Question

Ray is 23 years old. She will retire in 40 years. Ray believes that she will live for 35 years after she retires. In order to live comfortably, she thinks she will need to withdraw $35,000 every month. These withdrawals will be made at the end of each period during retirement.

Ray wishes to establish a scholarship in Toronto. The scholarship will make annual payments. The first payment will be made 5 years after Ray retires. The amount of the first payment will be $50,000. The number of payments will increase by 2% each year. Ray wants the payments to be made in perpetuity.

She currently has $25,000 in their investment account that earns 6% interest compounded semi-annually.

In order to fund her around the scholarship, and her retirement income Ray is prepared to make monthly payments into her RRSP. The monthly payments will be made at the beginning of each month.

Ray expects to earn 8% compounded annually on her RRSP contributions prior to retirement. During retirement, Ray expects to earn 5% compounded annually.

a)             How much ray need when she retires?

b)            How much will Ray have when she retires?

c)             How much will she be short?

d)            What is the amount of the monthly payments Ray must make to fund her retirement and the scholarship?

Answer 1

It is assumed that regarding scholarship, the words "....... number of payments will increase by 2% each year" actually stand for "...amount of payments will...".

Monthly equivalent of interest rates before and after retirement are as follows:

Part (a):

Amount required for scholarship as on retirement is the PV of perpetuity (commencing 5^th year) after 4 years from retirement and further discounted for 4 years.

=(P/(r-g)/(1+r)^4

Where P= first payment (given as $50,000), r= interest rate (5%) and g= growth rate (2%)

=(50000/(5%-2%))/(1+5)^4 = $ 1,666,666.67/1.05^4 = $1,371,170.79

Amount required for retirement withdrawals is the PV of monthly annuity for 35 years= $6,254,956.56

Calculation as below:

Total amount required when Ray retires= $1,371,170.79 + $6,254,956.56= $7,626,127.35

Part (b):

Amount available when Ray retires= Future value of savings

=P*(1+r/t)^(n*t)

Where P= Present savings (given as $25,000), r= interest rate per year (5%), t= number of times compounded a year (2) and n= number of years till maturity (35 years)

=25000*(1+6%/2)^(35*2) = 25000*1.03^70 = $197,945.55

Part (c):

Amount with which she be short= $7,626,127.36 -$197,945.55 = $7,428,181.81

Part (d):

Amount of monthly payments Ray should make to fund the shortfall= $3,444.79 as follows: