Fundamentals of Financial Instruments. Sunil K. Parameswaran

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The present and future value of the series is depicted in Table 2.4. As can be seen, we have simply computed the present and future value of each term in the series, and summed up the values.

TABLE 2.4 Present and Future Values of the Cash Flows

      While computing the present value of each cash flow we have to discount the amount so as to obtain the value at time “0.” Thus the first year's cash flow has to be discounted for one year, whereas the fifth year's cash flow has to be discounted for five years. On the other hand, while computing the future value of a cash flow we have to find its terminal value as at the end of five years. Consequently, the cash flow arising after one year has to be compounded for four years, whereas the final cash flow, which is received at the end of five years, does not have to be compounded.

There is a relationship between the present value of the vector of cash flows as a whole and its future value. It may be stated as:

      In this case

THE INTERNAL RATE OF RETURN

      Consider a deal where we are offered the vector of cash flows depicted in Table 2.4, in return for an initial investment of $30,000. The question is, what is the rate of return that we are being offered? The rate of return r is obviously the solution to the following equation.

      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

Consider the series of cash flows depicted in Table 2.5. Assume that the annual rate of interest is 8%.

TABLE 2.5 Vector of Cash Flows

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

      Similarly the future value will be

      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 . The present value will then be given by

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