Question #70454

Suppose there are two investors: A and B. Both plan to retire after T years but save for their
retirement in very different 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 first 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 0 ≤ N ≤ 40 years.
Repeat the same exercise for r = 0.04/year and r = 0.06/year.

Expert's answer

a) Amount of Investor A = 1*(1+r)0+…+1*(1+r)T-1;

Amount of Investor B = contributions * ((1+r)N+…+(1+r)T-1).

Then Contributions of Investor B = (1*(1+r)0+…+1*(1+r)T-1 )/((1+r)N+…+(1+r)T-1),

if Amount of Investor A = Amount of Investor B.

b) Contributions of Investor B = (1*(1+r)0+…+1*(1+r)T-1 )/((1+r)N+…+(1+r)T-1)= (1*(1+0,02)0+…+1*(1+0,02)60-1 )/((1+r)40+…+(1+r)60-1), if r = 0,02.

If r = 0,04, Contributions of Investor B = (1*(1+0,04)0+…+1*(1+0,04)60-1) / ((1+0,04)40+…+(1+0,04)60-1).

If r = 0,06, Contributions of Investor B = (1*(1+0,06)0+…+1*(1+0,06)60-1) / ((1+0,06)40+…+(1+0,06)60-1).

Amount of Investor B = contributions * ((1+r)N+…+(1+r)T-1).

Then Contributions of Investor B = (1*(1+r)0+…+1*(1+r)T-1 )/((1+r)N+…+(1+r)T-1),

if Amount of Investor A = Amount of Investor B.

b) Contributions of Investor B = (1*(1+r)0+…+1*(1+r)T-1 )/((1+r)N+…+(1+r)T-1)= (1*(1+0,02)0+…+1*(1+0,02)60-1 )/((1+r)40+…+(1+r)60-1), if r = 0,02.

If r = 0,04, Contributions of Investor B = (1*(1+0,04)0+…+1*(1+0,04)60-1) / ((1+0,04)40+…+(1+0,04)60-1).

If r = 0,06, Contributions of Investor B = (1*(1+0,06)0+…+1*(1+0,06)60-1) / ((1+0,06)40+…+(1+0,06)60-1).

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