Question #70427

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.

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) The one year spot rate r1: 1,08/(1+r1)= $1018.772/$1000=> r1=6,01%

The two year spot rate r2: 1/(1+r2)2=$907,209/$1000=> r2=3,11%

The two year spot rate r3: $50*(1+r3)3= $136.967=> r3=40%.

b) The one-two year forward rate r0,1: 1,08/(1+r1)= $1018.772/$1000=> r1=6,01%.

The one-two year forward rate r1,2: 0/(1+r1)+1/(1+r1)(1+r1,2)=0,907209=>r1,2≈4%.

The one-two year forward rate r2,3: 0,4/(1+r1)+0,4/(1+r1)(1+r1,2)+1,4//(1+r1)(1+r1,2)(1+r2,3)= 0,136967=>r2,3=10%.

The two year spot rate r2: 1/(1+r2)2=$907,209/$1000=> r2=3,11%

The two year spot rate r3: $50*(1+r3)3= $136.967=> r3=40%.

b) The one-two year forward rate r0,1: 1,08/(1+r1)= $1018.772/$1000=> r1=6,01%.

The one-two year forward rate r1,2: 0/(1+r1)+1/(1+r1)(1+r1,2)=0,907209=>r1,2≈4%.

The one-two year forward rate r2,3: 0,4/(1+r1)+0,4/(1+r1)(1+r1,2)+1,4//(1+r1)(1+r1,2)(1+r2,3)= 0,136967=>r2,3=10%.

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