alt="StartLayout 1st Row 1st Column Blank 2nd Column PVIFA left-parenthesis 8 comma 25 right-parenthesis equals 10.6748 2nd Row 1st Column Blank 2nd Column PVIFA Subscript upper A upper D Baseline left-parenthesis 8 comma 25 right-parenthesis equals 10.6748 times 1.08 equals 11.5288 EndLayout"/>
Thus the present value of the annuity due is:
Future Value
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.
EXAMPLE 2.19
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:
PERPETUITIES
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.
EXAMPLE 2.20
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 METHOD
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.
Take the case of a loan which is repaid in N installments of $A each. We will denote the original loan amount by L, and the periodic interest rate by r. Thus this is an annuity with a present value of L, which is repaid in N installments.
The interest component of the first installment
The
