Question

Your client is deciding between two investment choices: one that pays $100 per year in perpetuity, and another that pays $100 per year for 100 years. The current market interest rate for investments of similar risk is at 10% p.a. What is the present value of these two investments? Are they similar? Explain.

Answer 1

The present value of a perpetuity

$$PV=\displaystyle\sum_{k=1}^\infin\dfrac{C}{(1+r)^k}=\dfrac{C}{r}, r>0$$

where:

PV=present value

C=cash flow

r=discount rate

$$PV=\dfrac{\$100}{0.1}=\$1000$$

Present value of annuity

$$PV=\displaystyle\sum_{k=1}^n\dfrac{C}{(1+r)^k}=C\dfrac{1-(1+r)^{-n}}{r}, r>0$$

where:

PV=present value

n=number of periods

C=cash flow

r=discount rate

$$PV=\$100\cdot\dfrac{1-(1+0.1)^{-100}}{0.1}=\$999.93<\$1000$$

The present value of a perpetuity is greater than the present value of annuity.