Business Funding & Financial Awareness

Time Value of Money – The Role of Interest Rates in Decision Taking

J R DaviesMay 2011

Time Value of Money – The Role of Interest Rates in Decision Taking

May 2011

Dick Davies

3

Investment/Financing Decisions The time dimension –

An investment decision involves the commitment of resources on the expectation of future benefits

0 1 2 3

Costs Benefits Benefits Benefits……Time

Many financial and investment decisions involve costs and benefits spread out over time

This implies it is necessary to allow for •The time value of money•The impact of risk and uncertainty

4

Time Value of Money• A pound today is worth more than a pound to-morrow………

even in the absence of – Risk and uncertainty– Inflation

• The time value of money stems from the interest rate – effectively the price that balances the supply and demand for loans- and this will positive in a world of constant prices and no uncertainty

5

Determining Interest Rates – A Simple Model

Saving = Deposits = Lending

Investment = Borrowing

Saving and Investment

r

Interest rate

6

Interest rates adjust to changes in savings and investment – eg an increase in savings

Saving = Deposits = Lending

Investment = Borrowing

Saving and Investment

rr1

Interest rate

7

Interest Rates• The riskless real rate of interest (r0): the rate of interest that can

be expected in the absence of

– Risk and uncertainty

– Inflation

• A premium is added to the “real” rate of interest for

– Risk and uncertainty – this will vary across borrowers

– Inflation.

15.005.006.004.00

0

ufrr

per centemium is 5he risk prent, and tis 6 per c

ationte of inflnt, the ras 4 per ceeal rate ie if the rfor exampl

ufrr

M

M

8

Time Dimension – Investment/Financing Decisions

Capital Budgeting (Real Investments)

0 1 2 3 4 5

-C0 +C1 +C2 +C3 +C4 +C5

-C0 +C1 +C2 +C3 +C4 +C5

0 1 2 3 4 5

Share Purchase (Financial Investment)

Loan (Financing Decision)+C0 -C1 -C2 -C3 -C4 -C5

Time

Time

Time

0 1 2 3 4 5

9

Evaluating Cash Flows Arising at Different Pointsin Time

• Cash flows that occur at different points in time cannot be summed to determine the net benefit position

• If a cash payment is made now on the expectation of receiving cash inflows in the future it is necessary to

• borrow the funds to make the payment now, and this implies an interest cost will be incurred, or

• use your own funds - and this implies foregoing the interest income that could have been earned on these funds

• in either case there is an interest cost to consider.

10

Adjusting Values to Allow for Interest (1)

Assume the interest rate is 10 % ie 0.10 What are the equivalent values at the end of one year, year two, etc to a sum of £100 available today ?

100 ? ? ? 0 1 2 3

time

11

Adjusting Values to Allow for Interest (2)

Given an interest rate of 10 % what is the equivalent value at the end of one year of £100 that is available today ?

100 (10) 110 ? ?

0 1 2 3

The original sum (£100) plus interest that can be earned over one year (£10).

time

12

Adjusting Values to Allow for Interest (3)

Given an interest rate of 10 % what is the equivalent value at the end of two years of £100 that is available today ?

100 (10) 110 (11) 121 ? 0 1 2 3

The original sum (£100) plus interest (£10) for year one, and interest of £10 for year two on the initial £100, plus £1 of interest on the interest of £10 earned in period one.

Time

13

Adjusting Values to Allow for Interest (4)

Given an interest rate of 10 % £100.00 today£110.00 next year

£121.00 two years from now£133.10 three years from now

all have the same real value (in principle) and are equally acceptable

(assume no risk and no inflation for simplicity).

14

Future Value Factors

To obtain the equivalent value at a point in time in the future of a sum available today we must multiply this sum by a future value factor – also referred to as a compound interest factor, or more simply as the interest factor – to allow for interest that can be earned on the sum available to-day:

FVFn/r = (1 + r)n

where r is the rate of interest

n is the number of time periods in the future

15

Developing Future Value Factors

)1(

)1(

)1)(1(

)1(

)1(

0

0

0

1112

0001

2

nn rVV

rV

rrV

rVrVVV

rVrVVV

16

Developing Future Value Factors - An Example

10.133331.1 100)10.01(100

)10.01( )]10.01)(10.01(100[

.10)0 1(121

.100 121121

121)10.01(110

)10.01)(10.01(100

)10.01(110.100 110110

110)10.01(100.100 100100

3

3

2

1

2

x

x

xV

xV

xV

17

Developing Future Value Factors

303

02

01

)1(

)1(

)1(2

rVV

rVV

rVV

This implies one more interest factor is introduced for each added time period and the value at the end of period n is given by

)1)....(1)(1)(1()1(0

rrrrrVV n

n

Multiply together n interest factors

Using Future Values (1)What will £800 deposited in a bank account at an interest rate of 12 per cent grow to by the end of year 5 if all interest income is reinvested?

.

19

Determining the Future Value of a Sum (2)

87.1409

)7623.1(800

)12.01(800

)1(5

6

0

V

rVV nn

What will £800 invested at interest rate is 12 per cent grow to by the end of year 5 ?

The interest rate for these calculations must be written in decimal form.In principle this implies that £800 today is of equivalent value to £1410to be received after five years.

Use factors taken from table 1

20

Determining the Future Value of a Sum (3)

70.1259

)2597.1(1000

)08.01(1000

)1(3

5

2525

V

rVV

You expect to receive £1000 at the end two years and you expect to be able to invest this to earn an interest rate of 8 per cent. What can the sum be expected to grow to by the end of year 5 ?

The interest rate for these calculations must be written in decimal form.In principle this implies that £1000 after two years is of equivalent valueto £1259.70 to be received after five years.

Use factors taken from table 1

21

Annuities

An annuity is a constant payment at the end in each time period for a specified number of periods.

A constant periodic NCF

0 1 2 3 4 5 6 7 8 …

A A A A A A A A

Constant net cash flows

22

An Example of an Annuity

500 500 5000 1 2 3

An annuity of £500 for three years

An annuity is a constant payment at the end (or the start) of each time period for a specified number of time periods.A constant periodic NCF

Annuities

23

Investment Example and the use of Annuity Factors

1655]3100.3[ 500

)]000.1()100.1()2100.1[( 500

)]0000.1()1000.1()1000.1[( 500

)0.1( 500)1.1( 500)1.1( 500)3(

3

3

123

123

xV

xV

xV

xxxVFV

Future value annuity factorfor three years at 10 per cent

24

Future Value Annuity Factors

25

Using Future Value Annuity Factors

What will be the accumulated value of annual savings of £1200 deposited in a savings account at the end of each of the next 8

years if the interest rate is 7 percent ?

0 1 2 3 4 5 6 7 8

1200 1200 1200 1200 1200 1200 1200 1200

Accumulated Value ?

26

Using Future Value Annuity Factors (2)What will be the value of annual savings of £1200 for the next 8 years if the interest rate is 7 percent ? (Interest being reinvested at 7 per cent.)

YearOpening

Value InterestClosing Value Savings

1 0.00 0.00 £0.00 1200

2 1200.00 84.00 1,284.00 1200

3 2484.00 173.88 2,657.88 1200

4 3857.88 270.05 4,127.93 1200

5 5327.93 372.96 5,700.89 1200

6 6900.89 483.06 7,383.95 1200

7 8583.95 600.88 9,184.83 1200

8 10384.83 726.94 11,111.76 1200

9 12311.76

FV (8) = 1200 times FVAF8/0.07

= 1200 times 10.2598 = 12311.76

27

Example: Using Time Value Concepts (3)

Determining Pension Income

An individual pays £3,000 per annum into a pension fund (a defined contribution scheme) for thirty years. The scheme guarantees a minimum return of 5 per cent.

How much will have been accumulated in the fund by the end of the 30 year period.

28

Assessing Pension Payments

0 1 2 ………………...…………………… 30

Period for contributions

V30 = £3000 times FVAF30/.05

= £3000 x 66.4388 = £199,317

3000 3000……. 3000

29

Using Future Value Annuity Factors (5)

Hendy Hotels Ltd

Hendy Hotels is a family owned concern that avoids the use of external funding. The owners recognise that they will have to undertake a major investment five years from now to meet EU safety regulations. This investment will cost £600,000 and the company’s management intend putting aside funds at the end of each of the next five years so as to be able to cover the expenditure. The funds can be invested at 7 per cent until needed. If the same amount is saved each year how much has to saved on an annual basis to cover the expenditure?

30

Using Future Value Annuity Factors (5)

Hendy Hotels Ltd

Hendy Hotels is a family owned concern that avoids the use of external funding. The owners recognise that they will have to undertake a major investment five years from now to meet EU regulations. This investment will cost £600,000 and the company’s management intend putting aside funds at the end of each of the next five years so as to be able to cover the expenditure. The funds can be invested at 7 per cent until needed. If the same amount is saved each year how much has to saved on an annual basis to cover the expenditure?

FV (5) = X times FVAF5/0.07 = £600,000 = X times 5.7507 = £600,000 X = 600,000 / 5.7507 = £104,334

31

Present Value or Discount Factors (1)

To derive the value today, the present value, of a sum expected in the future this future sum must be multiplied by a present value or discount factor.

nr)1(

1

This has a value of less than one as the denominator (1+r) is greaterthan one when r is positive, and applying this to a future NCF willallow for the loss of interest as a result of the delay in the receipt of the cash..

32

Discount Rates - Terminology

• The discount rate• The opportunity cost of funds – interest

foregone by waiting.• The required rate of return.• The cost of capital.

33

Present Value Factors

Time0 1 2 n

nrn rPVF

)1(

1/

34

Present Value Factors

nn

nn

n

n

rVV

Vr

xV

rVV

)1(

1

)1(

1

)1(

0

0

0

All financial arithmetic is based on the future value equation.

If a future value is known the equivalent value today is derivedby multiplying the future value by the discount factor, one over the interest factor

i.e.

35

Present Value Factors at 10 %

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

YEARS

Interest lost in the delay in receiving

cash.

36

Determining Present Values

73.438

6750.0650

)14.01(

1650

)1(

1

30

0

V

rVV

nn

What is the equivalent value today of £650 to be received three years from now if the interest rate (discount rate) is 14 percent ?

37

Present Value Factors

38

Net Present Value of an Investment

• The surplus expected from a project, measured in today’s values ….after appropriate allowances have been made for the– recovery the capital outlay– the interest charges

• It can also be defined as the increment of wealth generated created by an investment

39

Net Present Value Equation

....)1(

1

)1(

12210 r

Cr

CINPV

40

Assessing Investment Proposals Using NPV

Time NCF PVF(10%) PV

0 -1,200 1.000 -1,200.01 500 0.909 454.52 500 0.826 413.03 500 0.751 375.5

NPV = 43.0

An investment of 1200 is expected to produce cash flows of 500 at the end of years 1, 2 and 3. The required rate of return is 10 per cent. Determine the investment’s NPV

41

Present Value Annuity Factors at 10%

0.0000

2.0000

4.0000

6.0000

8.0000

10.0000

12.0000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Annuity Factors

Discount Factors

Years

42

YearPresent Value

Factor 10%Sum of Present Value Factors

1 0.9091 0.90912 0.8264 1.73553 0.7513 2.48694 0.6830 3.16995 0.6209 3.79086 0.5645 4.35537 0.5132 4.86848 0.4665 5.33499 0.4241 5.759010 0.3855 6.1446

Present Value Annuity Factors As The Sumof Discount Factors

43

Using Present Value Annuity Factors

What is the equivalent value today of £840 to be received at the end of each year for the next seven years if the interest rate is 6 percent ?

YearCash Flow

Present Value Factor 6% Present Value

1 840 0.9434 792.452 840 0.8900 747.603 840 0.8396 705.284 840 0.7921 665.365 840 0.7473 627.706 840 0.7050 592.177 840 0.6651 558.65

Present Value = 4689 20Using PVAF

1 to 7 840 5.5824 4689.20

44

Example: Using Time Value Concepts (1)

Arrangements for repaying a bank loan

A bank makes a loan at £10,000 at a fixed interest rate of 12 per cent and this is to be repaid in five equal instalments. (Each instalment covers repayment of the loan as well as the interest on the outstanding balance of the loan.

Determine the annual instalment. (Convert a capital sum into a constant cash flow.)

The instalment is the equivalent constant annual cash flow to a capital sum.

45

Bank Loan – the Required Annual Payments

Loan = Present Value of Repayments at 12 per cent

10,000 = X . PVAF5|.12

10,000 = X times 3.6048

X = 10,000/3.6048

= 2,774

46

Internal Rate of Return

....)1(

1

)1(

10

2210 iC

iCINPV

The rate of discount at which the NPV is equal to zero. This may be interpreted as the highest rate of interest that can be paid on a loan used to finance a project without making a loss.

47

Investment Appraisal (IRR)

Time NCF PVF12%) PV

0 -1,200 1.000 -1,200.0

1 500 0.893 446.4

2 500 0.797 398.6

3 500 0.712 355.9

NPV = 0

48

Loan Analysis (2)

Period Loan at Outset

Interest (12%)

Loan at End of Year

Repayment

1

1200

144

1,344

500

2 844 101 945 500 3 445 55* 500 500

Surplus 0

49

NPV and IRR

NPV

PRODUCTIVITY OF CAPITAL (IRR)

SIZE OF THEINVESTMENT

50

Investment Appraisal (IRR)

Time NCF PVF(12%) PV 0 -2,400 1.0000 -2,400.0 1 1000 0.8928 892.8 2 1000 0.7972 797.2 3 1000 0.7118 711.8 NPV = 0

Consider the simple investment considered earlier - an outlay of 1200 that is expected to produce three annual NCFs of 500 and a discount rate of 10 per cent. The NPV was 43 and the IRR was 12 per cent. Now double the size of all the NCFs – the NPV doubles but the IRR remains at 12 per cent.

Time NCF PVF(10%) PV 0 -2,400 1.000 -2,400 1 1000 0.909 909 2 1000 0.826 826 3 1000 0.751 751

NPV = 86

Determining the IRR (1)An investment of £400,000 is expected to a constant annual NCF of £102,865 for the next 6 years. Determine the approximate value of the investment’s IRR.

0.14i

6)able, row (Look up t3.8887 PVAF

3.88872,863400,000/10PVAF

PVAF102,863 x 400,0000NPV

6/0.14

6/i

i

|6

Determining the IRR (2)An investment of £750,000 is expected to produce NCFs at the end of years 1 to 5 of £150,000, £200,000, £300,000, £300,000 and £100,000 respectively.

Determine the approximate value of the investment’s IRR.

54321

|5|4|3|2|1

)1(

1000,100

)1(

1000,300

)1(

1000,300

)1(

1000,250

)1(

1000,15075

000,100000,300000,300000,250000,15075

i x

i x

i x

i x

i x 0,0000NPV

x PVF x PVF x PVF x PVF x PVF0,0000NPV iiiii

Time NCF0 -750,0001 100,0002 250,0003 300,0004 300,0005 100,000

IRR = 12%

Use Excel!!