Question

A person A contributes to an RRSP at the age of 20. She deposits $ 3,000 per year for 10 years, but no longer makes any deposits afterwards. A Person B waits until age 30 to contribute to an RRSP. She deposits $ 3,000 per year for 30 years. There is no initial investment in neither cases. (Suppose the interest is compounded continuously.)

a) Assume a rate of return of 8%. What the RRSP balance at age 60 for persone A and B?

b) For a constant rate of return r but not specified, determine the RRSP balance of each at the age of 60 based on r.

c) Draw the graph of the difference between the two balances from what you obtained in part b) for 0 <= r <= 0.10.

d) determine the rate of return for which the two RRSPs will have an equal value at the age of 60 years old.

Answer 1

a) For person A , there are 10 deposits, 1st deposit earning interest for 40 years , 2nd for 39 years and so on

Final balance at the age of 60 (if the interest rate is r)

= 3000* (exp(40*r)+exp(39*r)+...+exp(31*r))

=3000*exp(31*r) * (1+exp(r)+exp(r*2)+...+exp(r*9))

=3000*exp(31*r)* (exp(r*10) -1)/(exp(r)-1)

For r=8%

=3000*exp(31*0.08)*(exp(10*0.08)-1)/(exp(0.08)-1)

=$527135

For person B , there are 30 deposits, 1st deposit earning interest for 30 years , 2nd for 39 years and so on

Final balance at the age of 60 (if the interest rate is r)

= 3000* (exp(30*r)+exp(39*r)+...+exp(1*r))

=3000* exp(r)* (1+exp(r)+exp(r*2)+...+exp(r*29))

=3000*exp(r)* (exp(r*30) -1)/(exp(r)-1)

For r=8%

=3000*exp(0.08)*(exp(30*0.08)-1)/(exp(0.08)-1)

=$391104

b) For constant rate of return r but not specified, the RRSP balance of A =3000*exp(31*r)* (exp(r*10) -1)/(exp(r)-1)

the RRSP balance of A =3000*exp(r)* (exp(r*30) -1)/(exp(r)-1)

c) The graph is as follows

d) From the graph it can be seen that the rate at which balance of A and B are equal at 60 (Difference is 0) is somewhere between 6% and 7%

Using Solver , it can be found that the rate = 6.09378% for which the two RRSPs will have an equal value at the age of 60 years old.