Fundamentals of Financial Instruments. Sunil K. Parameswaran

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Fundamentals of Financial Instruments - Sunil K. Parameswaran


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      The solution to this equation is termed as the Internal Rate of Return. It can be obtained using the IRR function in EXCEL. In this case the solution is 11.6106%.

       Note 5: A Point About Effective Rates

      Let us assume that we are asked to compute the present value or future value of a series of cash flows arising every six months, and are given a rate of interest quoted in annual terms, without the frequency of compounding being specified. The normal practice is to assume semiannual compounding. That is, we would divide the annual rate by two to determine the periodic interest rate for discounting or compounding. In other words, the quoted interest rate per annum will be treated as the nominal rate and not as the effective rate.

      EXAMPLE 2.15

Period Cash Flow
6 months 2,000
12 months 2,500
18 months 3,500
24 months 7,000

      The present value will be calculated as

normal upper P period normal upper V period equals StartFraction 2 comma 000 Over left-parenthesis 1.04 right-parenthesis EndFraction plus StartFraction 2 comma 500 Over left-parenthesis 1.04 right-parenthesis squared EndFraction plus StartFraction 3 comma 500 Over left-parenthesis 1.04 right-parenthesis cubed EndFraction plus StartFraction 7 comma 000 Over left-parenthesis 1.04 right-parenthesis Superscript 4 Baseline EndFraction equals dollar-sign 13 comma 329.5840

      Similarly the future value will be

StartLayout 1st Row 1st Column normal upper F period normal upper V period 2nd Column equals 2 comma 000 times left-parenthesis 1.04 right-parenthesis cubed plus 2 comma 500 times left-parenthesis 1.04 right-parenthesis squared plus 3 comma 500 times left-parenthesis 1.04 right-parenthesis plus 7 comma 000 2nd Row 1st Column Blank 2nd Column equals dollar-sign 15 comma 593.7280 EndLayout

      However, if it were to be explicitly stated that the effective annual rate is 8%, then the calculations would change. The semiannual rate that corresponds to an effective annual rate of 8% is left-parenthesis 1.08 right-parenthesis Superscript 0.5 Baseline equals 1.039230. The present value will then be given by

normal upper P period normal upper V period equals StartFraction 2 comma 000 Over left-parenthesis 1.039230 right-parenthesis EndFraction plus StartFraction 2 comma 500 Over left-parenthesis 1.039230 right-parenthesis squared EndFraction plus StartFraction 3 comma 500 Over left-parenthesis 1.039230 right-parenthesis cubed EndFraction plus StartFraction 7 comma 000 Over left-parenthesis 1.039230 right-parenthesis Superscript 4 Baseline EndFraction equals dollar-sign 13 comma 359.1103

      Similarly, the future value will then be given by

StartLayout 1st Row 1st Column normal upper F period normal upper V period 2nd Column equals 2 comma 000 times left-parenthesis 1.039230 right-parenthesis cubed plus 2 comma 500 times left-parenthesis 1.039230 right-parenthesis squared plus 3 comma 500 times left-parenthesis 1.039230 right-parenthesis plus 7 comma 000 2nd Row 1st Column Blank 2nd Column equals dollar-sign 15 comma 582.0372 EndLayout

      The present value is higher when we use an effective annual rate of 8% for discounting. This is because the lower the discount rate, the higher will be the present value; obviously, an effective annual rate of 8% is lower than a nominal annual rate of 8% with semiannual compounding. Because the interest rate that is used is lower, the future value at the end of four half-years is lower when we use an effective annual rate of 8%.

      Kapital Markets is offering an instrument that will pay $25,000 after four years in return for an initial investment of $12,500. Alfred is a potential investor, who requires a rate of return of 12% per annum. The issue is, is the offer attractive from his perspective? There are three ways of approaching this problem.

      The Future Value Approach

      Let us assume that Alfred buys this instrument for $12,500. If the rate of return received by him were to be 12%, he would have to receive a future value of $19,669. This can be stated as:

normal upper F period normal upper V period equals 12 comma 500 times left-parenthesis 1.12 right-parenthesis Superscript 4 Baseline equals dollar-sign 19 comma 669

      If Alfred were to receive a higher terminal payment, his rate of return would be higher than 12%, else it would be lower. Because the instrument offered to him promises a terminal value of $25,000, which is greater than the required future value of $19,669, the investment is attractive from his perspective.

      The Present Value Approach

      The present value of $25,000 using a discount rate of 12% per annum is:

normal upper P period normal upper V period equals StartFraction 25 comma 000 Over left-parenthesis 1.12 right-parenthesis Superscript 4 Baseline EndFraction equals 25 comma 000 times 0.6355 equals dollar-sign 15 comma 888.15

      The rate of return, if one were to make an investment of $15,888.15 in return for a payment of $25,000 four years hence, is 12%. If the investor were to pay a lower price at the outset, he would earn a rate of return that is higher than 12%, whereas if he were to invest more, he would obviously earn a lower rate of return. In this case Alfred is being asked to invest $12,500, which is less than $15,288.15. Consequently, the investment is attractive from his perspective.

      The Rate of Return Approach

      If Alfred were to pay $12,500 in return for a cash flow of $25,000 after four years, his rate of return may be computed as:

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