Which of the following expressions correctly represents the present value of a perpetuity paying C per period at rate r?

• A. PV = C * r
• B. PV = C / (1 + r)
• C. PV = C / r^2
• D. PV = C / r

Answer: (D)

The key idea is to sum an infinite series of payments that are discounted each period. If a perpetuity pays C every period starting one period from now, its present value is the sum C/(1+r) + C/(1+r)^2 + C/(1+r)^3 + ... This is a geometric series with first term a = C/(1+r) and ratio q = 1/(1+r). The formula for such a sum is a/(1−q), which gives [C/(1+r)] / [1 − 1/(1+r)] = [C/(1+r)] / [r/(1+r)] = C/r. So the present value is C divided by the discount rate r.

This relies on r being the per-period discount rate and the payments beginning next period. If the payments started today, the value would be different (a perpetuity due). The other expressions don’t match the infinite discounted sum: multiplying by r, or discounting only one period, or using r^2, all fail to capture the full series.