In: Finance
A woman turns 30 today and wishes to withdrawal a perpetuity of $1000 at the beginning of each month starting on her 65th birthday. Starting today, she makes monthly contributions of X to a fund. The last contribution is one month before her 65th birthday. The fund earns a nominal rate of 8% compounded monthly during contribution period and 5% compounded monthly during withdrawal period.
Calculate X.
The current age is 30 years and the age is retirement is 65 years. Thus, the remaining working period is 35 years. The deposits are made at the end of the month and the last deposit one month before the retirement age. Hence, the total number of deposits is 419.
Compute the monthly interest rate during the contribution period, using the equation as shown below
Monthly interest = Annual interest/ 12 months
= 8%/ 12
= 0.66666666666%
Hence, the monthly interest rate is 0.6666666666%.
Compute the monthly interest rate during the withdrawal period, using the equation as shown below
Monthly interest = Annual interest/ 12 months
= 5%/ 12
= 0.41666666666%
Hence, the monthly interest rate is 0.4166666666%.
Compute the value of withdrawals at the end of the last deposit, using the equation as follows:
Value of withdrawals = Monthly withdrawals/ Rate of interest
= $1,000/ 0.4166666666%
= $240,000
Hence, the value of withdrawals is $240,000.
Compute the present value of withdrawals, using the equation as shown below:
Present value = Value of withdrawals/ (1 + Monthly rate)Time
= $240,000/ (1 + 0.00666666666)419
= $240,000/ 16.1846521714
= $14,828.8636331
Hence, the present value of deposits is $14,828.8636331.
Compute the present value annuity factor, using the equation as shown below:
PVIFA = {1 – (1 + Rate)-Number of periods}/ Rate
= {1 – (1 + 0.00666666666)-419}/ 0.6666666666%
= 140.73196037
Hence, the present value annuity factor is 140.73196037.
Compute the monthly contrition to fund, using the equation as shown below:
Monthly contribution = Present value of deposits/ Present value annuity factor
= $14,828.8636331/ 140.73196037
= $105.369552119
Hence, the monthly contribution (X) to fund is $105.369552119.