Save Early

Many financial advisors would suggest that you save early for your retirement. The

reason is relatively few years of contributions to your retirement plan made early in

your life are equivalent to many more years of contributions made later in your life.

In order to understand this point better, we consider the following exercise. Suppose

there are two investors: A and B. Both plan to retire after T years but save for their

retirement in very dierent ways. Investor A puts $1 into his retirement account at

the beginning of each year for T years (i.e., at t = 0; 1; ...; T- 1). Investor B does

not make any contributions for the rst N years, and try to make it up with more

contributions at the start of each year for the remaining T ????N years, i.e., he will make

contributions at t = N;N + 1; ...T -1.

(a) Suppose Investor B wants to have the same amount of money as Investor A when

both of them retire. What is the annual contribution that Investor B has to make

in the remaining T- N years. Express your answer as a function of r, N and T.

(b) Suppose r = 0.02/year and T = 60 years. Plot the annual contribution that

Investor B has to make in part (a) as a function of N for 0N40 years.

Repeat the same exercise for r = 0.04/year and r = 0.06/year.