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

$PV= \sum^{\infty}_{k=1}$ $\frac {C}{(1+r)^k}=\frac {C}{r}, >0$

Where PV= Present value

C=Cash flow

r= discount rate.

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

PV of Annuity

$PV= \sum^{n}_{k=1}$ $\frac {C}{(1+r)^k}$ = $C$ $\frac {1-(1+r)^ {-n}}{r}, r>0$

Where:

PV= Present value

C=Cash flow

r= discount rate.

n= number of periods

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

Thus: $999.93<$1000.

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