Assume that your friend Mary is 30 years old and wishes to provide for her retirement. Suppose that she invests Ksh. 50,000 per year at an interest rate of 9% per annum for the next 30 years with the 1st deposit accruing one year from now. At the age of 60 she will tour around the world for five years and on returning to Kenya, she wants to withdraw Ksh. 530,000 per annum for the next 15 years. Assuming that the 9% return remains constant, compute the maximum amount she can consume each year during her world tour. (10 Marks)
Make a time graph, with a focal time at 60 years.
My interpretation:
There will be 30 annual deposits of 50,000, then 5 withdrawals of x followed by
15 withdrawals of 530,000
Pick a focal point at age 60, and reduce all payments by a factor
of 1000 to keep the numbers smaller
amount of her 30 deposits = 50(1.09^30 - 1)/.09 = 6,815.376927 thousands
present value of her 15 withdrawals of 530 at age 60
= 530(1 - 1.09^-15)/.09 (1.09)^-5 = 2,776.614035 thousands
amount available for her at age 60 = 6,815.376927 - 2,776.614035 = 4,038.762892
This becomes the "present value" of 5 annuities of x
x(1 - 1.09^-5)/.09 = 4,038.76...
x = 1,038.335475 thousands or 1,038,335.48 Ksh.
Checking by doing it another way again reducing all monies by a factor of 1000:
let focal time be at age 80 (when she should run out of money)
value of her 30 deposits then = 50(1.09^30 - 1)/.09 (1.09)^20 = 38,196.17184
at age 80, value of her last 15 withdrawals = 530(1.09^15 - 1)/.09 = 15561.2856
value of her travel money at age 80
= 38,196.17184 - 15561.2856
= 22,634.88624
at age 80, value of her 5 withdrawals of x
= x(1.09^5 -1)/.09 (1.09^15) = x(21.79920342)
thus: x(21.79920342) = 22,634.88624
x = 1,038.335475 thousands = 1,038,335.48
One more try: focal point now
pv of deposits = 50(1-1.09^-30)/.09 = 513.6827022
Pv of her final 15 deposits = 530(1 - 1.09^-15)/.09 (1.09^-35) = 209.2765544
pv of her 5 withdrawals of x = x(1 - 1.09^-5)/.09 (1.09)^-30 = x(.293167434)
x(.293167434) + 209.2765544 = 513.6827022
x = 1,038.335475 thousands = 1,038,335.48 , wow! looks like I have the right answer
I dont understand which formulae you used to get the answer for the 15 withdrawals
To compute the maximum amount Mary can consume each year during her world tour, we need to perform the following steps:
1. Calculate the future value of Mary's retirement investments:
Using the formula for compound interest, we can calculate the future value of Mary's annual investments over the 30-year period. The formula is:
FV = P * (1 + r)^n
Where:
- FV is the future value of the investment.
- P is the annual investment amount (Ksh. 50,000 in this case).
- r is the interest rate (9% or 0.09 as a decimal).
- n is the number of years (30 years).
Substituting the values into the formula, we have:
FV = 50,000 * (1 + 0.09)^30
Calculating this gives:
FV = 50,000 * (1.09)^30
2. Calculate the present value of Mary's annual withdrawals:
Using the formula for present value of an annuity, we can calculate the present value of Mary's annual withdrawals over the 15-year period. The formula is:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
- PV is the present value of the annuity.
- C is the annual withdrawal amount (Ksh. 530,000 in this case).
- r is the interest rate (9% or 0.09 as a decimal).
- n is the number of years (15 years).
Substituting the values into the formula, we have:
PV = 530,000 * [1 - (1 + 0.09)^(-15)] / 0.09
Calculating this gives:
PV = 530,000 * [1 - (1.09)^(-15)] / 0.09
3. Calculate the maximum amount Mary can consume each year:
To calculate the maximum amount Mary can consume each year during her world tour, we subtract the present value of her withdrawals from the future value of her investments.
Maximum amount = FV - PV
Substituting the calculated values, we have:
Maximum amount = 50,000 * (1.09)^30 - 530,000 * [1 - (1.09)^(-15)] / 0.09
Calculating this gives us the maximum amount Mary can consume each year during her world tour.