By Sunil K. Parameswaran
A level annuity is a series of identical cash flow payments made at equally spaced intervals of time. There are many such examples in real life. Salaries and rent till they are revised. Interest on fixed deposits and coupon payments from fixed rate bonds. Instalments paid on loans such as housing loans, automobile loans, educational loans and personal loans are also examples of annuities.
An N period annuity makes its first payment after one period, and its final payment after N periods. On the other hand, an N period annuity due will make the first payment immediately, that is, at time zero, and its final payment after
N-1 periods. Examples of annuity dues include premiums on life and general insurance policies. As you will be aware, if an insurance policy is taken, the first year’s premium is payable upfront and not after one year. There are also growing annuities, where the payments increase at a rate that is usually assumed to be constant, year after year.
The present value of an annuity due is equal to that of an identical annuity, multiplied by a factor of (1+r), since each cash flow is discounted for one period less. If the future value of an N period annuity and an identical N period annuity due are computed at time N, the latter will have a future value that is greater by a factor of (1+r) since each cash flow is compounded for one period extra. Thus, the future value of an annuity due at N-1, will be equal to that of the annuity at time N.
A perpetuity is an annuity that pays forever. This sounds like a great deal, but cash flows beyond a point contribute insignificantly to the value of such a cash flow stream. If a perpetuity promises to pay Rs 10,000 per year, and the investor wants a rate of return of 8% per annum, he will pay 10,000/.08 = 1,25,000, despite the fact that the payments will never cease.
The future value of a perpetuity obviously cannot be computed since the payments will never stop. The present value of a perpetuity due is the present value of the perpetuity plus the initial cash flow at time zero. In this case it will be 1,25,000 + 10,000 = 1,35,000.
Coupon paying bonds have a related statistic called the duration, which captures their interest rate sensitivity. For plain vanilla bonds the duration is also a measure of the effective time to maturity of the cash flows. Duration can be computed for both annuities and perpetuities. Despite the fact that the cash flow stream is endless, the duration of a perpetuity is (1+r)/r, where r is the periodic interest rate.
The duration of a perpetuity due is 1/r, which is lower than that of the corresponding perpetuity. The reason is that while the first cash flow of a perpetuity due is received immediately, and hence the weight corresponding to its first cash flow is multiplied by zero while computing the average time to maturity, it will have a higher price than the corresponding perpetuity due to the additional cash flow at the outset. Consequently, the weights attached to each cash flow are lower than in the case of an equivalent perpetuity, and hence the duration is lower.
The writer is CEO, Tarheel Consultancy Services