Sunil K. Parameswaran - Fundamentals of Financial Instruments

Similarly, we can compute the future value of a level annuity that makes N payments, by compounding each cash flow until the end of the last payment period.

Therefore,

is called the Future Value Interest Factor Annuity (FVIFA) . This is the future value of an annuity that pays $1 per period for N periods, where interest is compounded at the rate of r % per period. The advantage once again is that if we know the factor, we can calculate the future value of any annuity that pays $ A per period.

Paula Baker expects to receive $2,500 per year for the next 25 years, starting one year from now. Assuming that the cash flows can be reinvested at 8% per annum, how much will she have at the point of receipt of the last cash flow?

Thus the future value is:

The difference between an annuity and an annuity due is that in the case of an annuity due the cash flows occur at the beginning of the period. An N period annuity due that makes periodic payments of $A may be depicted as follows

FIGURE 2.2 Timeline for an Annuity Due

Therefore,

The present value of an annuity due that makes N payments is obviously greater than that of a corresponding annuity that makes N payments, because in the case of the annuity due, each of the cash flows has to be discounted for one period less. Consequently, the present value factor for an N period annuity due is greater than that for an N period annuity by a factor of (1 + r ).

An obvious example of an annuity due is an insurance policy, because the first premium has to be paid as soon as the policy is purchased.

David Mathew has just bought an insurance policy from MetLife. The annual premium is $2,500, and he is required to make 25 payments. What is the present value of this annuity due if the discount rate is 8% per annum?

Thus the present value of the annuity due is:

Therefore,

Hence

The future value of an annuity due that makes N payments is higher than that of a corresponding annuity that makes N payments, if the future values in both cases are computed at the end of N periods. This is because, in the first case, each cash flow has to be compounded for one period more.

Note 6:It should be reiterated that the future value of an N period annuity due is greater than that of an N period annuity if both the values are computed at time N that is after N periods. The future value of an annuity due as computed at time N − 1 will be identical to that of an ordinary annuity as computed at time N .

In the case of Mathew's MetLife policy, the cash value at the end of 25 years can be calculated as follows.

Thus the cash value of the annuity due is:

An annuity that pays forever is called a perpetuity. The future value of a perpetuity is obviously infinite. But it turns out that a perpetuity has a finite present value. The present value of an annuity that pays for N periods is

The present value of the perpetuity can be found by letting N tend to infinity. As follows:

Thus, the present value of a perpetuity is A/r .

Let us consider a financial instrument that promises to pay $2,500 per year for ever. If investors require a 10% rate of return, the maximum amount they would be prepared to pay may be computed as follows.

Thus, although the cash flows are infinite, the security has a finite value. This is because the contribution of additional cash flows to the present value becomes insignificant after a certain point in time.

The amortization process refers to the process of repaying a loan by means of regular installment payments at periodic intervals. Each installment includes payment of interest on the principal outstanding at the start of the period and a partial repayment of the outstanding principal itself. In contrast, an ordinary loan entails the payment of interest at periodic intervals, and the repayment of principal in the form of a single lump-sum payment at maturity. In the case of an amortized loan, the installment payments form an annuity whose present value is equal to the original loan amount. An Amortization Schedule is a table that shows the division of each payment into a principal component and an interest component and displays the outstanding loan balance after each payment.