Present value of a perpetuity paying C per period at rate r?

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

Answer: (C)

The key idea is valuing a never-ending stream of payments by summing a geometric series. Each period you receive C and these payments are discounted back t periods at rate r, so the present value of the payment at period t is C/(1+r)^t. Summing from t = 1 to infinity gives C times the infinite sum of (1/(1+r))^t. That sum equals C/r, so the present value is C/r, assuming payments start one period from now (a standard perpetual annuity).

If the payments started immediately, the value would be higher, specifically C(1 + r)/r, but that isn’t the case here. The other expressions don’t reflect summing an infinite geometric series: they either don’t capture the full stream, or don’t converge to the correct perpetuity value.