Perpetuity

Study guide: Exam FM-2

A perpetuity is an annuity that pays 1 per period forever. For a perpetuity-immediate (payments at end of each period) the present value is obtained by letting n \to \infty in a_{\overline{n}|} = (1-v^n)/i; since v^n \to 0:

a_{\overline{\infty}|} = \frac{1}{i}

• For a perpetuity-due (payments at the start of each period), each payment comes one period earlier than under the perpetuity-immediate, so its value is (1+i)\,a_{\overline{\infty}|}:

\ddot{a}_{\overline{\infty}|} = \frac{1}{d}

= \frac{1+i}{i}

• The relationship \ddot{a}_{\overline{\infty}|} = 1 + a_{\overline{\infty}|} holds because the perpetuity-due can be seen as an immediate payment of 1 followed by a perpetuity-immediate.
• Perpetuities model payments meant to continue indefinitely, such as a fixed dividend on a stock or the payout of an endowment fund.

Endowment Fund Perpetuity A university endowment must pay $50{,}000 per year in perpetuity, with the first payment one year from now. If the fund earns i = 4\% per year, how much must be deposited today?

Answer This is a perpetuity-immediate, so: PV = \frac{50{,}000}{i} = \frac{50{,}000}{0.04} = \$1{,}250{,}000