1

Tarheel Consultancy Services

Manipal, Karnataka

2

Corporate Training and Consulting

3

Course on Fixed Income Securities

For XIM -Bhubaneshwar

4

For

PGP-II 2003-2005 Batch

Term-V: September-December 2004

5

Module-I

Part-IITime Value of Money:

Concepts & Illustrations

6

Introduction If we were to be given a choice

between receiving one rupee today, and receiving it one year later, would we be indifferent?

The answer is no, because if we were to receive it today, we can invest it at a rate of interest, so as to have more than one rupee tomorrow.

7

Introduction (Cont…) Thus, in order for us to be indifferent,

we would have to be offered more than a rupee next year.

Consequently, a rupee today is worth more than a rupee to be offered in the future.

Reversing the argument, a rupee to be received in the future, is worth less than a rupee today.

8

Introduction (Cont…) This is an illustration of the concept

of Time Value of Money. What matters is not only how much

you receive in monetary terms, but also when you receive the cash flow.

9

Interest What is interest? It is the compensation paid by the

borrower of capital to the lender for permitting him to use it.

Technically, it is the economic rent paid by the borrower to compensate the lender for the loss of opportunity to use the amount, when it is on loan.

10

Rationale for Interest One of the main reasons for charging

interest is Time Preference. Everyone would rather have a given amount

of money today, than the same amount at a future point in time.

This is because, money received today can be used to meet current needs, whereas if the same amount is received later, it can only be used for deferred needs, and that too in an uncertain future.

11

Rationale (Cont…) Thus interest can be defined as the price

that is adequate to cause individuals to overcome their time preferences to retain money, and lend it instead.

Thus far we have looked at the supply side of funds.

On the demand side, all firms need access to capital, most of which is borrowed.

12

Rationale (Cont…) For a firm to be successful, the

return on capital employed should be greater than the cost of funds.

The prospect of high returns on capital employed, causes firms to compete for scarce capital, by offering the market rate of interest.

13

Determinants of Interest In a free market economy the interest

rate is determined by the demand for and the supply of capital.

The factors which influence the interest rate are the following.

1. There is what is called a Pure or base rate of interest. It is the rate of return that would prevail on a risk-less investment, in the absence of inflation.

14

Determinants (Cont…) It can also be defined as the Real

risk-less rate of return, where the word `real’ connotes that there is no loss in purchasing power.

For instance assume that a banana is currently worth 2 rupees.

Assume that an investment of Rs 2 in a risk-less asset will yield Rs 2.20 next period.

15

Determinants (Cont…) If Rs 2.20 is adequate to purchase 1.10

bananas next period, then we can say that the real rate of return is 10%.

2. Inflation Premium In the real world, the prices of goods do

not remain constant. For most people, however, wealth is in

the form of financial and not physical assets.

16

Determinants (Cont…) So what is observed in practice is

not the `real’ but the `nominal’ or `money’ rate of interest.

In our example above the nominal rate is 10%.

In this case the real rate is equal to the nominal rate because there is no inflation.

17

Determinants (Cont…) However, in practice, to calculate

the real rate of return from an investment, we would have to estimate and factor in the rate of inflation.

18

Real versus Nominal Rates Let us consider a simple economy where there

is only one physical good. Assume that the current price of the good is

P0. So Rs 1 can buy 1/P0 units of the good today. Assume that the price of the good next period

is P1, which is known with certainty today. If so, a Rupee can buy 1/P1units of the good

after one period.

19

Real & Nominal Rates (Cont…) Assume that there is financial bond

that will pay 1+R rupees next period per rupee invested now.

Also assume that there is a `goods bond’ which will pay 1+r units of the good in the next period, per unit of the good invested now.

20

Real & Nominal Rates (Cont…) If you invest one rupee in the financial

asset you will receive 1+R rupees next period, which will be adequate to buy (1+R)/P1 units of the good.

Similarly if you invest Rs 1 in the goods bond, which amounts to an investment of 1/P0 in goods terms, you will get (1+r)/P0 units of the good next period.

21

Real & Nominal Rates (Cont…) In order for the market as a whole to be in

equilibrium, an investor must be indifferent between the two bonds, which means that the two bonds should yield an identical rate of return.

So we require that:1+R = 1+r P1(1+r) = (1+R)----- ----- ---------

P1 P0 P0

22

Real & Nominal Rates (Cont…) Let us define (P1 – P0)/P0 as , which

represents the rate of inflation. So (1+)(1+r) = (1+R) This relationship between the

nominal or money rate of return, the real or the `goods’ rate of return, and the rate of inflation is called the Fisher Hypothesis.

23

Real & Nominal Rates (Cont…) If r and are very small, we can

write the approximate relationship asR = r +

In other words, the nominal rate of return is equal to the real rate of return plus the rate of inflation.

24

Uncertain Inflation In order to derive the Fisher relationship,

we assumed that the inflation rate was known with certainty.

In real life inflation is a random variable. Hence we can only have an expectation

of it. Thus in real life, even risk-less securities

do not provide a guaranteed real rate of return.

25

Uncertain Inflation (Cont…) We will first assume that investors

do not demand a risk premium for bearing inflation risk, even though the rate of inflation is uncertain.

If the real rate of return required by them is r, then the nominal rate demanded by them will be:R = r + E()

26

Uncertain Inflation (Cont…) Thus the required nominal rate of return in

this case will equal the required real rate of return plus the rate of inflation.

Once the nominal rate is specified, it will not change.

However the actual real rate of return could differ from the expected real rate, because the actual rate of inflation can be different from the expected rate of inflation.

27

Uncertain Inflation (Cont…) Technically we say that the ex-ante

inflation measure need not equal the ex-post inflation measure.

The term ex-ante refers to a forecasted measure, whereas the term ex-post refers to a measure based on actual results.

Consequently the realized or ex-post real rate will differ from the expected real rate.

28

Illustration Assume that the current inflation is 8%. Assume that the probability distribution

for inflation is as follows: Inflation Probability

4% .256% .258% .2510% .25

29

Illustration (Cont…) The expected rate of inflation is:

.04x.25 + .06x.25 + .08x.25 + .10x.25 = .07

If the required real rate of return is 3%, then the nominal rate demanded will be 10%.

However if the actual rate of inflation is 8%, then the realized real rate will be:.10 - .08 = .02 2%

30

Risk Premia One of the fundamental assumptions made

in Finance, is that investors are risk averse. Since inflation is uncertain, a higher than

expected rate of inflation, could result in a lower than expected real rate of return.

Hence in order to induce investors to purchase financial assets, they must be offered a nominal rate that factors in this uncertainty, besides the expected inflation rate.

31

Risk Premia (Cont…) If we denote the premium for

bearing this risk as R.P, then the Fisher hypothesis can be rewritten as:R = r + E() + R.P.

Since investors are assumed to be risk averse, the premium will be positive.

32

Illustration We will assume that investors expect

to be compensated by the difference between the actual rate of inflation and the expected rate of inflation, whenever the actual rate is higher.So:R.P = (.08-.07)x.25 + (.10-.07)x.25 = .01

33

Illustration (Cont…) According to our assumption the risk

premium demanded is 1%. So the required nominal rate will be:

R = .03 + .07 + .01 = .11 11% Thus when investors are not

indifferent to uncertainty, what matters is not just the expected inflation, but the possibility of deviations from the expected value.

34

Other Factors Influencing the Rate of Interest3. Risk and Uncertainty Inflation risk is obviously one

source of risk. And risk averse investors will

demand a risk premium whenever confronted with a source of risk.

35

Other Factors (Cont…) In practice, most investments,

other than those in government securities, carry a risk of default.

Default Risk, also known as Credit Risk, is the possibility that the borrower may not make principal and/or interest payments as promised at the outset.

36

Other Factors (Cont…) Hence, in real life, whenever an

investor is contemplating an investment, he will factor in a default risk premium.

The premium demanded will be a function of the perceived creditworthiness of the borrower.

37

Other Factors (Cont…)4. Length of Investment The rate of return demanded by an

investor will depend on the time to maturity of the financial claim that he receives in return.

Lenders in general like to lend short-term, whereas borrowers like to borrow long-term.

38

Other Factors (Cont…) Thus in order to raise long-term funds,

borrowers have to tempt lenders with a higher nominal rate of return.

5. Quality of Information In practice lenders and borrowers may not

have full information about each other. If so, the rate of return required/offered may not be consistent with what is suggested by economic factors alone.

39

Other Factors (Cont…)6. Legal Restrictions In reality many rates of interest

are subject to government control. This has tended to be true of

command economies. But even a country like the U.S.

has had legislations like Regulation Q in the past.

40

Other Factors (Cont…) To give an example from India,

although the general level of interest rates has come down, the government has not reduced the rate of return on contributions to the Employees Provident Fund (EPF).

Thus the EPF rate does not reflect market realities, and is artificially maintained at a high level.

41

Other Factors (Cont…)7. Fiscal and Monetary Policies These have an impact on interest rates. The RBI controls the money supply in

India. If it wants to increase the supply of

money it can buy T-bills from the market. It can also reduce the SLR/CRR which will

increase the funds availability.

42

Other Factors (Cont…) An increase in money supply is

designed to stimulate the economy, but will invariably have an impact on the inflation rate.

On the contrary if the economy is perceived to be over-heated, the RBI can increase the reserve requirements, or sell T-bills as a part of its Open-Market operations.

43

Other Factors (Cont…) The budgetary deficits of the

government also impact the system. The higher the shortfalls in tax

collections, the greater will be the need for the government to resort to borrowings.

8. Random Fluctuations Finally, over and above all the objective

causes, interest rates also exhibit random fluctuations.

44

Simple Interest According to this principal, whenever we

deposit money, the interest accrued per period is based only on the original principal deposited.

That is, interest earned at an intermediate stage, does not itself earn any further interest.

Thus the amount of interest accrued every period is a constant.

45

Simple Interest (Cont…) Consider an investor who deposits $ P in an

account that pays interest at the rate of r% per period, for a duration of N periods.

At the end of the first period the balance would be P(1+r).

At the end of two periods it would be P(1+2r). In general, after N periods, it would be

P(1+rN). N need not be an integer. That is, interest may

accrue for a fractional period.

46

Illustration-1 Caroline deposits $ 10,000 with a

bank for 3 years. Assume that the bank pays simple

interest at the rate of 10% per annum.

What will be her balance at the end of 3 years?

At the end of one year she would have 10000(1+.10) = 11000.

47

Illustration (Cont…) During the second period, only the

original principal of $ 10,000 will earn interest.

So the balance after two years will be 11000 + 1000 = 12000.

Finally, at the end of 3 years, the balance will be 12000 + 1000 = 13000

13000 = 10000(1+.10x3) = P(1+rN)

48

Illustration-2 Gulati deposited Rs 10000 with ICICI

Bank 5 years and 6 months ago. If the bank pays 8% per annum on a

simple interest basis, how much can he withdraw?

P(1+rN) = 10000(1+.08x5.5) = 14,400

49

Compound Interest According to the principle of compound

interest, when an amount is deposited, it is not only the original principal which earns interest every period, but the interest accrued at the end of an intermediate period continues to earn interest till the maturity of the deposit.

Thus each time interest is accrued, it is automatically reinvested at the same rate for the next compounding period.

50

Compound Interest (Cont…) Thus if an investment of $ P is made

for N periods, with a bank which pays interest on a compounded basis at the rate of r% per period, then the terminal balance will be:P(1+r)N

Once again, N need not be an integer.

51

Illustration-1 Assume that Caroline deposits $

10000 for 3 years with a bank that pays interest at the rate of 10% per annum, compounded annually.

She will have 10000x1.10 = 11000 after one year.

During the second year this entire amount will earn interest.

52

Illustration-1 (Cont…) So the terminal value at the end of 2

years will be: 11000(1.10) = 12,100 By the same logic, at the end of 3

years, the balance will be:12,100x 1.10 = 13,310

It can be verified that 13,310 = 10000(1.10)3

53

Illustration-2 Assume that Gulati deposits Rs 1000 for 5.5

years at a rate of 10% per annum compounded annually.

The terminal balance will be:10000(1.10)5.5 = 15,269.7053

In the earlier case when we calculated using simple interest, the balance was 14,400.

So compounding can lead to substantial benefits.

54

Length of The Time Period Assume that interest is paid at the

rate of r% per period. If the length of investment is equal

to 1 period, then it does not matter as to which method we use.

That is, both simple as well as compound interest principles will yield the same result.

55

Time Period (Cont…) If the investment is for less than

one period then the accumulated value using simple interest will be greater.

That is:(1+rN) > (1+r)N if N < 1

56

Time Period (Cont…) If the length of investment is

greater than one period, then the compound interest principle will yield a greater accumulated value.

That is:(1+rN) < (1+r)N if N > 1

57

Simple or Compound? In financial transactions covering a

period of one year or more, we usually use only the compound interest principle.

At times even for transactions which are less than one year in duration, this principle is used.

Simple interest is usually used for investments for less than one year.

58

Simple or Compound? (Cont…) Simple interest is also at times used

as an approximation for compound interest over fractional time periods.

We will illustrate this with an example.

59

Illustration Let us go back to Gulati’s investment

for 5.5 years. Assume that interest accrues on a

compounding basis for the first 5 years, and then on a simple interest basis for the last 6 months.

The final balance will be:10000(1.08)5(1+0.5x.08) = 15281.01

60

Illustration (Cont…) Previously when we assumed

compounding for 5.5 years, we got a figure of 15269.7053.

The accumulated amount is greater in the second case because using simple interest for .5 periods (<1) will obviously yield greater returns than using compound interest.

61

Future Value When an amount is deposited in an

interest bearing account for a given time period, the amount accrued at the end of the deposit period is called the Future Value of the original investment.

So if $P is invested for N periods at a rate of r% per period, then: F.V. = P(1+r)N

62

Future Value (Cont…) (1+r)N is the future value of $ 1 if it

is invested at a rate of r% per period for N periods.

It is called the FVIF (Future Value Interest Factor).

It can be read off a standard table for integer values of r and N. Otherwise you need a calculator.

63

Future Value (Cont…) For a given value of the principal

amount, denoted by P, we can calculate the future value by multiplying the amount by the corresponding FVIF.

Therefore: F.V. = P x FVIF(r,N) The process of finding the future

value, given today’s investment, is called compounding.

64

Illustration Suhasini deposits $ 1000 for 5

years in an account that pays interest at the rate of 10% per annum compounded annually.

What is the future value? FVIF(10,5) = 1.6105 So F.V = 1000x 1.6105 = 1610.50

65

Illustration Syndicate Bank is offering the

following scheme. If you deposit Rs 10000 for 10 years, you will earn interest at the rate of 10% per annum for the first 5 years, and at the rate of 12% per annum for the next 5 years.

What will be the future value after 10 years?

66

Illustration (Cont…) The future value after 5 years will be:

FV5 = 10000xFVIF(10,5) = 16105 The future value after another 5 years

will be:FV10 = 16105x FVIF(12,5) = 16105x1.7623= 28381.84

67

Present Value

Let us consider a deposit where the assured terminal value at the end of N periods is F.

If interest is compounded at the rate of r% per period, how much must we deposit today?

The answer of course will depend on whether we use the simple interest technique or the compound interest method.

68

Present Value (Cont…) If we assume that interest is paid

on a simple basis, then, if we denote the required principal by P:P(1+rN) = F P = F

------------

(1+rN)

69

Illustration Venkat wants to ensure that he has $

12000 in his bank account at the end of 4 years.

If the bank pays 5% interest per annum on a simple interest basis, how much must he deposit today?

P = 12000 ------------ = 10000 (1+.05x4)

70

Present Value (Cont…) The amount required to be deposited

can also be calculated assuming that the deposit pays compound interest.

So if F is the terminal value, the amount required to be deposited, P is given by:

P(1+r)N = F P = F ---------- (1+r)N

71

Illustration Priyanka wants to have $ 15000 in her

bank account at the end of 3 years. If the bank pays interest at the rate of

10% per annum compounded annually, how much should she deposit?

P = 15000 -------- = 11269.7220 (1.10)3

72

Present Value (Cont…) For a given amount F, that is payable

after N periods, the equivalent amount P that has to be deposited now is called the Present Value of F.

The process of finding the present value of a future payoff is called discounting.

The interest rate used is called the discount rate.

73

Present Value (Cont…) Consider the relationship: P = F

_____

(1+r)N

1/(1+r)N is the amount that we have to deposit today if we want to have $ 1 after N periods, and if the investment pays compound interest at the rate of r% per period.

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Present Value (Cont…) It is called PVIF (Present Value Interest

Factor). It can also be read off a table for integer

values of r and N. Relationship between PVIF and FVIF: PVIF = 1 1

------ = ------

(1+r)N FVIF

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A Series of Cash Flows In order to compute the present value

of a series of cash flows, we should compute the present value of each component, and then add up the amounts.

Similarly to find the future value of a series of payments, we must compound each component, and then add up the future values so obtained.

76

A Series (Cont…) Thus both present value and future

value are additive. We will illustrate these principles with

the help of an example. An investor is contemplating the

purchase of an instrument that promises the following series of cash flows. If he wants a 5% rate of return, and the current price is $ 1300, should he buy it?

77

IllustrationYears From Now Promised

Payment1 1002 1003 1004 1005 1100

78

Illustration (Cont…)Year Payment Present

Value1 100 95.242 100 90.703 100 86.384 100 82.275 1100 861.88

TOTAL 1216.47

79

Illustration (Cont…) The present value of the cash flow stream, if

a discount rate of 5% is used, is $ 1216.47. If he pays exactly this amount, and gets the

promised cash flows, then he will earn a rate of return of 5%.

If he pays less he will earn a higher rate of return, whereas if he pays more, he will earn a lower rate of return.

Since $ 1300 > $ 1216.47, he should not buy it.

80

Principle of Equivalency Two rates of interest, compounded

at different intervals of time, are said to be equivalent if a given amount of principal invested for the same total length of time at each of the rates, produces the same accumulated value at the end.

81

Effective versus Nominal Rates We will define the payment period as

the length of time at the end of which interest is calculated and credited to the depositor’s account.

We will define the interest conversion period as the length of time over which interest is compounded once.

82

Effective/Nominal Rates (Cont…) The effective rate of interest per payment

period, i, is the amount of money that $ 1invested at the beginning of the period will earn during the period, if interest is assumed to be compounded once, at the end of the period, and credited thereupon.

The nominal rate of interest, r, is the quoted rate of interest for a payment period, which consists of one or more interest conversion periods.

83

Effective/Nominal Rates (Cont…) The two rates will obviously be equal

if interest is actually compounded only once during the payment period.

In other words, the effective rate will be equal to the nominal rate if the length of the payment period is equal to the length of the interest conversion period.

84

Effective/Nominal Rates (Cont…) However if the interest conversion

period is shorter than the payment period, or in other words interest is compounded more than once per payment period, then the effective rate will exceed the nominal rate.

85

Illustration ICICI Bank is quoting 9% per annum

compounded annually on its deposits. HDFC Bank is quoting 8.75% per

annum compounded quarterly. In the first case the nominal as well as

the effective rates are 9% per annum. In the second case the nominal rate is

8.75% per annum.

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Illustration (Cont…) What is the effective rate? 8.75% per annum corresponds to

8.75/4 = 2.1875% per quarter. If Rs 1 were to be deposited with

HDFC, it would accumulate to (1.021875)4 after 4 quarters.

The effective annual rate is given by(1+i) = (1.021875)4 i = 9.0413%

87

Effective/Nominal Rates (Cont…) In general, if a nominal rate of r is

quoted per period, with the frequency of compounding during the period being m, then by the principal of equivalency the effective periodic interest rate, i, is given by:(1+i) = (1 + r/m)m

88

Effective/Nominal Rates (Cont…) We can also calculate the

equivalent nominal rate, if the effective rate is known.

If the effective rate is i, then the equivalent nominal rate is given by:r = m[(1+i)1/m – 1]

89

Illustration HDFC Bank is offering 10% per

annum compounded quarterly. If you deposit Rs 10000, how much

will you have after 1 year.10/4 = 2.5%

F.V = 10000(1.025)4 = 11038 The effective annual rate is 10.38%.

90

Illustration HDFC Bank wants to offer an

effective annual rate of 10% per annum with quarterly compounding.

What nominal rate should it quote? r = 4[(1.10).25 – 1] = .0964

9.64%

91

Continuous Compounding Consider an amount P that is invested

for N periods at a nominal rate of r% per period.

If interest is compounded m times per period, then:

F.V = P(1 + r/m)mN

As m we get the case of continuous compounding.

Limit of F.V is erN, where e = 2.71828…

92

Illustration Narasimha Rao has deposited Rs

10000 in Corporation Bank for 5 years at a rate of 10% per annum compounded continuously.

What will be the final account balance?

F.V. = 10000e.10x5 = 10000x1.6487 = 16487

93

Continuous Compounding (Cont…) Continuous compounding is the

limit of the compounding process as we go from annual, to semi-annual, on to quarterly, monthly, daily, and even shorter intervals.

We will illustrate this using an example.

94

Illustration Sangeeta has deposited Rs 100 with ICICI

Bank for one year. What will be the account balance at the end, if interest is compounded using a nominal annual rate of 10% on:

An annual basis A semi-annual basis A quarterly basis A monthly basis A daily basis A continuous basis

95

Illustration (Cont…)Method BalanceAnnual 110.0000Semi-annual 110.2500Quarterly 110.3813Monthly 110.4713Daily 110.5156Continuously 110.5171

96

Annuity An annuity is a series of identical

payments made at equal intervals of time.

Examples include house rent, automobile loan repayments, or for that matter any EMI payment.

In the case of an ordinary annuity the first payment is made one period from now.

97

Annuity (Cont…) We will assume that the interval

between successive payments, the payment period, is equal to the length of time over which interest is compounded once, the conversion period, and that the points in time at which payments are made coincide with the points in time at which interest is compounded.

98

Annuity (Cont…) An annuity which makes periodic payments

of $ A for N periods, can be depicted as follows:

Time 0 is today.

99

Annuity (Cont…) The present value is the sum of a

geometric series:

100

Annuity (Cont…)

Is called PVIFA(r, N), or the Present Value Interest Factor Annuity.

101

Annuity (Cont…) It is the present value of an annuity

that pays $ 1 per period for N periods, starting one period from now, using a discount rate of r%.

It can be read off a table for integer values of r and N.

The present value of any annuity that pays $ A per period is given by: Ax PVIFA(r,N)

102

Illustration Apex Corporation is offering an

instrument that promises to pay Rs 1000 per year for 20 years, starting one year from now.

The price of the instrument is Rs 10000. If the required rate of return is 5%, is

the instrument worth buying. 1000xPVIFA(5,20) = 1000x12.4622

= 12462.20

103

Illustration (Cont…) If the investor pays this amount,

he will get a rate of return of exactly 5%, whereas if he pays less he will get a higher rate of return.

Since the quoted price is Rs 10000, the instrument is worth buying.

104

Future Value of an Annuity The future value of an annuity is given by:

is called the FVIFA or Future Value Interest Factor Annuity.

105

Future Value (Cont…) It too can be read off a table and

represents the future value of an annuity that pays

Rs 1 per period for N periods, computed using an interest rate of r% per period.

106

Illustration Pooja expects to receive Rs 10000

per year for the next 5 years, starting one year from now.

If the cash flows can be invested at 10% per annum, how much will she have after 5 years?

F.V = 10000xFVIFA(10,5) = 10000x6.1051= Rs 61051

107

Relationship Between PVIFA and FVIFA It can be seen that: PVIFA = FVIFA x PVIF Or conversely FVIFA = PVIFA x FVIF

108

An Annuity Due The only difference between an ordinary

annuity and an annuity due, is that in the latter case the cash flows occur at the beginning of the period rather than at the end.

Consequently in the case of an annuity due that is scheduled to make N payments, the first payment will occur at time 0, and the last at time N-1.

109

Present Value The present value of an annuity due that

makes N payments of $ 1 each is given by:

PVIFAAD(r,N) = PVIFA(r,N) x (1+r) Thus we have to multiply the present

value of a corresponding N period annuity by 1+r.

The present value of the annuity due is greater because each cash flow has to be discounted for one period less.

110

Illustration David has just bought an insurance

policy for which the annual premium is Rs 12,000/-.

He is required to make 25 payments in all, and the first payment is due immediately.

What is the present value if the discount rate is 10%?

P.V. = 12,000x9.0770x1.10=119,816.40

111

Future Value The future value of an annuity due that

makes N payments, as calculated at the end of N periods from now, is given by:

FVIFAAD(r,N) = FVIFA(r,N)x(1+r) The future value is greater as

compared to an ordinary annuity, because each cash flow has to be compounded for one period more.

112

Illustration What will be the cash value of

David’s policy at the end of 25 years?

F.V = 12000x98.3471x1.1=12,98,181.72

113

Perpetuities A perpetuity is an annuity that pays

forever. The future value of a perpetuity is

obviously infinite. But the present value can be computed. We know that for an N period annuity:

114

Perpetuities (Cont…) As N , 1/(1+r)N 0 Thus the present value of a perpetuity

is A/r. For instance consider an instrument

that promises to pay Rs 1000 per year forever.

If you require a 20% rate of return, how much should you be willing to pay?

Price = 1000/.20 = Rs 5000

115

Amortization The term Amortization refers to the

process of repaying a loan by means of installment payments at periodic intervals.

The installment payments constitute an annuity, whose present value is equal to the original loan amount.

Each installment will consist of a partial repayment of principal, and payment of interest on the outstanding loan balance.

116

Amortization (Cont…) An amortization schedule is a table

that shows the division of each payment into a principal component, and an interest component, together with the outstanding loan balance after each payment is made.

We will illustrate the process with the help of an example.

117

Illustration Srividya has borrowed Rs 10000 from

Syndicate Bank and wants to repay in 5 equal annual installments.

If the interest rate is 10% per annum on the outstanding balance, what is the installment amount?

Draw up an amortization schedule.

118

Illustration (Cont…) Let us denote the unknown

installment amount by A. We know that: 10000 = A x PVIFA(10,5) = A x

3.7908 A = 2637.97 The amortization schedule is as

depicted.

119

Amortization ScheduleYear Payment Interest Principal

Repayment

Outstanding

Principal0 - - - 100001 2637.9

71000 1637.9

78362.0

32 2637.9

7836.20 1801.7

76560.2

63 2637.9

7656.03 1981.9

44578.3

24 2637.9

7457.83 2180.1

42398.1

85 2637.9

7239.82 2398.1

5.03

120

Analysis Let us analyze a few entries in the table. At time 0, the outstanding balance is

10000. After one year, the first payment of

Rs 2637.97 is made. The interest due for the first year is 10%

of 10000 which is Rs 1000. So the excess of 1637.97 represents a

repayment of principal.

121

Analysis (Cont…) Once this amount is adjusted, the

outstanding balance at the end of the first year is 8362.03.

At the end of the second year, another payment of 2637.97 is made.

The interest due for this period is 10% of 8362.03 which is 836.20.

122

Analysis (Cont…) The excess, which is 1801.77,

constitutes a partial repayment of principal.

The outstanding balance at the very end should be zero.

The difference is due to rounding off errors.