An annuity consists of a series of payments, starting at time t = 1 period and ending at time t = n periods. This annuity will have n payments. We assume that each of the payments is 1. The annuity will therefore have n equal payments of 1, each one period apart from the previous and from the next payment, and starting in one period. For payments unequal to 1, simply multiply the present value of the annuity by the amount of the payment.
The present value of each payment, made at time j, will be vj, with j = 1, . . . , n, and vj indicating v raised to the power j, equal to the present value of the payment of 1 due at time t = j.
Therefore, the present value of the annuity will be an = v1 +... + vn (summing the present value of the n payments of 1)
Using the algebraic factorization
(an - bn) = (a - b)(an-1 + an-2b +... + abn-2 + bn-1) If a = 1, and b = a, and dividing through by (a - b) (a n b and 1 n v), then
(1 + v +... + vn 1 ) = (1 - vn )/(1 - v) Equation 5.3
Substituting into Equation. 5.2, we have v(1 - vn )
(Note: an is called a angle n and indicates an annuity certain with n payments.) Substituting v = 1/(1 + i), we have
Multiplying numerator and denominator by (1 + i) (i n — 1), we obtain
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