Question

Derive a = [1-(1+i)^-n]/i as the difference between the discounted value of an ordinary simple perpetuity of $1 per period and the discounted value of an ordinary simple perpetuity of $1 per period deferred for n periods.

Answer 1

Interest Rate = i%

Present Value of Ordinary Simple Perpetuity of $1 = 1/(1+i) + 1/(1+i)^2 + 1/(1+i)^3 + ....
Present Value of Ordinary Simple Perpetuity of $1 = 1 / i

Present Value of Ordinary Simple Perpetuity of $1 deferred for n period = 1/(1+i)^(n+1) + 1/(1+i)^(n+2) + 1/(1+i)^(n+3) + ...
Present Value of Ordinary Simple Perpetuity of $1 deferred for n period = [1/(1+i)^n] * [1/(1+i) + 1/(1+i)^2 + 1/(1+i)^3 + ....]
Present Value of Ordinary Simple Perpetuity of $1 deferred for n period = [1/(1+i)^n] * [1 / i]

a = Present Value of Ordinary Simple Perpetuity of $1 - Present Value of Ordinary Simple Perpetuity of $1 deferred for n period
a = [1 / i] - [1/(1+i)^n] * [1 / i]
a = [1 / i] * [1 - (1/(1+i)^n]
a = [1 / i] * [1 - (1+i)^(-n)]
a = [1 - (1+i)^(-n)] / i