E i

as ti are independent 5

5 1 tE S E i

1 1t tE i E i = 1.035

55 1.035E S 1.187686

5 52 22

51 1

1 1t tt t

E S E i E i (using independence)

5 5 52 2 2

52

1 1 2 1 2

1 2

t t t t t

t t t

E i E i i E i E i

E i Var i E i

225 5 5

52 101 2 1t t t t

Var S E S E S

E i Var i E i E i

2

0.035

0.03

t

t

E i

Var i

5 102 25 1 2 0.035 0.03 0.035 1.035

1.416534 1.410598

0.0059356

Var S

Mean value of the accumulation of premiums is: 5

5425000 425000(1.03) 425000 1.187686 425000 1.15927

997458

E S

Standard deviation is 5425000 425000 0.0059356 32743.21SD S

Candidates who obtained slightly different answers by first deriving the parameters of the lognormal distribution received full credit.


September 2005

Examiners Report

Page 7

(ii) Investing all premiums in the risky assets is likely to be more risky because, although there may be a higher probability of the assets accumulating to more than £1 million, the standard deviation would be twice as high so the probability of a large loss would be greater.

9 (i) Bond yields are determined by investors expectations of future short-term interest rates, so that returns from longer-term bonds reflect the returns from making an equivalent series of short-term investments

(ii) (a) Let it be the spot yield over t years:

One year: yield is 8% therefore i1 = 0.08

two years: (1 + i2)2 = 1.08

1.07 therefore i2 = 0.074988

three years: (1 + i3)3 = 1.08

1.07

1.06 therefore i3 = 0.06997

four years: (1 + i4)4 = 1.08

1.07

1.06

1.05 therefore i4 = 0.06494

(b) Price of the bond is 5[(1.08) 1 + (1.074988) 2 + (1.06997) 3] + 105

(1.06494) 4 = 13.03822 + 81.6373 = 94.67552

Find gross redemption yield from

94.67552 = 54

a

+ 100v4

try 7%; 4

a

= 3.3872; v4 = 0.76290

gives RHS = 93.226

GRY must be lower, try 6%; 4

a

= 3.4651; v4 = 0.79209

gives RHS = 96.5345

interpolate between 6% and 7%. i = 0.07 0.01 (94.67552

93.226)/(96.5345

93.226) i = 0.07 0.0043812 = 0.06562

(c) Present value of the dividend is 4v calculated at 8% per annum effective = 3.70370.

Therefore forward price is F = (400

3.70370)

1.08

1.07 = 457.9600


September 2005

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Page 8

10 (i) Price paid by first investor is P1

5%

(2) 151 15

(2)

15

15

1

4 100

1.012348

0.48102

10.3797

4 1.012348 10.3797 100 0.48102

42.0315 48.1020 90.1335

P a v

i

i

v

a

P

(ii) (a)

22

21 1.06 0.0591262

ii

11 0.04 0.75 0.03g t

211i t g

Capital gain on contract

Price paid by second investor is P2

(2) 7 72 6% 2 6%7 6%

(2)7 72 6% 6%7 6%

(2)

7

7

2

0.75 4 100 0.4 100

1 0.4 0.75 4 0.6 100

1.014782

0.66506

5.5824

0.75 4 1.014782 5.5824 60 0.66506

1 0.4 0.66506

77.5207

P a v P v

P v a v

i

i

v

a

P


September 2005

Examiners Report

Page 9

(b) Rate of return earned by the first investor is the solution to:

(2) 88

(2)

8

8

(2)

8

8

90.1335 0.75 4 77.5207

2%

1.004975

0.85349

7.3255

88.2490

1.5%

1.003736

0.88771

7.4859

91.357590.1335 88.2490

0.02 0.005 1.697% 1.7%91.3575 88.2490

a v

i

i

i

v

a

RHS

i

i

i

v

a

RHS

i

11 (i) (a) An equation of value expresses the equality of the present value of positive and negative (or incoming and outgoing) cash flows that are connected with an investment project, investment transaction etc.

(b) The discounted payback period from an investment project is the first time at which the net present value of the cash flows from the project is positive.


September 2005

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Page 10

(ii) Consider first the NPV at 9% per annum effective. Working in £million.

Present value of cash outflows:

(4) 3 3 4 2 5 11 149% 9% 9% 9% 9%3 9% 12 9%

12 12

1.5 0.3 1.05 1.05 1.05

1.5 1.044354 2.5313 0.3 1.055644 7.1607 0.77218

1 1.050.77218 5.71647 7.60679 13.32326

1 1.05

a a v v v v v

v

v

Present value of cash inflows:

159%6 9% 3 9% 9 9% 6 9% 15 9% 9 9%

1515 9 6 3

1.9 2.5 8

2.5 0.6 0.9 8

1.044354 2.5 8.0607 0.6 5.9952 0.9 4.4859 2.5313 8 0.27454

12.6253

a a a a a a v

a a a a v

Hence NPV of project @ 9% = 12.6253 13.3233 = £0.698 million so the IRR is less than 9% p.a. effective

To find whether the discounted payback period is less than 12 years at 7% per annum effective, we need to find the NPV @ 7% of first twelve years cashflows

Present value of cash outflows:

(4) 3 3 4 2 5 8 117% 7% 7% 7% 7%3 7% 9 7%

9 9

1.5 0.3 1.05 1.05 1.05

1.5 1.034605 2.6243 0.3 1.043380 6.5152 0.81630

1 1.050.81630 5.73739 6.82096 12.55835

1 1.05

a a v v v v v

v

v


September 2005

Examiners Report

Page 11

Present value of cash inflows:

6 7% 3 7% 9 7% 6 7% 12 7% 9 7%

12 9 6 3

1.9 2.5

2.5 0.6 0.9

1.034605 2.5 7.9427 0.6 6.5152 0.9 4.7665 2.6243

9.3461

a a a a a a

a a a a

NPV is negative so the discounted payback period is more than 12 years.

Project fulfils neither the discounted payback period criterion nor the internal rate of return criterion.



EXAMINATION

4 April 2006 (am)

Subject CT1













Faculty of Actuaries CT1 A2006 Institute of Actuaries

CT1 A2006 2

1 An investment is discounted for 28 days at a simple rate of discount of 4.5% per annum. Calculate the annual effective rate of interest. [3]

2 An annuity certain with payments of £150 at the end of each quarter is to be replaced by an annuity with the same term and present value, but with payments at the beginning of each month instead.

Calculate the revised payments, assuming an annual force of interest of 10%. [3]

3 At time t = 0 the n-year spot rate of interest is equal to (2.25 + 0.25n)% per annum effective (1 5).n

(a) Calculate the 2-year forward rate of interest from time t = 3 expressed as an annual effective rate of interest.

(b) Calculate the 4-year par yield.

(c) Without performing any further calculations, explain how you would expect the gross redemption yield of a 4-year bond paying annual coupons of 3.5% to compare with the par yield calculated in (b).

[7]

4 An investor, who is liable to income tax at 20% but is not liable to capital gains tax, wishes to earn a net effective rate of return of 5% per annum. A bond bearing coupons payable half-yearly in arrear at a rate 6.25% per annum is available. The bond will be redeemed at par on a coupon date between 10 and 15 years after the date of issue, inclusive. The date of redemption is at the option of the borrower.

Calculate the maximum price that the investor is willing to pay for the bond. [5]

5 A share currently trades at £10 and will pay a dividend of 50p in one month s time. A six-month forward contract is available on the share for £9.70. Show that an investor can make a risk-free profit if the risk-free force of interest is 3% per annum. [4]

6 An actuarial student has created an interest rate model under which the annual effective rate of interest is assumed to be fixed over the whole of the next ten years. The annual effective rate is assumed to be 2%, 4% and 7% with probabilities 0.25, 0.55 and 0.2 respectively.

(a) Calculate the expected accumulated value of an annuity of £800 per annum payable annually in advance over the next ten years.

(b) Calculate the probability that the accumulated value will be greater than £10,000.

[4]


7 A company has entered into an interest rate swap. Under the terms of the swap the company makes fixed annual payments equal to 6% of the principal of the swap. In return, the company receives annual interest payments on the principal based on the prevailing variable short-term interest rate which currently stands at 5.5% per annum.

(a) Describe briefly the risks faced by a counterparty to an interest rate swap.

(b) Explain which of the risks described in (a) are faced by the company. [4]

8 An ordinary share pays annual dividends. A dividend of 25p per share has just been paid. Dividends are expected to grow by 2% next year and by 4% the following year. Thereafter, dividends are expected to grow at 6% per annum compound in perpetuity.

(i) State the main characteristics of ordinary shares. [4]

(ii) Calculate the present value of the dividend stream described above at a rate of interest of 9% per annum effective from a holding of 100 ordinary shares. [4]

(iii) An investor buys 100 shares in (ii) for £8.20 each. He holds them for two years and receives the dividends payable. He then sells them for £9 immediately after the second dividend is paid.

Calculate the investor s real rate of return if the inflation index increases by 3% during the first year and by 3.5% during the second year assuming dividends grow as expected. [4]

[Total 12]

9 The force of interest ( )t is a function of time and at any time t, measured in years, is given by the formula:

2

0.04

( ) 0.008

0.005 0.0003

t t

t t

0 5

5 10

10

t

t

t

(i) Calculate the present value of a unit sum of money due at time t = 12. [5]

(ii) Calculate the effective annual rate of interest over the 12 years. [2]

(iii) Calculate the present value at time t = 0 of a continuous payment stream

that is paid at the rate of 0.05te per unit time between time t = 2 and time t = 5. [3]

[Total 10]

CT1 A2006 4

10 A piece of land is available for sale for £5,000,000. A property developer, who can lend and borrow money at a rate of 15% per annum, believes that she can build housing on the land and sell it for a profit. The total cost of development would be £7,000,000 which would be incurred continuously over the first two years after purchase of the land. The development would then be complete.

The developer has three possible project strategies. She believes that she can sell the completed housing:

in three years time for £16,500,000

in four years time for £18,000,000

in five years time for £20,500,000

The developer also believes that she can obtain a rental income from the housing between the time that the development is completed and the time of sale. The rental income is payable quarterly in advance and is expected to be £500,000 in the first year of payment. Thereafter, the rental income is expected to increase by £50,000 per annum at the beginning of each year that the income is paid.

(i) Determine the optimum strategy if this is based upon using net present value as the decision criterion. [9]

(ii) Determine which strategy would be optimal if the discounted payback period were to be used as the decision criterion. [2]

(iii) If the housing is sold in six years time, the developer believes that she can obtain an internal rate of return on the project of 17.5% per annum. Calculate the sale price that the developer believes that she can receive. [6]

(iv) Suggest reasons why the developer may not achieve an internal rate of return of 17.5% per annum even if she sells the housing for the sale price calculated in (iii). [2]

[Total 19]


11 An actuarial student has taken out two loans.

Loan A: a five-year car loan for £10,000 repayable by equal monthly instalments of capital and interest in arrear with a flat rate of interest of 10.715% per annum.

Loan B: a five-year bank loan of £15,000 repayable by equal monthly instalments of capital and interest in arrear with an effective annual interest rate of 12% for the first two years and 10% thereafter.

The student has a monthly disposable income of £600 to pay the loan interest after all other living expenses have been paid.

Freeloans is a company which offer loans at a constant effective interest rate for all terms between three years and ten years. After two years, the student is approached by a representative of Freeloans who offers the student a 10-year loan on the capital outstanding which is repayable by equal monthly instalments of capital and interest in arrear. This new loan is used to pay off the original loans and will have repayments equal to half the original repayments.

(i) Calculate the final disposable income (surplus or deficit) each month after the loan payments have been made. [5]

(ii) Calculate the capital repaid in the first month of the third year assuming that the student carries on with the original arrangements. [5]

(iii) Estimate the capital repaid in the first month of the third year assuming that the student has taken out the new loan. [5]

(iv) Suggest, with reasons, a more appropriate strategy for the student. [2] [Total 17]

CT1 A2006 6

12 A pension fund has liabilities of £3 million due in 3 years time, £5 million due in 5 years time, £9 million due in 9 years time, and £11 million due in 11 years time. The fund holds two investments, X and Y. Investment X provides income of £1 million payable at the end of each year for the next five years with no capital repayment. Investment Y is a zero coupon bond which pays a lump sum of £R at the end of n years (where n is not necessarily an integer). The interest rate is 8% per annum effective.

(i) Investigate whether values of £R and n can be found which ensure that the fund is immunised against small changes in the interest rate.

You are given that 5

2

1

40.275t

t

t v at 8%. [8]

(ii) (a) The interest rate immediately changes to 3% per annum effective. Calculate the revised present values of the assets and liabilities of the fund.

(b) Explain your answer to (ii)(a). [4] [Total 12]

END OF PAPER


EXAMINATION

April 2006

Subject CT1


EXAMINERS REPORT

Introduction

The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable.

M Flaherty Chairman of the Board of Examiners

June 2006

Comments

Individual comments are shown after each question.

General comments

As is in some recent diets, the questions requiring verbal reasoning (such as Q3(c), Q7(b), Q10(iv) and Q11(iv)) tended not to be well answered with candidates producing vague statements which did not demonstrate that they understood the relevant points.

Please note that different answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this.

However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown.



April 2006

Examiners Report

Page 2

1 Annual rate of interest is i where1

28/365281 1

365

di

This gives

365/ 2828 0.045

1 1 4.611%365

i

Comments on question 1: This was generally well answered.

2 We require X where:

(4) (12)(4) (12)

(12) (4)600 12 50 50n

n nn

a da Xa X

ia

12

4

1/12(12)

1/ 4(4)

12 1 1 12 1 0.099584

4 1 1 4 1 0.101260

d d e

i i e

Hence 49.1724X or £49.17

Comments on question 2: Candidates were not penalised for assuming that the annuities were for a specific term even though this was not needed for the calculations.

3 (a) 55

2 53,2 3,23 3

3

1.03511 4.255%

1 1.03

yf f

y

(b) Par yield is 4yc where 1 2 3 4 4

2 3 4 44 1y y y y yyc v v v v v

Thus 1 2 3 4 44 1.025 1.0275 1.03 1.0325 1.0325 1yc

40.12009

3.230%3.71785

yc

(c) The par yield is equal to the gross redemption yield for a par yield bond. Coupons for the 3.5% bond are higher than for the par yield bond. Thus a lower proportion of the total proceeds are included within the redemption payment which is when spot yields/discount rates are highest. The present value of the proceeds of the 3.5% bond will be higher and so the gross redemption yield will be lower than that of the par yield bond and thus less than the par yield.


Examiners Report

Page 3

Comments on question 3: Part (a) was answered well but some candidates struggled with the calculation of the par yield in part (b). In part (c) the marks were awarded for a clear explanation. Many candidates, who just stated their conclusion, were unable to explain their reasoning clearly and so failed to score full marks on this part.

4 2 0.049390i

11 0.0625 0.80 0.05g t

211i t g

Capital loss on contract

Assume redeemed as early as possible (i.e.: after 10 years) to obtain minimum yield.

Price of stock per £100 nominal, P:

P 2

10100 0.0625 0.80 a 10100 at 5%v

2 1010

5 100P a v

5 1.012348 7.7217 100 0.61391

39.0852 61.3910 £100.4762

Comments on question 4: Well answered although some candidates who recognised that the investor faced a capital loss did not recognise that this meant that the minimum yield would be obtained if the bond was redeemed at the earliest possible date.


April 2006

Examiners Report

Page 4

5 An investor can borrow £10 at the risk-free rate, buy one share for £10, enter into the forward contract to sell the share in six months time.

The initial cashflow is zero.

After one month the 50p dividend from the share is invested at the risk-free rate. After six months the share can be sold for £9.70, the dividend proceeds are worth

5120.030.5e and the borrowing is repaid at 10 0.015e . This gives a net cashflow of 9.7

+5120.030.5e 10 0.015e = 0.0552

The investor has made a deal with zero initial cost, no risk of future loss and a risk-free future profit.

Comments on question 5: The majority of candidates were able to calculate the non-arbitrage forward price by use of the appropriate formula. However, marks were lost for not clearly explaining how a risk-free profit could thus be made.

6 (a) Expected accumulated value

10 0.02 10 0.04 10 0.07

11 0.02 11 0.04 11 0.07

800 0.25 0.55 0.2

800 0.25 1 0.55 1 0.2 1

800 0.25 11.1687 0.55 12.4864 0.2 14.7836

0.25 8934.96 0.55 9989.12 0.2 11826.88

£10,093.13

s s s

s s s

(b) Accumulation is only over £10,000 if the interest rate is 7% p.a. which has probability 0.2

Comments on question 6: The most poorly answered question on the paper. This model of interest rates had not been examined recently and the majority of candidates assumed instead that the interest rate changed each year (in line with previous examination questions on this topic).


Examiners Report

Page 5

7 (a) The counterparty faces market risk which is the risk that market conditions will change so that the present value of the net outgo under the agreement increases.

The counterparty also faces credit risk which is the risk that the other counterparty will default on its payments.

(b) The company still faces the market risk since the interest rates could fall further which will make the value of the swap even more negative to the company.

The company does not currently face a credit risk since the value of the swap is positive to the other counterparty.

Comments on question 7: Part (a) was answered well but many candidates failed to recognise in (b) that the company would not currently face credit risk in this example.

8 (i) Main characteristics of ordinary shares:

Issued by commercial undertakings and other bodies.

Entitle holders to receive all net profits of the company in the form of dividends after interest on loans and other fixed interest stocks has been paid.

Higher expected returns than for most other asset classes

but risk of capital losses

and returns can be variable.

Lowest ranking form of finance.

Low initial running yield but dividends should increase with inflation.

Marketability varies according to size of company.

Voting rights in proportion to number of shares held.

(ii) Present value of future dividends

2 3 2 4100 0.25 1.02 1.02 1.04 1.02 1.04 1.06 1.02 1.04 1.06v v v v

2 2 2

2

25 1.02 25 1.02 1.04 1 1.06 1.06

1.0925 1.02 25 1.02 1.04

0.03

23.3945 811.0092 834.4037 £834.40

v v v v

v v


April 2006

Examiners Report

Page 6

(iii) Real rate of return is i such that:

2

2

2

100 100 100820 100 0.25 1.02 100 0.25 1.02 1.04

103 103 103.5

100 100900

103 103.5

24.7573 869.1150

v v

v

v v

224.7573 24.7573 4 869.1150 8200.95719

2 869.1150v

(taking positive root)

Hence i = 4.47%

Comments on question 8: Despite being a bookwork question, part (i) was answered patchily with few students getting all of the required points. Part (ii) was answered well. In part (iii), it was expected that students would solve the quadratic equation. However, full credit was given to students who used interpolation methods.

9 (i) 5 50 0

0.04 0.04 0.2(0,5) 1.22140dt t

A e e e

10210

5 50.0040.008 0.3(5,10) 1.34986

ttdtA e e e

1212 2 3210 10

0.0025 0.00010.005 0.0003 0.1828(10,12) 1.20057t tt t dt

A e e e

Required present value

1 1 1

0,5 5,10 10,12 1.22140 1.34986 1.20057 1.97941A A A

= 0.50520

(ii) Equivalent effective annual rate is i where 12

1 1.97941 5.855%i i


Examiners Report

Page 7

(iii) Present Value at time t = 0

0

5 50.040.05 0.05 0.04

2 2

55 0.09 0.18 0.450.09

2 2

2.19600.09 0.09

tdst t t

tt

e e dt e e dt

e e ee dt

Comments on question 9: Well answered.

10 (i) Net present value of costs

2 25,000,000 3,500,000 5,000,000 3,500,000

5,000,000 3,500,000 1.073254 1.6257 11,106,762

ia a

Net present value of benefits

(4)42 222

450,000 50,000 nnnn

v a v Ia S v

2 22 2(4) (4)

450,000 50,000 nnn n

i iv a v Ia S v

d d

where n is the year of sale and nS are the sale proceeds if the sale is made in

year n.

If n = 3 the NPV of benefits

450,000 0.75614 1.092113 0.86957

50,000 0.75614 1.092113 0.86957

16,500,000 0.65752

323,137 35,904 10,849,080 11, 208,121

Hence net present value of the project is 11,208,121 11,106,762 = 101,359

Note that if n = 4 the extra benefits in year 4 consist of an extra £1.5 million on the sale proceeds and an extra £650,000 rental income. This is clearly less than the amount that could have been obtained if the sale had been made at the end of year 3 and the proceeds invested at 15% per annum. Hence selling in year 4 is not an optimum strategy.


April 2006

Examiners Report

Page 8

If n = 5 the NPV of benefits

450,000 0.75614 1.092113 2.2832

50,000 0.75614 1.092113 4.3544

20,500,000 0.49718

848, 450 179,791 10,192,190 11, 220,431

Hence net present value of the project is 11,220,431 11,106,762 = 113,669

Hence the optimum strategy if net present value is used as the criterion is to sell the housing after 5 years.

(ii) If the discounted payback period is used as the criterion, the optimum strategy is that which minimises the first time when the net present value is positive. By inspection, this is when the housing is sold after 3 years.

(iii) We require

(4)42 22 22

5,000,000 3,500,000 450,000 50,000 nnnn

ia v a v Ia S v at 17.5%

LHS20.175

0.175

1 1 0.724315,000,000 3,500,000 5,000,000 3,500,000

0.16127

v

10,983,227

RHS 440.1752 2 60.175 4

0.175 0.175 6 0.1754 4

41450,000 50,000

a vvv v S v

d d

144

0.175 4 1 0.15806d v

4

4

13.1918

va

d

Therefore we have on the RHS

6

6

3.1918 2.0985450,000 0.72431 3.0076 50,000 0.72431 0.37999

0.15806

980, 296 250,502 0.37999

S

S


Examiners Report

Page 9

For equality 610,983,227 1,230,798

£25,665,0000.37999

S

(iv) Reasons investor may not achieve the internal rate of return:

Allowance for expenses when buying/selling which may be significant.

There may be periods when the property is unoccupied and no rental income is received.

Rental income may be reduced by maintenance expenses.

Tax on rental income and/or sale proceeds

Comments on question 10: A significant number of candidates assumed that the development costs amounted to £7 million per annum and subsequently found that no strategy would lead to a profit. Otherwise the calculations were performed well. In part (iv), credit was given for other valid answers. Despite this, few students scored full marks on this part.

11 (i) Let ,A BX X be the monthly repayments under Loans A and B respectively.

For loan A:

Flat rate of interest = 10.715% 60 60 10000

£255.965 50000

A A AA

A

X L XX

L

For loan B:

(12) (12)212%2 12% 310%

212%2 312 12

12% 10%

15000 12

1,250

1,250

1.053875 1.6901 0.79719 1.045045 2.4869

B B

B

L X a v a

Xi i

a v ai i

£324.43BX

Hence student s overall surplus = 600 A BX X = £19.61


April 2006

Examiners Report

Page 10

(ii) Effective rate of interest under loan A is i % where

12 12

5 512 255.96 10000 3.2557a a

Try i = 20%: 12

5a = 3.2557

So capital outstanding after 24 months is 12 255.9612

3a at 20%

12 255.96 1.088651 2.1065 7043.74

Capital outstanding under B is 12 324.4312

3a at 10%

12 324.43 1.045045 2.4869 10118.02

So interest paid in month 25 under loans A and B

20% 10%12 12

7043.74 10118.02 107.84 80.68 £188.5212 12

i i

and capital repaid

255.96 107.84 324.43 80.68 148.12 243.75 £391.87

(iii) Under the new loan the capital outstanding is the same as under the original arrangement = 17161.76.

The monthly repayment 255.96 324.43

£290.202

The effective rate of interest on the new loan A is i where

12 12

10 1012 290.20 17161.76 4.9281a a

Try i = 20%: 12

104.5642a

Try i = 15%: 12

105.3551a

By interpolation 5.3551 4.9281

15% 20% 15% 17.7%5.3551 4.5642

i


Examiners Report

Page 11

Hence interest paid in month 25

17.7%12

17161.76 234.6612

i

and capital repaid is £290.20 £234.66 = £55.54

(iv) The new strategy reduces the monthly payments but repays the capital more slowly. The student could consider the following options:

Keeping loan B and taking out a smaller new loan to repay loan A (which has the highest effective interest rate).

Taking out the new loan for a shorter term to repay the capital more quickly.

Comments on question 11: In part (i) some candidates struggled to deal with the flat rate of Loan A whilst others failed to deal with the change in interest rate of Loan B. Part (ii) was answered well. In part (iii), different answers for the effective rate of interest (and hence the interest paid) for the new loan could be obtained according to the actual interpolation used and full credit was given for a range of answers. If calculated exactly, the effective rate of interest is actually 17.5%. In part (iv), credit was again given for any valid strategy suitably explained.

12 (i) We will consider three conditions necessary for immunisation

(1) A LV V (all expressions in terms of £m)

5

nAV a Rv at 8%

3.9927 nRv

3 5 9 113 5 9 11LV v v v v at 8%

15.0044

11.0117nRv

(2) ' 'A LV V where ' '&A L

A L

V VV V

'

5

11.3651

nA

n

V Ia nRv

nRv

' 3 5 9 119 25 81 121

116.5741LV v v v v


April 2006

Examiners Report

Page 12

105.2090nnRv

9.5543

105.20909.5543

11.0117

11.0117 1.08 £22.9720

n

R m

Alternatively:

' 'A LV V where ' '&A L

A L

V VV V

i i

' 1

5

1

1

11.3651

10.5233

nA

n

n

V v Ia nRv

v nRv

nRv

' 4 6 10 129 25 81 121

107.9389LV v v v v

1 97.4156nnRv

9.5543

97.41569.5543

11.0117

11.0117 1.08 £22.9720

nv

R m

(3) '' ''A LV V

(where 2 2

'' ''2 2

&A LA L

V VV V )

5'' 2 2

1

2 9.554340.275 9.5543 22.9720

1045.483

t nA

t

V t v n Rv

v

'' 3 5 9 1127 125 729 1331

1042.031LV v v v v


Examiners Report

Page 13

Alternatively (differentiating with respect to i):

5'' 2 2

1

52 2 2 2

51

11.5543

1 1

1

0.85734 40.275 0.85734 11.3651 9.5543 10.5543 22.9720

34.53 9.74 952.00 996.27

t nA

t

t n

t

V t t v n n Rv

v t v v Ia n n Rv

v

'' 5 7 11 133 3 4 5 5 6 9 9 10 11 11 12

993.32LV v v v v

Thus 9.5543, £22.9720n R m will satisfy all three conditions and so will achieve immunisation.

(ii) (a) Value of assets at 3% 9.5543

54.5797 22.9720 £21.900na Rv v m

Value of liabilities at 3% = 3 5 9 113 5 9 11 £21.903v v v v m

Hence fund has a deficit of approximately £3,000.

(b) Immunisation will only enable to be a fund to be protected against a small change in interest rates. It will not be necessarily protected against sudden large changes as in this case.

Comments on question 12: Part (i) was answered surprisingly poorly, given that it required the same techniques as those required in previous examination questions on the same topic. Full credit was given to students who observed directly that the spread of the assets around the mean term was greater than the spread of the liabilities. Few students answered part (ii) fully and the examiners felt that students should have recognised that immunisation would not protect the fund against such a large change in interest rates even if they had not answered part (i) correctly.



EXAMINATION


Subject CT1













Faculty of Actuaries CT1 S2006 Institute of Actuaries

CT1 S2006 2

1 (a) Distinguish between a future and an option. (b) Explain why convertibles have option-like characteristics.

[3]

2 An individual makes an investment of £4m per annum in the first year, £6m per annum in the second year and £8m per annum in the third year. The investments are made continuously throughout each year. Calculate the accumulated value of the investments at the end of the third year at a rate of interest of 4% per annum effective.

[3]

3 An individual has invested a sum of £10m. Exactly one year later, the investment is worth £11.1m. An index of prices has a value of 112 at the beginning of the investment and 120 at the end of the investment. The investor pays tax at 40% on all money returns from investment. Calculate:

(a) The money rate of return per annum before tax. (b) The rate of inflation. (c) The real rate of return per annum after tax.

[4]

4 An investor is able to purchase or sell two specially designed risk-free securities, A and B. Short sales of both securities are possible. Security A has a market price of 20p. In the event that a particular stock market index goes up over the next year, it will pay 25p and, in the event that the stock market index goes down, it will pay 15p. Security B has a market price of 15p. In the event that the stock market index goes up over the next year, it will pay 20p and, in the event that the stock market index goes down, it will pay 12p.

(i) Explain what is meant by the assumption of no arbitrage used in the pricing of derivative contracts. [2]

(ii) Find the market price of B, such that there are no arbitrage opportunities and assuming the price of A remains fixed. Explain your reasoning. [2]

[Total 4]

5 (i) Calculate the time in days for £3,600 to accumulate to £4,000 at:

(a) a simple rate of interest of 6% per annum (b) a compound rate of interest of 6% per annum convertible quarterly (c) a compound rate of interest of 6% per annum convertible monthly

[4]

(ii) Explain why the amount takes longest to accumulate in (i)(a) [1] [Total 5]


6 The rate of interest is a random variable that is distributed with mean 0.07 and variance 0.016 in each of the next 10 years. The value taken by the rate of interest in any one year is independent of its value in any other year. Deriving all necessary formulae calculate:

(i) The expected accumulation at the end of ten years, if one unit is invested at the beginning of ten years. [3]

(ii) The variance of the accumulation at the end of ten years, if one unit is invested at the beginning of ten years. [5]

(iii) Explain how your answers in (i) and (ii) would differ if 1,000 units had been invested. [1]

[Total 9]

7 A life insurance fund had assets totalling £600m on 1 January 2003. It received net income of £40m on 1 January 2004 and £100m on 1 July 2004. The value of the fund was:

£450m on 31 December 2003; £500m on 30 June 2004; £800m on 31 December 2004.

(i) Calculate, for the period 1 January 2003 to 31 December 2004, to three decimal places:

(a) The time weighted rate of return per annum.

(b) The linked internal rate of return, using sub intervals of a calendar year.

[8]

(ii) Explain why the linked internal rate of return is higher than the time weighted rate of return. [2]

[Total 10]

8 The force of interest ( )t

at time t is 2at bt where a and b are constants. An amount of £100 invested at time t = 0 accumulates to £150 at time t = 5 and £230 at time t = 10.

(i) Calculate the values of a and b. [5]

(ii) Calculate the constant force of interest that would give rise to the same accumulation from time t = 0 to time t = 10. [2]

(iii) At the force of interest calculated in (ii), calculate the present value of a

continuous payment stream of 0.0520 te paid between from time t = 0 to time t = 10. [4]

[Total 11]

CT1 S2006 4

9 An individual took out a loan of £100,000 to purchase a house on 1 January 1980. The loan is due to be repaid on 1 January 2010 but the borrower can repay the loan early if he wishes. The borrower pays interest on the loan at a rate of 6% per annum convertible monthly, paid in arrears. The loan instalments only cover the interest on the loan. At the same time, the borrower took out a thirty-year investment policy, which was expected to repay the loan, and into which monthly premiums were paid, in advance, at a rate of £1,060 per annum. The individual was told that premiums in the investment policy were expected to earn a rate of return of 7% per annum effective. After twenty years, the individual was informed that the premiums had only earned a rate of return of 4% per annum effective and that they would continue to do so for the final ten years of the policy. The borrower agrees to increase his monthly payments into the investment policy to £5,000 per annum for the final ten years.

(a) Calculate the amount to which the investment policy was expected to accumulate at the time it was taken out.

(b) Calculate the amount by which the investment policy would have fallen short of repaying the loan had extra premiums not been paid for the final ten years.

(c) Calculate the amount of money the individual will have, after using the proceeds of the investment policy to repay the loan, after allowing for the increase in premiums.

(d) Suggest another course of action the borrower could have taken which would have been of higher value to him, explaining why this higher value arises.

(e) Calculate the level annual instalment that the investor would have had to pay from outset if he had repaid the loan in equal instalments of interest and capital.

[11]

10 A financial regulator has brought in a new set of regulations and wishes to assess the cost of them. It intends to conduct an analysis of the costs and benefits of the new regulations in their first twenty years.

The costs are estimated to be as follows:

The cost to companies who will need to devise new policy terms and computer systems is expected to be incurred at a rate of £50m in the first year increasing by 3% per annum over the twenty year period.

The cost to financial advisers who will have to set up new computer systems and spend more time filling in paperwork is expected to be incurred at a rate of £60m in the first year, £19m in the second year, £18m in the third year, reducing by £1m every year until the last year, when the cost incurred will be at a rate of £1m.

The cost to consumers who will have to spend more time filling in paperwork and talking to their financial advisers is expected to be incurred at a rate of £10m in the first year, increasing by 3% per annum over the twenty year period.


The benefits are estimated as follows:

The benefit to consumers who are less likely to buy inappropriate policies is estimated to be received at a rate of £30m in the first year, £33m in the second year, £36m in the third year and so on, rising by £3m per year until the end of twenty years.

The benefit to companies who will spend less time dealing with complaints from customers is estimated to be received at a rate of £12m per annum for twenty years.

Calculate the net present value of the benefit or cost of the regulations in their first twenty years at a rate of interest of 4% per annum effective. Assume that all costs and benefits occur continuously throughout the year.

[12]

11 (i) Describe the characteristics of an index-linked government bond. [3]

(ii) On 1 July 2002, the government of a country issued an index-linked bond of term seven years. Coupons are paid half-yearly in arrears on 1 January and 1 July each year. The annual nominal coupon is 2%. Interest and capital payments are indexed by reference to the value of an inflation index with a time lag of eight months.

You are given the following values of the inflation index.

Date Inflation index

November 2001 110.0 May 2002 112.3 November 2002 113.2 May 2003 113.8

The inflation index is assumed to increase continuously at the rate of 2½% per annum effective from its value in May 2003.

An investor, paying tax at the rate of 20% on coupons only, purchased the stock on 1 July 2003, just after a coupon payment had been made.

Calculate the price to this investor such that a real net yield of 3% per annum convertible half yearly is obtained and assuming that the investor holds the bond to maturity. [10]

[Total 13]

CT1 S2006 6

12 A pension fund has the following liabilities: annuity payments of £160,000 per annum to be paid annually in arrears for the next 15 years and a lump sum of £200,000 to be paid in ten years. It wishes to invest in two fixed-interest securities in order to immunise its liabilities. Security A has a coupon rate of 8% per annum and a term to redemption of eight years. Security B has a coupon rate of 3% per annum and a term to redemption of 25 years. Both securities are redeemable at par and pay coupons annually in arrear.

(i) Calculate the present value of the liabilities at a rate of interest of 7% per annum effective. [2]

(ii) Calculate the discounted mean term of the liabilities at a rate of interest of 7% per annum effective. [4]

(iii) Calculate the nominal amount of each security that should be purchased so that both the present value and discounted mean terms of assets and liabilities are equal. [7]

(iv) Without further calculation, comment on whether, if the conditions in (iii) are fulfilled, the pension fund is likely to be immunised against small, uniform changes in the rate of interest. [2]

[Total 15]

END OF PAPER


EXAMINATION

September 2006

Subject CT1 — Financial Mathematics Core Technical

EXAMINERS’ REPORT

Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. M A Stocker Chairman of the Board of Examiners November 2006

© Faculty of Actuaries © Institute of Actuaries

Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’Report

Page 2

Comments As in many recent diets, the questions requiring verbal reasoning (e.g. Question 4(i)) tended not to be well answered with candidates producing vague statements which did not demonstrate that they understood the relevant points Please note that differing answers may be obtained from those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. Comments on solutions presented to individual questions for this September 2006 paper are given below. Question 1 Generally well answered. To gain full marks candidates were required to specify the difference between futures and options rather than just defining each contract separately. Question 2 Well answered. This was a question where some candidates were penalised if answers had been rounded excessively. Question 3

Generally well answered. Another possible solution is to use 1 0.6 1 0.6 0.1111 1.07143

ijf

+ + ×+ = =

+

which leads to the same answer. Question 4 For full marks in part (i), an answer should have included a description of the ‘risk-free’ concept (rather than just saying arbitrage profits are impossible). Many students had difficulty with part (ii). Question 5 Full marks were given if either 365 or 365.25 days were used in the calculation. Most students scored well on this question. Question 6 This question was well answered. For full marks, candidates were required to show detailed steps in deriving the result required including a definition of the initial terms used and a correct explanation of the relevance of the independence assumption.

Subject CT1 (Financial Mathematics Core Technical) — September 2006 — Examiners’ Report

Page 3

Question 7 This question was poorly answered to the surprise of the examiners. Many candidates struggled to deal with the linked internal rate of return. Question 8 Well answered. Question 9 This question appeared to reward candidates who had a good understanding of the topic. Whilst the best candidates usually scored close to full marks on this question, weaker or less-prepared candidates often scored very badly. Whilst the question did state that payments were made monthly, the examiners recognised that there was some potential for misinterpretation as to the frequency of the loan repayments in part (e) and took this into account. Thus students who used the formula 30 100,000Xa =

with ( )12i = 6% & i =6.168% to get an answer of £7,396 in this part were awarded full marks. Question 10 Generally well answered. Question 11 This was the worst answered question on the paper by some margin with very few candidates scoring close to full marks. This may be because this type of question has not appeared in recent diets. Candidates needed to show that they could derive logically the amounts that will be paid, the real values of those amounts and their present values in real terms. Appropriate formulae then needed to be developed. Question 12 Many candidates answered this question well although a minority scored very badly (possibly due to time pressure).


Page 4

1 (i) A future is a contract binding buyer and seller to deliver or take delivery of an asset at a given price at a given time in the future. An option is a contract that gives the buyer the option to deliver or take delivery of the asset at the given price. The seller of the option must deliver/take delivery if the buyer of the option wishes to exercise the option.

(ii) Convertibles have option-like characteristics because they give the holder the

option to purchase equity in a company on pre-arranged terms. 2 The accumulated value is 3 2 14 2 2s s s+ +

( )3 2 14 2 2i s s s= + +δ

( )0.04 4 3.1216 2 2.0400 20.039221

18.9352

= × + × +

=

3 (a) The money rate of return is i where (1+i) = 11.1/10 i = 0.11 or 11%

(b) The rate of inflation is f where (1+f) = 120/112 f = 0.07143 or 7.143%

(c) The net real rate of return per annum is j

where 0.6 0.6 0.11 0.071431 1.07143

i fjf− × −

= =+

= −0.005068 or −0.5068%

4 (i) The no arbitrage assumption means that it is assumed that an investor is unable

to make a risk-free trading profit.

(ii) In all states of the world, security B pays 80% of A. Therefore its price must be 80% of A’s price, or the investor could obtain a better payoff by only purchasing one security and make risk-free profits by selling one security short and buying the other. The price of B must therefore be 16p.


Page 5

5 (i) (a) Let the answer be t days 3,600(1 + 0.06 × t/365) = 4,000 t = 675.9 days (b) Let the answer be t days

3,600 ( )4

3650.0641

t+ = 4,000

(4t/365) ln(1.015) = ln (4,000/3,600) t = 645.7 days (c) Let the answer be t days

3,600 ( )12

3650.06121

t+ = 4,000

(12t/365) ln(1.005) = ln (4,000/3,600) t = 642.5 days (ii) (i)(a) takes longest because, under conditions of simple interest, interest does

not earn interest. 6 (i) Let ti be the (random) rate of interest in year t . Let 10S be the accumulation of

the unit investment after 10 years: ( ) ( )( ) ( )10 1 2 101 1 1E S E i i i⎡ ⎤= + + +⎣ ⎦… ( ) [ ] [ ] [ ] { }10 1 2 101 1 1 as tE S E i E i E i i= + + +… are independent [ ]tE i j= ( ) ( )10 10

10 1 1.07 1.96715E S j∴ = + = =

(ii) ( ) ( )( ) ( ) 2210 1 2 101 1 1E S E i i i⎡ ⎤⎡ ⎤= + + +⎣ ⎦⎢ ⎥⎣ ⎦

…

( ) ( ) ( )2 2 2

1 2 101 1 1E i E i E i= + + +… (using independence) ( ) ( ) ( )2 2 2

1 1 2 2 10 101 2 1 2 1 2E i i E i i E i i= + + + + + +…


Page 6

( ) 1021 2 t tE i i⎡ ⎤= + +⎣ ⎦ ( )102 21 2 j s j= + + +

as [ ] [ ]22 2 2

i t tE i V i E i s j⎡ ⎤ = + = +⎣ ⎦

[ ] ( ) ( )10 202 2Var 1 2 1nS j s j j∴ = + + + − +

( ) ( )10 2021 2 0.07 0.016 0.07 1.07 0.5761= + × + + − =

(iii) If 1,000 units had been invested, the expected accumulation would have been

1,000 times bigger. The variance would have been 1,000,000 times bigger.

7 (i) (a) ( )2 450 500 8001 1.015%600 450 40 500 100

i i+ = ⇒ =+ +

(b) First sub-interval is first year. Money weighted rate of return is 1i

where ( )1 14501 25%600

i i+ = ⇒ = −

Second sub-interval is second year. Money weighted rate of return is 2i

where ( ) ( )1

22 2490 1 100 1 800i i+ + + =

Then ( )1

22

2100 100 4 490 ( 800) 100 1256.18471

2 490 980i

− ± − × × − − ±+ = =

×

= 1.17978 (taking positive root) ( )2 21 1.39188 39.188%i i+ = ⇒ = Linked internal rate of return is i where ( )21 0.75 1.39188 2.1719%i i+ = × ⇒ = (ii) The linked IRR is higher because it relies on two money weighted rates of

return. With the calculation of the second money weighted rate of return, there is more money in the fund when the fund is performing well (in the second half of the year).


Page 7

8 (i) ( ) [ ]5 52 2 31 1

2 3 00

150 100exp 100exp 100exp 12.5 41.667at bt dt at bt a b⎧ ⎫⎪ ⎪ ⎡ ⎤= + = + = +⎨ ⎬ ⎣ ⎦⎪ ⎪⎩ ⎭∫

( ) [ ]10 102 2 31 1

2 3 00

230 100exp 100exp 100exp 50 333.333at bt dt at bt a b⎧ ⎫⎪ ⎪ ⎡ ⎤= + = + = +⎨ ⎬ ⎣ ⎦⎪ ⎪⎩ ⎭∫

ln(1.5) 12.5 41.667a b= + ln(2.3) 50 333.333a b= + The second expression less four times the first expression gives: ln(2.3) 4ln(1.5) 166.667 0.0047337b b− = ⇒ = −

ln(2.3) 333.333 0.0047337 0.048216250

a − ×−= =

(ii) 10100 230 10 ln 2.3 0.08329e δ = ⇒ δ = ⇒ δ =

(iii) Present Value 10

0.05 0.08329

0

20 t te e dt−= ∫

10

0.03329

0

20 te dt−= ∫

100.03329

0

200.03329

te−⎡ ⎤= ⎢ ⎥

−⎢ ⎥⎣ ⎦

20 8.5058 170.116= × =

9 (a) Premiums were expected to accumulate to ( )12

301,060s at 7% ( ) 3012

1,060 1,060 1.037525 94.4608 £103,885.77i sd

= = × × =

(b) Premiums would have accumulated to ( )12

301,060s at 4% ( ) 3012

1,060 1,060 1.021537 56.0849 £60,730.37i sd

= = × × =

The shortfall is 100,000 – 60,730.37 = £39,269.63


Page 8

(c) Accumulation will be

( ) ( ) ( )1012 1220 4% 10 4%

1,060 1.04 5,000s s+

( ) ( ) ( )10

20 1012 121,060 1.04 5,000

1,060 1.021537 29.7781 1.48024 5,000 1.021537 12.0061

£109,053.12

i is sd d

= +

= × × × + × ×

=

Therefore the excess is £9,053.12 (d) The investor has earned a return of 4 % by investing extra premiums in the

investment policy. The investor could have obtained a lower present value of total payments on the loan by paying off part of the loan instead. This is because the interest being paid on the loan was greater than the interest he was earning on his premiums.

(e) If he had repaid the loan by a level annuity, the annual instalment would have

been X where

360 100,00012X a = at 0.5% (or ( )12

30100,000Xa = with ( )12i = 6% & i = 6.168%)

360

12 100,000 1,200,000 £7,194.61166.7916

Xa×

= = =

10 Present value of companies’ and consumers’ costs is (in £ million)

( )( )2 2 3 19 2050 10 1.03 1.03 1.03i v v v v+ + + + +δ

…

( ) ( )( )

( )( )

2 19

20

60 1 1.03 1.03 1.03

1 1.03 1 1.80611 0.4563960 1.019869 60 0.961541 1.03 1 1.03 0.96154

1.019869 60 0.96154 18.27680 1075.383

i v v v v

vi vv

= + + + +δ

− − ×⎛ ⎞= = × × ×⎜ ⎟δ − − ×⎝ ⎠

= × × × =

…


Page 9

Present value of costs to financial advisors (in £ million)

( )2 3 2060 19 18i v v v v+ + + +δ

…

( )

( ) ( )

2 3 20

20 20 20 20

40 20 19 18

40 21 40 21

iv i v v v v

iv i ia Ia v a Ia

= + + + + +δ δ

= + − = + −δ δ δ

…

( )1.019869 40 0.96154 21 13.5903 125.1550

1.019869 198.7029 202.651

= × × + × −

= × =

Total PV of all costs = £1278.034 million Present value of benefits (in £ million)

( )2 3 202030 33 36 87 12i iv v v v a+ + + + +

δ δ…

( )

( )( )

( )

2 3 2020 20

2020

27 3 6 9 60 12

3 39

1.019869 3 125.1550 39 13.59031.019869 905.4867

i a v v v v a

i Ia a

+ + + + + +δ

= +δ

= × + ×

= ×

…

= 923.478 Net present value of costs = PV(costs) – PV(benefits) = 1278.034 – 923.478 = £354.556 million


Page 10

11 (i) • Payments guaranteed by government. • Can be various different indexation provisions but, in general, protection is

given against a fall in the purchasing power of money. • Fairly liquid (i.e. large issue size and ability to deal in large quantities)

compared with corporate issues, but not compared with conventional issues.

• Normally coupon and capital payments both indexed to increases in a given price index with a lag.

• Low volatility of return and low expected real return. • More or less guaranteed real return if held to maturity (can vary due to

indexation lag). • Nominal return is not guaranteed.

(ii) The first coupon the investor will receive will be on 31st December 2003. The

net coupon per £100 nominal will be:

0.8 1× × (Index May 2003/Index November 2001) = 113.80.8 1110

× ×

In real present value terms, this is ( )0.5

113.80.8110 1

v

r+

where r = 2.5% per annum and v is calculated at 1.5% (per half year) The second coupon on 30th June 2004 per £100 nominal will be

( )0.5113.80.8 1 1110

r× × +

In real present value terms, this is ( ) ( )2

0.5 113.80.8 1110 1

vrr

++

The third coupon on 31st December 2004 per £100 nominal will be

( )113.80.8 1 1110

r× × +

In real present value terms, this is ( )( )

3

1.5113.80.8 1110 1

vrr

++

Continuing in this way, the last coupon payment on 30 June 2009 per £100

nominal will be ( )5.5113.80.8 1 1110

r× × +

In real present value terms, this is ( )( )

125.5

6113.80.8 1110 1

vrr

++


Page 11

By similar reasoning, the real present value of the redemption payment is

( )( )

125.5

6113.8100 1110 1

vrr

++

The present value of the succession of coupon payments and the capital

payment can be written as:

( )( )( )

( )

( )

2 12 120.5

121.5%12 1.5%

1 113.8 0.8 1001101

1 113.8 0.8 1001.0124224 110

1.02185 0.8 10.9075 100 0.83639

94.3833

P v v v vr

a v

= + + + ++

= +

= × × + ×

=

…

12 (i) Present value of liabilities is 1015160,000 200,000a v+ at 7%

160,000 9.1079 200,000 0.50835£1,558,934

= × + ×=

(ii) Discounted mean term (DMT) of liabilities is

( )2 15 10

1015

1 160,000 2 160,000 15 160,000 200,000 10

160,000 200,000

v v v v

a v

× × + × × + + × × + × ×=

+

…

( ) 10

1510

15

160,000 200,000 10

160,000 200,000

Ia v

a v

× + × ×=

+

160,000 61.5540 200,000 10 0.508351,558,934

× + × ×=

10,865,3401,558,934

= = 6.9697 years (½ mark deducted for no units)


Page 12

(iii) Let the nominal amounts in each security equal A and B respectively. If the present values of assets and liabilities are to be equal then: ( ) ( )8 25

8 250.08 0.03 1,558,934A a v B a v+ + + = (1)

If the DMTs of the assets and liabilities are equal, then:

( )( ) ( )( )8 25

8 250.08 8 0.03 256.9697

1,558,934

A Ia v B Ia v+ + +=

or ( )( ) ( )( )8 25

8 250.08 8 0.03 25 10,865,340A Ia v B Ia v+ + + = (2)

From (1)

( ) ( )0.08 5.9713 0.58201 0.03 11.6536 0.18425 1,558,9341.059714 0.533858 1,558,934

A BA B

× + + × + =

⇒ + =

From (2)

( ) ( )0.08 24.7602 8 0.58201 0.03 112.3301 25 0.18425 10,865,3406.636896 7.976153 10,865,340

A BA B

× + × + × + × =

⇒ + =

Therefore

1,558,934 0.5338586.636896 7.976153 10,865,3401.059714

6.636896 0.533858 6.636896 1,558,9347.976153 10,865,3401.059714 1.059714

1,101,872.85 £237,8504.632647

B B

B

B

−⎛ ⎞ + =⎜ ⎟⎝ ⎠

× ×⎛ ⎞⇒ − = −⎜ ⎟⎝ ⎠

⇒ = =

1,558,934 0.533858 £1,351,2661.059714

BA −⎛ ⎞= =⎜ ⎟⎝ ⎠


Page 13

(iv) It appears that the asset payments are more spread out than the liability payments. The third condition for immunisation is that that convexity of the assets is greater than that of the liabilities, or that the asset times are more spread around the discounted mean term than the liability times. From observation is appears likely that this condition is met.

END OF EXAMINERS’ REPORT


EXAMINATION

12 April 2007 (am)





2. You must not start writing your answers in the booklet until instructed to do so by the

supervisor. 3. Mark allocations are shown in brackets. 4. Attempt all 11 questions, beginning your answer to each question on a separate sheet. 5. Candidates should show calculations where this is appropriate.





© Faculty of Actuaries CT1 A2007 © Institute of Actuaries

CT1 A2007—2

1 An investor pays £400 every half-year in advance into a 25-year savings plan. Calculate the accumulated fund at the end of the term if the interest rate is 6% per

annum convertible monthly for the first 15 years and 6% per annum convertible half-yearly for the final 10 years. [5]

2 The force of interest ( )tδ is a function of time and at any time, measured in years, is

given by the formula:

( )( )( )

0.04 0.01 0 4

0.12 0.01 4 8

0.06 8

t t t

t t t

t t

δ = + ≤ ≤

δ = − < ≤

δ = <

Calculate the present value at time t = 0 of a payment stream, paid continuously from

time t = 9 to t = 12, under which the rate of payment at time t is 0.0150 .te [6] 3 An ordinary share pays annual dividends. The next dividend is due in exactly eight

months’ time. This dividend is expected to be £1.10 per share. Dividends are expected to grow at a rate of 5% per annum compound from this level and are expected to continue in perpetuity. Inflation is expected to be 3% per annum. The price of the share is £21.50.

Calculate the expected effective annual real rate of return for an investor who

purchases the share. [7] 4 An investor entered into a long forward contract for a security five years ago and the

contract is due to mature in seven years’ time. The price of the security was £95 five years ago and is now £145. The risk-free rate of interest can be assumed to be 3% per annum throughout the 12-year period.

Assuming no arbitrage, calculate the value of the contract now if: (i) The security will pay dividends of £5 in two years’ time and £6 in four years’

time. [3] (ii) The security has paid and will continue to pay annually in arrear a dividend of

2% per annum of the market price of the security at the time of payment. [3] [Total 6]

CT1 A2007—3 PLEASE TURN OVER

5 In a particular bond market, n-year spot rates per annum can be approximated by the function 0.10.08 0.04 ne−− .

Calculate: (i) The price per unit nominal of a zero coupon bond with term nine years. [2] (ii) The four-year forward rate at time 7 years. [3] (iii) The three-year par yield. [3] [Total 8] 6 A fund had a value of £21,000 on 1 July 2003. A net cash flow of £5,000 was

received on 1 July 2004 and a further net cash flow of £8,000 was received on 1 July 2005. Immediately before receipt of the first net cash flow, the fund had a value of £24,000, and immediately before receipt of the second net cash flow the fund had a value of £32,000. The value of the fund on 1 July 2006 was £38,000.

(i) Calculate the annual effective money weighted rate of return earned on the

fund over the period 1 July 2003 to 1 July 2006. [3] (ii) Calculate the annual effective time weighted rate of return earned on the fund

over the period 1 July 2003 to 1 July 2006. [3] (iii) Explain why the values in (i) and (ii) differ. [2] [Total 8] 7 An insurance company has liabilities of £87,500 due in 8 years’ time and £157,500

due in 19 years’ time. Its assets consist of two zero coupon bonds, one paying £66,850 in four years’ time and the other paying £X in n years’ time. The current interest rate is 7% per annum effective.

(i) Calculate the discounted mean term and convexity of the liabilities. [5] (ii) Determine whether values of £X and n can be found which ensure that the

company is immunised against small changes in the interest rate. [5] [Total 10]

CT1 A2007—4

8 A company has borrowed £800,000 from a bank. The loan is to be repaid by level instalments, payable annually in arrear for 10 years from the date the loan is made. The annual repayments are calculated at an effective rate of interest of 8% per annum.

(i) Calculate the amount of the level annual payment and the total amount of

interest which will be paid over the 10 year term. [3] (ii) At the beginning of the eighth year, immediately after the seventh payment

has been made, the company asks for the term of the loan to be extended by two years. The bank agrees to do this on condition that the rate of interest is increased to an effective rate of 12% per annum for the remainder of the term and that payments are made quarterly in arrear.

(a) Calculate the amount of the new quarterly payment. (b) Calculate the capital and interest components of the first quarterly

instalment of the revised loan repayments. [6] [Total 9] 9 A property developer is constructing a block of offices. It is anticipated that the

offices will take six months to build. The developer incurs costs of £40 million at the beginning of the project followed by £3 million at the end of each month for the following six months during the building period. It is expected that rental income from the offices will be £1 million per month, which will be received at the start of each month beginning with the seventh month. Maintenance and management costs paid by the developer are expected to be £2 million per annum payable monthly in arrear with the first payment at the end of the seventh month. The block of offices is expected to be sold 25 years after the start of the project for £60 million.

(i) Calculate the discounted payback period using an effective rate of interest of

10% per annum. [7] (ii) Without doing any further calculations, explain whether your answer to (i)

would change if the effective rate of interest were less than 10% per annum. [3] [Total 10]

CT1 A2007—5

10 A loan is issued bearing interest at a rate of 9% per annum and payable half-yearly in arrear. The loan is to be redeemed at £110 per £100 nominal in 13 years’ time.

(i) The loan is issued at a price such that an investor, subject to income tax at

25%, and capital gains tax at 30%, would obtain a net redemption yield of 6% per annum effective. Calculate the issue price per £100 nominal of the stock. [5]

(ii) Two years after the date of issue, immediately after a coupon payment has

been made, the investor decides to sell the stock and finds a potential buyer, who is subject to income tax at 10% and capital gains tax at 35%. The potential buyer is prepared to buy the stock provided she will obtain a net redemption yield of at least 8% per annum effective.

(a) Calculate the maximum price (per £100 nominal) which the original

investor can expect to obtain from the potential buyer. (b) Calculate the net effective annual redemption yield (to the nearest 1%

per annum effective) that will be obtained by the original investor if the loan is sold to the buyer at the price determined in (ii) (a).

[10] [Total 15] 11 £80,000 is invested in a bank account which pays interest at the end of each year.

Interest is always reinvested in the account. The rate of interest is determined at the beginning of each year and remains unchanged until the beginning of the next year. The rate of interest applicable in any one year is independent of the rate applicable in any other year.

During the first year, the annual effective rate of interest will be one of 4%, 6% or 8%

with equal probability. During the second year, the annual effective rate of interest will be either 7% with

probability 0.75 or 5% with probability 0.25. During the third year, the annual effective rate of interest will be either 6% with

probability 0.7 or 4% with probability 0.3. (i) Derive the expected accumulated amount in the bank account at the end of

three years. [5] (ii) Derive the variance of the accumulated amount in the bank account at the end

of three years. [8] (iii) Calculate the probability that the accumulated amount in the bank account is

more than £97,000 at the end of three years. [3] [Total 16]

END OF PAPER


EXAMINATION

April 2007


EXAMINERS’ REPORT

Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. M A Stocker Chairman of the Board of Examiners June 2007


Subject CT1 (Financial Mathematics Core Technical) — April 2007 — Examiners’ Report

Page 2

Comments Please note that different answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. Q1. Whilst most candidates made a good attempt at this question on basic compound interest accumulation, comparatively few students completed the question without error. Q2. Well answered. Q3. Most students answered this question well although candidates were expected to note that the sum of the geometric progression would only converge if the rate of return was below the dividend growth rate. Depending on the interpolation used, the final answer can justifiably vary from that given. Q4. This proved to be the most difficult question on the paper. Other related methods to determine the answers were available e.g. calculating the forward price of each contract and working out the present value of the difference in these prices. Q5. Well answered. Q6. The calculations in parts (i) and (ii) were generally well done. Again, depending on the interpolation used, the final answer can justifiably vary from that given although the examiners penalised the use of too wide a range of interpolation. The explanation in part (iii) was very poorly handled. In such cases, the examiners are not simply looking for a statement lifted directly from the Core Reading. Instead, candidates are expected to apply the relevant theory to the actual situation described in the question.


Page 3

Q7. Generally well answered. Q8. This was the best answered question on the paper. Q9. Many candidates struggled with this question, firstly in determining when the various costs/payments would be made and then in manipulating the resulting equation(s). A common error was not to recognise that the DPP should be expressed as a whole number of months since payments at the relevant time were being made at monthly intervals. In part (ii) little credit was given for a correct conclusion without any accompanying explanation. Q10. This question seemed to provide a significant differentiation between candidates with many scoring well and a sizeable minority scoring very badly. This seemed surprising given that this topic is regularly examined. A common omission on part (ii)(b) was not to state whether a capital gain had been made. Q11. The workings for parts (i) and (ii) were often too brief (the questions said ‘Derive…’). Note that the final answer in part (ii) can justifiably vary significantly according to the rounding used in intermediate calculations. Part (iii) was poorly done with many candidates assuming a lognormal distribution for this discrete example.


Page 4

1 Fund after 25 years =

( )*% 20 3%

30 20400 1.03 400iS S× +

where ( )61 * 1.005i+ = 3.03775%i∗⇒ = per ½-year

( )30

301.0303775 1

@ 3.03775% 1.03037750.0303775

s⎡ ⎤−⎢ ⎥= ×⎢ ⎥⎣ ⎦

= 49.3215

( )20

201.03 1

@3% 1.030.03

S⎡ ⎤−⎢ ⎥= ×⎢ ⎥⎣ ⎦

= 27.6765

Hence fund = ( )20400 49.3215 1.03 400 27.6765× × + × = 35632.06 + 11070.60 = £46,702.66

2 (i) ( )012 0.01

950 .

t t dttPV e e dt− δ∫= ∫

where

( ) ( ) ( )4 8

0 0 4 80.04 0.01 0.12 0.01 0.06

t tt dt t dt t dt dtδ = + + − +∫ ∫ ∫ ∫

[ ]4 82 2

80 40.04 0.005 0.12 0.005 0.06 tt t t t t⎡ ⎤ ⎡ ⎤= + + − +⎣ ⎦ ⎣ ⎦

[ ] [ ] [ ]0.24 0.64 0.40 0.06 0.48t= + − + − = 0.06t


Page 5

Hence

12 0.01 0.069

50 .t tPV e e dt−= ∫

0.0512

950

te dt−

= ∫

12

0.05

9

500.05

te−−⎡ ⎤= ⎢ ⎥⎣ ⎦

= –548.812 + 637.628 = 88.816 3 Let i = money rate of return i′= real rate of return ( )( )1 1 1.03i i′⇒ + = + here

( ) ( )( )412 22 321.50 1 1.10 1.05 1.10 1.05 1.10i v v v= + ⋅ + × + × +

( )( )

( )4

121.0511.051

11 1.10

1i

i

i v∞

+

+

⎛ ⎞−⎜ ⎟= + × ⎜ ⎟−⎜ ⎟⎝ ⎠

( ) ( )8

12 1.051

1 11.1011 ii +

= ×−+

assuming i > 0.05

( )

812 1.05

1

1 119.545511 ii +

= ×−+

Try i = 10% RHS = 20.6456 11% RHS = 17.2566

20.6456 19.54550.10 0.01 0.1032520.6456 17.2566

i −⇒ = + × =

−

1.10325 comes from 1 7.1% p.a.1.03

i i i′ ′ ′⇒ + = ⇒ =


Page 6

4 (i) The current value of the forward price of the old contract is: ( ) ( ) ( )5 2 495 1.03 5 1.03 6 1.03− −× − − whereas the current value of the forward price of a new contract is: ( ) ( )2 4145 5 1.03 6 1.03− −− − Hence, current value of old forward contract is: ( )5145 95 1.03 £34.87− = (ii) The current value of the forward price of the old contract is: ( ) ( )12 595 1.02 1.03 86.8376− = whereas the current value of the forward price of a new contract is: ( ) 7145 1.02 126.2312− = ⇒ current value of old forward contract is: 126.23 - 86.84 = £39.39 5 (i) Let kY = spot rate for k year term kP = Price per unit nominal for k year term 9 0.063737Y =

9

99

1 0.573441

PY

⎛ ⎞= =⎜ ⎟+⎝ ⎠


Page 7

(ii) ( )0.1 77 0.08 0.04 0.060137Y e−= − =

( )0.1 11

11 0.08 0.04 0.066685Y e−= − =

( ) ( )( )

( )( )

11 114 11

7,4 7 77

1 1.0666851

1 1.060137

Yf

Y

++ = =

+

= 1.35165 ∴ 4-year forward rate is 7.824% at time 7. (iii) 1 2 30.04381, 0.04725, 0.05037Y Y Y= = = ( )( )3 1 2 3 3

1 2 3 31 c Y Y Y YY v v v v= + + +

30.05016cY = i.e. 5.016% p.a.

6 (i) Work in £000’s MWRR is i such that:

( ) ( ) ( )3 221 1 5 1 8 1 38i i i+ + + + + = Try i = 5%, LHS = 38.223 i = 4%, LHS = 37.350 By interpolation i = 4.74% p.a. (ii) TWRR is i such that:

( )3 24 32 381 6.21% p.a.21 29 40

i i+ = × × ⇒ =

(iii) MWRR is lower than TWRR because of the large cash flow on 1/7/05; the

overall return in the final year is much lower than in the first 2 years, and the payment at 1/7/05 gives this final year more weight in the MWRR, but does not affect the TWRR.


Page 8

7 Let LPV be PV of liabilities, LDMT be DMT of liabilities, LC be convexity of liabilities.

(i) 8 1987,500 157,500LPV v v= + at 7% = 94,475.86

8 1987,500 8 157,500 19

94,475.86Lv vDMT × + ×

⇒ = at 7%

1, 234,857.5694,475.86

=

= 13.070615 years

10 2187,500 8 9 157,500 19 2094,475.86L

v vC × × + × ×= at 7%

17,657,158.7894,475.86

=

= 186.895985 (ii) Firstly, PVs should be equal: 466,850 94, 475.86nv Xv⇒ + = at 7%

43, 476.31507nXv⇒ = Secondly, DMTs should be equal

466,850 4 1, 234,857.56nv Xnv⇒ × + =

1,030,859.38nXnv⇒ = 23.710827n⇒ = years 43, 476.31507 1.07nX⇒ = × = 216,255.12


Page 9

Lastly, verify 3rd condition

( ) ( )( )2666,850 4 5 216,255.12 1 / 94,475.86nAC v n n v += × × + +

= 23,140,343.20/94,475.86 = 244.93393 > CL Hence, immunisation is achieved. 8 (i) 8%

10800,000 6.7101P a P= = × 119,223.26P⇒ = Total amount of interest = 10 × 119,223.26 – 800,000

= £392,232.60 (ii) (a) Capital o/s at start of 8th year = 119,223.26 8%

3 119, 223.26 2.5771 307, 250.26a = ∗ = Let new payment be P′ per annum, then (4)

512%

1.043938 3.6048 307, 250.26P a P′ ′= ∗ ∗ =

81,646.28P′⇒ = q'ly payment 20,411.57⇒ = (b) Capital o/s after 7 years = 307,250.26

Interest in 1st q'ly payment⇒ ( )1430,7250.26 1.12 1 8,829.56⎛ ⎞= ∗ − =⎜ ⎟

⎝ ⎠ capital component = 20,411.57 8,829.56 = 11,582.01⇒ −


Page 10

9 (i) The discounted payback period is the first point at which the present value of the income exceeds the present value of the outgoings. The present value of all payments and income up to time t is given by (working in £m)

( ) ( ) ( )

( ) ( ) ( )

1 12 2

1 1 1 12 2 2 12

1 12 2

1 1 12 2 2

12 12 12

12 12 12112

40 36 2 12

40 36 2 12

t t

t t

PV a v a v a

a v a v a

− − +

− −

= − − − +

⎛ ⎞= − − − + +⎜ ⎟⎝ ⎠

= ( )( )

1 1 0.52 2

1212

12 140 36 10tv

ia v v

−−− − + +

( )( )

12

12

1212

1 at 10%vai

−=

1 0.9534626 0.486340.0956897−

= =

⇒ 0.56758 = 1 - vt-0.5 ⇒ vt-0.5 = 0.43242

⇒ t = ( )( )

log 0.43242log 0.90909 0.5+

⇒ t ≥ 9.296

Hence, the discounted pay back period is 9 years and 4 months. (ii) If the effective rate of interest were less than 10% p.a. then the present values

of the income and outgo would both increase. However, the bigger impact would be on the present value of the income since the bulk of the outgo occurs in the early years when discounting has less effect. Hence, the DPP would decrease.

10 (i) ( )2 0.059126i =

( )10.091 0.75=0.061361.10

g t− = ×

( ) ( )2

11i t g⇒ < − ⇒ No capital gain


Page 11

Price of £100 nominal stock

( )2 1313

0.75 9 +110 at 6%a v= ×

= 0.75 × 9 × 1.014782 × 8.8527 + 110 × 0.46884 = 60.639 + 51.572 = £112.21 (ii) (a) ( )2 0.078461i =

( )10.091 0.90= 0.0736361.10

g t− = ×

( ) ( )2

11i t g⇒ > − ⇒ Capital gain

Price, P = 0.90 × 9 × ( ) ( )( )2 1111

110 110 0.35a P v+ − − × at 8%

0.90 9 1.019615 7.1390 0.65 110 0.428881 0.35 0.42888

P × × × + × ×⇒ =

− ×

89.62508 105.4550.849892

= =

(b) No capital gain made

112.21 = 0.75 × 9 × ( )2 22

105.455a v+

Try i = 3%, RHS = 112.41 i = 4%, RHS = 110.36 ⇒ yield = 3% p.a. to nearest 1%


Page 12

11 (i) Let 3 Accumulated fund after 3 years of investment of 1 at time 0S = ti = Interest rate for year t Then, fund after 3 years = 80,000 ( )( )( )3 1 2 380000 1 1 1S i i i= + + + ( ) ( )1

1 3 0.04+0.06+0.08 =0.06E i = ( )2 0.75 0.07+0.25 0.05=0.065E i = × × ( )3 0.7 0.06 + 0.3 0.04 = 0.054E i = × × Then: [ ] [ ]3 380000 80,000E S E S= ( )( )( )1 2 380,000 1 1 1E i i i⎡ ⎤= + + +⎣ ⎦ ( ) ( ) ( )1 2 380,000 1 . 1 . 1E i E i E i= + + + since 'ti s are independent = 80,000 × 1.06 × 1.065 × 1.054 = £95,188.85 (ii) Var[ ] [ ]2

3 380000 80,000 VarS S= ×

where [ ] [ ]( )223 3 3Var S E S E S⎡ ⎤= −⎣ ⎦

( ) ( ) ( )2 2 223 1 2 31 1 1E S E i i i⎡ ⎤⎡ ⎤ = + + +⎢ ⎥⎣ ⎦ ⎣ ⎦

( ) ( ) ( )2 2 21 2 31 . 1 . 1E i E i E i⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + + +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

using independence

[ ]( ) [ ]( )2 21 1 2 21 2 . 1 2E i E i E i E i⎡ ⎤ ⎡ ⎤= + + + +⎣ ⎦ ⎣ ⎦ [ ]( )2

3 31 2 E i E i⎡ ⎤⋅ + + ⎣ ⎦


Page 13

Now, ( ) ( )2 2 2 21

1 3 0.04 0.06 0.08 0.0038667E i = + + =

( )2 2 2

2 0.75 0.07 0.25 0.05E i = × + × 0.0043=

( )2 2 2

3 0.7 0.06 0.3 0.04 0.0030E i = × + × =

Hence, 2

3E S⎡ ⎤⎣ ⎦

( )1 2 0.06 0.0038667= + × + ( )1 2 0.065 0.0043× + × + ( )1 2 0.054 0.003× + × + =1.41631 Hence Var[ ]380,000S [ ]2

380,000 Var S=

( )( )2280,000 1.41631 1.18986= −

= 3,476,355 (iii) Note: 80,000 × 1.08 × 1.07 × 1.06 = 97,995 > 97,000 But, if in any year, the highest interest rate for the year is not achieved then the

fund after 3 years falls below £97,000. Hence, answer is probability that highest interest rate is achieved in each year

1 0.75 0.7 0.1753

= × × =



EXAMINATION












© Faculty of Actuaries CT1 S2007 © Institute of Actuaries

CT1 S2007—2

1 A 90-day government bill is purchased for £96 at the time of issue and is sold after 45 days to another investor for £97.90. The second investor holds the bill until maturity and receives £100.

Determine which investor receives the higher rate of return. [2] 2 An investor purchases a share for 769p at the beginning of the year. Halfway through

the year he receives a dividend, net of tax, of 4p and immediately sells the share for 800p. Capital gains tax of 30% is paid on the difference between the sale and the purchase price.

Calculate the net annual effective rate of return the investor obtains on the investment.

[4] 3 An insurance company offers a customer two payment options in respect of an

invoice for £456. The first option involves 24 payments of £20 paid at the beginning of each month starting immediately. The second option involves 24 payments of £20.50 paid at the end of each month starting immediately. The customer is willing to accept a monthly payment schedule if the annual effective interest rate per annum he pays is less than 5%.

Determine which, if any, of the payment options the customer will accept. [4] 4 State the characteristics of an equity investment. [4]

5 A one-year forward contract is issued on 1 April 2007 on a share with a price of 900p

at that date. Dividends of 50p per share are expected on 30 September 2007 and 31 March 2008. The 6-month and 12-month spot, risk-free rates of interest are 5% and 6% per annum effective respectively on 1 April 2007.

Calculate the forward price at issue, stating any assumptions. [4]

6 The annual effective forward rate applicable over the period t to t + r is defined as

,t rf where t and r are measured in years. 0,1f = 4%, 1,1f = 4.25% 2,1f = 4.5%, 2,2f = 5%. Calculate the following: (i) 3,1f [1] (ii) All possible zero coupon (spot) yields that the above information allows you

to calculate. [4] (iii) The gross redemption yield of a four-year bond, redeemable at par, with a 3%

coupon payable annually in arrears. [6] (iv) Explain why the gross redemption yield from the four-year bond is lower than

the one-year forward rate up to time 4, 3,1f [2] [Total 13]

CT1 S2007—3 PLEASE TURN OVER

7 The force of interest, ( )tδ , is a function of time and at any time t (measured in years) is given by

0.04 0.01 for 0 10

( )0.05 for 10

t tt

t+ ≤ ≤⎧

δ = ⎨ >⎩

(i) Derive, and simplify as far as possible, expressions for ( )v t where ( )v t is the

present value of a unit sum of money due at time t. [5]

(ii) (a) Calculate the present value of £1,000 due at the end of 15 years. (b) Calculate the annual effective rate of discount implied by the

transaction in (a). [4] (iii) A continuous payment stream is received at a rate of 0.0120 te− units per

annum between t = 10 and t = 15. Calculate the present value of the payment stream.

[4] [Total 13]

8 A pension fund makes the following investments (£m):

1 January 2004

1 July 2004 1 January 2005 1 January 2006

12.5 6.6 7.0 8.0

The rates of return earned on money invested in the fund were as follows:

1 January 2004 to 30 June 2004

1 July 2004 to 31 December 2004

1 January 2005 to 31 December 2005

1 January 2006 to 31 December 2006

5% 6% 6.5% 3%

You may assume that 1 January to 30 June and 1 July to 31 December are precise half year periods.

(i) Calculate the linked internal rate of return per annum over the three years from

1 January 2004 to 31 December 2006, using semi-annual sub-intervals. [3]

(ii) Calculate the time weighted rate of return per annum over the three years from 1 January 2004 to 31 December 2006. [3]

(iii) Calculate the money weighted rate of return per annum over the three years

from 1 January 2004 to 31 December 2006. [4]

(iv) Explain the relationship between your answers to (i), (ii) and (iii) above. [2] [Total 12]

CT1 S2007—4

9 The expected effective annual rate of return from a bank’s investment portfolio is 6% and the standard deviation of annual effective returns is 8%. The annual effective returns are independent and (1+ ti ) is lognormally distributed, where ti is the return in year t.

Deriving any necessary formulae:

(i) calculate the expected value of an investment of £2 million after ten years. [6]

(ii) calculate the probability that the accumulation of the investment will be less

than 80% of the expected value. [3] [Total 9]

10 A government is holding an inquiry into the provision of loans by banks to consumers

at high rates of interest. The loans are typically of short duration and to high risk consumers. Repayments are collected in person by representatives of the bank making the loan. Campaigners on behalf of the consumers and campaigners on behalf of the banks granting the loans are disputing one particular type of loan. The initial loans are for £2,000. Repayments are made at an annual rate of £2,400 payable monthly in advance for two years.

The consumers’ association case The consumers’ association asserts that, on this particular type of loan, consumers who make all their repayments pay interest at an annual effective rate of over 200%. The banks’ case The banks state that, on the same loans, 40% of the consumers default on all their remaining payments after exactly 12 payments have been made. Furthermore half of the consumers who have not defaulted after 12 payments default on all their remaining payments after exactly 18 payments have been made. The banks also argue that it costs 30% of each monthly repayment to collect the payment. These costs are still incurred even if the payment is not made by the consumer. Furthermore, with inflation of 2.5% per annum, the banks therefore assert that the real rate of interest that the lender obtains on the loan is less than 1.463% per annum effective. (i) (a) Calculate the flat rate of interest paid by the consumer on the loan

described above.

(b) State why the flat rate of interest is not a good measure of the cost of borrowing to the consumer. [4]

(ii) Determine, for each of the cases above, whether the assertion is correct. [10]

[Total 14]

CT1 S2007—5

11 A pension fund has liabilities to pay pensions each year for the next 60 years. The pensions paid will be £100m at the end of the first year, £105m at the end of the second year, £110.25m at the end of the third year and so on, increasing by 5% each year. The fund holds government bonds to meet its pension liabilities. The bonds mature in 20 years time and pay an annual coupon of 4% in arrears.

(i) Calculate the present value of the pension fund’s liabilities at a rate of interest

of 3% per annum effective. [4]

(ii) Calculate the nominal amount of the bond that the fund needs to hold so that the present value of the assets is equal to the present value of the liabilities. [3]

(iii) Calculate the duration of the liabilities. [6]

(iv) Calculate the duration of the assets. [4]

(v) Using your calculations in (iii) and (iv), estimate by how much more the value of the liabilities would increase than the value of the assets if there were a reduction in the rate of interest to 1.5% per annum effective. [4]

[Total 21]

END OF PAPER


EXAMINATION

September 2007


MARKING SCHEDULE

Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. M A Stocker Chairman of the Board of Examiners December 2007



Page 2

Comments Please note that different answers may be obtained from those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. It should be noted that the rubric of the examination paper does ask for candidates to show their calculations where this is appropriate. Candidates often failed to show sufficient clarity and detail in their working and lost marks as a result. Q1. Well answered. Q2. Well answered. Q3. Whilst this question was generally answered well, some candidates lost marks by not stating the conclusions that arose from their calculations i.e. that neither deal was acceptable. Q4. This question was very poorly answered which was disappointing given that this was a bookwork question. Q5. Reasonably well answered but some candidates failed to obtain full marks by not stating the required assumption. Q6. Parts (i) and (ii) were well answered but part (iii) was a good differentiator with weaker candidates failing to recognise the correct method for calculating the gross redemption yield. As with many previous diets, many candidates in part (iv) had great difficulty in giving a clear explanation of their calculations.


Page 3

Q7. Generally well answered. Some candidates lost marks by not giving an explicit formula for v(t) when t ≤ 10. Q8. This question was very poorly answered to the surprise of the examiners who felt that the question should have been relatively straightforward. Q9. Part (i) can be done much more simply than by using the method given in this report but the calculations given would still need to be done for part (ii). Q10. This question was the worst answered on the paper. Part (ii) did successfully differentiate between candidates with weaker candidates appearing to struggle to apply the theory to a real-life situation. Q11. The first three parts were generally answered well by the candidates who attempted the question. Many struggled to complete part (iv) although it is possible that this was due to time pressure. When calculating DMTs, candidates were expected to give the answer in terms of the correct units.


Page 4

1 The first investor receives the higher rate of return if:

97.9 10096 97.9

>

This inequality does not hold, therefore the second investor receives the higher rate of

return. 2 Start by working in half years. The half yearly effective return is i such that:

769 = 4v + 800v – 0.3(800 – 769)v 769 = (804 - 240 + 230.7)v

v = 769 0.967661794.7

= therefore i = 3.3420%

Annual effective rate is (1.033422 – 1) = 6.7957%

3 The annual rate of payment for the first deal is 240.

This deal is acceptable if:

240 ( )122

a < 456 at a rate of interest of 5%

240 ( )122

a = 240×1.8594×1.026881 = 458.252

Therefore first deal is not acceptable

The annual rate of payment on the second deal is 246. This deal is acceptable if:

246 ( )122

a = 246×1.8594×1.022715 = 467.803

Therefore second deal is also not acceptable

4 Main characteristics of equity investments:

• Issued by commercial undertakings and other bodies. • Entitle holders to receive all net profits of the company in the form of

dividends after interest on loans and other fixed interest stocks has been paid. • Higher expected returns than for most other asset classes …


Page 5

• …but risk of capital losses • … and returns can be variable. • Lowest ranking form of finance. • Low initial running yield… • … but dividends should increase with inflation. • Marketability varies according to size of company. • Voting rights in proportion to number of shares held.

5 Assuming no arbitrage: Present value of dividends is (in£): 0.5v1/2 (at 5%) + 0.5v (at 6%) = 0.5(0.97590+0.94340) = 0.95965 Hence forward price is: F = (9-0.95965)×1.06 = £8.5228 6 (i) 3,1f is such that 1.045× 3,1f = 1.052. Therefore 3,1f = 5.5024% (ii) One-year spot rate is same as one-year forward rate = 4%

Two-year spot rate is i2 such that (1+ 2i )2 = 1.04×1.0425. Therefore 2i = 4.1249% Three-year spot rate is i3 such that (1+ 3i )3 = 1.04×1.0425×1.045. Therefore 3i = 4.2498% Four year spot rate is such that (1+ 4i )4 = 1.04×1.0425×1.045×1.055024 Therefore 4i = 4.5615%

(iii) Present value of the payments from the bond is:

P = 3(1.04-1 + 1.041249-2 + 1.042498-3 + 1.045615-4) + 100×1.045615-4 Therefore P = 3(0.96154 + 0.92234 + 0.88262 + 0.83659) + 100×0.83659 = 94.468 Equation of value to find the gross redemption yield from the bond is such that: 94.468 = 3 4a + 100v4 Try i = 4.5% v4 = 0.83856, 4a = 3.58753, RHS = 94.619


Page 6

Try i = 5% v4 = 0.82270, 4a = 3.5460, RHS = 92.908 Interpolation: Yield = 0.045 + 0.005× (94.619 – 94.468) /(94.619 – 92.908) = 4.544%

(iv) The yield from the bond is lower than the one-year forward rate up to time 4

because the bond can be seen to be a series of zero coupon bonds (1 year, 2 years etc.) each with lower yields than the forward rate. The gross redemption yield from the bond is, in effect, an average of spot rates that are themselves a weighted average of earlier forward rates.

7 (i) For t ≤ 10

( )2 2

0 00.04 0.0050.04 0.01 0.04 0.005

tt s ssds t tv t e e e⎡ ⎤− +− + − −⎣ ⎦∫= = =

For t > 10

( ) ( ) [ ] ( ) ( )10 100.05 0.05 0.05 10 0.4 0.050.9 0.910

t tds s t tv t v e e e e e e− − − − − +− −∫= = = = (ii) (a) Present value ( )0.4 0.05 15 1.151000 1000 316.637e e− + × −= = = (b) 151000(1 ) 316.637 7.380%d d− = ⇒ =

(iii) Present value 15 (0.4 0.05 ) 0.0110

20t te e dt− + −= ∫

15 0.4 0.0610

20 te e dt− −= ∫

( )150.06

0.4 0.4

10

20 20 6.77616 + 9.14686 31.7830.06

tee e−

− −⎡ ⎤= = − =⎢ ⎥

−⎢ ⎥⎣ ⎦


Page 7

8 (i) Linked internal rate of return is found by linking the money weighted rate of return from the sub-periods.

(LIRR)3 = 1.05×1.06×1.065×1.03

Therefore LIRR = 0.06879 or 6.879%

(ii) The TWRR requires the value of the fund every time a payment is made. Size of the fund after six months is: 12.5× (1.05) = 13.125 Size of the fund after one year is: (13.125 + 6.6) ×1.06 = 20.909 Size of the fund after two years is: (20.909 + 7) ×1.065 = 29.723 Size of the fund after three years is: (29.723 + 8) × 1.03 = 38.855

The TWRR is i where i is the solution to: (1+i)3 = (13.125/12.5) × [20.909/(13.125+6.6)] × [29.723/(20.909+7)]

× [38.855/(29.723+8)] or just use the rates of return given to give: (1+i)3 = 1.05×1.06×1.065×1.03 giving i = 6.879%

(iii) For MWRR, we need to know the size of the fund at the end of the period. We can use the values above to give: MWRR is solution to: 12.5(1+i)3 + 6.6(1+i)2.5 + 7(1+i)2 + 8(1+i) = 38.855

Solve by iteration and interpolation, starting with i = 7%.

i = 7% gives LHS = 39.704 i = 6% gives LHS = 38.868 i = 5.5% gives LHS = 38.454

Interpolate between 5.5% and 6%. i = 0.055 + 0.005× (38.855-38.454)/(38.868-38.454) = 5.98%

(iv) (i) and (ii) are the same because there are no cash flows within sub-periods to “distort” the LIRR away from the TWRR. The MWRR is lower because the fund has a smaller amount of money in it at the beginning when rates of return are higher.


Page 8

9 (i) ( ) ( )21 ~ ,ti Lognormal+ μ σ

( ) ( )2ln 1 ~ ,ti N+ μ σ

( ) ( ) ( ) ( ) ( )10 2ln 1 ln 1 ln 1 ln 1 ~ 10 ,10t t t ti i i i N+ = + + + + + + μ σ…

since 'ti s are independent

( ) ( )10 21 ~ 10 ,10ti Lognormal+ μ σ

[½] for correct use of independence assumption

( )

( ) ( ) ( )

( )

2

2 2 2

22 2

2

1 exp 1.062

1 exp 2 exp 1 0.08

0.08 exp 1 0.00567981.06

t

t

E i

Var i

⎛ ⎞σ+ = μ + =⎜ ⎟⎜ ⎟

⎝ ⎠

⎡ ⎤+ = μ +σ σ − =⎣ ⎦

⎡ ⎤= σ − ∴σ =⎣ ⎦

0.0056798 0.0056798exp 1.06 ln1.06 0.0554292 2

⎛ ⎞μ + = ⇒ μ = − =⎜ ⎟⎝ ⎠

10 0.55429μ = , 210 0.056798σ = Let S10 be the accumulation of one unit after 10 years:

( )100.056798exp 0.55429 1.790848

2E S ⎛ ⎞= + =⎜ ⎟

⎝ ⎠

Expected value of investment ( )102,000,000 £3.5817E S m= = (ii) We require [ ]10 0.8 1.790848 1.4327P S < × =

[ ]10ln ln1.4327P S < where 10ln S ~N(0.55429,0.056798)

( ) ln1.4327 0.554290,10.056798

P N −⎡ ⎤⇒ <⎢ ⎥

⎣ ⎦

( )0,1 0.8171P N⎡ ⎤⇒ < −⎣ ⎦ = 0.207 21%≈


Page 9

10 (i) (a) The flat rate of interest is: (2×2,400 – 2,000)/(2×2,000) = 70% (b) The flat rate of interest is not a good measure of the cost of borrowing

because it takes no account of the timing of payments and the timing of repayment of capital.

(ii) If the consumers’ association is correct, then the present value of the

repayments is greater than the loan at 200%

i.e. ( ) 2122,000 2, 400 i a

d<

i =2; 2a = 0.44444; ( )12d = 1.04982 gives RHS = 2,032 The consumers’ association is correct.

If the banks are correct, then the present value of the payments received by the bank, after expenses, is less than the amount of the loan at a nominal (before inflation) rate of interest of (1.01463×1.025 -1) per annum effective = 0.04.

i.e. ( ) ( ) ( ) ( )2 1.5 1 212 12 12 122,000 720 720 960 0.3 2, 400i i i ia a a a

d d d d> + + − ×

( )12i

d = 1.021529; 2a = 1.8861; 1a = 0.9615;

1.5

1.51 1.04 1.4283

0.04a

−−= =

So RHS = 720×1.021529×1.8861+ 720×1.021529×1.4283 + 960×1.021529×0.9615 – 0.3×2,400×1.021529×1.8861 = 1,387.23+ 1,050.52 + 942.91 – 1,387.23 = 1,993.43 Therefore, the banks are also correct.

11 (i) Present value of the fund’s liabilities (in £m) is:

( )( ) ( )( )

2 2 3 59 60

2 59

100 1.05 1.05 1.05

100 1 1.05 1.05 1.05

v v v v

v v v v

+ + + +

= + + + +

…

…

( ) ( )( )

6060 1.051.03

1.051.03

1-1 1.05100 100 0.97087

1 1.05 1-

97.087 111.7795 £10,852

vv

v

m

⎛ ⎞⎛ ⎞− ⎜ ⎟⎜ ⎟= = × ⎜ ⎟⎜ ⎟− ⎜ ⎟⎝ ⎠ ⎝ ⎠= × =

(ii) Let the nominal holding of bonds = N in £m

The present value of the bonds must equal £10,852m


Page 10

Therefore 20200.04 10,852Na Nv+ = at 3%

20a = 14.8775, v20 = 0.55368 So 10,852 = 0.04N×14.8775 + N×0.55368 N = 10,852 /(0.04×14.8775 + 0.55368) = £9,446.54m

(iii) The numerator for the duration of the liabilities can be expressed as follows:

100v (1×1 + 1.05v×2 + 1.052v2 ×3+…+1.0559v59×60)

= 1.031.05

100 v (1.05v×1 + 1.052v2×2 + 1.053v3 ×3+…+1.0560v60×60)

The part inside the brackets can be regarded as ( )60Ia evaluated at a rate of interest i such that v = 1.05/1.03; the discount factor outside the brackets should be evaluated at 3% 1.031.05

100 v = 1001.05

= 95.2381

For the ( )60Ia function, v = 1.019417; i = -0.019048; ( ) 601 i a+ = 111.7727

( )60

60111.7727 60 1.019417 = 4118.567

0.019048Ia − ×

=−

Therefore numerator for duration is: 95.2381×4118.567 = 392,244 Therefore the duration is: 392,244/10,852 = 36.1 years.

(iv) The duration of the assets can be expressed as the sum of payments times time

of receipt times present value factors divided by total present value. The equation for the numerator is 0.04×9,446.54 ( )20Ia + 9,446.54×20×v20 at 3% ( )20Ia = 141.6761, v20 = 0.55368 Numerator is: 158,141 Therefore the duration is: 158,141/10,852 = 14.6 years.

(v) Duration of the liabilities is 36.1 years. Therefore volatility of the liabilities is:

36.1/1.03 = 35. If there were a reduction in interest rates to 1.5%, the liabilities would increase in value by approximately 35×1.5 = 52.5%


Page 11

Duration of the assets is 14.6 years. Therefore volatility of the assets is: 14.6/1.03 = 14.2. If there were a reduction in interest rates to 1.5%, the assets would increase in value by approximately 14.2×1.5 = 21.3%.

The liabilities would increase in value by an additional 31.2% of their original value i.e. by £3,386 more than the value of the assets.



EXAMINATION

15 April 2008 (am)










In addition to this paper you should have available the 2002 edition of the Formulae and Tables and your own electronic calculator from the approved list.


CT1 A2008—2

1 An eleven month forward contract is issued on 1 March 2008 on a stock with a price of £10 per share at that date. Dividends of 50 pence per share are expected to be paid on 1 April and 1 October 2008.

Calculate the forward price at issue, assuming a risk-free rate of interest of 5% per

annum effective and no arbitrage. [4] 2 Describe the characteristics of the following investments: (a) Eurobonds (b) Certificates of deposit [4] 3 A mortgage company offers the following two deals to customers for twenty-five year

mortgages. Product A A mortgage of £100,000 is offered with level repayments of £7,095.25 made annually

in arrear. There are no arrangement or exit fees. Product B A mortgage of £100,000 is offered whereby a monthly payment in advance is

calculated such that the customer pays an effective rate of return of 4% per annum ignoring arrangement and exit fees. In addition the customer also has to pay an arrangement fee of £6,000 at the beginning of the mortgage and an exit fee of £5,000 at the end of the twenty-five year term of the mortgage.

Compare the annual effective rates of return paid by customers on the two products. [8]

4 A loan of nominal amount £100,000 is to be issued bearing coupons payable quarterly

in arrear at a rate of 7% per annum. Capital is to be redeemed at 108% on a coupon date between 15 and 20 years after the date of issue, inclusive. The date of redemption is at the option of the borrower.

An investor who is liable to income tax at 25% and capital gains tax at 35% wishes to

purchase the entire loan at the date of issue. Calculate the price which the investor should pay to ensure a net effective yield of at

least 5% per annum. [8]


5 The n –year spot rate of interest, ,ni is given by: ni a bn= − for 1,2 and 3,n = and where a and b are constants.

The one-year forward rates applicable at time 0 and at time 1 are 6.1% per annum effective and 6.5% per annum effective respectively. The 4–year par yield is 7% per annum.

Stating any assumptions: (i) calculate the values of a and b. [4] (ii) calculate the price per £1 nominal at time 0 of a bond which pays annual

coupons of 5% in arrear and is redeemed at 103% after 4 years. [5] [Total 9]

6 (i) An investor is considering the purchase of an annuity, payable annually in

arrear for 20 years. The first payment is £500. Using a rate of interest of 8% per annum effective, calculate the duration of the annuity when:

(a) the payments remain level over the term. (b) the payments increase at a rate of 8% per annum compound. [6]

(ii) Explain why the answer in (i)(b) is higher than the answer in (i)(a). [2] [Total 8] 7 The shares of a company currently trade at £2.60 each, and the company has just paid

a dividend of 12p per share. An investor assumes that dividends will be paid annually in perpetuity and will grow in line with a constant rate of inflation. The investor estimates the assumed inflation rate from equating the price of the share with the present value of all estimated future gross dividend payments using an effective interest rate of 6% per annum.

(i) Calculate the investor’s estimation of the effective inflation rate per

annum based on the above assumptions. [4] (ii) Suppose that the actual inflation rate turns out to be 3% per annum effective

over the following twelve years, but that all the investor’s other assumptions are correct.

Calculate the investor’s real rate of return per annum from purchase to sale, if

she sold the shares after twelve years for £5 each immediately after a dividend has been paid. You may assume that the investor pays no tax. [6]

[Total 10]

CT1 A2008—4

8 An investor is considering investing in a capital project.

The project requires an outlay of £500,000 at outset and further payments at the end of each of the first 5 years, the first payment being £100,000 and each successive payment increasing by £10,000. The project is expected to provide a continuous income at a rate of £80,000 in the first year, £83,200 in the second year and so on, with income increasing each year by 4% per annum compound. The income is received for 25 years. It is assumed that, at the end of 15 years, a further investment of £300,000 will be required and that the project can be sold to another investor for £700,000 at the end of 25 years.

(i) Calculate the net present value of the project at a rate of interest of 11% per

annum effective. [9] (ii) Without doing any further calculations, explain how the net present value

would alter if the interest rate had been greater than 11% per annum effective. [3] [Total 12]

9 The force of interest, ( )tδ , is a function of time and at any time t, measured in years, is given by the formula:

( )0.06 0 40.10 0.01 4 70.01 0.04 7

tt t t

t t

≤ ≤⎧⎪δ = − < ≤⎨⎪ − <⎩

(i) Calculate the value at time t = 5 of £1,000 due for payment at time t = 10. [5] (ii) Calculate the constant rate of interest per annum convertible monthly which

leads to the same result as in (i) being obtained. [2] (iii) Calculate the accumulated amount at time t = 12 of a payment stream, paid

continuously from time t = 0 to t = 4, under which the rate of payment at time t is ( ) 0.02100 tt eρ = . [6] [Total 13]

CT1 A2008—5

10 An insurance company holds a large amount of capital and wishes to distribute some of it to policyholders by way of two possible options.

Option A £100 for each policyholder will be put into a fund from which the expected annual

effective rate of return from the investments will be 5.5% and the standard deviation of annual returns 7%. The annual effective rates of return will be independent and (1+ ti ) is lognormally distributed, where ti is the rate of return in year t. The policyholder will receive the accumulated investment at the end of ten years.

Option B £100 will be invested for each policyholder for five years at a rate of return of 6% per

annum effective. After five years, the accumulated sum will be invested for a further five years at the prevailing five-year spot rate. This spot rate will be 1% per annum effective with probability 0.2, 3% per annum effective with probability 0.3, 6% per annum effective with probability 0.2, and 8% per annum effective with probability 0.3. The policyholder will receive the accumulated investment at the end of ten years.

Deriving any necessary formulae:

(i) Calculate the expected value and the standard deviation of the sum the

policyholders will receive at the end of the ten years for each of options A and B. [17]

(ii) Determine the probability that the sum the policyholders will receive at the

end of ten years will be less than £115 for each of options A and B. [5] (iii) Comment on the relative risk of the two options from the policyholders’

perspective. [2] [Total 24]

END OF PAPER



EXAMINERS’ REPORT

April 2008

Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. M A Stocker Chairman of the Board of Examiners June 2008



Page 2

Comments Comments on solutions presented to individual questions for this April 2008 paper are given below. Please note that different answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. Question 1 Well answered. Question 2 As has often been the case when words rather than numbers have been

required, this bookwork question was answered poorly. Question 3 Generally well answered, although some students treated the fees on Product

B paid by the customer as a cost to the mortgage company. Question 4 Well answered although many candidates’ working was unclear when

performing the CGT test. Question 5 Part (i) was answered well but in part (ii) many candidates failed to recognise

the need to calculate the 4-year spot rate before calculating the bond price. Question 6 Part (i) of this question did appear to differentiate between stronger

candidates who often scored very well and weaker candidates who often failed to score at all. As with many previous diets, many candidates in part (ii) had difficulty in giving a clear explanation of their results.

Question 7 This question was answered relatively poorly with, particularly in part (ii),

candidates often appearing confused between real and money rates of interest. Question 8 Most candidates managed to make a reasonable attempt at this question

although marks were often lost in part (i) through a combination of calculation errors and insufficient working being shown. Candidates generally made a better attempt at the explanation required in part (ii) when compared to similar questions both on this paper and in previous diets.

Question 9 Well answered. Question 10 Part (i) (for Option A) can be done much more simply than by using the

method given in this report but the calculations given would still need to be done for part (ii). It was disappointing to see many candidates incorrectly calculate the mean accumulated value for Option B by using the mean rate of interest. Few candidates brought together the answers from (i) and (ii) to fully answer part (iii).


Page 3

1 The present value of the dividends, I, is: ( )71

12 120.5 0.5 0.5 0.99594 0.97194 0.98394I v v= + = + = calculated at i = 5%

Hence forward price is (again calculated at i = 5%):

( )( )11

1210 0.98394 1 9.42845£9.43

F i= − + =

=

2 (a) Eurobonds form of unsecured medium or long-term borrowing issued in a currency other than the issuer's home currency outside the

issuer's home country pay regular interest payments and a final capital repayment at par. issued by large companies, governments and supra-national organisations. yields depend upon the issuer and issue size but will typically be slightly

lower than for the conventional unsecured loan stocks of the same issuer. issuers have been free to add novel features to their issues in order to

make them appeal to different investors. usually issued in bearer form

(b) Certificates of Deposit a certificate stating that some money has been deposited issued by banks and building societies terms to maturity are usually in the range 28 days to 6 months. interest is payable on maturity security and marketability will depend on the issuing bank active secondary market

3 For Product A, the annual rate of return satisfies the equation:

25

25

7,095.25 100,000

14.0939

a

a

=

⇒ =

This equates to the value of 25

a at 5%. Hence the annual effective rate of return is 5%.

For Product B, the annual rate of payment is X such that:

( )1225

100,000Xa = at 4%

( )( )

12251225

1.021537 15.6221 = 15.95855

100,000= 6,266.2315.95855

ia ad

X

= = ×

⇒ =


Page 4

The equation of value to calculate the rate of return from Product B is:

( )25

25126,000 + 5,000 6,266.23 100,000iv a

d+ =

Clearly the rate of return must be greater than 4%. Try 5%.

6,000 5,000 0.29530 + 6,266.2335 1.026881 14.0939 = 98,166LHS = + × × × At 5% the present value of the payments is less than the amount of the loan at 5% so the rate of return must be less than 5%. Try 4%:

6,000 5,000 0.37512 + 100,000 = 107,876LHS = + × Interpolate between 4% and 5% to get the effective rate of return, i:

107,876 100,0000.04 0.01 4.81%107,876 98,166

i −⎛ ⎞= + ≈⎜ ⎟−⎝ ⎠

(actual answer is 4.80%)

Therefore Product B charges a lower effective annual return than Product A.

4 ( )

( )44

41 1.05 0.0490894

i i⎛ ⎞+ = ⇒ =⎜ ⎟⎜ ⎟

⎝ ⎠

( )10.071 0.75 0.048611.08

g t− = × =

( ) ( )4

11i t g⇒ > − ⇒ Capital gain on contract and we assume loan is redeemed as late as possible (i.e. after 20 years) to obtain minimum yield. Let Price of stock = P

( )420

0.07 100,000 0.75P a= × × × ( )( ) 20108,000 0.35 108,000 at 5%P v+ − −

( )4 2020

20

5250 70, 2001 0.35a v

Pv

+⇒ =

−

5250 1.018559 12.4622 70,200 0.37689

1 0.35 0.37689× × + ×

=− ×

= 107,245.38


Page 5

5 Assuming no arbitrage. (i) ( ) ( )( )2

1 0 2 1 1 and 1 1 1 .i f i i f= + = + + Hence a – b = 0.061 0.061a b⇒ = +

( )21 2 1.061 1.065

1 2 1.061 1.065

a b

a b

+ − = ×

⇒ + − = ×

0.002

0.059ba

⇒ = −⇒ =

(ii) Firstly, find the 4-year spot rate. Consider £1 nominal: 1 = 0.07 ( )1 2 3 4 4

2 3 4 4i i i i iv v v v v+ + + +

= 0.07 ( )4

1 2 3 41.061 1.063 1.065 1.07 iv− − −+ + + ×

( )441 1.31429212i⇒ + =

4 7.0713% .i p a⇒ = Let bond price per £1 nominal be P. Then ( )1 2 3 4 4

2 3 4 40.05 1.03i i i i iP v v v v v= + + + +

= ( )21 3 40.05 1.061 1.063 1.065 1.08 1.070713−− − −+ + + ×

= 0.9545 i.e. 95.45 pence per £1 nominal

6 (i) (a) The duration is:

( )

( )

2 3 20

2 3 20

20

20

500 2 3 20at 8%

500( )

78.9079 8.037 years9.8181

v v v v

v v v vIaa

+ + + +

+ + + +

= = =

…

…


Page 6

(b) The duration is:

( ) ( ) ( )( ) ( ) ( )

( ) ( )

2 2 3 19 20

2 2 3 19 20

12

500 1.08 2 1.08 3 1.08 20 at 8%

500 1.08 1.08 1.08

20 211 2 3 2010.5 years

20 20

v v v v

v v v v

vv

⎡ ⎤+ × + × + + ×⎣ ⎦

⎡ ⎤+ + + +⎣ ⎦

×+ + + += = =

…

…

…

(ii) The duration in (i)(b) is higher because the payments increase over time so

that the weighting of the payments is further towards the end of the series. 7 (i) ( ) ( ) ( )( )2 32 3260 12 1 1 1 ......v e v e v e= + + + + + +

where v = 11.06

and e denotes inflations rate.

Then,

260 = 1 112 % where 1 1

ea at jj i∞

+=

+ + i.e. 0.06

1ej

e−

=+

12260

0.0461538460.01324 i.e 1.324% pa

jje

⇒ =

⇒ =⇒ =

(ii) ( )2 2 12 12260 12 1.03 1.03 ..... 1.03v v v= + + + 12500v+

1212 %%

12 500ij

a v= + where 0.031.03

ij −=

Try 10%, RHS 255.67 i = = 9%, RHS 279.35i = =

Hence, 279.35 2600.09 0.01279.35 255.67

i −= + ×

−

= 0.098 Let i′ = real return Then ( )( )1 1 1i e i′+ + = +

1.098211.03

i′⇒ + = 6.62%i pa′⇒ =


Page 7

8 (i) Working in £000s

Outlay ( )5 5

500 90 10Pv a Ia= + + @11%

5

5

1 3.6958970.11

va −= =

( )5 5

55

5 1.11 3.695897 50.11 0.11

a v vIa− × −

= =

= 10.319900 500 90 3.695897 10 10.3199PV⇒ = + × + × = 935.8297

Income

( ) ( )( )2 242 24

1 1 1 180 1.04 1.04 1.04PV a v a v a v a= + + + +

( )( )

25

1

1 1.0480

1 1.04v

av

⎡ ⎤−= × ⎢ ⎥

−⎢ ⎥⎣ ⎦

where 1

0.11 1. . 0.949589ln1.11 1.11

ia vδ

= = =

80 0.949589 12.74554 968.2421PV⇒ = × × =

PV of cost of further investment

15300 62.7013v= = PV of sale 25700 51.5257v= =

Hence NPV = 968.2421 + 51.5257 - 935.8297 - 62.7013 = 21.2368 (£21,237)

(ii) If interest > 11% then 11 i+

decreases.

PV⇒ of both income and outgo ↓


Page 8

However, PV of outgo is dominated by initial outlay of £500k at time 0 which is unaffected.

PV⇒ of income decreases by more than decrease in PV of outgo NPV PV⇒ = of income – PV of outgo would reduce (and possibly become negative)

9 (i) ( ) ( )10 7

7 51,000 exp 0.01 0.04 exp 0.10 0.01pv t dt t dt⎡ ⎤ ⎡ ⎤= ∗ − − ∗ − −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫

10 72 2

7 5

0.01 0.011000 exp 0.04 exp 0.102 2

t tt t⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟= ∗ − − ∗ − −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠

0.01 51 0.01 241000 exp 0.04 3 exp 0.10 2

2 2⎛ ∗ ⎞ ⎛ ∗ ⎞⎡ ⎤ ⎡ ⎤= ∗ − − × ∗ − ∗ −⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠

( )1000 exp 0.255 0.12 0.20 0.12= ∗ − + − +

( )1000 exp 0.215= ∗ − = 806.54

(ii) Required interest rate p.a. convertible monthly is given by

( ) 12 512

806.54 1 1,00012i

×⎛ ⎞+ =⎜ ⎟⎜ ⎟

⎝ ⎠

( )12 4.3077% . . convertible monthlyi p a⇒ =

(iii) Accumulated amount = ( ) ( )

4 127

744 0.06 0.01 0.040.10 0.010.02

0100 t

dr r drr drte e e e dt−−∫ ∫∫× × ×∫

[ ]127 22 0.010.014

2274

0.040.104 0.060.02

0100

rr

trrrte e e e dt

⎡ ⎤⎡ ⎤ −− ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦= × × ×∫

( ) ( ) ( )4 0.24 0.06 0.30 0.165 0.475 0.2000.02

0100 tte e e e dt− − −= ∫

=

40.24 0.135 0.275 0.04

0100 te e e e dt−∫

= 40.04

0.65

0

1000.04

tee−⎡ ⎤−

⎢ ⎥⎣ ⎦

= ( )0.65 0.162,500 1e e−−


Page 9

= 2,500 * 1.915540829 * 0.1478562 = 708.06

10 (i) Option A: ( ) ( )21 ~ ,ti Lognormal+ μ σ

( ) ( )2ln 1 ~ ,ti N+ μ σ

( ) ( ) ( ) ( ) ( )10 2ln 1 ln 1 ln 1 ln 1 ~ 10 ,10t t t ti i i i N+ = + + + + + + μ σ…

since 'ti s are independent

( ) ( )10 21 ~ 10 ,10ti Lognormal+ μ σ

( )

( ) ( ) ( )( )

2

2 2 2

22 2

2

1 exp 1.0552

1 exp 2 exp 1 0.07

0.07 exp 1 0.00439281.055

t

t

E i

Var i

⎛ ⎞σ+ = μ + =⎜ ⎟⎜ ⎟

⎝ ⎠⎡ ⎤+ = μ +σ σ − =⎣ ⎦

⎡ ⎤= σ − ∴σ =⎣ ⎦

0.0043928 0.0043928exp 1.055 ln1.055 0.0513442 2

⎛ ⎞μ + = ⇒μ = − =⎜ ⎟⎝ ⎠

10 0.51344μ = , 210 0.043928σ = Let S10 be the accumulation of one unit after 10 years:

( )100.043928exp 0.51344 1.70814

2E S ⎛ ⎞= + =⎜ ⎟

⎝ ⎠

Accumulated sum is ( )10100 £170.81E S = Option B:

The accumulated sum at the end of five years is: 5100 1.06 = 100 1.33823 = £133.823× ×


Page 10

The expected value of the accumulated sum at the end of ten years is: ( )5 5 5 5133.823 0.2 1.01 0.3 1.03 0.2 1.06 0.3 1.08× + × + × + ×

( )133.823 0.2 1.05101 0.3 1.15927 0.2 1.33823 0.3 1.46933£169.48

= × + × + × + ×

=

Option A:

( ) ( ) ( )10 exp 2 0.51344 0.043928 exp 0.043928 1

2.91776 0.04491 0.13103

Var S ⎡ ⎤= × + −⎣ ⎦= × =

Therefore standard deviation of £100 is 100 0.13103 £36.20= Option B: Here we need to find the expected value of the square of the accumulation as follows:

( )2 2 2 2 2133.823 0.2 1.05101 +0.3 1.15927 +0.2 1.33823 +0.3 1.46933

= 29,189.86

× × × ×

The variance of the accumulation is therefore: 2 229,189.86 169.48 £ 467.54− =

and the standard deviation is £21.62 (ii) For option A we require [ ]10 1.15P S <

[ ]10ln ln1.15P S < where 10ln S ~N(0.51344,0.043928)

( ) ln1.15 0.513440,10.043928

P N −⎡ ⎤⇒ <⎢ ⎥

⎣ ⎦

( )0,1 1.7829P N⎡ ⎤⇒ < −⎣ ⎦ = 0.0373 4%≈

For option B we first examine the lowest payout possible. There is a probability of 0.2 that the amount will be 5 5100 1.06 1.01× × or less which equals133.823 1.05101 £140.65× = . Therefore the probability of a payment of less than £115 is zero.

(iii) Option A is riskier both from the perspective of having a higher standard

deviation of return and also a higher probability of a very low value.



EXAMINATION













CT1 S2008—2

1 A 91-day government bill is purchased for £95 at the time of issue and is redeemed at the maturity date for £100. Over the 91 days, an index of consumer prices rises from 220 to 222.

Calculate the effective real rate of return per annum. [3] 2 (i) State the strengths and weaknesses of using the money-weighted rate of return

as opposed to the time-weighted rate of return as a measure of an investment manager’s skill. [3]

(ii) An investor had savings totalling £41,000 in an account on 1 January 2006.

He invested a further £12,000 in this account on 1 August 2006. The total value of the account was £45,000 on 31 July 2006 and was £72,000 on 31 December 2007.

Assuming that the investor made no further deposits or withdrawals in relation

to this account, calculate the annual effective time-weighted rate of return for the period 1 January 2006 to 31 December 2007. [2]

[Total 5] 3 (i) A forward contract with a settlement date at time T is issued based on an

underlying asset with a current market price of B. The annualised risk-free force of interest applying over the term of the forward

contract is δ and the underlying asset pays no income. Show that the theoretical forward price is given by ,TK Be= δ assuming no arbitrage. [3]

(ii) An asset has a current market price of 200p, and will pay an income of 10p in

exactly three months’ time.

Calculate the price of a forward contract to be settled in exactly six months, assuming a risk-free rate of interest of 8% per annum convertible quarterly. [3] [Total 6]

4 Describe the characteristics of commercial property (i.e. commercial real estate) as an

investment. [5] 5 A bank offers two repayment alternatives for a loan that is to be repaid over ten years.

The first requires the borrower to pay £1,200 per annum quarterly in advance and the second requires the borrower to make payments at an annual rate of £1,260 every second year in arrears.

Determine which terms would provide the best deal for the borrower at a rate of

interest of 4% per annum effective. [5]


6 A pension fund holds an asset with current value £1 million. The investment return on the asset in a given year is independent of returns in all other years. The annual investment return in the next year will be 7% with probability 0.5 and 3% with probability 0.5. In the second and subsequent years, annual investment returns will be 2%, 4% or 6% with probability 0.3, 0.4 and 0.3, respectively.

(i) Calculate the expected accumulated value of the asset after 10 years, showing

all steps in your calculations. [3]

(ii) Calculate the standard deviation of the accumulated value of the asset after 10 years, showing all steps in your calculations. [4]

(iii) Without doing any further calculations explain how the mean and variance of

the accumulation would be affected if the returns in years 2 to 10 were 1%, 4%, or 7%, with probability 0.3, 0.4 and 0.3 respectively. [2] [Total 9]

7 The force of interest, ( )tδ , is a function of time and at any time t (measured in years)

is given by

0.05 0.02 for 0 5

( )0.15 for 5

t tt

t+ ≤ ≤⎧

δ = ⎨ >⎩

(i) Calculate the present value of £1,000 due at the end of 12 years. [5]

(ii) Calculate the annual effective rate of discount implied by the transaction in (i).

[2] [Total 7] 8 A tax advisor is assisting a client in choosing between three types of investment. The

client pays tax at 40% on income and 40% on capital gains. Investment A requires the investment of £1m and provides an income of £0.1m per

year in arrears for ten years. Income tax is deducted at source. At the end of the ten years, the investment of £1m is returned.

In Investment B, the initial sum of £1m accumulates at the rate of 10% per annum

compound for ten years. At the end of the ten years, the accumulated value of the investment is returned to the investor after deduction of capital gains tax.

Investment C is identical to Investment B except that the initial sum is deemed, for tax

purposes, to have increased in line with the index of consumer prices between the date of the investment and the end of the ten-year period. The index of consumer prices is expected to increase by 4% per annum compound over the period.

(i) Calculate the net rate of return expected from each of the investments. [7] (ii) Explain why the expected rate of return is higher for Investment C than for

Investment B and is higher for Investment B than for Investment A. [3] [Total 10]

CT1 S2008—4

9 Three bonds, paying annual coupons in arrears of 6%, are redeemable at £105 per £100 nominal and reach their redemption dates in exactly one, two and three years’ time respectively. The price of each of the bonds is £103 per £100 nominal.

(i) Calculate the gross redemption yield of the three-year bond. [3]

(ii) Calculate to three decimal places all possible spot rates, implied by the information given, as annual effective rates of interest. [4]

(iii) Calculate to three decimal places all possible forward rates, implied by the

information given, as annual effective rates of interest. [4] [Total 11]

10 An insurance company is considering two possible investment options.

The first investment option involves setting up a branch in a foreign country. This will involve an immediate outlay of £0.25m, followed by investments of £0.1m at the end of one year, £0.2m at the end of two years, £0.3m at the end of three years and so on until a final investment is made of £1m in ten years’ time. The investment will provide annual payments of £0.5m for twenty years with the first payment at the end of the eighth year. There will be an additional incoming cash flow of £5m at the end of the 27th year. The second investment option involves the purchase of 1 million shares in a bank at a price of £4.20 per share. The shares are expected to provide a dividend of 21p per share in exactly one year, 22.05p per share in two years and so on, increasing by 5% per annum compound. The shares are expected to be sold at the end of ten years, just after a dividend has been paid, for £5.64 per share.

(i) Determine which of the options has the higher net present value at a rate of

interest of 7% per annum effective. [9] (ii) Without doing any further calculations, determine which option has the higher

discounted mean term at a rate of interest of 7% per annum effective. [2] [Total 11]


11 A company has a liability of £400,000 due in ten years’ time.

The company has exactly enough funds to cover the liability on the basis of an effective interest rate of 8% per annum. This is also the interest rate on which current market prices are calculated and the interest rate earned on cash. The company wishes to hold 10% of its funds in cash, and to invest the balance in the following securities: • a zero-coupon bond redeemable at par in twelve years’ time • a fixed-interest stock which is redeemable at 110% in sixteen years’ time bearing

interest at 8% per annum payable annually in arrear (i) Calculate the nominal amounts of the zero-coupon bond and the fixed-interest

stock which should be purchased to satisfy Redington’s first two conditions for immunisation. [10]

(ii) Calculate the amount which should be invested in each of the assets mentioned

in (i). [2] (iii) Explain whether the company would be immunised against small changes in

the rate of interest if the quantities of stock in part (i) are purchased. [2] [Total 14]

CT1 S2008—6

12 An individual takes out a 25-year bank loan of £300,000 to purchase a house. The individual agrees to pay only the interest payments, monthly in arrear, for the first

15 years whereupon he repays half of the capital as a lump sum. He then pays only the interest for the remaining 10 years, quarterly in arrear, and repays the other half of the capital as a lump sum at the end of the term.

(i) Calculate the total amount of interest paid by the individual, assuming an

effective rate of interest of 8½% p.a. [5] (ii) The individual believes that he can earn a nominal rate of interest convertible

half-yearly of 9% p.a. from a separate savings account. Calculate the level contribution he must make monthly in advance to the

savings account in order to repay half the capital after 15 years. [4] (iii) The individual made the monthly contributions calculated in (ii) to the savings

account. However, over the first 15 years, the effective rate of return earned on the savings account was 10% per annum.

The individual used the proceeds at that time to repay as much of the loan as

possible and then decided to repay the remainder of the loan by level instalments of interest and capital. After the first 15 years, the effective rate of interest changed to 7% per annum.

Calculate the level payment he must make, payable monthly in arrear, to repay

the loan over the final 10 years of the loan. [5] [Total 14]

END OF PAPER



EXAMINERS’ REPORT

September 2008

Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. R D Muckart Chairman of the Board of Examiners November 2008



Page 2

Comments Please note that different answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. Candidates appeared to be less well prepared than in previous recent diets. As has often been the case when words rather than numbers have been required, Q4 was answered relatively poorly despite only involving bookwork with a wide range of available points that could be made. Many candidates also struggled with the first part of Q2 where explanation rather than calculation was required. The remainder of the shorter questions were answered well with candidates scoring particularly highly on Q7. The more application styled questions (especially Qs 8, 11 and 12) tended to act as a clear discriminator between stronger and weaker candidates with a significant minority of candidates scoring very few marks on these questions. By contrast, Q9 on spot and forward yields was answered relatively well compared to questions in previous diets on this topic.


Page 3

1 If j = real rate of return then equation of value in real terms is:

( )91/365 22095 1 100222

j+ =

( )91/3651 1.04315j+ = therefore j = 18.465% 2 (i) MWRR

• Requires less information compared to TWRR But • Affected by amount and timing of net cashflows, which may not be in the

manager’s control and less fair measure than TWRR • More difficult equation to solve than TWRR • Also: equation may not have unique (or any) solution

(ii) Let TWRR = i

Then

2 45 72(1 )41 57

1.38639281117.745% p.a.

i

i

+ = ×

=⇒ =

3 (i) Consider two portfolios A and B at time 0. Portfolio A: - buy forward at price of K - deposit TKe−δ in risk-free asset Portfolio B: - buy asset at price of B

Then, at maturity, both portfolios have the same value (i.e. hold the underlying asset).

Thus, by the no-arbitrage principle, both portfolios must have same value at

time 0. T TKe B K Be−δ δ⇒ = ⇒ =


Page 4

(ii) 2%i = per quarter ( )2200 1.02 10 1.02 197.88K⇒ = × − × =

( )( )1using T tTK Be Ceδ −δ= −

4 Main characteristics of commercial property investments:

• Many different types of properties available for investment, e.g. offices, shops and

industrial properties. • Return comes from rental income and from the proceeds on sale. • Total expected return higher than for gilts • Rents and capital values are expected to increase broadly with inflation in the long

term • Neither rental income nor capital values are guaranteed – capital values in

particular can fluctuate in the short term… • …but rental income more secure than dividends • Rents and capital values expected to increase when the price level rises (though

the relationship is far from perfect). • Rental terms are specified in lease agreements. Typically, rents increase every

three to five years, Some leases have clauses which specify upward-only adjustments of rents.

• Large unit sizes, leading to less flexibility than investment in shares • Each property is unique… • …. so can be difficult to value. • Valuation is expensive, because of the need to employ an experienced surveyor • Marketability and liquidity are poor because of uniqueness … • …and because buying and selling incurs high costs. • Rental income received gross of tax. • Net rental income may be reduced by maintenance expenses • There may be periods when the property is unoccupied, and no income is

received. • The running yield from property investments will normally be higher than that for

ordinary shares.

5 Present value in first case is

( ) 1041,200 1200 1.024877 8.1109 = £9,975.210i a

d× × = × ×

Present value in second case is:

( )( )

102 4 10 2

2

12,520 ( ) 2,520

1

vv v v v

v

−× + +…+ = × ×

−


Page 5

( )( )1 0.67556

2,520 0.92456 £10,020.011 0.92456−

= × × =−

Therefore first option is better for the borrower. 6 (i) Let ti = investment return for year t

Then, the expected value of the accumulation ( )10S is given by (in £ millions):

( )10E S = ( )10

11 t

tE i

=

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠∏

( )10

11 t

tE i

=

= +∏ using independence

( )( )10

11 t

tE i

=

= +∏

Now, ( ) ( )1 0.5 0.07 0.03 0.05E i = × + =

and for ( ) ( )1, 0.3 0.02 0.4 0.04 0.3 0.06tt E i≠ = × + × + × = 0.04

So the expected value of the accumulation is

91.05 1.04 1.494477 (i.e. £1,494,477)× =

(ii) The variance of the accumulation is

( ) ( )( )22 210 101,000,000 E S E S× −

where ( ) ( )10

2210

11 t

tE S E i

=

⎛ ⎞= +⎜ ⎟⎜ ⎟

⎝ ⎠∏

= ( )10

2

11 2 t t

tE i i

=

⎛ ⎞+ +⎜ ⎟⎜ ⎟

⎝ ⎠∏

= ( ) ( )( )10

2

11 2 t t

tE i E i

=

+ +∏ from independence


Page 6

Now ( ) ( )2 2 21 0.5 0.07 0.03 0.0029E i = × + =

for ( )2 2 2 21, 0.3 0.02 0.4 0.04 0.3 0.06tt E i≠ = × + × + ×

= 0.00184 Hence, ( ) ( ) ( )92

10 1 0.1 0.0029 1 0.08 0.00184E S = + + × + +

= 2.238739 Standard deviation of the accumulation is

( )1

221,000,000 2.238739 1.494477 £72,646× − =

(iii) The mean would remain unchanged as the expected rate of return in years 2-10

is unchanged. The variance of the rate in years 2-10 has increased and this will lead to an increase in the variance of the 10 year accumulation.

7 (i) Discounting from t = 12 to t = 5

( )

[ ]

12

5

12 1.055

12,5 exp 0.15

exp 0.15 0.34994

v ds

s e−

⎛ ⎞= −⎜ ⎟⎝ ⎠

= − = =

∫

Discounting from t = 5 to t = 0

( )

5

0

52 0.50

5,0 exp 0.05 0.02

exp 0.05 0.01 0.60653

v sds

s s e−

⎛ ⎞= − +⎜ ⎟⎝ ⎠

⎡ ⎤= − − = =⎣ ⎦

∫

Hence present value of £1,000 at time t = 12 ( ) ( )1,000 12,5 5,0 1,000 0.34994 0.60653 £212.25v v= = × × =


Page 7

(ii) The annual effective rate of discount is d such that:

( )112

121000 1 212.25

1 0.21225 12.117%

d

d

− =

⇒ = − =

8 (i) Investment A: the gross rate of return per annum effective is clearly 10%. The

net return is therefore ( )1-0.4 10% = 6%× per annum effective. Investment B: the investment will accumulate to 10£1 1.1 = £2.5937m m× at the

end of the ten years. The equation of value is:

( ) ( )( )( )

( )

10 10

10

10

1 2.59374 1 0.4 2.59374 -1 1

1.95625 1

1 1.95625 6.94%

i i

i

ii

− −

−

= + − +

= +

⇒ + =

⇒ =

Investment C: again the investment will accumulate to £2.5937m at the end of ten years. However, the indexed purchase price is subtracted from the value of the investment in this case. Thus the equation of value is:

( ) ( )( )

( ) ( ) ( )( )

( )

10 1010

10 10 1010

10

10

1 2.59374 1 0.4 2.59374 -1 1.04 1

2.5937 1 0.4 2.59374 1 0.4 1.04 1

2.14834 1

1 2.148347.95%

i i

i i i

i

ii

− −

− − −

−

= + − × +

= + − × + + × × +

= +

⇒ + =

⇒ =

(ii) All investments give a gross return of 10% per annum effective. Investment B

gives a higher return than A because the tax is deferred until the end of the investment as capital gains tax is paid and not income tax. [However, candidates might note that tax is paid on the interest earned by deferral of tax]. Investment C gives a higher return than investment B because the tax is only paid on the real return over the ten year period which is lower than the nominal return.


Page 8

9 (i) 33103 6 105a v= +

try i = 6%: 3

3 2.6730 0.83962a v= = RHS = 104.1981

try i = 7%: 3

3 2.6243 0.81630a v= = RHS = 101.4573

Using linear interpolation:

( )( )

104.1981 1030.06 0.01 0.06437 6.44%

104.1981 101.4573i

−= + × = =

−

(ii) Let ni = spot yield for term n Then ( )1 1103 1 111 7.767%i i+ = ⇒ =

( ) ( )1 22 2103 6 1.07767 111 1 6.736%i i− −= + + ⇒ =

( ) ( ) ( )1 2 33 3103 6 1.07767 6 1.06736 111 1 6.394%i i− − −= + + + ⇒ =

(iii) First year forward rate is 7.767% (same as spot rate). Forward rate from time one to time two is i such that: ( ) 21.07767 1 1.06736 5.715%i i+ = ⇒ = Forward rate from time two to time three is i such that: ( )2 31.06736 1 1.06394 5.713%i i+ = ⇒ = Forward rate from time one to time three is i such that: ( )2 31.07767 1 1.06394 5.714%i i+ = ⇒ =

Forward rate from time zero to two and from time zero to three are the same as the respective spot rates (no additional marks for this point).

10 (i) NPV of first project in £m is:

( ) ( )( )

2727 7 100.5 5 0.1 0.25 at 7%

0.5 11.9867 5.3893 5 0.16093 0.1 34.7391 0.25 £0.379

a a v Ia

m

− + − −

= − + × − × −

=


Page 9

The NPV of second project in £m is:

2 2 3 9 10 10

10 1010

0.21 0.21(1.05) 0.21(1.05) 0.21(1.05) 5.64 4.2

1 1.05 0.21 5.64 4.21 1.05

v v v v v

vv vv

+ + + + + −

⎛ ⎞−= + −⎜ ⎟⎜ ⎟−⎝ ⎠

…

100.93458 0.50835v v= = Therefore NPV 1.8055 5.64 0.50835 4.2 £0.473m= + × − = The second project has the higher net present value at 7% per annum effective.

(ii) The second project clearly has a discounted mean term of less then ten years.

However, the discounted mean term of the first project must be greater than ten years because the undiscounted incoming cash flows are less than the undiscounted outgoing cash flows after ten years.

11 (i) Working in ‘000s

Let X = Nominal amount of Zero Coupon Bond Y = Nominal amount of 8% bond 10400 185.2774LV v= = 12

1618.52774 0.08AV Xv Ya= + + 161.1Yv+ Then, since A LV V= (1st condition)

166.74966 0.39711 0.08 8.8514 0.32108X Y Y⇒ = + × +

( )166.74966 0.39711 1.02919 ...... 1X Y⇒ = + 2nd condition is ' '

A LV V= ' 104000 1852.7740LV v= = ( )' 12

1612 0.08 .AV X v Ia Y= + 161.1 16 Y v+ ∗ 4.76537 0.08 61.1154 5.13727X Y Y= + ∗ + ( )1852.7740 4.76537 10.0265 ..... 2X Y⇒ = + 148.2429 2.32391Y⇒ =


Page 10

( ) ( )4.76537from 2 10.39711

⎡ ⎤− ∗⎢ ⎥⎣ ⎦

Hence Y = 63,790 X = 254,583

(ii) Amount invested in X is 254,583 12v = 101,098 and amount invested in Y is: 185,277 - 18,528 - 101,098 = 65,651 (iii) The spread of the assets is clearly greater than the spread of the liability

(which is a single point). Hence, Redington’s 3rd condition is satisfied and the fund is immunised.

12 (i) First 15 years:

Interest paid each month

( ) ( ) 1212 12300,000 where1.085 1

12 12i i⎛ ⎞

⎜ ⎟= × = +⎜ ⎟⎝ ⎠

( )120.0068215

12i

⇒ =

⇒ monthly interest = 0.0068215 × 300,000 = £2,046.45 After repayment of £150,000 after 15 years: Interest paid each quarter

( ) ( ) 44 4150,000 where1.085 1

4 4i i⎛ ⎞

⎜ ⎟= × = +⎜ ⎟⎝ ⎠

( )40.020604

4i

⇒ =

⇒Quarterly interest = 0.020604 × 150,000 = £3,090.66


Page 11

Total interest paid over the 25 years = (2046.45 × 12 × 15) + (3090.66 × 4 × 10) = £491,987.40

(ii) 150,000 = ( )630

X s @ 4½ %

where X = Amount paid in each 6 month period

( ) ( )( )

306

630

1.045 1s

d

−=

where ( ) 661 1

1.045 6d⎛ ⎞

⎜ ⎟= −⎜ ⎟⎝ ⎠

(6) 0.043856d⇒ =

Hence ( )301.045 10.043856

150000 15000062.5985

X−

= =⎡ ⎤⎢ ⎥⎣ ⎦

= 2396.23

⇒ Monthly contribution = 2396.236

= £399.37 per month

(iii) Savings proceeds after 15 years:

( )121510%

12 399.37 s×

where ( )( )

12151215

is sd

= ×

1.0533781 31.772533.46845

= ×=

Hence, savings proceeds = 4792.44 × 33.46845 = 160,395.56 ⇒ Loan o/s after 15 years = 300,000 - 160,395.56 = 139,604.44


Page 12

Let Y = new monthly payment

139,604.44 = ( )12107%

12 Y a

0.0712 7.023580.06785

Y= ×

£1,605.50Y⇒ = per month



EXAMINATION

21 April 2009 (am)












CT1 A2009—2

1 Describe the characteristics of Government Bills. [3] 2 Describe the characteristics of:

(a) an interest-only loan (or mortgage); and (b) a repayment loan (or mortgage). [4]

3 A loan is to be repaid by an annuity payable annually in arrear. The annuity starts at a

rate of £300 per annum and increases each year by £30 per annum. The annuity is to be paid for 20 years.

Repayments are calculated using a rate of interest of 7% per annum effective. Calculate: (i) The amount of the loan. [3] (ii) The capital outstanding immediately after the 5th payment has been made. [2] (iii) The capital and interest components of the final payment. [2] [Total 7]

4 (i) Explain what is meant by the “no arbitrage” assumption in financial mathematics. [2]

An investor entered into a long forward contract for £100 nominal of a security eight

years ago and the contract is due to mature in four years’ time. The price per £100 nominal of the security was £94.50 eight years ago and is now £143.00. The risk-free rate of interest can be assumed to be 5% per annum effective throughout the contract.

(ii) Calculate the value of the contract now if it were known from the outset that

the security will pay coupons of £9 two years from now and £10 three years from now. You may assume no arbitrage. [5]

[Total 7]


5 A company’s required return for a particular investment project can be expressed as a force of interest, δ(t). This force of interest is a function of time and at any time t, measured in years, is given by the formula:

( ) 0.05 0.002 0 5( ) 0.06 5t t tt t

δ = + ≤ ≤δ = <

The expenditure required for this project is a payment of £100,000 at t = 0 and a

further payment of £80,000 at t = 2. The income received from the project is a payment stream paid continuously from

t = 8 to t = 12 under which the annual rate of payment at time t is 0.001£100,000 te . Calculate the discounted payback period for this project. [8] 6 A pension fund purchased an office block nine months ago for £5 million. The pension fund will spend a further £900,000 on refurbishment in two months time. A company has agreed to occupy the office block six months from now. The lease

agreement states that the company will rent the office block for fifteen years and will then purchase the property at the end of the fifteen year rental period for £6 million.

It is further agreed that rents will be paid quarterly in advance and will be increased

every three years at the rate of 4% per annum compound. The initial rent has been set at £800,000 per annum with the first rental payment due immediately on the date of occupation.

Calculate, as at the date of purchase of the office block, the net present value of the

project to the pension fund assuming an effective rate of interest of 8% per annum. [8]

7 A fund had a value of £150,000 on 1 July 2006. A net cash flow of £30,000 was

received on 1 July 2007 and a further net cash flow of £40,000 was received on 1 July 2008. The fund had a value of £175,000 on 30 June 2007 and a value of £225,000 on 30 June 2008. The value of the fund on 1 January 2009 was £280,000.

(i) Calculate the time-weighted rate of return per annum earned on the fund

between 1 July 2006 and 1 January 2009. [3] (ii) Calculate the money-weighted rate of return per annum earned on the fund

between 1 July 2006 and 1 January 2009. [4] (iii) Explain why the time-weighted rate of return is more appropriate than the

money-weighted rate of return when comparing the performance of two investment managers over the same period of time. [2] [Total 9]

CT1 A2009—4

8 An insurance company has liabilities consisting of eleven annual payments of £1 million, with the first payment due to be made in 10 years’ time and the last payment due to be made in 20 years’ time. The rate of interest is 6% per annum effective.

(i) Show that the discounted mean term of these liabilities, to four significant

figures, is 14.42 years. [3] The insurance company holds two zero-coupon bonds, one paying £X in 10 years’

time and the other paying £Y in 20 years’ time. (ii) Find values of X and Y such that Redington’s first two conditions for

immunisation from small changes in the rate of interest are satisfied. [6] (iii) Explain, without making any further calculations, whether you would expect

Redington’s third condition for immunisation to be satisfied for the values of X and Y calculated in (ii). [2]

[Total 11] 9 Two bonds paying annual coupons of 5% in arrear and redeemable at par have terms

to maturity of exactly one year and two years, respectively. The gross redemption yield from the 1-year bond is 4.5% per annum effective; the

gross redemption yield from the 2-year bond is 5.3% per annum effective. You are informed that the 3-year par yield is 5.6% per annum.

Calculate all zero-coupon yields and all one-year forward rates implied by the yields

given above. [12] 10 A loan pays coupons of 11% per annum quarterly on 1 January, 1 April, 1 July and

1 October each year. The loan will be redeemed at 115% on any 1 January from 1 January 2015 to 1 January 2020 inclusive, at the option of the borrower. In addition to the redemption proceeds, the coupon then due is also paid.

An investor purchased a holding of the loan on 1 January 2005, immediately after the

payment of the coupon then due, at a price which gave him a net redemption yield of at least 8% per annum effective. The investor pays tax at 30% on income and 25% on capital gains.

On 1 January 2008 the investor sold the holding, immediately after the payment of the

coupon then due, to a fund which pays no tax. The sale price gave the fund a gross redemption yield of at least 9% per annum effective.

Calculate the following: (i) The price per £100 nominal at which the investor bought the loan. [6] (ii) The price per £100 nominal at which the investor sold the loan. [4] (iii) The net yield per annum convertible quarterly that was actually obtained by

the investor during the period of ownership of the loan. [5] [Total 15]

CT1 A2009—5

11 An individual wishes to receive an annuity which is payable monthly in arrears for 15 years. The annuity is to commence in exactly 10 years at an initial rate of £12,000 per annum. The payments increase at each anniversary by 3% per annum. The individual would like to buy the annuity with a single premium 10 years from now.

(i) Calculate the single premium required in 10 years’ time to purchase the

annuity assuming an interest rate of 6% per annum effective. [5] The individual wishes to invest a lump sum immediately in an investment product

such that, over the next 10 years, it will have accumulated to the premium calculated in (i). The annual effective returns from the investment product are independent and (1 )ti+ is lognormally distributed, where ti is the return in the tth year. The expected annual effective rate of return is 6% and the standard deviation of annual returns is 15%.

(ii) Calculate the lump sum which the individual should invest immediately in

order to have a probability of 0.98 that the proceeds will be sufficient to purchase the annuity in 10 years’ time. [9]

(iii) Comment on your answer to (ii). [2] [Total 16]

END OF PAPER



EXAMINERS’ REPORT

April 2009

Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. R D Muckart Chairman of the Board of Examiners June 2009



Page 2

Comments Please note that different answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. There were some excellent performances and well-prepared candidates scored well across the whole paper. However, the comments below on each question concentrate on areas where candidates could have improved their performance. Q1, Q2. As has often been the case when words rather than numbers have been required, these bookwork questions were answered relatively poorly (although Q2 was answered better than Q1). Q3. Well answered. Q4. Defining an arbitrage profit correctly was also acceptable as an answer to (i) although a description of both possible arbitrage scenarios was required for full marks. Many candidates performed the calculations well although the methodology being used was not always clear. Q5. The question required an ability to bring together two separate elements of the syllabus and less well-prepared candidates seemed to struggle with this. Q6. This was another question where students scored relatively poorly with many candidates having difficulty with the income calculation. A common error was to assume that the income rose by 4% every three years. Q7. This was answered much better than questions on the same topic in previous exams. However, some candidates did confuse the money-weighted and time-weighted rates of return.


Page 3

Q8. It was particularly disappointing to see many candidates using the wrong formula for DMT in part (i) but ending their proof with‘=14.42 QED’ in the final line. This suggests a lack of professionalism, honesty and integrity which are key attributes of the actuarial profession. Part (ii) was well-answered with various different methods leading to the correct answer. Q9. This was the worst-answered question on the paper although it was still possible to score significant marks by calculating forward rates using the correct formula even if the spot rates had been calculated incorrectly. Q10. Part (i) was answered well but many candidates lost marks in part (ii) by not realising that a separate test was required to ascertain the worst time to redemption. Many candidates calculated the annual effective yield rather than the yield per annum convertible quarterly in part (iii). Q11. Many candidates seemed confused as to what to calculate in part (i) and failed to distinguish between the premium needed in 10 years’ time and the present value of that premium. Part (ii) was answered well (although some candidates appeared to be short of time at this stage). Part (iii) was answered very poorly with many candidates not appreciating the effects of the high variance.


Page 4

1 Characteristics of government bills:

• short-dated securities issued by governments to fund their short-term spending requirements.

• issued at a discount and redeemed at par with no coupon. • mostly denominated in the domestic currency, although issues can be made in

other currencies. • yield is typically quoted as a simple rate of discount for the term of the bill • absolutely secure • often highly marketable despite being unquoted. • often used as a benchmark risk-free short-term investment.

2 (a) An interest-only loan requires the borrower only to pay interest on the entire loan in each time period. The loan does not reduce over time so the interest remains constant. A separate investment or savings account can be established in which payments are made to extinguish the whole loan at the end of the term.

(b) A repayment loan involves level repayments of capital and interest. The first part of the payment is used to pay interest on any remaining capital. The remaining part of the payment is then used to repay capital so that the capital gradually reduces over the term of the loan.

3 (i) ( )20 19300 30 at 7%a v a+ Ι = ( ) 1

1.07300 10.594 30 82.9347 5503.47+ × × =

(ii) Capital outstanding after 5 payments: ( )15 15420 30a a+ Ι

420 9.1079 30 61.5540 5671.94= × + × = (iii) Cap o/s after 19 payments = 870v @ 7% = £813.08 = Capital in the final payment Interest in the final payment = 870 – 813.08 = £56.92


Page 5

4 (i) The “no arbitrage” assumption means that neither of the following applies: (a) an investor can make a deal that would give her or him an immediate

profit, with no risk of future loss; nor (b) an investor can make a deal that has zero initial cost, no risk of future

loss, and a non-zero probability of a future profit.

(ii) The forward price at the outset of the contract was: ( ) ( )1210 11

5% 5%94.5 9 10 1.05 149.29v v− − × =

The forward price that should be offered now is: ( ) ( )42 3

5% 5%143 9 10 1.05 153.39v v− − × =

Hence the value of the contract now is: ( ) 4

5%153.39 149.29 3.37v− =

Note: This result can also be obtained directly from: ( )8143 94.5 1.05 3.38− × = since the coupons are irrelevant in this calculation.

5 Working in £000’s

PV of outgo = ( )20

0.05 0.002100 80 t dte− +∫+

= 220

0.05 0.001100 80

t te

⎡ ⎤− +⎣ ⎦+ = 0.104100 80 172.10e−+ = DPP is value of T for which: PV (income paid up to T) = PV (outgo)


Page 6

Where

PV (income paid up to T) = ( )0.0018

100T te v t dt∫

and v(t) ( )5

0 50.05 0.002 0.06tt dt dt

e⎡ ⎤− + +⎢ ⎥⎣ ⎦∫ ∫

=

( )520

0.05 0.001 0.06 0.30.t t te e

⎡ ⎤− +⎣ ⎦ − −= 0.275 0.06 0.30. te e e− −= 0.025 0.06te e−=

( ) 0.001 0.025 0.068

income paid up to 100T t tPV T e e e dt−⇒ = ∫

0.025 0.0598

100T te e dt−= ∫

0.025 0.059 0.059 81000.059

Te e e− − ×⎡ ⎤= −⎣ ⎦−

0.0591737.8222 1083.97Te−= − + DPP is such thatT⇒ 0.059172.10 1737.8222 1083.97Te−= − +

0.059 0.52472

0.059 (0.52472) 10.93 years

TeT Ln T

−⇒ =⇒ − = ⇒ =


Page 7

6 Working in 000’s PV of costs =

11125000 900v+ at 8%

= 5838.695

PV of income = ( ) ( ) ( ) ( )( )312 124 4 41 3 3 12

3 3 3800 1.04 1.04v a v a v a+ + +

= ( ) ( ) ( )( )312 3 1241

3800 1 1.04 1.04v a v v+ +

= ( )( )

151.041.08

31.041.08

1800 0.908281 1.049519 2.5771

1

⎛ ⎞−⎜ ⎟× × × ×⎜ ⎟⎜ ⎟−⎝ ⎠

= 1965.3133 4.038121× = 7936.173 PV of proceeds from sale

312166000 1717.969v= =

NPV of project = 7936.173+1717.969 – 5838.695 = 3815.447 (i.e. £3,815,447) 7 Working in 000’s (i) TWRR is i such that

( )1

221 i+ = 175 225 280150 175 30 225 40

× ×+ +

175 225 280 1.352968150 205 265

= × × =

12.85%i∴ = p.a.


Page 8

(ii) MWRR is i such that

( ) ( ) ( )1 1 1

2 2 22 1150 1 30 1 40 1 280i i i+ + + + + = Try: i = 12%, LHS = 277.02 i = 12.5%, LHS = 279.58 i = 13%, LHS = 282.16

( )28 27.95812.5% 0.5%

(28.216 27.958)i

−∴ = + ×

−

= 12.58% p.a. (iii) The TWRR is better for comparing 2 investment manager’s performances as it

is not sensitive to cash flow amounts and timing of payments. The MWRR is sensitive to both.

8 (i) Working in £m Discounted mean term =

10 11 12 20

10 11 12 2010 11 12 ............... 20

.................v v v v

v v v v+ + + ++ + + +

2 3 11

2 3 1110 11 12 ............ 20

................v v v vv v v v+ + + +

=+ + + +

( ) ( )11 11 11

11 11

99

a a aa a+ Ι Ι

= = + at 6%

( )11 42.7571

42.75719 14.421287.8869

to 4 significent figures DMT = 14.42

a

DMT

Ι =

⇒ = + =

⇒

(ii) First condition: pv assets = pv liabilities 10 20 9

11 *1Xv Yv v a⇒ + = at 6%. 0.55839 0.31180 0.59190*7.8869X Y∗ + ∗ = (using tables) = 4.668256 ………….(1)


Page 9

2nd condition: DMT assets = DMT liabilities

10 20

10 20*10 *20 14.42128X v Y v

Xv Yv+

⇒ =+

(use of 14.42 from (i) will be

accepted) ( )10 20*5.5839 *6.236 14.42128X Y Xv Yv⇒ + = ∗ +

= 14.42128*4.668256 from (1) = 67.3222 (or 67.3163 if DMT of 14.42 is used)…………(2) Equ n (2) – 10* Equ n (1) ⇒ *6.236 *3.1180 67.3222 10*4.668256Y Y− = −

20.639667 6.6195 (or 6.6176 if DMT of 14.42 is used)3.1180

Y⇒ = =

[or ' 'A LV V= (differentiating with respect to i)

( )( )

11 21 11 12 21

1011 11

10 20 10 11 20

9

Xv Yv v v v

v a Ia

+ = + + +

= +

…

5.2679 5.8831 63.5112X Y⇒ + = ………….(2)

Equ n (2) – 5.26795.8831

×Equ n (1)

⇒ 2.94155 19.4711Y = 6.6193Y⇒ = ] Equ n (1) ⇒ X * 0.55839 = 4.668256 – 6.6195 * 0.31180 ⇒ X = 4.6639 (or 4.6650 if DMT of 14.42 is used) [check, in equ n (2). 4.6639 * 5.5839 + 6.6195 * 6.236 = 67.3222] (iii) For the third condition to be satisfied, it is necessary for the spread of the

assets to exceed the spread of the liabilities. This appears to be the case given that the liabilities occur in equal annual amounts at durations from 10 years to 20 years, whereas the assets are concentrated in two lumps at the two most extreme durations, 10 years and 20 years.


Page 10

9 Let the 1-year and 2-year zero-coupon yields (spot rates) be 2andii i respectively.

1

105 105 @ 4.5%1

vi=

+

1 0.045i∴ = For the 2-year spot rate:

( )

25.3%2 5.3%2

1 2

5 105 5 1001 1

a vi i+ = +

+ +

( )

2

2 22

115 105 1001.05351.045 0.053 1.0531 i

⎛ ⎞−⎜ ⎟⎝ ⎠+ = +

+

= 9.257681 + 90.186858 = 99.444539

( )22

105 599.4445391.0451 i

= −+

( )221051

94.659850i⇒ + =

2 5.3202%i⇒ = p.a. For the 3-year spot rate: The 3-year par yield is 5.6% p.a.

( ) ( ) ( )2 3 3

1 2 3 3

1 1 1 11 0.0561 1 1 1i i i i

⎛ ⎞⎜ ⎟⇒ = + + +⎜ ⎟+ + + +⎝ ⎠

( ) ( )3 2

3

1.056 0.056 0.05611.0451 1.053202i

⇒ = − −+

( )331.0561

0.895926i⇒ + =

3 5.6324% p.a.i⇒ =


Page 11

1-year forward rates: 0 1 4.5% p.a.f i= = ( )( ) ( )21 1 21 1 1i f i+ + = +

2

11.0532021

1.045f⇒ + =

1 6.1468% p.a.f⇒ = ( ) ( ) ( )2 3

2 2 31 1 1i f i+ + = +

( )( )

3

2 21.056324

11.053202

f⇒ + =

2 6.2596% p.a.f⇒ = 10 (i) check for capital gain:

( ) ( )10.111 1 0.31.15

g t− = ∗ −

= 0.06696 ( )48% 0.077706i i= ⇒ = ( ) ( )4

11i g t⇒ > − ⇒ There’s a capital gain and thus loan should be assumed to be redeemed at

the latest possible date. Let P be price at which the investor bought the loan. Then

( ) ( )4 15 1515

11 0.7 115 0.25 115P a v P v= × + − − at 8%

7.7 1.029519 8.5595 0.75 115 0.315241 0.25 0.31524

P × × + × ×⇒ =

− ×

= £103.17 per £100 nominal


Page 12

(ii) check for capital gain:

( )10.111 0.0956521.15

g t− = =

( )

( ) ( )

4

41

9% 0.087113

1

i i

i g t

= ⇒ =

⇒ < −

⇒ There’s no capital gain and thus loan should be assumed to be redeemed at

the earliest possible date. Let 'P be the price at which the investor sold the loan. Then

( )4 77

' 11 115P a v= + at 9%

11 1.033144 5.033 115 0.54703= × × + × £120.1064= per £100 nominal (iii) Let j be the yield per quarter. Then

1212

11103.17 0.7 120.10644

a v= × + ( ) 120.25 120.1064 103.17 v− − at j %

12

12103.17 1.925 115.8723a v⇒ = + Try j = 3%: RHS = 100.4319638 j = 2.5%: RHS = 105.9042724 Linear interpolation:

( )

( )103.17 105.9042724

0.025 0.005100.4319638 105.9042724

j−

= + ×−

= 0.02749828 Hence, net yield is 11% p.a. (or 10.99931% p.a.) payable quarterly.


Page 13

11 (i) In 10 years’ time the single premium P is

P = ( ) ( ) ( ) ( ) ( ) ( )( )2 1412 12 12 122 141 1 1 1

12000 1.03 1.03 ... 1.03a a v a v v a+ + + +

= ( )2 14

121

1.03 1.03 1.0312000 1 ...1.06 1.06 1.06

a⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟+ + + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

= ( )

15

121

1.0311.0612000 1.0311.06

a

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎜ ⎟−⎜ ⎟⎜ ⎟

⎝ ⎠

where ( )( )

12121ia v

i=

1.027211 0.9690671.06

= =

0.349914612000 0.9690670.0283019

P⇒ = × ×

= 143,774.45

(ii) ( )2

21 1.06tE i eσμ++ = =

( ) ( ) ( )2 22 21 0.15 . 1tVar i e eμ+σ σ+ = = −

Then ( )

22

20.15 11.06

eσ= −

2 0.01982706⇒σ =

0.019827061.062

n∴μ = −

0.04835538= ( )10 0.4835538,0.1982706S LN⇒ ∼ Let X be the amount to be invested at time 0


Page 14

We want ( )10Pr . 143,774.45 0.98X S ≥ =

so 10143,774.45Pr 0.98S

X⎛ ⎞≥ =⎜ ⎟⎝ ⎠

so 143774.45

2

101

10XLn⎛ ⎞− μ

−Φ⎜ ⎟⎜ ⎟σ⎝ ⎠ = 0.02

143774.45

2

102.0537

10XLn − μ

⇒ = −σ

So 143774.45 2.0537 0.1982706LnX

= − × 0.4835538+

0.430909= −

143774.45 0.6499179X

⇒ =

£221,219.41X⇒ = (iii) It might seem odd that the initial investment needs to be substantially higher

than the single premium required in 10 years’ time to have a 98% probability of accumulating to the single premium.

This strange result is explained by the fact that the variance of the interest rate

is so high relative to the mean. There is therefore a significant risk that the investment will decrease in value over the next 10 years.



EXAMINATION













CT1 S2009—2

1 A 182-day government bill, redeemable at £100, was purchased for £96 at the time of issue and was later sold to another investor for £97.89. The rate of return received by the initial purchaser was 5% per annum effective.

(a) Calculate the length of time in days for which the initial purchaser held the

bill. (b) Calculate the annual simple rate of return achieved by the second investor. [4] 2 List the characteristics of an equity investment. [4]

3 An investor bought a number of shares at 78 pence each on 31 December 2005. She

received dividends on her holding on 31 December 2006, 2007 and 2008. The rate of dividend per share is given in the table below:

Date Rate of dividend per share

Retail price index

31.12.2005 31.12.2006 31.12.2007 31.12.2008

- - - - - - 4.1 pence 4.6 pence 5.1 pence

147.7 153.4 158.6 165.1

On 31 December 2008, she sold her shares at a price of 93 pence per share. Calculate, using the retail price index values shown in the table, the effective annual

real rate of return achieved by the investor [7] 4 A fixed-interest security has just been issued. The security pays half-yearly coupons

of 5% per annum in arrear and is redeemable at par 20 years after issue. (i) Calculate the price to provide an investor with a net redemption yield of 6%

per annum effective. The investor pays tax at a rate of 20% on income and is not subject to capital gains tax. [3]

(ii) Determine the annual effective gross redemption yield of this security

assuming the price calculated in (i) is paid. [5] (iii) Determine the real annual effective gross redemption yield of this security if

the rate of inflation is constant over the twenty years at 3% per annum. [2] [Total 10]


5 The force of interest ( )tδ at time t is 2a bt+ where a and b are constants. An amount of £100 invested at time t = 0 accumulates to £130 at time t = 5 and £200 at time t = 10. (i) Calculate the values of a and b. [6]

(ii) Calculate the constant rate of interest per annum convertible monthly that

would give rise to the same accumulation from time t = 0 to time t = 5. [2] (iii) Calculate the constant force of interest that would give rise to the same

accumulation from time t = 5 to time t = 10. [2] [Total 10]

6 (i) Distinguish between a future and an option. [2]

An investor wishes to purchase a one year forward contract on a risk-free bond which has a current market price of £97 per £100 nominal. The bond will pay coupons at a rate of 7% per annum half yearly. The next coupon payment is due in exactly six months and the following coupon payment is due just before the forward contract matures. The six-month risk-free spot interest rate is 5% per annum effective and the 12-month risk-free spot interest rate is 6% per annum effective.

(ii) Stating all necessary assumptions: (a) Calculate the forward price of the bond.

(b) Calculate the six-month forward rate for an investment made in six

months’ time. (c) Calculate the purchase price of a risk-free bond with exactly one year

to maturity which is redeemed at par and which pays coupons of 4% per annum half-yearly in arrears.

(d) Calculate the gross redemption yield from the bond in (c).

(e) Comment on why your answer in (d) is close to the one-year spot rate.

[10] [Total 12]

CT1 S2009—4

7 A member of a pensions savings scheme invests £1,200 per annum in monthly instalments, in advance, for 20 years from his 25th birthday. From the age of 45, the member increases his investment to £2,400 per annum. At each birthday thereafter the annual rate of investment is further increased by £100 per annum. The investments continue to be made monthly in advance for 20 years until the individual’s 65th birthday.

(i) Calculate the accumulation of the investment at the age of 65 using a rate of

interest of 6% per annum effective. [6] At the age of 65, the scheme member uses his accumulated investment to purchase an annuity with a term of 20 years to be paid half-yearly in arrear. At this time the interest rate is 5% per annum convertible half-yearly. (ii) Calculate the annual rate of payment of the annuity. [3] (iii) Calculate the discounted mean term of the annuity, in years, at the time of

purchase. [3] [Total 12]

8 A bank offers a customer two different repayment options on a loan of £50,000 as

follows:

Option 1 – level instalments of capital and interest are paid annually in arrear over a period of 20 years. Option 2 – over the 20-year term the customer pays only interest on the loan, annually in arrear at a rate of 5.5% per annum with the whole of the capital amount payable at the end of the term. The customer will take out a separate savings policy which involves making monthly payments in advance such that the proceeds will be sufficient to repay the loan at the end of its term. The payments into the savings policy accumulate at a rate of interest of 4% per annum effective.

(i) Determine the effective rate of interest per annum that would be paid by the

customer on the loan under Option 1, given that the level annual instalment on this loan is £4,012.13. [3]

(ii) Determine the annual effective rate of interest paid by a customer under

Option 2. [7] [Total 10]


9 A life insurance company is issuing a single premium policy which will pay out £20,000 in twenty years time. The interest rate the company will earn on the invested funds over the first ten years of the policy will be 4% per annum with a probability of 0.3 and 6% per annum with a probability of 0.7. Over the second ten years the interest rate earned will be 5% per annum with probability 0.5 and 6% per annum with probability 0.5.

(i) Calculate the premium that the company would charge if it calculates the

premium using the expected annual rate of interest in each ten year period. [2] (ii) Calculate the expected profit to the company if the premium is calculated as in

(i). The rate of interest in the second ten year period is independent of that in the first ten year period. [3]

(iii) Explain why, despite the company using the expected rate of interest to

calculate the premium, there is a positive expected profit. [2] (iv) By considering each possible outcome in (ii):

(a) Find the range of possible profits. (b) Calculate the standard deviation of the profit to the company. [7]

[Total 14]

CT1 S2009—6

10 A group of experts is analysing options to try to avert problems caused by climate change. They agree on the following expected costs and benefits of climate change over the next 50 years, starting from the current time. All figures are given in 2009 dollars.

Costs of climate change: Serious events will occur once every three years, in arrear, each giving rise to

costs of $30bn, incurred immediately on the date of the event. Communities affected by climate change will incur costs of $20bn per annum

incurred continuously, increasing at a continuous rate of 1% per annum. Other costs, assumed to be $40bn per annum, will be incurred annually in

arrear.

Benefits arising from climate change: Benefits from higher crop yields and lower heating costs are assumed to be

$10bn per annum, incurred annually in arrear. The experts are considering whether to recommend investment in a carbon storing technology which, it is believed, will reduce all the costs and benefits listed above to zero. The technology requires a one-off investment immediately of $440bn. Costs are then assumed to be $50bn per annum incurred annually in arrear for 50 years. The experts do not agree about the appropriate rate of interest at which to evaluate the options available. One group believes that the net present value of using the carbon storage technology should be evaluated at a real rate of return of 4% per annum effective. A second group believe that it should be evaluated at a real rate of return of 1% per annum effective. (i) Define what is meant by the discounted payback period of an investment and

indicate its main disadvantage as an investment decision criterion. [3] (ii) Explain why the project must have a discounted payback period when the

interest rate is 1.5% and the internal rate of return is higher than 1.5%. [2] (iii) Calculate the net present value of the carbon storing technology at a real rate

of interest of 1% per annum effective. [5] (iv) Calculate the net present value of the carbon storing technology at a real rate

of interest of 4% per annum effective. [5] ` (v) Comment on whether the investment in the carbon storing technology should

go ahead. [2] [Total 17]

END OF PAPER


Faculty of Actuaries

Institute of Actuaries

Subject CT1 — Financial Mathematics.

Core Technical.

September 2009 examinations

EXAMINERS REPORT

Introduction

The attached subject report has been written by the Principal Examiner with the aim of

helping candidates. The questions and comments are based around Core Reading as the

interpretation of the syllabus to which the examiners are working. They have however given

credit for any alternative approach or interpretation which they consider to be reasonable.

R D Muckart

Chairman of the Board of Examiners

December 2009

Comments for individual questions are given with the solutions that follow.

Subject CT1 (Financial Mathematics. Core Technical) — September 2009 —Examiners’ Report

Page 2

Please note that different answers may be obtained to those shown in these solutions depending

on whether figures obtained from tables or from calculators are used in the calculations but

candidates are not penalised for this. However, candidates may be penalised where excessive

rounding has been used or where insufficient working is shown. Well-prepared candidates scored well across the whole paper. However, the comments below on

each question concentrate on areas where candidates could have improved their performance.

1

a. 97.89

96 1.05 97.89 1.0596

t t

97.8996ln

0.400 years or 146 daysln 1.05

t

b. Second investor held the bill for 36 days. Therefore

36 365 100

97.89 1 100 1 21.854%365 36 97.89

i i

This was answered well except by the very weakest candidates.

2

Issued by corporations.

Holders entitled to a distribution (dividend) declared from profits.

Potential for high returns relative to other asset classes.

Commensurate risk of capital losses.

Lowest ranking finance issued by companies.

Initial running yield low but has potential to increase with dividend growth.

Dividends and capital values have the potential to grow in nominal terms during times of inflation.

Return made up of income return and capital gains.

Marketability depends on the size of the issue.

Ordinary shareholders receive voting rights in proportion to their holding.

This question was not answered as well as the examiners would have expected given that

the topic is standard bookwork.

3

We convert all cash flow to amounts in time 0 values:


Page 3

Dividend paid at 147.7

1:10000 0.041 394.77153.4

t


2 :10000 0.046 428.39158.6

t


3:10000 0.051 456.25165.1

t

Sale proceeds at 147.7

3:10000 0.93 8319.87165.1

t

Equation of value involving v where 1

1v

r

and r = real rate of return:

2 37800 394.77 428.39 8776.17 .....v v v (1)

[To estimate r:

Approx nominal rate of return is

93 78

4.6 / 78 12.3% p.a.3

Average inflation over 3 year period comes from

13165.1

1 3.8 %147.7

p.a.

Approx real return: 1.123

1 8.2 % p.a.1.038

]

Try 8%, RHS of (1) 7699.61r

7%, RHS of (1) 7907.09r

7907.09 7800

7% 1%7907.69 7699.61

r

= 7.52 % p.a.


Page 4

Some candidates seemed to struggle to derive the equation of value based on a real rate

of return and multiplied (rather than divided) the payments by the increase in the

inflation index.

4

(i) Let required price = P:

2 20

201 0.2 5 100 at 6%P a v

2 20

20220

0.06 = 11.4699 11.6394; 0.311805

0.059126

ia a v

i

Therefore

1 0.2 5 11.6394 100 0.311805

46.5576 31.1805 77.7381

P

(ii) The equation of value for the gross rate of return is:

2 20

2077.7381 5 100 a v

If i = 8%

2 20

20220 = 1.019615 9.8181 10.0107; 0.21455

ia a v

i

RHS = 50.0534 + 21.4550 = 71.5084

If i = 7%

2 20

20220 = 1.017204 10.5940 10.7763; 0.25842

ia a v

i

RHS = 53.8813 + 25.8420 = 79.7233

Interpolating gives 79.7233 77.7381

0.07 0.01 7.24% 7.2% say79.7233 71.5084

i

(iii) If the nominal rate of return is 7.2% per annum effective and inflation is 3% per

annum effective, then the real rate of return is calculated from:

1.0721 4.1%

1.03

This question was answered very well.


Page 5

5

(i)

55

2 313 0

0

130 100exp 100exp 100exp 5 41.667a bt dt at bt a b

1010

2 313 0

0

200 100exp 100exp 100exp 10 333.333a bt dt at bt a b

ln 1.3 5 41.667a b

ln 2 10 333.333a b

The second expression less twice times the first expression gives:

ln(2) 2ln(1.3) 250 0.0006737b b

ln(2) 333.333 0.0006737

0.0468610

a

(ii)

160

6012

12 12130100 1 130 12 1 5.259% p.a.

12 100

ii i

(iii) 5 200130 200 5 ln 8.616% p.a.

130e

This question was answered very well.

6

(i) A future is a contract which obliges the parties to deliver/take delivery of a

particular quantity of a particular asset at a particular time at a fixed price.

An option is the right to buy or sell a particular quantity of a particular asset at (or

before) a particular time at a given price.

(ii) Assume no arbitrage

a. Buying the forward is exactly the same as buying the bond except that the

forward will not pay coupons and the forward does not require immediate

settlement.


Page 6

Let the forward price = F. The equation of value is:

12

1.06 97 1.06 3.5 3.5

1.05

102.82 3.62059 3.5 95.6994

F

b. Let six month forward interest rate 1

20.5,0.5

1.061 3.4454%

1.05f

This does not have to be expressed as a rate of interest per annum

effective, though it could be.

c. 0.5 1

2 1.05 102 1.06 1.9518 96.2264 98.1782P

d. Gross redemption yield is i such that

0.5 1

98.1782 2 1 102 1i i

Using the formula for solving a quadratic (interpolation will do):

0.5

1 0.97133i . Therefore, i ≈ 6% (in fact 5.99%).

e. Answer is very close to 6% (the one-year spot rate) because the payments

from the bond are so heavily weighted towards the redemption time in one

year.

This was generally well-answered apart from part (e). A common error in parts (c) and

(d) was to assume that the coupon payments were 4% per half-year.

7 .

(i) The accumulation is 20 12 2012 12

20 20 201200 1.06 2300 100 1.06s s Ia

20 20

20 20 20121200 1.06 2300 100 1.06

1,200 36.7856 3.20714 2,300 36.78561.032211

100 98.7004 3.20714

1.032211 141,571.88 84,606.88 31,654.60

266,138

is s Ia

d


Page 7

(ii) Let half-yearly payment = X

40266,138 at 2.5%Xa

266,138

10,601.9425.1028

X

Therefore, annual rate of payment = £21,203.88

(iii) Work in half-years. Discounted mean term is:

2 4010,601.94 v + 2v + +40v /266,138

Numerator = 40

10,601.94 Ia at 2.5% per half year effective.

10,601.94 433.3248 4,584,075

Therefore DMT = 17.26 half years or 8.63 years.

In part (i), many candidates developed the correct formula although calculation errors

were common. In such cases, candidates also lost marks for not showing and explaining

their working fully. Part (ii) was answered well but many candidates surprisingly had

trouble calculating the DMT in part (iii). In this part, candidates often lost marks for not

showing the units properly at the end of the answer; indeed, in many cases, showing the

units may well have alerted candidates to possible mistakes.

2

(i) The equation of value for the borrower is 20

4,012.13 50,000a .

Therefore 20

50,000a = = 12.4622

4,012.13

From inspection of tables, i = 5%

(ii) The second customer pays interest of 0.055 50,000 = £2,750 per annum, annually in arrear.

The annual rate of monthly payments in advance from the savings policy is X such that:


Page 8

12

20

20 12

=50,000 at 4%

50,000

50,000£1,643.69

29.7781 1.021537

Xs

iXs

d

X

The equation of value for this borrower is:

12

20 20

20 2012

50,000 2,750 1,643.686

2,750 1,643.686

a a

ia a

d

Try i = 6%: RHS = 51,002.41

Try i = 7%: RHS = 47,200.14

By interpolation i = 6.3%

Part (i) was well answered but weaker candidates failed to recognise the need to

calculate separately the payments into the savings policy in part (ii).

3

(i) The expected annual interest rate in the first ten years is 0.3 0.04 + 0.7 0.06 =

0.054. The expected interest rate in the second ten years is clearly 5.5%.

If the premium is calculated on the basis of these interest rates, then the premium will be P such

that:

10 1020,000 1.054 1.055

20,000 2.89022 6,919.89

P

P P

(ii) The expected accumulation factor in the first ten years is:

10 100.3 1.04 0.7 1.06 1.69767

The expected accumulation factor in the second ten years is:

10 100.5 1.05 1.06 1.70987

As they are independent, we can multiply the accumulation factors together and multiply by the

premium to give an expected accumulation of: 6,919.89 1.69767 1.70987 = 20,087.04.


Page 9

The expected profit is 87.04.

(iii) There is an expected profit because (in general) the accumulation of a sum of money at the

expected interest rate is not equal to the expected accumulation when the interest rate is a random

variable.

(iv) The highest possible outcome for the accumulation factor is:

10 101.06 1.06 = 3.20714 with probability 0.7 0.5 = 0.35

The lowest possible outcome is:

10 101.04 1.05 = 2.41116 with probability 0.3 0.5 = 0.15.

The range is therefore: 6,919.89 (3.20714 – 2.41116) = 5,508.05.

The other two possible outcomes are:

10 101.06 1.05 = 2.91710 with probability 0.7 0.5 = 0.35

and 10 101.04 1.06 = 2.65089 with probability 0.3 0.5 = 0.15

The mean accumulation factor is: 1.69767 1.70987 = 2.90280

The variance of the accumulation from one unit of investment is:

0.35(3.20714-2.90280)2 + 0.15(2.41116-2.90280)

2

+ 0.35 (2.91710-2.90280)2 +0.15 (2.65089-2.90280)

2

= 0.03241 + 0.03626 + 0.00007 + 0.00952 = 0.07826.

Standard deviation is 0.07826 = 0.27976.

Standard deviation of the accumulation of the whole premium is: 6,919.89 0.27976 = £1,935.88

which is also the standard deviation of the profit.

This was the worst answered question on the paper with many candidates not recognising

that the accumulation of a sum of money at the expected interest rate is not equal to the

expected accumulation when the interest rate is a random variable. The calculation of the

standard deviation of the accumulation was generally only calculated correctly by the

strongest candidates.

4

(i) The discounted payback period is the first time at which the accumulated profit

from/net present value of the cash flows from a project is positive at a given

interest rate.


Page 10

It is an inappropriate decision criterion because it does not tell us anything about

the overall profitability of the project.

(ii) If the internal rate of return were greater than 1.5% then the net present value of

the project at 1.5% must be greater than zero. As such, there must be a discounted

payback period as the discounted payback period is the first time at which the net

present value is greater than zero: such a time must exist.

(iii) Returns are real rates of return and figures are in 2009 dollar terms so we are

automatically working with real rather than nominal values. All figures below are

in $bn.

The net benefits from using the technology are the $30 every three years; $20 incurred

continuously increasing at 1% per annum and $30 per annum incurred annually in arrears.

The costs of the technology are $440 incurred immediately and $50 incurred annually in arrears.

The net present value of the project at 1% per annum effective is:

3 6 48

50 5030 50 20 30 440 50v v v a a

The 20 does not need to be discounted because the cash flows are growing at the same rate as they

are being discounted.

48

3

503

130 560 20

1

vv a

v calculated at 1%

1 0.6202630 0.97059 560 20 39.1961

1 0.97059

= 375.967 560 783.922

152.045

(iv) The net present value of the project at 4% per annum effective is:

3 6 48 '

50 50 5030 20 30 440 50v v v a a a

All are calculated at 4% except '

50a which is calculated at

1.04 - 1 2.97%

1.01i

48

3 '

50 503

130 20 440 20

1

v iv a a

v

1 0.1521930 0.88900 20 1.014779 25.8755 440 20 21.4822

1 0.88900


Page 11

203.704 525.158 440 429.644

140.790

(v) Whether the investment should go ahead would depend on the choice of the interest rate – it is

clearly a crucial assumption (students could make a choice themselves and indicate whether it

should go ahead on the basis of that rate but there must be some justification for the choice).

This question was also poorly answered possibly because project appraisal using real

interest rates has rarely been examined in the past (and also possibly because of time

pressure). Whilst some parts of the question were challenging (e.g. the treatment of the

increasing costs of climate change), it was disappointing that many candidates failed to

recognise that the costs of climate change no longer incurred would be a benefit of the

carbon storing technology project and so failed to score many marks.



EXAMINATION

27 April 2010 (am)







Graph paper is NOT required for this paper.





CT1 A2010—2

1 (i) Explain the difference (a) between options and futures (b) between call options and put options [4] A security is priced at £60. Coupons are paid half-yearly. The next coupon is due in

two months’ time and will be £2.80. The risk-free force of interest is 6% per annum. (ii) Calculate the forward price an investor should agree to pay for the security in

three months’ time assuming no arbitrage. [3] [Total 7] 2 In January 2008, the government of a country issued an index-linked bond with a term

of two years. Coupons were payable half-yearly in arrear, and the annual nominal coupon rate was 4%. Interest and capital payments were indexed by reference to the value of an inflation index with a time lag of six months.

A tax-exempt investor purchased £100,000 nominal at issue and held it to redemption.

The issue price was £98 per £100 nominal. The inflation index was as follows:

Date Inflation Index

July 2007 110.5 January 2008 112.1 July 2008 115.7 January 2009 119.1 July 2009 123.2

(i) Calculate the investor’s cashflows from this investment and state the month when each cashflow occurs. [3]

(ii) Calculate the annual effective money yield obtained by the investor to the

nearest 0.1% per annum. [3] [Total 6]


3 A company issues ordinary shares to an investor who is subject to income tax at 20%. Under the terms of the ordinary share issue, the investor is to purchase 1,000,000

shares at a purchase price of 45p each on 1 January 2011. No dividend is expected to be paid for 2 years. The first dividend payable on

1 January 2013 is expected to be 5p per share. Dividends will then be paid every 6 months in perpetuity. The two dividend payments in any calendar year are expected to be the same, but the dividend payment is expected to increase at the end of each year at a rate of 3% per annum compound.

Calculate the net present value of the investment on 1 January 2011 at an effective

rate of interest of 8% per annum. [5] 4 An investor is considering purchasing a fixed interest bond at issue which pays half-

yearly coupons at a rate of 6% per annum. The bond will be redeemed at £105 per £100 nominal in 10 years’ time. The investor is subject to income tax at 20% and capital gains tax at 25%.

The inflation rate is assumed to be constant at 2.8571% per annum. Calculate the price per £100 nominal if the investor is to obtain a net real yield of 5%

per annum. [7] 5 Let tf denote the one-year forward rate of interest over the year from time t to time

( )1t + . The current forward rates in the market are:

time, t 0 1 2 3

one-year forward rate, tf 4.4% p.a. 4.7% p.a. 4.9% p.a. 5.0% p.a.

A fixed-interest security pays coupons annually in arrear at the rate of 7% per annum

and is redeemable at par in exactly four years. (i) Calculate the price per £100 nominal of the security assuming no arbitrage. [3] (ii) Calculate the gross redemption yield of the security. [3] (iii) Explain, without doing any further calculations, how your answer to part (ii)

would change if the annual coupon rate on the security were 9% per annum (rather than 7% per annum). [2]

[Total 8]

CT1 A2010—4

6 The annual returns, i, on a fund are independent and identically distributed. Each year, the distribution of 1 + i is lognormal with parameters μ = 0.05 and σ2 = 0.004, where i denotes the annual return on the fund.

(i) Calculate the expected accumulation in 25 years’ time if £3,000 is invested in

the fund at the beginning of each of the next 25 years. [5] (ii) Calculate the probability that the accumulation of a single investment of £1

will be greater than its expected value 20 years later. [5] [Total 10] 7 A pension fund has to pay out benefits at the end of each of the next 40 years. The

benefits payable at the end of the first year total £1 million. Thereafter, the benefits are expected to increase at a fixed rate of 3.8835% per annum compound.

(i) Calculate the discounted mean term of the liabilities using a rate of interest of

7% per annum effective. [5] The pension fund can invest in both coupon-paying and zero-coupon bonds with a

range of terms to redemption. The longest-dated bond currently available in the market is a zero-coupon bond redeemed in exactly 15 years.

(ii) Explain why it will not be possible to immunise this pension fund against

small changes in the rate of interest. [2] (iii) Describe the other practical problems for an institutional investor who is

attempting to implement an immunisation strategy. [3] [Total 10] 8 A loan is repayable by annual instalments paid in arrear for 20 years. The first

instalment is £4,650 and each subsequent instalment is £150 greater than the previous instalment.

Calculate the following, using an interest rate of 9% per annum effective: (i) the amount of the original loan [3] (ii) the capital repayment in the tenth instalment [4] (iii) the interest element in the last instalment [2] (iv) the total interest paid over the whole 20 years [2] [Total 11]


9 A company is undertaking a new project. The project requires an investment of £5m at the outset, followed by £3m three months later.

It is expected that the investment will provide income over a 15 year period starting

from the beginning of the third year. Net income from the project will be received continuously at a rate of £1.7m per annum. At the end of this 15 year period there will be no further income from the investment.

Calculate at an effective rate of interest of 10% per annum: (i) the net present value of the project [3] (ii) the discounted payback period [4] A bank has offered to loan the funds required to the company at an effective rate of

interest of 10% per annum. Funds will be drawn from the bank when required and the loan can be repaid at any time. Once the loan is paid off, the company can earn interest on funds from the venture at an effective rate of interest of 7% per annum.

(iii) Calculate the accumulated profit at the end of the 17 years. [4] [Total 11]

10 A pension fund’s assets were invested with two fund managers. On 1 January 2007 Manager A was given £120,000 and Manager B was given

£100,000. A further £10,000 was invested with each manager on 1 January 2008 and again on 1 January 2009.

The values of the funds were: 31 December 2007 31 December 2008 31 December 2009 Manager A £130,000 £135,000 £180,000 Manager B £140,000 £145,000 £150,000 (i) Calculate the time-weighted rates of return earned by Manager A and Manager

B over the period 1 January 2007 to 31 December 2009. [4] (ii) Show that the money-weighted rate of return earned by Manager A over the

period 1 January 2007 to 31 December 2009 is approximately 9.4% per annum. [2]

(iii) Explain, without performing further calculations, whether the money-weighted

rate of return earned by Manager B over the period 1 January 2007 to 31 December 2009 was higher than, lower than or equal to that earned by Manager A. [3]

(iv) Discuss the relative performance of the two fund managers. [3] [Total 12]

CT1 A2010—6

11 The force of interest δ(t) is a function of time and at any time t, measured in years, is given by the formula

0.04 0.02 0 5

( )0.05 5

t tt

t+ ≤ <⎧

δ = ⎨ ≤⎩.

(i) Derive and simplify as far as possible expressions for v(t), where for v(t) is the

present value of a unit sum of money due at time t. [5] (ii) (a) Calculate the present value of £1000 due at the end of 17 years. (b) Calculate the rate of interest per annum convertible monthly implied

by the transaction in part (ii)(a). [4] A continuous payment stream is received at a rate of 10e0.01t units per annum between

t = 6 and t = 10.

(iii) Calculate the present value of the payment stream. [4] [Total 13]

END OF PAPER


EXAMINERS’ REPORT

April 2010 Examinations


Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. R D Muckart Chairman of the Board of Examiners July 2010



Page 2

Comments Please note that different answers may be obtained to those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. Well-prepared candidates scored well across the whole paper and the examiners were pleased with the general standard of answers. However, questions that required an element of explanation or analysis were less well answered than those which just involved calculation. The comments below concentrate on areas where candidates could have improved their performance. Q2. A common error was to divide the nominal payments by the increase in the index factor (rather than multiplying). Q3. Many candidates made calculation errors in this question but may have scored more marks if their working had been clearer. Q6. Many candidates assumed that the accumulation in part (i) was for a single payment. Q7. The calculation was often performed well. In part (ii), many explanations were unclear and some candidates seemed confused between DMT and convexity although a correct explanation could involve either of these concepts. Q9. A common error was to assume that income only started after three years rather than ‘starting from the beginning of the third year’. Q10. This question was answered well but examiners were surprised by the large number of candidates who used interpolation or other trial and error methods in part (ii) when the answer had been given in the question. The examiners recommend that students pay attention to the details given in the solutions to parts (iii) and (iv). For such questions, candidates should be looking critically at the figures given/calculated and making points specific to the scenario rather than just making general statements taken from the Core Reading.


Page 3

1 (i) (a) Options – holder has the right but not the obligation to trade Futures – both parties have agreed to the trade and are obliged to do so. (b) Call Option – right but not the obligation to BUY specified asset at

specified price at specified future date. Put Option – right but not the obligation to SELL specified asset at

specified price at specified future date. (ii)

3 112 120.06 0.0660 2.80 60.90678 2.81404 £58.09K e e× ×= − = − =

2 (i) Cash flows: Issue price: Jan 08 0.98 100,000− × = –£98,000

Interest payments: July 08 112.10.02 100,000110.5

× × = £2,028.96

Jan 09 115.70.02 100,000110.5

× × = £2,094.12

July 09 119.10.02 100,000110.5

× × = £2,155.66

Jan 10 123.20.02 100,000110.5

× × = £2,229.86

Capital redeemed: Jan 10 123.2100,000110.5

× = £111,493.21

(ii) Equation of value is:

1 12 21 2 298000 2028.96 2094.12 2155.66 2229.86 111493.21v v v v v= + + + +

At 11%, RHS = 97955.85 ≈ 98000


Page 4

3 Purchase price = 0.45 × 1,000,000 = £450,000

PV of dividends = ( )1 1 12 2 22 3 42 3 2 450000 1 0.2 1.03 1.03v v v v v v⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞× − × + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

= 1222 2 240000 1 1.03 1.03v v v v⎛ ⎞ ⎡ ⎤+ + + +⎜ ⎟ ⎣ ⎦⎝ ⎠

@ 8%

= 140000 1.68231 1,453,5161 1.03 1.08⎛ ⎞

× × =⎜ ⎟−⎝ ⎠

⇒ NPV = 1,453,516 – 450,000 = £1,003,516 4 Let i = money yield 1 1.0285714 1.05 1.08i⇒ + = × = 8%i⇒ = p.a. Check whether CGT is payable: compare ( )2i with ( )1 t g−

( ) 61 0.8 0.04571105

t g− = × =

From tables, ( )2 7.8461%i = ( ) ( )2 1i t g⇒ > − ⇒ CGT is payable

P ( )2 10 1010

0.8 6 105 0.25(105 )a v P v= × + − − @ 8%

( )2 1010

10

0.8 6 0.75 105

1 0.25

a v

v

× + ×=

−

4.8 1.019615 6.7101 78.75 0.463191 0.25 0.46319

× × + ×=

− ×

= £78.39


Page 5

5 (i) Let P denote the current price (per £100 nominal) of the security. Then, we have:

7 7 7 107 108.08721.044 1.044 1.047 1.044 1.047 1.049 1.044 1.047 1.049 1.05

P = + + + =× × × × × ×

(ii) The gross redemption yield, i , is given by: % 4

%4108.09 7 100iia v= × + ×

Then, we have:

( )5% 107.0919 108.0872 108.96880.045 0.05 0.045 0.04734.5% 108.9688 107.0919 108.9688

i RHSi

i RHS= ⇒ = ⎫ −⎛ ⎞⇒ ≈ + − × =⎬ ⎜ ⎟= ⇒ = −⎝ ⎠⎭

(iii) The gross redemption yield represents a weighted average of the forward rates

at each duration, weighted by the cash flow received at that time. Thus, increasing the coupon rate will increase the weight applied to the cash

flows at the early durations and, as the forward rates are lower at early durations, the gross redemption yield on a security with a higher coupon rate will be lower than above.

Note to markers: no marks for simply plugging 9% pa in, and providing no

explanation for result. 6 (i) ( )

2121E i eμ+ σ+ =

120.05 0.004e + ×=

1.0533757= [ ] ( ) ( )0.0533757 since 1 1E i E i E i∴ = + = + Let A be the accumulation at the end of 25 years of £3,000 paid annually in

advance for 25 years.


Page 6

Then [ ] 253000 at rate 0.0533757E A S j= =

( )( )

( )251 1

3000 1j

jj

+ −= × +

( )251.0533757 1

3000 1.05337570.0533757

−= ×

£158,036.43= (ii) Let the accumulation be 20S 20S has a log-normal distribution with parameters 20μ and 220σ

[ ]21

220 2020E S e μ+ × σ∴ =

( ){ }20or 1 j+

( )exp 20 0.05 10 0.004= × + × 1.04 2.829217e= = In ( )2

20 ~ 20 , 20S N μ σ

⇒ In ( )20 ~ 1, 0.08S N ( ) ( )20 20Pr 2.829217 Pr 1n S 1n 2.829217S > = >

( )1n 2.829217-1Pr where 0,10.08

N⎛ ⎞= Ζ > Ζ⎜ ⎟

⎝ ⎠∼

( ) ( )Pr 0.14 1 0.14Z= > = −Φ = 1 – 0.55567 = 0.44433 i.e. 44.4%


Page 7

7 (i) DMT of liabilities is given by:

( ) ( ) ( )

( ) ( ) ( )

2 392 3 407% 7% 7% 7%

2 392 3 407% 7% 7% 7%

1 1 2 1.038835 3 1.038835 40 1.038835

1 1.038835 1.038835 1.038835

v v v v

v v v v

× × + × × + × × + + × ×

× + × + × + + ×

…

…

= ( )

( )

2 3 401

2 3 401

1.038835 1.038835 1.038835 1.0388351.038835 2 3 401.07 1.07 1.07 1.07

1.038835 1.038835 1.038835 1.0388351.0388351.07 1.07 1.07 1.07

−

−

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞× + × + × + + ×⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞× + + + +⎢⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣

…

…⎤⎥⎦

= * * * *

* * * *

2 3 40

2 3 40

2 3 40i i i i

i i i i

v v v v

v v v v

+ × + × + + ×

+ + + +

…

…

= ( )

*

*40

40

i

i

Ia

a

where **

*1 1.038835 1.07 0.07 0.0388351 0.03

1.07 1.038835 1.0388351iv ii

−≡ = ⇒ = − = =

+.

Hence, DMT of liabilities is:

( )3%

403%40

384.8647 16.6523.1148

Ia

a= = years

(Alternative method for DMT formula

( )3%3% 3%2 2 39 39

40 40 402 2 39 39 3% 3% 3%

40 40 40

( ) ( )(1 2 3 40 )(1 )

Iav Ia Iav gv g v g vDMTv gv g v g v va a a+ + + +

= = = =+ + + +

where 1.038835g = .)

(ii) Even if the fund manager invested entirely in the 15-year zero-coupon bond,

the DMT of the assets will be only 15 years (and, indeed, any other portfolio of securities will result in a lower DMT).

Thus, it is not possible to satisfy the second condition required for

immunisation (i.e. DMT of assets = DMT of liabilities). Hence, the fund cannot be immunised against small changes in the rate of

interest.


Page 8

(iii) The other problems with implementing an immunisation strategy in practice include:

• the approach requires a continuous re-structuring of the asset portfolio to

ensure that the volatility of the assets remains equal to that of the liabilities over time

• for most institutional investors, the amounts and timings of the cash flows

in respect of the liabilities are unlikely to be known with certainty • institutional investor is only immunised for small changes in the rate of

interest

• the yield curve is unlikely to be flat at all durations

• changes in the term structure of interest rates will not necessarily be in the form of a parallel shift in the curve (e.g. the shape of the curve can also change from time to time)

8 (i) Loan = ( )20 204500 150a Ia+ at 9% ⇒Loan = 4500 9.1285 150 70.9055× + × 41,078.25 10,635.83 51,714.08= + = (ii) Loan o/s after 9th year = ( ) ( )11 114500 1350 150a Ia+ + at 9% Loan o/s = 5,850 6.8052 150 35.0533× + × 39,810.42 5258.00 45,068.42= + = Repayment = 6000 45,068.42 0.09 £1,943.84− × =

(Alternative solution to (ii) (ii) Loan o/s after 9th year = ( ) ( )11 114500 1350 150a Ia+ + at 9% = 5,850 6.8052 150 35.0533 45,068.42× + × = as before Loan o/s after 10th year = ( ) ( )10 104500 1500 150a Ia+ + at 9% = 6,000 6.4177 150 30.7904 43,124.76× + × = Repayment = 45,068.42 43,124.76 £1,943.66− = )


Page 9

(iii) Last instalment = 4650 19 150 7500+ × = Loan o/s = 17500 7500a v= Interest = 7500 0.91743 0.09 £619.27× × =

(iv) Total payments 120 4650 19 20 1502

= × + × × ×

93,000 28,500 121,500= + = Total interest = 121,500 – 51,714.08 = £69,785.92 9 (i) NPV =

14 2

155 3 1.7v a v− − + @10%

NPV = 155 3 0.976454 1.7 0.82645 @10%i a− − × + × ×δ

5 2.929362 1.404965 1.049206 7.6061= − − + × × 7.929362 11.21213458= − + 3.282772575= NPV = £3.283m (ii) DPP is 2t + such that

1421.7 5 3 1.474097708 7.929362 @10%t ta v v a= + ⇒ =

1 1.1 5.379129 1 1.1 0.53791290.1

tt

−−−

= ⇒ − =

0.4620871 1.1 1n 0.4620871 1n 1.1t t−⇒ = ⇒ = − 8.100t⇒ = 10.1DPP∴ = years


Page 10

(iii) Accumulated profit 17 years from start of project:

( )6.9

6.9 7%

1.07 11.7 1.7 @ 7%s

−= = ×

δ

( )6.91.07 1

1.70.067659

−= ×

1.7 8.79346= × £14.95m= 10 (i) The values of the funds before and after the cash injections are:

Manager A

Manager B

1 January 2007 120,000 100,000 31 December 2007 130,000 140,000 140,000 150,000 31 December 2008 135,000 145,000 145,000 155,000 31 December 2009 180,000 150,000

Thus, TWRR for Manager A is given by:

( )3 130 135 1801 0.0905 or 9.05%120 140 145

i i+ = × × ⇒ =

And, TWRR for Manager B is given by:

( )3 140 145 1501 0.0941 or 9.41%100 150 155

i i+ = × × ⇒ =

(ii) MWRR for Manager A is given by: ( ) ( ) ( )3 2120 1 10 1 10 1 180i i i× + + × × + × + = Then, putting 0.094 gives 180.03i LHS= = which is close enough to 180. (iii) Both funds increased by 50% over the three year period and received the same

cashflows at the same times. Since the initial amount in fund B was lower, the cash inflows received

represent a larger proportion of fund B and hence the money weighted return earned by fund B over the period will be lower, particularly since the returns were negative for the 2nd and 3rd years.


Page 11

[Could also note that for fund B: ( ) ( ) ( )3 2100 1 10 1 10 1 150i i i× + + × × + × + = So by a proportional argument ( ) ( ) ( )3 2120 1 12 1 12 1 180i i i× + + × × + × + = which when compared with the equation for fund A in (ii) clearly shows that

the return for B is lower.] (iv) The money weighted rate of return is higher for fund A, whilst the time

weighted return is higher for fund B. When comparing the performance of investment managers, the time weighted

rate of return is generally better because it ignores the effects of cash inflows or outflows being made which are beyond the manager’s control.

In this case, Manager A’s best performance is in the final year, when the fund

was at its largest, whilst Manager B’s best performance was in the first year, where his fund was at its lowest.

Overall, it may be argued that Manager B has performed slightly better than

Manager A since Manager B achieved the higher time weighted return. 11 (i) 5t <

( ) ( )00.04 0.02t s ds

v t e− +∫=

= 2

00.04 0.01

ts s

e⎡ ⎤− +⎣ ⎦

= 20.04 0.01t t

e⎡ ⎤− +⎣ ⎦

5t ≥

( )v t = ( ){ }5

0 50.04 0.02 0.05ts ds ds

e− + +∫ ∫

= ( ) ( )0.05 55 tv e ⎡ ⎤− −⎣ ⎦×

= ( ) [ ]0.05 5 0.05 0.20.45 t te e e⎡ ⎤− − − +− ⎣ ⎦× =


Page 12

(ii) (a) [ ]0.05 17 0.2 1.051,000PV e e− × + −= = = 349.94

(b) ( ) 20412

1000 1 349.9412

i−

⎛ ⎞⎜ ⎟+ =⎜ ⎟⎝ ⎠

( )12 6.1924%i⇒ =

(iii) [ ]10 0.05 0.250.45 0.016

10t tPV e e e dt− −−= ∫

= 100.2 0.046

10 te e dt− −∫

= 100.04

0.2

6

100.04

tee−

− ⎡ ⎤−⎢ ⎥⎢ ⎥⎣ ⎦

= 8.18733 2.90769× = 23.806 (Alternative Solution to (iii) Accumulated value at time t = 10

[ ]( )

10 100.016

10 100.016

10 100.01 0.5 0.05 0.5 0.046 6

100.5 0.04

6

10 exp 0.05

10 exp 0.05

10 10

10 276.293 324.233 47.9400.04

tt

tt

t t t

t

e ds dt

e s dt

e e dt e dt

e

− −

−

⎛ ⎞= ⎜ ⎟⎝ ⎠

=

= =

⎡ ⎤= = − + =⎢ ⎥

−⎢ ⎥⎣ ⎦

∫ ∫

∫

∫ ∫

Present value = ( ) [ ]0.05 10 0.2510 47.940 0.63763 47.940 23.806v e− × −× = × =



EXAMINATION

7 October 2010 (am)







Graph paper is NOT required for this paper.





CT1 S2010—2

1 A bond pays coupons in perpetuity on 1 June and 1 December each year. The annual coupon rate is 3.5% per annum. An investor purchases a quantity of this bond on 20 August 2009.

Calculate the price per £100 nominal to provide the investor with an effective rate of

return per annum of 10%. [3] 2 A bond is redeemed at £110 per £100 nominal in exactly four years’ time. It pays

coupons of 4% per annum half-yearly in arrear and the next coupon is due in exactly six months’ time. The current price is £110 per £100 nominal.

(i) (a) Calculate the gross rate of return per annum convertible half-yearly

from the bond. (b) Calculate the gross effective rate of return per annum from the bond. [2] (ii) Calculate the net effective rate of return per annum from the bond for an

investor who pays income tax at 25%. [2] [Total 4] 3 The annual rates of return from an asset are independently and identically distributed.

The expected accumulation after 20 years of £1 invested in this asset is £2 and the standard deviation of the accumulation is £0.60.

(a) Calculate the expected effective rate of return per annum from the asset,

showing all the steps in your working. (b) Calculate the variance of the effective rate of return per annum. [6] 4 A six-month forward contract was issued on 1 April 2009 on a share with a price of

700p at that date. It was known that a dividend of 20p per share would be paid on 1 May 2009. The one-month spot, risk-free rate of interest at the time of issue was 5% per annum effective and the forward rate of interest from 1 May to 30 September was 3% per annum effective.

(i) Calculate the forward price at issue, assuming no arbitrage, explaining your

working. [3]

It has been suggested that the forward price cannot be calculated without making a judgement about the expected price of the share when the forward contract matures.

(ii) Explain why this statement is not correct. [2] (iii) Comment on whether the method used in part (i) would still be valid if it was

not known with certainty that the dividend due on 1 May 2009 would be paid. [1]

[Total 6]


5 (a) Describe the characteristics of Eurobonds. (b) Describe the characteristics of convertible bonds. [6]

6 On 1 January 2001 the government of a particular country bought 200 million shares in a particular bank for a total price of £2,000 million. The shares paid no dividends for three years. On 30 June 2004 the shares paid dividends of 10 pence per share. On 31 December 2004, they paid dividends of 20 pence per share. Each year, until the end of 2009, the dividend payable every 30 June rose by 10% per annum compound and the dividend payable every 31 December rose by 10% per annum compound. On 1 January 2010, the shares were sold for their market price of £3,500 million.

(i) Calculate the net present value on 1 January 2001 of the government’s

investment in the bank at a rate of interest of 8% per annum effective. [5] (ii) Calculate the accumulated profit from the government’s investment in the

bank on the date the shares are sold using a rate of interest of 8% per annum effective. [1] [Total 6]

7 (i) State the three conditions that are necessary for a fund to be immunised from small, uniform changes in the rate of interest. [2]

(ii) A pension fund has liabilities of £10m to meet at the end of each of the next

ten years. It is able to invest in two zero-coupon bonds with a term to redemption of three years and 12 years respectively. The rate of interest is 4% per annum effective.

Calculate:

(a) the present value of the liabilities of the pension fund

(b) the duration of the liabilities of the pension fund

(c) the nominal amount that should be invested in the zero-coupon bonds to ensure that the present values and durations of the assets and liabilities is the same

[7]

(iii) One year later, just before the pension payment then due, the rate of interest is 5% per annum effective. (a) Determine whether the duration of the assets and the liabilities are still

equal. (b) Comment on the practical usefulness of the theory of immunisation in

the context of the above result. [6]

[Total 15]

CT1 S2010—4

8 The force of interest, δ(t), is a function of time and at any time t, measured in years, is given by the formula

( )

0.05 0.001 0 200.05 20

t tt

t+ ≤ ≤⎧

δ = ⎨ >⎩

(i) Derive and simplify as far as possible expressions for v(t), where v(t) is the

present value of a unit sum of money due at time t. [5] (ii) (a) Calculate the present value of £100 due at the end of 25 years.

(b) Calculate the rate of discount per annum convertible quarterly implied by the transaction in part (ii)(a). [4]

(iii) A continuous payment stream is received at rate 30e−0.015t units per annum

between t = 20 and t = 25. Calculate the accumulated value of the payment stream at time t = 25. [4]

[Total 13] 9 The government of a particular country has just issued three bonds with terms to

redemption of exactly one, two and three years respectively. Each bond is redeemed at par and pays coupons of 8% annually in arrear. The annual effective gross redemption yields from the one, two and three year bonds are 4%, 3% and 3% respectively.

(i) Calculate the one-year, two-year and three-year spot rates of interest at the

date of issue. [8]

(ii) Calculate all possible forward rates of interest from the above spot rates of interest. [4]

An index of retail prices has a current value of 100. (iii) Calculate the expected level of the retail prices index in one year, two years’

and three years’ time if the expected real spot rates of interest are 2% per annum effective for all terms. [5]

(iv) Calculate the expected rate of inflation per annum in each of the next three

years. [2] [Total 19]

CT1 S2010—5

10 On 1 April 2003 a company issued securities that paid no interest and that were to be redeemed for £70 after five years. The issue price of the securities was £64. The securities were traded in the market and the market prices at various different dates are shown in the table below.

Date Market price

of securities (£)

1 April 2003 64 1 April 2004 65 1 April 2005 60 1 April 2006 65 1 April 2007 68 1 April 2008 70

(i) Explain why the price of the securities might have fallen between 1 April 2004

and 1 April 2005. [1]

Two investors bought the securities at various dates. Investor X bought 100 securities on 1 April 2003 and 1,000 securities on 1 April 2005. Investor Y bought 100 securities every year on 1 April from 2003 to 2007 inclusive. Both investors held the securities until maturity. (ii) Construct a table showing the nominal amount of the securities held and the

market value of the holdings for X and Y on 1 April each year, just before any purchases of securities. [5]

(iii) (a) Calculate the effective money weighted rate of return per annum for X

for the period from 1 April 2003 to 1 April 2008. (b) Calculate the effective time weighted rate of return per annum for X

for the period from 1 April 2003 to 1 April 2008. [6] (iv) (a) Determine whether the effective money weighted rate of return for Y is

lower or higher than that for X for the period from 1 April 2003 to 1 April 2008.

(b) Determine the effective time weighted rate of return per annum for Y

for the period from 1 April 2003 to 1 April 2008. [7] (v) Discuss the relationship between the different rates of return that have been

calculated. [3] [Total 22]

END OF PAPER

INSTITUTE AND FACULTY OF ACTUARIES

EXAMINERS’ REPORT

September 2010 Examinations


Introduction The attached subject report has been written by the Principal Examiner with the aim of helping candidates. The questions and comments are based around Core Reading as the interpretation of the syllabus to which the examiners are working. They have however given credit for any alternative approach or interpretation which they consider to be reasonable. T J Birse Chairman of the Board of Examiners December 2010

© Institute and Faculty of Actuaries


Page 2

Comments Please note that different answers may be obtained from those shown in these solutions depending on whether figures obtained from tables or from calculators are used in the calculations but candidates are not penalised for this. However, candidates may be penalised where excessive rounding has been used or where insufficient working is shown. Candidates also lose marks for not showing their working in a methodical manner which the examiner can follow. This can particularly affect candidates on the pass/fail borderline when the examiners have to make a judgement as to whether they can be sure that the candidate has communicated a sufficient command of the syllabus to be awarded a pass. The general standard of answers was noticeably lower than in previous sessions and there were a significant number of very ill-prepared candidates. As in previous exams, questions that required an element of explanation or analysis were less well answered than those which just involved calculation. Comments on individual questions, where relevant, can be found after the solution to each question. These comments concentrate on areas where candidates could have improved their performance.


Page 3

1 Working in half years:

The present value of the security on 1st June would have been ( )23.5

i

20 August is 80 days later so the present value is ( ) ( )80

3652

3.5 1 ii

+

Hence the price per £100 nominal is ( )80

3653.5 1.1 £36.6110.097618

=

2 (i) (a) Gross rate of return convertible half yearly is simply 4/110 = 0.03636 or 3.636%.

(b) Gross effective rate of return is 20.036361 1

2⎛ ⎞+ − =⎜ ⎟⎝ ⎠

0.03669 or 3.669%

(ii) The net effective rate of return per half year is 0.036360.75 0.0136352

× = .

The net effective rate of return per annum is therefore: ( )21.013635 1− = 0.02746 or 2.746%. A common error was to divide the nominal payments by the increase in the index factor (rather than multiplying). 3 (a) Let 20S be the accumulation of the unit investment after 20 years: ( ) ( )( ) ( )20 1 2 201 1 1E S E i i i⎡ ⎤= + + +⎣ ⎦… ( ) [ ] [ ] [ ] { }20 1 2 201 1 1 as tE S E i E i E i i= + + +… are independent [ ]tE i j= ( ) ( )20

20 1 2E S j∴ = + =

1202 1 3.5265%j⇒ = − =


Page 4

(b) The variance of the effective rate of return per annum is 2s where

[ ] ( )( ) ( )202 402 2Var 1 1 0.6nS j s j= + + − + =

( )( ) ( )

( )

120

1 120 10

220 22 2

2 2

0.6 1 1

0.6 2 2 0.004628

s j j⎡ ⎤

= + + − +⎢ ⎥⎣ ⎦

= + − =

Many candidates made calculation errors in this question but may have scored more marks if their working had been clearer. 4 (i) Assuming no arbitrage, buying the share is the same as buying the forward

except that the cash does not have to be paid today and a dividend will be payable from the share.

Therefore, price of forward is:

( ) ( ) ( )5 51

12 12 12700 1.05 1.03 20 1.03 − = 711.562 – 20.248 = 691.314 (ii) The no arbitrage assumption means that we can compare the forward with the

asset from which the forward is derived and for which we know the market price. As such we can calculate the price of the forward from this, without knowing the expected price at the time of settlement. [It could also be mentioned that the market price of the underlying asset does, of course, already incorporate expectations].

(iii) If it was not known with certainty that the dividend would be received we

could not use a risk-free interest rate to link the cash flows involved with the purchase of the forward with all the cash flows from the underlying asset.

5 (a) Eurobonds

• Medium-to-long-term borrowing. • Pay regular coupon payments and a capital payment at maturity. • Issued by large corporations, governments or supranational organisations. • Yields to maturity depend on the risk of the issuer. • Issued and traded internationally (not in core reading). • Often have novel features. • Usually unsecured • Issued in any currency • Normally large issue size • Free from regulation of any one government


Page 5

(b) Convertible Securities

• Generally unsecured loan stocks. • Can be converted into ordinary shares of the issuing company. • Pay interest/coupons until conversion. • Provide levels of income between that of fixed-interest securities and equities. • Risk characteristics vary as the final date for convertibility approaches. • Generally less volatility than in the underlying share price before conversion. • Combine lower risk of debt securities with the potential for gains from

equity investment. • Security and marketability depend upon issuer • Generally provide higher income than ordinary shares and lower income than

conventional loan stock or preference shares

6 (i) Net present value (all figures in £m)

( )

( )3 0.5 1.5 2 2.5 5 5.5

3 2 2 3 5 6 9

2,000 0.1 200 1.1 1.1 1.1

0.2 200 1.1 1.1 1.1 3,500

v v v v v

v v v v v v

= − + × × + + + +

+ × × + + + + +

…

…

at 8% per annum effective.

( ) ( ) ( ) ( )( )

( )

2 3 62.5 3 9

2.5 3 ' 96

2002,000 0.1 0.2 1.1 1.1 1.1 1.1 3,5001.1

2002,000 0.1 0.2 3,5001.1

v v v v v v v

v v a v

= − + + + + + + +

= − + + +

…

where the annuity is evaluated at a rate of 0.08 0.1 1.818%1 0.1

−= −

+ per annum

effective.

( ) 6'6

1 1 0.0181816.4011

0.018181a

−− −= =

−

and so net present value is

( )2.5 3 92002,000 0.1 1.08 0.2 1.08 6.4011 3,500 1.08 £31.661.1

m− − −− + × + × × + × =

(ii) Accumulated profit at the time of sale is 931.66 1.08 £63.30m× = Many candidates assumed that the accumulation in part (i) was for a single payment.


Page 6

7 (i) The present value of the assets is equal to the present value of the liabilities. The duration of the assets is equal to the duration of the liabilities. The spread of the asset terms around the duration is greater than that for the liability terms (or, equivalently, convexity of assets is greater).

(ii) (a) Present value of liabilities (in £m)

1010 at 4% 10 8.1109 81.109a= = × =

(b) Duration is equal to( )10

10

10 41.9922at 4% 5.1773 years10 8.1109

Iaa

= =

(c) Let the amounts to be invested in the two zero coupon bonds be X and

Y.

3 12

3 12

81.109 (1)

3 12 419.922 (2)

Xv Yv

Xv Yv

+ =

+ =

(2) less 3 times (1) gives:

129 176.595

176.595 £31.4159 0.62460

Yv

Y m

=

⇒ = =×

Substituting back into (1) gives:

( )81.109 31.415 0.62460£69.164

0.88900X m

− ×= =

(iii) (a) In one year, the present value of the liabilities is:

910 10 at 5% 10 10 7.1078 81.078a+ = + × = Numerator of duration is ( )910 0 10 332.347Ia× + =

Duration of liabilities is therefore 332.347 4.0991 years81.078

=

Present value of assets is: 2 1169.164 31.415 69.164 0.90703 31.415 0.58468v v× + × = × + × 81.101=


Page 7

Duration of assets will be:

2 112 69.164 11 31.41581.101

2 69.164 0.90703 11 31.415 0.58468 4.0383 years81.101

v v× × + × ×

× × + × ×= =

(b) One of the problems of immunisation is that there is a need to

continually adjust portfolios. In this example, a change in the interest rate means that a portfolio that has a present value and duration equal to that of the liabilities at the outset does not have a present value and duration equal to that of the liabilities one year later.

The calculation was often performed well. In part (ii), many explanations were unclear and some candidates seemed confused between DMT and convexity although a correct explanation could involve either of these concepts. 8 (i) 20t ≤ :

( )

2

0

2

0

0.05 0.0005

exp 0.05 0.001

0.001exp 0.052

t

t

t t

v t sds

ss

e− −

⎛ ⎞= − +⎜ ⎟⎝ ⎠

⎧ ⎫⎡ ⎤⎪ ⎪= − +⎢ ⎥⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

=

∫

20t > :

( ) ( )

( ) [ ]{ }

20

0 20

20

1.2 1 0.05 0.2 0.05

exp 0.05

20 exp 0.05

t

t

t t

v t s ds ds

v s

e e e− − − −

⎧ ⎫⎛ ⎞= − δ +⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

= −

= =

∫ ∫

(ii) (a) ( ) 0.2 0.05 25PV = 100 25 100v e− − ×= 1.45100 £23.46e−= =


Page 8

(b) ( )

( )4 254

100 1 100 25 23.464

d v×

⎛ ⎞⎜ ⎟− = =⎜ ⎟⎝ ⎠

( ) ( )11004 4 1 0.2346 0.05758d⇒ = − =

(iii) PV = 25 0.015 (0.2 0.05 )20

30 t te e dt− − +∫

( )

0.2 25250.2 0.065 0.06520 20

0.21.625 1.3

30300.065

30 28.5750.065

t tee e dt e

e e e

−− − −

−− −

⎡ ⎤= = ⎣ ⎦−

= − =−

∫

Accumulated value = ( )

0.2 0.05 25 1.4528.575 28.575 28.575 121.8225

e ev

+ ×= = =

9 (i) The one-year spot rate of interest is simply 4% per annum effective. For two-year spot rate of interest First we need to find the price of the security, P:

228 100P a v= + at 3% per annum effective.

2

2 1.91347 0.942596a v= = 8 1.91347 100 0.942596 109.5673P⇒ = × + × =

Let the t-year spot rate of interest be it. We already know that 1i = 4%. 2i is such that:

( )22

8 108109.56736 1.04 1 i

= ++

( ) 2

21 0.943287i −⇒ + = 2 0.029623 or 2.9623%.i⇒ =


Page 9

For three-year spot rate of interest we need to find the price of the security P: 3

38 100P a v= + at 3% per annum effective. 3

3 2.8286 0.91514a v= = 8 2.8286 100 0.91514 114.1428P⇒ = × + × = 3i is such that:

( ) ( )2 3

3

8 8 108114.1428 +1.04 1.029623 1 i

= ++

( )33

108 114.1428 15.23860 98.90421 i

⇒ = − =+

3 0.02976 or 2.976%.i⇒ = (ii) The one year forward rate of interest beginning at the present time is clearly

4%. The forward rate for one year beginning in one year is 1,1f such that: ( ) 2

1,1 1,11.04 1 1.029623 0.01935 1.935%.f f+ = ⇒ = = The forward rate for one year beginning in two years is 2,1f such that: ( )2 3

2,1 2,11.029623 1 1.02976 0.03003 3.003%.f f+ = ⇒ = = The forward rate for two years beginning in one year is 1,2f such that:

( )231,21.02976 1.04 1 f= +

1,2 0.02468 2.468%f⇒ = =


Page 10

(iii) Let the t-year “spot rate of inflation” be et

For each term ( )( )

( )t t

tt

1 11.02 1 = 1.021

t t tt

t

i iee

+ +⎛ ⎞= ⇒ + ⎜ ⎟⎝ ⎠+

( )1 11.041 = 1.96%1.02

e e+ ⇒ =

and so the value of the retail price index after one year would be 101.96

( )2

22 2

1.0296231 = 0.943%1.02

e e⎛ ⎞+ ⇒ =⎜ ⎟⎝ ⎠

and so the value of the retail price index after two years would be

( )2100 1.00943 101.90=

( )3

33 3

1.029761 = 0.9569%1.02

e e⎛ ⎞+ ⇒ =⎜ ⎟⎝ ⎠

and so the value of the retail price index after three years would be

( )3100 1.009569 102.90=

(iv) The “spot” rates of inflation or the price index values could be used.

Clearly the expected rate of inflation in the first year is 1.96%.

The expected rate of inflation in the second year is:

101.90 101.96 = 0.06%.101.96

−−

The expected rate of inflation in the third year is:

102.90 101.90 0.98%101.90

−=

A common error was to assume that income only started after three years rather than “starting from the beginning of the third year”.


Page 11

10 (i) The price of the securities might have fallen because interest rates have risen or because their risk has increased (for example credit risk).

(ii)

Date Market price of

securities (£)

X Y No of

securities held

before purchases

Market value of holdings before

purchases (£)

No of securities

held before

purchases

Market value of holdings before

purchases (£)

1 April 2003 64 – – – – 1 April 2004 65 100 6,500 100 6,500 1 April 2005 60 100 6,000 200 12,000 1 April 2006 65 1,100 71,500 300 19,500 1 April 2007 68 1,100 74,800 400 27,200 1 April 2008 70 1,100 77,000 500 35,000

(iii) (a) Money weighted rate of return is i where: ( ) ( )5 36, 400 1 60,000 1 77,000i i+ + + =

try i = 5% LHS = 77,625.70 try i = 4% LHS = 75,278.42 interpolation implies that

77,625.70 77,0000.05 0.01 4.73%77,625.70 75,278.42

i −= − × =

−

(Note true answer is 4.736%)

(b) Time weighted rate of return is i where using figures in above table:

( )5 6,000 77,0001 1.09375. 6, 400 6,000 60,000

i+ = =+

1.808%i⇒ =


Page 12

(iv) (a) Money weighted rate of return is i where:

( ) ( ) ( ) ( ) ( )5 4 3 26,400 1 6,500 1 6,000 1 6,500 1 6,800 135,000

i i i i i+ + + + + + + + +

=

Put in i = 4.73%; LHS = 37,026.95 Therefore the money weighted rate of return for Y is less to make LHS

less. (b) Time weighted rate of return for Y uses the figures in the above table:

( )51 i+ = 6,500 12,000 19,500 27,200 35,0006,400 6,500 6,500 12,000 6,000 19,500 6,500 27,200 6,800+ + + +

= 1.09375.

1.808%i⇒ =

(Student may reason that the TWRRs are the same and can be derived

from the security prices in which case, time would be saved.)

(v) The money weighted rate of return was higher for X than for Y because there was a much greater amount invested when the fund was performing well than when it was performing badly.

The money weighted rate of return for X (and probably for Y) was more than

the time weighted rate of return because the latter measures the rate of return that would be achieved by having one unit of money in the fund from the outset for five years: both X and Y has less in the fund in the years it performed badly.

This question was answered well but examiners were surprised by the large number of candidates who used interpolation or other trial and error methods in part (ii) when the answer had been given in the question. The examiners recommend that students pay attention to the details given in the solutions to parts (iii) and (iv). For such questions, candidates should be looking critically at the figures given/calculated and making points specific to the scenario rather than just making general statements taken from the Core Reading.


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