# Tarheel Consultancy Services Bangalore 1. 2 Part-01:Interest Rates & The Time Value of Money.

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Tarheel Consultancy Services Bangalore 1

2 Part-01:Interest Rates & The Time Value of Money

3 Simple Interest & Compound Interest

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

5 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.

6 Nominal Rate of Interest The quoted rate of interest per measurement period is called  The NOMINAL rate of interest

7 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

8 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

9 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

10 Simple Interest Consider an investment of Rs P for N periods. According to this principle  Interest earned every period is a constant  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

11 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

12 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)

13 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

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

15 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.

16 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

17 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

18 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.

19 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

20 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

21 Properties Simple interest is usually used for short- term 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.

22 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.

23 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 8.75%  The effective rate is obviously higher

24 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

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

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

27 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%

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

29 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

30 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.

31 Illustration-9 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

32 Illustration-10 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

33 Illustration-11 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?

34 Illustration-12 (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

35 Illustration-13 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?

36 Illustration-13 (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.

37 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.

38 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.

39 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

40 Illustration-14 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?

41 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?

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

43 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

44 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.

45 Illustration-16 Consider the following vector of cash flows. The interest rate is 10% compounded annually. YEARCash Flow 12,500 24,000 35,000 47,500 510,000

46 Illustration-16 (Cont…)

47 Illustration-16 (Cont…) The relationship between the present and future values is given by FV = PV(1+r) N In this case

48 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?

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

50 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%

51 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.

52 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.

53 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

54 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:

55 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.

56 Annuities

57 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

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

59 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 semi- annual compounding

60 Present Value

61 Present Value (Cont…)

62 Present Value (Cont…)

63 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.

64 Illustration-17 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

65 Future Value

66 Future Value (Cont…)

67 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.

68 Illustration-18 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

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

70 Perpetuities (Cont…)

71 Illustration-19 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?

72 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.

73 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.

74 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.

75 Amortization (Cont…)

76 Amortization (Cont…)

77 Amortization (Cont…)

78 Amortization (Cont…)

79 Illustration-20 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?

80 Amortization Schedule

81 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.

82 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

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