Part 06

Tarheel Consultancy Services Corporate Training and Consulting

1

Interest Rates & The Time Value of Money 2

Interest Rates

3

Introduction • All of us have either paid and/or received interest at some point of time. – Those of us who have taken loans have paid interest to the lending institutions. – Those of us who have invested have received interest from the borrowers.

4

Introduction (Cont…) • Types of Loans – Educational Loans

– Housing Loans

5

Introduction (Cont…) – Automobile Loans

6

Introduction (Cont…) • Investments – Savings accounts, and – Fixed deposits (Time deposits) with banks

7

Introduction (Cont…) – Bonds & Debentures

8

Introduction (Cont…) • Definition of interest – Compensation paid by the borrower of capital to the lender • For permitting him to use his funds

• An economist’s definition – Rent paid by the borrower of capital to the lender • To compensate for the loss of opportunity to use the funds when it is on loan 9

Introduction (Cont…) • Concept of rent – When we decide not to live in an apartment/house owned by us • We let it out to a tenant

– The tenant pays a monthly rental • Because as long as he is occupying our property we are deprived of an opportunity to use it

10

Introduction (Cont…) – The same concept applies to a loan of funds • The difference is – Compensation in the case of property is called RENT – Compensation in the case of capital is called INTEREST

11

The Real Rate of Interest • In a free market – Interest rates are determined by • Demand for capital – And

• The supply of capital

12

The Real Rate (Cont…) • One of the key determinants of Interest is – The Pure rate of interest a.k.a – The Real rate of interest

13

The Real Rate (Cont…) • Definition of the Real rate: – The rate of interest that would prevail on a risk-less investment in the absence of inflation.

• Example of a risk-less investment – Loan to the Federal/Central government

• Such loans are risk-less because there is no risk of default – The central government of a country is the only institution authorized to print money 14

The Real Rate (Cont…) • But they say that certain governments (in Latin America etc.) have defaulted on debt – Yes they have defaulted on dollar denominated debt • The government of Argentina for instance can print its own currency but not U.S. dollars

15

The Real Rate Illustrated • The price of a banana is Rs 1

• Assume that the price of a banana next year will also be Rs 1 – That is, there is no inflation • In other words there is no erosion in the purchasing power of money

16

Illustration (Cont…) • Take the case of a person who lends Rs 10 to the Government of India (GOI) – Obviously there is no fear of non-payment

• If the GOI pays back Rs 11 after one year – The amount will be sufficient to buy 11 bananas.

17

Illustration (Cont…) • In this case a loan of Rs 10 has been returned with 10% interest in money terms • Since the investor is in a position to buy 10% more in terms of bananas – The return on investment in terms of the ability to buy goods is also 10% – The rate of interest as measured by the ability to buy goods and services is termed as • THE REAL RATE of INTEREST 18

The Real Rate (Cont…) • In the real world price levels are not constant. – Erosion in the purchasing power of money is a fact of life • This is termed as inflation

19

The Real Rate (Cont…) • Most people who invest do so by acquiring financial assets such as – Shares of stock – Shares of a mutual fund – Or bonds/debentures

• Many also keep deposits with commercial banks

20

The Real Rate (Cont…) • Financial assets give returns in terms of money – Without any assurance about the investor’s ability to acquire goods and services at the time of repayment.

• Financial assets therefore give a NOMINAL or MONEY rate of return. – In the example, the GOI gave a 10% return on an investment of Rs 10. 21

The Real Rate (Cont…) • In the example the 10% money rate of return was adequate to buy 10% more in terms of bananas. – This was because we assumed that the price of a banana would remain fixed at Rs 1.

22

The Real Rate (Cont…) • But what if the price of a banana after a year is Rs 1.05. – Rs 11 can then acquire only

23

The Real Rate (Cont…) • In this case the nominal rate of return is 10% • But our ability to buy goods has been enhanced only by 4.80% – Thus the REAL rate of return is only 4.80%

• The relationship between the nominal and real rates of return is called the FISHER hypothesis – Because it was first postulated by Irving Fisher.

24

The Fisher Equation • Consider a hypothetical economy – It consists of one good – say BANANAS

• The current price of a banana is Rs P0 – So Rs 1 can buy

bananas.

25

The Fisher Equation (Cont…) • Assume that the price of a banana next period is P1. – P1 is known with certainty today but need not be equal to P0 – In other words although we are allowing for inflation, we are assuming that there is no uncertainty regarding the rate of inflation. – So one rupee will be adequate to buy bananas after one period 26

The Fisher Equation (Cont…) • Assume that the economy has two types of bonds available – We have FINANCIAL bonds and GOODS bonds – If we invest Rs 1 in a Financial bond, we will get Rs (1+R) after one period. – If we invest 1 banana in a Goods bond we will get (1+r) bananas after one period. 27

The Fisher Equation (Cont…) • Rs 1 in a Financial bond → Rs (1+R)→

Rs 1 in a goods bond →

bananas →

bananas after one period.

28

The Fisher Equation (Cont…) • In order for the economy to be in equilibrium both the bonds must yield identical returns. • Therefore it must be true that:

29

The Fisher Equation (Cont…) • Let us denote inflation or the rate of change in the price level by π

30

The Fisher Equation (Cont…) • This is the Fisher equation. – R or the rate of return on a financial bond is the nominal rate of return – r or the rate of return on a goods bond is the real rate of return

31

The Fisher Equation (Cont…) • If r and π are very small, then the product of the two will be much smaller. – For instance if r = 0.03 and π = 0.03, the product is 0.0009

• If we ignore the product we can rewrite the expression as R=r+π • This is the approximate Fisher equation. 32

Uncertainty • Thus far we have assumed that the rate of inflation is known with certainty. – In real life inflation is uncertain – Consequently it is a random variable

• In the case of random variables – We do not know the exact outcome in advance – All we know is the expected value of the variable • Which is a probability weighted average of the values that the variable can take.

33

Uncertainty (Cont…) Inflation

Probability

2.50%

0.20

5.00%

0.20

7.50%

0.20

10.00%

0.20

12.50%

0.20 34

Uncertainty (Cont…) • The expected value is given by

35

Uncertainty (Cont…) • The Fisher equation can therefore be rewritten as – R = r + E(π)

• Thus when inflation is uncertain – The actual real rate that we will eventually get is unpredictable and uncertain

36

Uncertainty (Cont…) • Assume that the required real rate is 4.50% • Since the expected inflation is 7.50% – an investor will demand a nominal rate of return of 12%

37

Uncertainty (Cont…) • Once the nominal rate is fixed, it will not vary • But there is no guarantee that the realized rate of inflation will equal the expected rate – In this case if the realized inflation is 9%, the realized real rate will be only 3%

38

Uncertainty (Cont…) • Thus in real life even a default free security will not give an assured real rate. – It will give an assured nominal rate – But the real rate that is eventually obtained will depend on the actual rate of inflation

39

Ex-ante versus Ex-post • An economist will say that the ex-ante rate of inflation need not equal the ex-post rate – Ex-ante means anticipated or forecasted value – Ex-post connotes actual or realized value

• Obviously the ex-ante real rate of interest need not equal the ex-post real rate

40

Uncertainty & Risk Aversion • In the real world investors are characterized by RISK AVERSION. – This does not mean that they will not take risk – What does it mean therefore? • To induce an investor to take a greater level of risk he must be offered a higher expected rate of return.

41

Risk Aversion (Cont…) • Given a choice between two investments with the same expected rate of return – The investor will choose the less risky option

• In the case of inflation – The investor will not accept the expected inflation as compensation

• Why? – The actual inflation could be higher than anticipated • Which implies that the actual real rate could be lower than anticipated. 42

Risk Aversion (Cont…) • To tolerate the inflation risk – The investor will demand a POSITIVE risk premium • That is, compensation over and above the expected rate of inflation

• The Fisher equation may be restated as – R = r + E(π) + R.P. – Where R.P is the risk premium 43

Risk Aversion (Cont…) • Does the provision of a risk premium guarantee that the ex-ante real rate = ex-post real rate • NO! • Suppose the required real rate is 4.5%, that E(π) = 7.5%, and that R.P = 1.5% – Then the required nominal rate will be 13.50%

44

Risk Aversion (Cont…) • In the absence of a risk premium – A rate of inflation > 7.5% implies a realized real rate < 4.5%

• But when a risk premium is factored in – A rate of inflation > 9% implies a realized real rate < 4.5%

• So the risk premium provides a bigger cushion against inflation – But it does not guarantee a minimum ex-post real rate 45

Other Determinants • Besides – the required real rate – the expected inflation – and the inflation risk premium

the following factors impact the required nominal rate • Length of the investment • Credit Risk 46

Length of the Investment • Lender like to lend short term • Borrowers like to borrow long-term • So how do we induce a lender to lend for a longer period – Offer a HIGHER nominal rate of return

47

Typical Interest Rate Schedule in a Bank Period of Investment

Interest Rate

< 1 year

7.50%

More than 1 year but < 2years More than 2 years but < 3 years More than 3 years but < 5 years More than 5 years

8.00% 8.50% 9.00% 9.25%

48

Credit Risk • We have focused on default free investments – Loans to a Central Government

• In reality most investments are fraught with risk – Interest may not be paid – Principal may not be repaid

49

Credit Risk (Cont…) • This is called credit risk – Applies to all investments except Central government securities

• There is a difference between inflation risk and credit risk – Inflation is an economy wide phenomenon – Credit risk however varies from borrower to borrower 50

Credit Risk (Cont…) • Because of credit risk – The rate of return demanded by a lender will vary from borrower to borrower

• Which is why – For a given real rate – For a given tenor of the loan – For a given rate of inflation

a bank will charge different rates of interest on loans made to different borrowers. 51

Simple Interest & Compound Interest

52

Measurement Period • The unit in which time is measured is called the Measurement Period – The most common measurement period is One Year.

53

Interest Conversion Period • The unit of time over which interest is paid once and is reinvested to earn additional interest is called – The Interest Conversion Period

• The interest conversion period is typically less than or equal to the measurement period.

54

Nominal Rate of Interest • The quoted rate of interest per measurement period is called – The NOMINAL rate of interest

55

Effective Rate of Interest • The interest that a unit of currency invested at the beginning of a measurement period would have earned by the end of the period is called – The EFFECTIVE Rate of Interest

56

Effective Rate (Cont…) • If the length of the interest conversion period is equal to the measurement period – The effective rate will be equal to the nominal rate

• If the interest conversion period is shorter than the measurement period – The effective rate will be greater than the nominal rate 57

Variables and Symbols • P ≡ principal invested at the outset • N ≡ # of measurement periods for which the investment is being made • r ≡ nominal rate of interest per measurement period • i ≡ effective rate of interest per measurement period • m ≡ # of interest conversion periods per measurement period 58

Simple Interest • Consider an investment of Rs P for N periods. • According to this principle – Interest earned every period is a constant – That is, every period interest is computed and credited only on the original principal – No interest is payable on any interest that has been accumulated at an intermediate stage 59

Simple Interest (Cont…) • If r is the nominal rate of interest – P → P(1+r) after one period→P(1+2r) after 2 periods →P(1+rN) after N periods

• So every period interest is paid only on the original principal • N need not be an integer – Investments can be made for fractional periods 60

Illustration-1 • Caroline has deposited Rs 10,000 with Corporation Bank for 3 years • The bank pays simple interest at the rate of 10% per annum • 10,000 will become 10000x1.1 = 11,000 after one year →10000x1.1 + 1,000 = 12,000 after two years → 13,000 after 3 years • 13,000 = 10,000(1+ .10x 3) ≡P(1+rN) 61

Illustration-2 • Amit Gulati deposits Rs 10,000 with ICICI Bank for 5 years and 6 months. • Bank pays simple interest at 8% per annum. • Maturity value = 10,000(1+.08x5.5) = Rs 14,400 – Notice: N need not be an integer

62

Compound Interest • Consider an investment of Rs P for N periods. • Assume that the interest conversion period is equal to the measurement periods – That is, the effective rate is equal to the nominal rate

63

Compound Interest (Cont…) • In the case of compound interest – Every time interest is earned it is automatically reinvested at the same rate for the next conversion period. – So interest earned every period is not a constant • It steadily increases

• P→P(1+r) after one period →P(1+r)2 after two periods→P(1+r)N after N periods. 64

Illustration-3 • Caroline has deposited Rs 10,000 with Corporation Bank – Bank pays 10% per annum compounded annually

• Rs 10,000→11,000 after one year→ 11000x 1.1 = 12,100 after 2 years → 12,100x1.1= 13,310 after 3 years – 13,310 = 10,000x (1.10)3 65

Illustration-4 • Gulati deposited Rs 10,000 with ICICI Bank for 5 years and 6 months. – Bank has been paying 8% compounded annually

• P(1+r)N = 10,000(1.08)5.5 = Rs 15,269.71

66

Compound Interest (Cont…) • Compounding yields greater benefits than simple interest – The larger the value of N the greater is the impact of compounding – Thus, the earlier one starts investing the greater are the returns.

67

Illustration-5 • The East India Company came to India in 1600. • Consider an investment of Rs 10 in 1600 with a bank which pays 3% per annum compounded annually. – The balance in 2000 = 10x(1.10)400 = Rs 1,364,237.18

68

Properties • If N=1, that is, the investment is for one period, both simple as well as compound interest will give the same accumulated value. • If N < 1, the accumulated value using simple interest will be higher. That is: – (1+rN) > (1+r)N if N < 1

• If N > 1, the accumulated value using compound interest will be greater. That is: – (1+rN) < (1+r)N if N > 1 69

Properties • Simple interest is usually used for shortterm transactions – investments of one year or less – It is the norm for money market transactions

• For capital market securities – medium to long term debt and equities – compound interest is the norm.

70

Illustration-6 • Amit Gulati deposited Rs 10,000 with ICICI Bank for 5 years and six months. – The bank pays compound interest at 8% for the first 5 years and simple interest at 8% for the last six months. – 10,000(1+.08)5 = 14,693.28 – 14,693.28(1 + .08x.5) = Rs 15,281.01 – On the other hand 10000(1.1)5.5 = 15,269.71 • The difference is because for the last six months simple interest yields more than compound interest. 71

Effective versus Nominal Rates • ICICI Bank is quoting 9% per annum compounded annually • HDFC Bank is quoting 8.75% per annum compounded quarterly • In the case of ICICI – The nominal rate is 9% per annum – The effective rate is also 9% per annum

• In the case of HDFC – The nominal rate is 9% – The effective rate is obviously higher 72

Effective…(Cont…) • 8.75% per annum ≡ 2.1875% per quarter – So a deposit of Rs 1→(1.021875)4 = 1.090413

• So the effective rate offered by HDFC is 9.0413% per annum • Thus when the frequencies of compounding are different – Comparisons between alternative investments should be based on effective rates and not nominal rates 73

Effective (Cont…) • The nominal rate is r% per annum • Interest is compounded m times per annum • The effective rate is:

74

Effective…(Cont…) • We can also derive the equivalent nominal rate if the effective rate is given

75

Illustration-7 • HDFC Bank is paying 10% compounded quarterly. – If Rs 10,000 is deposited for a year what will be the terminal amount – The terminal value will be

The effective annual rate is 10.38% 76

Illustration-8 • Suppose HDC Bank wants to offer an effective annual rate of 10% with quarterly compounding – What should be the quoted nominal rate

77

Equivalency • Two nominal rates compounded at different time intervals are said to be Equivalent if the same principal invested for the same length of time produces the same accumulated value in either case. – In other words two nominal rates compounded at different intervals are equivalent if they yield the same effective rate 78

Equivalency (Cont…) • ICICI Bank is offering 9% per annum with semiannual compounding. • What should be the equivalent rate offered by HDFC Bank if it intends to compound quarterly.

79

Equivalency (Cont…) • The issue is, what will be the nominal rate that will give an effective annual rate of 9.2025% with quarterly compounding

Thus 9% with semi-annual compounding is equivalent to 8.90% with quarterly compounding. 80

Continuous Compounding • Consider Rs P that is invested for N periods at r% per period. • If interest is compounded m times per period, the terminal value will be

81

Continuous Compounding (Cont…) • What about the limit as m→∞

This is the case of continuous compounding.

82

Illustration-9 • Narasimha Rao has deposited Rs 10,000 with Corporation Bank for 5 years at 10% per annum compounded continuously. • The final balance is:

83

Illustration-10 • Canara Bank is quoting 10% per annum with quarterly compounding. • What should be the equivalent rate with continuous compounding? • Two nominal rates are equivalent if they give the same effective annual rate. • Let r be the nominal rate with quarterly compounding, and k the nominal rate with continuous compounding. 84

Illustration-10 (Cont…)

85

Illustration-10 (Cont…) • In this case:

86

The Limit • Continuous compounding is the limit as we go from – Annual • To semi-annual – Quarterly » Monthly » Daily » And shorter intervals

87

Illustration-11 • Sangeeta has deposited Rs 100 with ICICI Bank. • The interest rate is 10% per annum. • What will be the terminal balance under the following scenarios: – Annual compounding • Semi-annual compounding – Quarterly compounding » Monthly compounding » Daily compounding » Continuous compounding 88

Illustration-11 (Cont…) Frequency of Compounding Annual

Terminal Balance

Semi-annual

Rs 110.2500

Quarterly

Rs 110.3813

Monthly

Rs 110.4713

Daily

Rs 110.5156

Continuously

Rs 110.5171

Rs 110.0000

89

Future Value • When an amount is deposited for a time period at a given rate of interest – The amount that is accrued at the end is called the future value of the original investment – So if Rs P is invested for N periods at r% per period

90

FVIF • (1+r)N is the amount to which an investment of Rs 1 will grow at the end of N periods. • It is called FVIF – Future Value Interest Factor. – It is a function of r and N. – It is given in the form of tables for integer values of r and N – If the FVIF is known, the future value of any principal can be found by multiplying the principal by the factor. – The process of finding the future value is called Compounding. 91

Illustration-12 • Suhasini has deposited Rs 10,000 for 5 years at 10% compounded annually. • What is the Future Value?

Thus F.V. = 10,000 x 1.6105 = Rs 16,105

92

Illustration-13 • Swapna has deposited Rs 10,000 for 4 years at 10% per annum compounded semi-annually. • What is the Future Value? – 10% for 4 years is equivalent to 5% for 8 half-years

Thus F.V. = 10,000 x 1.4775 = Rs 14,775

93

Illustration-14 • GIC has collected a one time premium of Rs 10,000 from Suhasini and has promised to pay her Rs 23,000 after 10 years. • The company is in a position to invest the premium at 10% compounded annually. • Can it meet its obligation?

94

Illustration-14 (Cont…) • The future value of Rs 10,000 = 10,000 x 2.5937 = Rs 25,937 • This is greater than the liability of Rs 23,000 • So GIC can meet its commitment

95

Illustration-15 • Syndicate Bank is offering the following scheme – An investor has to deposit Rs 10,000 for 10 years – Interest for the first 5 years is 10% compounded annually – Interest for the next 5 years is 12% compounded annually – What is the Future Value? 96

Illustration-15 (Cont…) • The first step is to calculate the future value after 5 years:

The next step is to treat this as the principal and compute its terminal value after another 5 years.

97

Present Value • When we compute the future value we seek to determine the terminal value of an investment that has earned a given rate of interest for a specified period. • Now consider the issue from a different angle? – If we want a specified terminal value, how much should we invest at the outset, if the interest rate is r% and the number of periods is N. 98

Present Value (Cont…) • So instead of computing the terminal value of a principal we seek to compute the principal that corresponds to a given terminal value. • The principal amount that we compute is called the Present Value of the terminal amount.

99

The Case of Simple Interest • An investment yields Rs F after N periods. • If the interest rate is r%, what is the present value? • We know that: F = P.V.x(1+rN) So obviously

100

Illustration-16 • Venkatachalam wants to ensure that he has saved Rs 12,000 after 4 years. – So he deposits Rs P with a bank – If the bank pays 5% per annum on a simple interest basis, what should be P?

101

The Case of Compound Interest • An investment pays r% per period on a compound interest basis. • If we want Rs F after N periods, how much should we deposit today?

102

Illustration-17 • Priyanka wants to ensure that she has Rs 15,000 after 3 years. • The bank pays 10% compounded annually • How much should she deposit?

103

PVIF • 1/ (1+r)N is the amount that has to be deposited to yield Rs 1 after N periods if the periodic interest rate is r% – It is called the Present Value Interest Factor (PVIF) – It is a function of r and N – It is given in the form of tables for integer values of r and N – If we know the factor, we can find the present value of any terminal amount by multiplying the two. – The process of finding the principal value of a terminal amount is called Discounting – PVIF is the reciprocal of FVIF 104

The Additivity Principle • Suppose you want to find the present or future value of a series of cash flows, where the rate of interest is r%, and the last cash flow is received after N periods. • You have to simply find the present or the future value of each cash flow and add up the terms to compute the present or future value of the series. • Thus Present and Future Values are additive. 105

Illustration-18 • Consider the following vector of cash flows. • The interest rate is 10% compounded annually. YEAR 1 2 3 4 5

Cash Flow 2,500 4,000 5,000 7,500 10,000

106

Illustration-18 (Cont…)

107

Illustration-18 (Cont…) • The relationship between the present and future values is given by FV = PV(1+r)N • In this case

108

The Internal Rate of Return • Suppose that we are told that an investment of Rs 18,000 will entitle us to the following vector of cash flows. – The question is what is the rate of return?

109

The IRR (Cont…) • The rate of return is the solution to the following equation:

110

The IRR (Cont…) • The solution to this equation is called the Internal Rate of Return (IRR) • It can be obtained using the IRR function in EXCEL. – In this case, the solution is 14.5189%

111

Effective Rates • Suppose we are asked to calculate the present or future values of a series of cash flows arising every six months. • And we are given an annual rate of interest without specifying the compounding frequency. – The normal practice is to divide the interest rate by 2 to determine the periodic interest rate – That is, the quoted rate is treated as a nominal rate and not as an effective rate

112

Illustration-19 • Consider the following vector of cash flows. • Assume that the annual interest rate is given as 10%.

113

Illustration-19 (Cont…) • The Present Value will be calculated as:

•Similarly the future value will be

114

Illustration-19 (Cont…) • But what if it is explicitly stated that the effective annual rate is10%? – Then the calculations will change

•And the future value is given by

115

Effective Rates (Cont…) • The present value is greater when we use an effective annual rate of 10% for discounting. – This is because the lower the discount rate the higher will be the present value – And an effective rate of 10% per annum is lower than a nominal rate of 10% with semiannual compounding. – By the same logic the future value is lower when we use an effective annual rate of 10% 116

Evaluating an Investment • Kotak Mahindra is offering an instrument that will pay Rs 10,000 after 5 years. • The price that is quoted is Rs 5,000. • If the investor wants a 10% rate of return, should he invest. • The problem can be approached in three ways. 117

The Future Value Approach • Assume that the instrument is bought for 5,000. • If the rate of return is 10% the future value is 5,000 x 1.6105 = Rs 8,052.50 • Since the instrument promises a terminal value of Rs 10,000 which is greater than the required future value, the investment is attractive. 118

The Present Value Approach • The present value of Rs 10,000 using a discount rate of 10% is 10,000 x 0.6209 = Rs 6,209 • So if Rs 6,209 is paid at the outset the rate of return will be 10% • If we pay more at the outset, the rate of return will be lower and vice versa. • In this case the investment of Rs 5,000 is less than Rs 6,209 • So the investment is attractive 119

The Rate of Return Approach • Suppose you invest Rs 5,000 and receive Rs 10,000 after 5 years. • What is the rate of return? • It is given by:

120

The Rate…(Cont…) • The solution is 14.87% • Since the actual rate of return is greater than the required rate of 10%, the investment is attractive.

121

Annuities

122

Annuities (Cont…) • What is an annuity? – It is a series of identical payments made at equally spaced intervals of time

• Examples – House rent till it is revised – Salary till it is revised – Insurance premia – EMIs on housing/automobile loans 123

Annuities (Cont…) • In the case of an ordinary annuity – The first payment is made one period from now

124

Annuities (Cont…) • The interval between successive payments is called the – PAYMENT Period

• We will assume that the payment period is the same as the interest conversion period – That is, if the annuity pays annually, we will assume annual compounding – If it pays semi-annually we will assume semiannual compounding 125

Present Value

126

Present Value (Cont…)

127

Present Value (Cont…)

128

Present Value (Cont…)

Is called the Present Value Interest Factor Annuity (PVIFA) It is the present value of an annuity that pays Rs 1 per period The present value of annuity that pays a periodic cash flow of Rs A can be found by multiplying A by PVIFA. 129

Illustration-20 • Apex Corporation is offering an instrument that will pay Rs 1,000 per year for 20 years, beginning one year from now. • If the rate of interest is 5%, what is the present value? – 1,000xPVIFA(5,20) = 1,000 x 12.4622 = Rs 12,462.20

130

Future Value

131

Future Value (Cont…)

132

Future Value (Cont…)

Is called the Future Value Interest Factor Annuity (FVIFA) It is the future value of annuity that pays Rs 1 per period. For any annuity that pays Rs A per period, the future value can be found by multiplying A by the factor. 133

Illustration-21 • Pooja expects to receive Rs 10,000 per year for the next 5 years, starting one year from now. • If the cash flows can be invested at 10% per annum what is the Future Value? – F.V. = 10,000 x FVIFA(10,5) = 10,000 x 6.1051 = Rs. 61,051

134

Relationship Between PVIFA and FVIFA

135

Annuity Due • In the case of an Annuity Due, the cash flows occur at the beginning of the period.

136

Present Value

137

Present Value (Cont…)

138

Present Value (Cont…) • The present value of an annuity due that makes N payments is greater than that of an annuity that makes N payments • Why? – Because each cash flow has to be discounted for one period less.

• Example of an annuity due? – An insurance policy • The first premium has to be paid as soon as the policy is purchased. 139

Illustration-22 • David has bought an LIC policy • The annual premium is Rs 12,000 and he has to make 25 payments. • What is the present value if the discount rate is 10% per annum?

140

Future Value

141

Future Value (Cont…) • The future value of an annuity due that makes N payments, is greater than that of a corresponding annuity, if the future value is computed at the end of N periods. • Why? – Because each cash flow has to be computed for one period more.

142

Illustration-23 • If David takes an LIC policy with a premium of Rs 12,000 per year for 25 years, what is the cash value at the end of 25 years?

143

Perpetuities • An annuity that pays forever is called a PERPETUIY. • The future value of a perpetuity is obviously infinite. • But a perpetuity has a finite present value.

144

Perpetuities (Cont…)

145

Illustration-24 • A financial instrument promises to pay Rs 1000 per year forever. • If the investor requires a 20% rate of return, how much should he be willing to pay for it?

146

Amortization • The amortization process refers to the process of repaying a loan by means of equal installment payments at periodic intervals. • The installments obviously form an annuity. – The present value of the annuity is the loan amount. 147

Amortization (Cont…) • Each installment consists of – Partial repayment of principal – And payment of interest on the outstanding balance

• An amortization schedule shows the division of each payment into a principal component and interest component, together with the outstanding loan balance after the payment is made. 148

Amortization (Cont…) • Consider a loan which is repaid in N installments of Rs A each. • The original loan amount is Rs L, and the periodic interest rate is r.

149

Amortization (Cont…)

150

Amortization (Cont…)

151

Amortization (Cont…)

152

Amortization (Cont…)

153

Illustration-25 • Srividya has borrowed Rs 10,000 from Syndicate Bank and has to pay it back in five equal annual installments. • The interest rate is 10% per annum on the outstanding balance. • What is the installment amount?

154

Amortization Schedule

155

Analysis • At time 0, the outstanding principal is 10,000 • After one period an installment of Rs 2,637.97 is made. – The interest due for the first period is 10% of 10,000 or Rs 1,000 – So the excess payment of Rs 1,637.97 is a partial repayment of principal. – After the payment the outstanding principal is Rs 8,362.03 – After another period a second installment is paid. – The interest for this period is 10% of 8,362.03 which is Rs 836.20. – The balance of Rs 1,801.77 constitutes a partial repayment of principal.

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Analysis (Cont…) • The value of the outstanding balance at the end should be zero. • After each payment the outstanding principal keeps declining. • Since the installment is constant – The interest component steadily declines – While the principal component steadily increases 157

Amortization with a Balloon Payment • Uttara has taken a loan of Rs 100,000 from ICICI Bank. • She has to pay in 5 equal annual installments along with a terminal payment of Rs 25,000 • The terminal payment which has to be made over and above the scheduled installment in year 5 – Is called a BALLOON payment. 158

Balloon (Cont…) • If the interest rate is 10% per annum, the annual installment may be calculated as

Obviously, the larger the balloon the smaller will be the periodic installment for a given loan amount. 159

Amortization Schedule

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Types of Interest Computation • Financial institutions employ a variety of different techniques to calculate the interest on the loans made by them. • The interest that is effectively paid on the loan may be very different from the rate that is quoted. – THUS WHAT YOU SEE IS NOT WHAT YOU ALWAYS GET 161

The Simple Interest Method • In this technique, interest is charged for only the period of time that a borrower has actually used the funds. – Each time principal is partly repaid, the interest due will decrease.

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Illustration • Alfred has borrowed $5,000 from the bank for a year. • The bank charges simple interest at the rate of 8% per annum. • If the loan is repaid at the end of one year: – Interest payable = 5000x0.08 = 400 – Total amount repayable = 5,400

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Illustration (Cont…) • Assume the loan is repaid in two equal semi-annual installments. – After six months principal of $2,500 is repaid. – Interest will however be charged on 5,000. – Amount repayable = 2500 + 5000x0.08x.5 = 2700

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Illustration (Cont…) • For the next six months interest will be charged only on $2,500. – The amount payable at the end of the second six-monthly period = 2500 + 2500x0.08x.5 = $2,600 – Total outflow on account of principal plus interest = 2700 + 2600 = 5300 – Obviously the more frequently the principal is repaid the lower is the interest. 165

The Add-on Rate Approach • In this case interest is first calculated on the full principal. • The sum of interest plus principal is then divided by the total number of payments in order to determine the amount of each payment. • In Alfred’s case if he repays in one annual installment, there will be no difference with this approach as compared to the simple interest approach.

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Add-on…(Cont…) • What if he repays in two installments? – Interest for the entire year = 400 – This will be added to the principal and divided by 2. – Thus each installment = (5000 + 400) ____________ 2 = 2700 167

Add-on…(Cont…) • The quoted rate is 8% per annum. • But the actual rate will be higher. • The actual rate is given by

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Add-on (Cont…) • The solution is i = 10.5758% – This is of course the nominal annual rate. – The effective annual rate is 10.8554%

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The Discount Method • In this approach the total interest is first computed on the entire loan amount. • This is then deducted from the loan amount. • The balance is lent to the borrower.

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Illustration • Alfred borrows 5000 at 8% for a year. • The interest for the year is 400. • So Alfred will be given 4600 and will have to repay 5000 at the end. • The effective rate of interest = (5000 – 4600) ___________ x 100 = 8.6957% 4600 171

Discount Loan (Cont…) • Such loans usually do not require installments and are settled in one lump sum at the end.

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Compensating Balances • Many banks require that borrowers keep a certain percentage of the loan amount with them as a deposit. • This is called a Compensating Balance. • It raises the effective interest rate – Since the borrower cannot use the entire amount that is sanctioned

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Illustration • Alfred is sanctioned $ 5,000 at the rate of 8%. • But he has to keep 10% of the loan amount with the bank for the duration of the loan. • So while he pays an interest of $400, the usable amount is only 5000x0.9 = $ 4,500 174

Illustration (Cont…) • The effective interest cost is 400 ________ x 100 = 8.8889% 4500 • Quite obviously – The higher the compensating balance, the greater will be the effective interest rate. 175

Annual Percentage Rate (APR) • The effective rate of interest that is paid by a borrower is a function of the type of loan that is offered to him. • Since different lenders used different loan structures, comparisons between competing loan offers can be difficult.

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APR (Cont…) • To ensure uniformity the U.S. Congress passed the – Consumer Credit Protection Act – This is commonly known as • The Truth-in-Lending Act

• The law requires institutions extending credit to use a prescribed method for computing the quoted rate. 177

APR (Cont…) • Every lending institution is required to compute the APR and report it before the loan agreement is signed. • The most accurate way to compute the APR is by equating the present value of the repayments made by the borrower to the loan amount.

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APR (Cont…) • For the examples that we have considered the precise APR would be: Loan Type

APR

Simple Interest-One Installment

8%

Simple Interest-Two Installments

7.90%

Add-on Method-Two Installments 10.5758% Discount Method-One Installment Compensating Balance-One Installment

8.6957% 8.8889% 179

The Approximate APR • There is a technique for calculating the approximate APR known as the ConstantRatio method. • The formula is: APR = 2xNo. Of pmts. Per yr.xAnn.Int.Costx100 ______________________________________ (Tot. No. of Loan Pmts. +1)xPrincipal Amount

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The Approximate APR (Cont…) Loan Type

Approximate APR

Simple Interest-One Installment Simple Interest-Two Installments Add-on Method-Two Installments Discount Method-One Installment Compensating BalanceOne Installment

8% 8% 10.67% 8.6957% 8.8889% 181

APR (Cont…) • The approximate formula gives the exact APR when the loan is repaid in one installment per year. • But it overstates the APR when loans are repaid by way of multiple installments in a year.

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