Which equation represents the present value of a perpetuity with payment PMT and discount rate r?

Which equation represents the present value of a perpetuity with payment PMT and discount rate r?

• A. PV = PMT / (r - g)
• B. PV = PMT / r
• C. PV = PMT * r
• D. PV = PMT

Answer: (B)

When payments are fixed and continue forever, the present value is found by summing the discounted payments: PMT/(1+r) + PMT/(1+r)^2 + PMT/(1+r)^3 + ... This is a geometric series with ratio 1/(1+r). The sum of an infinite geometric series gives PMT/(1+r) ÷ (1 - 1/(1+r)) = PMT/r. So the present value is PMT divided by the discount rate. This applies under the ordinary perpetuity assumption where payments occur at the end of each period. If payments grew at rate g, you’d use PMT/(r - g). If the payment started today (a perpetuity due), the value would be higher, specifically PMT * (1 + 1/r). The other expressions don’t reflect the fixed, infinite, end-of-period payments.

When payments are fixed and continue forever, the present value is found by summing the discounted payments: PMT/(1+r) + PMT/(1+r)^2 + PMT/(1+r)^3 + ... This is a geometric series with ratio 1/(1+r). The sum of an infinite geometric series gives PMT/(1+r) ÷ (1 - 1/(1+r)) = PMT/r. So the present value is PMT divided by the discount rate. This applies under the ordinary perpetuity assumption where payments occur at the end of each period. If payments grew at rate g, you’d use PMT/(r - g). If the payment started today (a perpetuity due), the value would be higher, specifically PMT * (1 + 1/r). The other expressions don’t reflect the fixed, infinite, end-of-period payments.