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Financial Mathematics
Financial Mathematics
Jonathan Ziveyi1
1University of New South Wales
Risk and Actuarial Studies, Australian School of Business
Module 3 Topic Notes
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Financial Mathematics
Plan
Module 3: Loan Valuation and Project Appraisal TechniquesIntroductionAllowing for TaxAnalysis of Loan Schedules and RepaymentsSinking FundsLoans at a Flat Rate of InterestLoan Valuation ExampleFixed Income Securities and BondsPricing BondsBond Valuation ExampleDefinitions of Yield, IRR and MIRR RatesInvestment Decision CriteriaSensitivity of Results and Duty of DisclosureProject Appraisal Example
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Introduction
Evaluation of a project Objective of a project appraisal:
value a given project: how much is it worth?
compare different projects based on certain criteria:which project is the best?
make a recommendation based on certain criteria:should we invest in that project?
This involves determining net cash flows:
gains: sales salvage value of assets
minus costs: expenses transaction costs taxes depreciation of assets cost of debt
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Introduction
Financing a project There are several ways of financing a project:
for an individual personal wealth personal loan
for a company equity (shares) debt (loans, bonds)
for a government taxes debt (treasury bonds)
The analysis of loans and bonds is necessary in order to be able tobuild the cash flow model. Note that bonds are nothing else thanlarger scale, tradable loans.
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Introduction
Plan of this module
1. Introduction
2. Allowing for Tax
3. Analysis of Loan Schedules and Repayments
4. Sinking Funds
5. Loans at a Flat Rate of Interest
6. Loan Valuation Example
7. Fixed Income Securities and Bonds
8. Pricing Bonds
9. Bond Valuation Example
10. Definitions of Yield, IRR and MIRR Rates
11. Investment Decision Criteria
12. Sensitivity of Results and Duty of Disclosure
13. Project Appraisal Example5/66
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Allowing for Tax
Allowing for tax Tax is a very important consideration whenanalysing a cash flow:
when and how much tax is paid influences the profitability of asecurity or project
the tax rate depends on the type of cash flows: income (e.g. interest, dividends, rents, . . . ), or capital gains (e.g. increase of the value of a share or property,
above par redemption payments, . . . )
the tax rate depends on the individual considered (person orcompany)
tax is usually paid with a lag that also depends on theindividual considered
income and capital losses are usually allowed to be offsetagainst gains to derive tax benefits
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Module 3: Loan Valuation and Project Appraisal Techniques
Allowing for Tax
Allowance for taxation in price/yield calculations
tax payments are nothing else than additional (negative) cashflows
in case of losses that can offset gains, tax benefits can beadded as positive cash flows(the government wont pay any money, but a loss means taxthat otherwise would have been paid will not be paid)
in many cases price and yield calculations allowing for tax canbe done analytically (using financial mathematics formulae),"by hand" and using a calculator
larger/more complicated models can be easily done using aspreadsheet model or other relevant software.
transaction costs are similar costs that need to be allowed for
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Module 3: Loan Valuation and Project Appraisal Techniques
Allowing for Tax
Depreciation Schedules and Tax Many projects involve aninvestment in capital equipment. For taxation purposes this isdepreciated usually on two (alternative) bases:
Prime Cost (level over life of equipment), or
Diminishing Value (constant percentage of written down valueWDV)
Taxable income is income minus expenses:
expenses include interest costs and depreciation
net cash flow is the cash payments less taxation expense
in case of deferral (or lag) for taxation payments, treat as twodifferent cash flows
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
Loans Definitions:
Consider a loan of amount L made at time 0 with repaymentsof K1,K2,. . . ,Kn at times 1, 2, . . . , n
Equation of value
L = K1v + K2v2 + . . . + Knv
n at effective rate i .
Each loan repayment Kt can be decomposed into a principal component (which amortises the loan) an interest component (which pays the interest due since the
last repayment)
The amount that still need to be reimbursed after a paymentis called the outstanding balance
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Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
Denote:
It the interest component of the tth payment
PRt the principal repaid in the tth payment
OBt the outstanding balance immediately after the tth
payment
Interest in tth payment is simply the previous outstanding balancemultiplied by the rate of interest
i OBt1
Principal repaid should just be the difference between the actualpayment and the interest component
Kt It
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Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
If we work recursively we have
at time 0OB0 = L
at time 1
I1 = iOB0 = iL
PR1 = K1 I1 = K1 iOB0
OB1 = OB0 (1 + i) K1 = OB0 (K1 iOB0)
= OB0 (K1 I1) = OB0 PR1
and then we move forward to the next time period
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Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
In general we have
It+1 = iOBt
PRt+1 = Kt+1 It+1
OBt+1 = OBt (1 + i) Kt+1 = OBt (Kt+1 It+1)
= OBt PRt+1
Total repayments
KT =n
1Kt
total interestIT =
n1It
and
L = KT IT =n
t=1
PRt
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
Numerical example Example: Consider a loan of $1000 repaid by 5equal installments of principal and interest at the end of each yearfor 5 years with an interest rate of 5%. Determine the repayments.
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Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
Loan Schedule In practice it is often much easier to set out all theinformation in a "loan schedule" providing information (for eachperiod) on:
Payments
Interest Due
Principal Repayments
Principal Outstanding
(and any other important items)
This is usually presented in a table computed with the help of R ora spreadsheet.
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Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
Example For the $1000 5 year loan with level repayments, what arethe interest and principal components in each year? Give arepayment schedule.
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Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
In order to determine a given line of the loan schedule, one needsonly the principal outstanding at the beginning (or the end) of theperiod. This can be determined directly via:
the prospective method,
OBt =
ns=t+1
Ksvst {= Kant i if repayments are equal}
or the retrospective method
OBt = L (1 + i)t
ts=0
Ks(1 + i)ts {= L (1 + i)t Kst i}
Both methods yield the same result.
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Analysis of Loan Schedules and Repayments
Numerical example (retrospective method) Consider a loan of$1000. For the first year the repayment was $200, and the interestcharged was 5%For the second and third years the repayment was $150 p.a., andinterest charged was 4% p.a. What is the loan outstanding at theend of the third year?
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Module 3: Loan Valuation and Project Appraisal Techniques
Sinking Funds
Sinking Funds Consider the following situation:
Company A has borrowed an amount L (from a bank, byissuing a bond, etc. . . ) and will need to reimburse the loanafter n years
in the mean time, it needs to pay interest at a rate i each yearto the lender(s)
Company A wants to set up payments to a fund that willaccumulate to the amount of the loan at time n in order toensure the reimbursement
this fund earns interest at a rate j not necessarily equal to i .Usually, j < i .
Such a fund is called a sinking fund.
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Module 3: Loan Valuation and Project Appraisal Techniques
Sinking Funds
In order to accumulate to L, level payments to the sinking fundneed to be equal to
L
sn j,
which means that the total payment for each time unit is
iL +L
sn j.
The first component is the interest component, paid to the lender,and the second is the principal component, paid to the sinkingfund.
Does a sinking fund lead to higher repayments than when theloan is reimbursed gradually using the amortisation method?
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Module 3: Loan Valuation and Project Appraisal Techniques
Sinking Funds
Sinking fund example A loan of $1000 is to be repaid by 5 annualpayments, beginning one year after the loan is made. The lenderwants annual payments of interest only at a rate of 7% andrepayments of the principal in a single lump sum at the end of 5years.The borrower can accumulate principal in a sinking fund earning anannual interest rate of 6%, and decides to do this with 5 leveldeposits starting one year after the loan is made. Determine therepayment and model the cash flows of this transaction in aspreadsheet.
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Module 3: Loan Valuation and Project Appraisal Techniques
Sinking Funds
Example
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Module 3: Loan Valuation and Project Appraisal Techniques
Loans at a Flat Rate of Interest
Loans at a Flat Rate of Interest The interest charge I is given by
I = L f n
where:
L is the loan amount
f is the flat rate of interest
n is the duration of loan (in time units of the flat rate ofinterest)
Loan Repayments R are given by
R =L + I
n
where n is the number of level instalments.
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Module 3: Loan Valuation and Project Appraisal Techniques
Loans at a Flat Rate of Interest
Numerical example A lawnmower worth $400 is offered for sale onthe following terms:10% deposit, flat interest of 10% p.a. with monthly repaymentsover 30 months.Determine the repayment and the effective annual rate of interest.
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Module 3: Loan Valuation and Project Appraisal Techniques
Loans at a Flat Rate of Interest
Usage Easier to understand, but presents serious problems:
the "real" rate of interest is usually much higher than whatthe flat rate suggests
flat rate loans do not encourage earlier payments (the amountof interest that has to be paid is fixed)
Flat rates of interest are not used everywhere:
because of the problems described above, it is forbidden issome countries (mainly developed, such as in Australia)
however, it is widely used in developing countries (mainly bymicrocredit institutions)
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Module 3: Loan Valuation and Project Appraisal Techniques
Loan Valuation Example
Example - Loan Valuation - Spreadsheet A loan of nominal amount$500,000 was issued bearing interest of 8% per annum payablequarterly in arrears. The loan will be repaid at $105% by 20 annualinstallments, each of nominal amount $25,000, the first repaymentbeing ten years after the issue date. An investor, liable to bothincome tax and capital gains tax, purchased the entire loan on theissue date at a price to obtain a net effective annual yield of 6%.Find the price paid, given that his rates of taxation for income andcapital gains are 40% and 30% respectively.What is the price paid allowing for taxation? Develop a spreadsheetmodel for the loan allowing for both income and capital gainstaxation.
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Module 3: Loan Valuation and Project Appraisal Techniques
Loan Valuation Example
Model with Tax on Interest and CGT This is much morecomplicated.
price depends on CGT and CGT depends on price. . .
capital repayments occur over time
We can solve this with a spreadsheet.
Set a dummy figure as the price
Given a price we can determine if a CGT is due for eachrepayment:
CGTt = 30%
(actual paymentt
face value reimbursedt500000
P
)+
we have then (net receipts):
CFt = APRt + It TIt CGTt
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Module 3: Loan Valuation and Project Appraisal Techniques
Loan Valuation Example
We solve then for the correct price:
Calculate the PV of the net receipts
Calculate in a cell the difference between the PV and the Price(which should be equal)
Using the solver, target a difference of 0
You may need to constraint the interest rate and the price tobe positive.
Note:(x y)+ = max(x y , 0).
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Module 3: Loan Valuation and Project Appraisal Techniques
Fixed Income Securities and Bonds
Fixed Income Securities Broad range of securities with fixedincome:
bonds (or notes, or debentures), issued by the government private companies
types of bonds short term (e.g. Australian Treasury note, or promissory note)
vs long term (e.g. Australian Treasury bond) virtually risk free to very risky (junk bonds) coupon bonds or zero-coupon bonds (ZCB) indexed bonds, or real return bonds
but also certificates of deposit (tradable or not) . . . (see readings)
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Module 3: Loan Valuation and Project Appraisal Techniques
Fixed Income Securities and Bonds
Government BondsGovernment Bonds:
borrow money from investors to fund spending plans
also provide low risk securities (liquidity on the market, anddetermination of the structure of interest)
both short and long term(in Australia: Treasury Notes and Treasury Bonds)
consist of both coupon and capital payments(in Australia: usually interest only until maturity)
for (Commonwealth) Government Bonds in Australia usually semiannual coupons coupons are paid on the 15th of each relevant month. yields are quoted as nominal p.a. with the same frequency as
the coupon payments
Other conventions: see Broverman and Sherris30/66
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Fixed Income Securities and Bonds
Bond basics Pays coupon (interest) to purchaser a certain numberof times a year, of amount
Fc
ppaid p times per year
where
F is the face value (par value)
c is the annual coupon rate
p is the frequency of payments
Payments will continue during the term to maturity of the bond(denoted by n).
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Module 3: Loan Valuation and Project Appraisal Techniques
Fixed Income Securities and Bonds
Maturity, redemption and principal amortisation
Maturity is the date when the bond is redeemed(reimbursed)
The redemption amount FR at maturity is not always equal tothe face value. We have R = 1: the bond is redeemed at par R < 1: the bond is redeemed below par R > 1: the bond is redeemed above par
Sometimes, principal is reimbursed before maturity. Again, theamount of face value can be reimbursed at par, orbelow/above par.
A bond is essentially a loan that is amortised in a single lumpsum payment (at maturity) and/or by earlier payments.
Being able to differentiate between face value redemption andcapital gain/loss (above/below par reimbursements)is important for accounting, yield and tax purposes.
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Module 3: Loan Valuation and Project Appraisal Techniques
Fixed Income Securities and Bonds
Numerical example A 3 year bond with a face value of $100,000pays annual coupons at a rate of 10% p.a. The bond is
1. entirely redeemed at maturity with a payment of $120,000;
2. redeemed by 2 payments of $65,000 each at the end of thesecond and third year, each for half of the bonds face value.
For both cases, establish a loan schedule showing interestpayments, principal repayments and capital gains.
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
The price of a bond
as usual for securities, the price of a bond is essentially thepresent value of its future cash flows
the rate, called yield, at which cash flows are discounted is acritical assumption
it is usually quoted along with the price of the bond (bothvalues are equivalent ways of quoting the price of a bond)
it is usually of the same type as the coupon rate (semiannualnominal for US/CA/AU, sometimes also annual in EU)
the yield usually depends on the current structure of interest,as well as the risk associated to the bond as perceived by themarket (note also some rating agencies rate bonds AAA to C)
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
On coupon dates For a bond at yield i (p) = pi whose redemptionand face values are the same (R = 1) we have:
P = Fcanp i + Fvnpi
= Fcanp i + F (1 ianp i)
= F + F (c i) anp i
A par bond must have c = i
A bond with c > i trades at a premium
A bond with c < i trades at a discount
If R 6= 1, the price is the PV of the future CF (simple application ofcompound interest techniques)
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
Between Coupon dates
in practice bonds are traded between coupon payment dates
the seller will require the interest accumulated since the lastcoupon date to be paid by the buyer
if the sale is too close to the next coupon payment (inAustralia, 7 days or less), the bond becomes ex-interest,which means that the next coupon payment will still be paidto the seller, even if the bond is not his property any more
the general pricing approach is to discount the bond cash flowsto the next coupon payment date (including the couponpayment at that date), and then further discount this presentvalue this to the (prior) sale date
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
The RBA formulaThe RBA uses the following formula to value Treasury Bonds whenmaturity is between n and n + 1 semesters:
P = vf
d
i [C + Gan i + 100vn]
where:
C is the next coupon payment (zero if ex-interest)
G is the regular semi annual coupon payment
f is the number of days until the next payment
d is the number of days in the current half year
(this is identical to the general formula given in Broverman)
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
Market price Two bonds with the same cash flows and the sameyield will have a different purchase price if coupons payment datesare different, which may be confusing. Hence, bonds are usuallyquoted at a market price. We distinguish:
Price-plus-accrued: the price with accrued interest (coupon) - see previous slide the purchase price also: "dirty price", "full price", or "flat price"
Market price price as quoted ( smoothed price) accrued interest is removed
market price = dirty price accrued interest = P tFc
also: "clean price"
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
Numerical example Consider a Government bond paying semiannual interest of 10% p.a. on 15-April and 15-October each year.It is redeemable at par on 15 Oct in 6 years time. Find the purchaseand market prices to yield 8.5%p.a. (semi-annual) on 30 June.
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Module 3: Loan Valuation and Project Appraisal Techniques
Pricing Bonds
Optional redemption dates In general the redemption date may:
be fixed (most of the bonds)
vary at borrowers option on or after certain date no final date (undated) between two dates
at lenders option
An uncertain redemption dates means that lenders (buyers) canteasily determine yields at the purchase date. In such a case, theycan still determine:
a maximum price, for given yield, or
a minimum yield, for given price
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Module 3: Loan Valuation and Project Appraisal Techniques
Bond Valuation Example
Example - Loan Valuation - "by hand" [UNSW Final Exam 2006] Abond with a nominal face value of $100, 000 is redeemable by twopayments, one in 5 years time and the other in 10 years time. Thepayment in 5 years time is for a nominal amount of $40, 000 and in10 years time for a nominal amount of $60, 000. Redemptionpayments are payable at $105 per $100 nominal face value.Coupons are paid on the bond at 6% p.a semi-annually based onthe nominal amount outstanding. Tax is paid on the coupons at arate of 30% and tax is paid on capital gains at a rate of 15%.Capital losses are assumed to be offset against other capital gainsof the investor.
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Module 3: Loan Valuation and Project Appraisal Techniques
Bond Valuation Example
1. Determine the price to be paid by an investor to earn a grossyield of 6.5% p.a. (semi-annual).
2. Determine the price to be paid by an investor to earn a net oftax (after tax) yield of 5% p.a. (semi-annual) allowing only fortax on the coupons.
3. Determine the price to be paid for the bond to yield an net oftax return of 4% pa. (semi-annual) allowing for tax oncoupons and capital gains.
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Module 3: Loan Valuation and Project Appraisal Techniques
Bond Valuation Example
1. Price to earn a gross yield of 6.5% p.a. (semi-annual) (note theyield and coupons are semi-annual so work in half years)
Price =0.06
2(40,000) a10 + 40,000 (1.05) v
10
+0.06
2(60,000) a20 + 60,000 (1.05) v
20 at6.5
2%
= 1,200 8.422395 + 42,000 0.726272
+1,800 14.539346 + 63,000 0.527471
= 10,106.874 + 30,503.431 + 26,170.823 + 33,230.689
= 100,011.82
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Bond Valuation Example
2. Gross yield of 5% p.a. (semi-annual), tax on the coupons
Price = (1 0.3)0.06
2(40,000) a10 + 40,000 (1.05) v
10
+ (1 0.3)0.06
2(60,000) a20 + 60,000 (1.05) v
20 at5.0
2%
= 840 8.752064 + 42,000 0.781198
+1260 15.589162 + 63,000 0.610271
= 7,351.7337 + 32,810.333 + 19,642.3445 + 38,447.0694
= 98,251.48
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Bond Valuation Example
After tax return of 4% p.a. (semi-annual), tax on coupons andcapital gains
Price = (1 0.3)0.06
2(40,000) a10
+40,000 (1.05) v10 0.15
(40,000 (1.05)
40000
100000P
)v10
+ (1 0.3)0.06
2(60,000) a20
+60,000 (1.05) v20 0.15
(60,000 (1.05)
60000
100000P
)v20
at4.0
2%
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Bond Valuation Example
We have
P = 840 8.982585 + [42,000 0.15 (42,000 0.4P)] 0.820348
+1260 16.351433 + [63,000 0.15 (63,000 0.6P)] 0.672971
and thus
(1 0.049221 0.060567)P = 7,545.371 + 29,286.4236
+20,602.8056 + 36,037.597
P =93,472.197
0.890212= 105,000.
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Definitions of Yield, IRR and MIRR Rates
Definitions Yield rate
effective "average" rate of interest over the whole (time)length of an investment:
yield rate =
(accumulated value
investment cost
)1/length of investment 1
Net Present Value (NPV)
present value of inflows (gains) minus outflows (expenses andinvestment costs), or net cash flows
must be calculated using a relevant rate of interest(reflecting risk and cost of capital)
Internal Rate of Return (IRR)
rate of interest such that the NPV is 0
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Definitions of Yield, IRR and MIRR Rates
Numerical example Consider the following two cash flows:
Option 1 Option 2
0 -1000.00 -1000.001 100.00 533.202 200.00 350.003 300.00 250.004 400.00 150.005 500.00 50.00
For option 1, the IRR is 12.01%. Consider the yield:
(Inflows accumulated @ IRR
1000
)1/51 =
(1762.90
1000
)1/51 = 12.01%
Yield = IRR!
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Definitions of Yield, IRR and MIRR Rates
What if it is not possible to reinvest inflows at a rate equal to IRR?For option 1
(Inflows accumulated @ 3%
1000
)1/51 =
(1561.37
1000
)1/51 = 9.32%
the yield is much lower
but the IRR does not change
if the reinvestment rate is different from the IRR, the yield isnot equal to the IRR!
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Module 3: Loan Valuation and Project Appraisal Techniques
Definitions of Yield, IRR and MIRR Rates
Numerical example
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Definitions of Yield, IRR and MIRR Rates
Reinvestment rates NPV and IRR
assume a homogeneous rate of interest for all cash flows:
(Accum. value @ IRR) = (Invmt cost)(1 + IRR)length of invmt
do not allow for a different reinvestment rate
Solution
MIRR:
(Accum. value @ reinv. rate) = (Invmt cost)(1+MIRR)length of invmt
MIRR is a modified yield that takes into account thereinvestment rates
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Definitions of Yield, IRR and MIRR Rates
Multiple IRR If cash flows are non conventional (change sign morethan once), there may be several IRR...
Example:
t CFt0 -591 1542 -99
= NPV(i):
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Investment Decision Criteria
Decision criteria
1. Payback period the amount of time until repayments accumulate (without
interest) to the initial investment
2. Discounted payback period the amount of time until discounted repayments have a higher
PV than the initial investment same idea as payback period, but taking the time value of
money into account
3. NPV (net present value) the present value of net cash flows
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Investment Decision Criteria
4. IRR (internal rate of return) the rate of interest such that the NPV is 0
5. MIRR a modified IRR that takes into account reinvestment rates
6. Profitability index the ratio
(PV of repayments) / (initial investment) remember the NPV is the difference:
(PV of repayments) - (initial investment)
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Investment Decision Criteria
7. Dollar-weighted rate of return The simple rate of interest such that the NPV is 0
8. Time-weighted rate of return returns over subsequent periods are compounded to yield an
average return particularly used by investment funds to transform monthly
returns into longer term returns (semesterly, annual, . . . )
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Investment Decision Criteria
Numerical exampleIn this example, what is the decision that the various decisioncriteria that were introduced would yield?
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Sensitivity of Results and Duty of Disclosure
Sensitivity of Results: Example Suppose your company isconsidering the purchase a small insurer. You forecast the followingcashflows for this insurer over the next 5 years:
Premiums: 100m p.a.
Claims: 80m p.a.
Expenses: 5m p.a, increasing at 3% p.a.
Assume that these are all incurred at the middle of the year onaverage.
At the end of the 5th year the business will be sold for a totalof 10m.
Find the NPV of this project at 6% p.a.
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Sensitivity of Results and Duty of Disclosure
Sensitivity of our results to
discount rate assumption?
expense increase rate?
premium and claim changes?58/66
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Sensitivity of Results and Duty of Disclosure
7. ReportingA Member must ensure that his or her reporting (whether oral orwritten) in respect of Professional Services provided:
(a) is appropriate, having regard to:
1. the intended audience;2. its fitness for the purposes for which such
reporting may be required or relevant;3. the likely significance of the reporting to its
intended audience;4. the capacity in which the Member is acting; and5. any inherent uncertainty and risks in relation to
the subject of the report;
(b) complies with any relevant Professional Standards.
Institute of Actuaries of Australia Code of Professional Conduct(November 2009, Section 7)
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Sensitivity of Results and Duty of Disclosure
Assess Extract of professional standards on economic valuations:
4.2 Scope of economic valuation[. . . ]The Member should ascertain the materiality limits that apply tothe economic valuation bearing in mind:
the quality of the data;
the intended use(s) of the economic valuation;
the degree of uncertainty; and
the sensitivity of the overall result to different assumptions.
Institute of Actuaries of Australia Guidance Note 552 on EconomicValuations (July 2004)
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Sensitivity of Results and Duty of Disclosure
And then communicate
3.3 TransparencyThe models, methods and assumptions used for the economicvaluation should, as far as practical, be transparent, enablingvaluation results and sensitivities in the results to changes inparticular assumptions to be understood by the intended users ofthe economic valuation.
Institute of Actuaries of Australia Guidance Note 552 on EconomicValuations (July 2004)
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Project Appraisal Example
Project Appraisal Example A company considers buying equipmentand then leasing it out to third parties. This project has thefollowing variables:
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Project Appraisal Example
Loan schedule
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Project Appraisal Example
Taxable income
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Project Appraisal Example
Net cash flows
NPV @ 18%: $161,745IRR: 22.06%
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Financial Mathematics
Module 3: Loan Valuation and Project Appraisal Techniques
Project Appraisal Example
NPV check
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