This question relates to the time value of money and deferred annuities. Ruth Br
ID: 2329054 • Letter: T
Question
This question relates to the time value of money and deferred annuities.
Ruth Bray is age 42 today and plans to retire on her 63rd birthday. With future inflation, Ruth estimates that she will require around $1,600,000 at age 63 to ensure that she will have a comfortable life in retirement. She is a single professional and believes that she can contribute $3,700 at the end of each month, starting in one month’s time and finishing on her 63rd birthday.
a. If the fund to which she contributes earns 4.8% per annum, compounded monthly (after tax), how much will he have at age 63? Will she have achieved her targeted sum? What is the surplus or the shortfall?
b. Using the entire fund balance, Ruth then wishes to commence a monthly pension payable by the fund starting one month after her 63rd birthday, and ending on her 87th birthday, after which she expects that the fund will be fully expended. If the fund continues to earn the above return of 4.8% per annum, compounded monthly, how much monthly pension will Ruth receive, if the fund balance reduces to zero as planned after the last pension payment on her 87th birthday?
Explanation / Answer
a. Since age of Ruth is 42 Years and savings ends with 63rd Birthday, there is 21 Years (252 months) now using FVA we can find the fund value at 63rd birthday
FVA = A x [(1+r)^n-1]/r
A = 3700, r = 0.4%(4.8%/12), n = 252
FVA = 3700 x [(1+0.004)^252-1]/0.004
FVA = 1,604,515.59
Yes, Her target is achieved with surplus of 4,515.59
b. retirement period is 87-63, 24 years(288 months)
pension amount will be calculated using this formula
PVA = A x [1-(1+r)^-n]/r
PVA = 1,604,515.59, r = 0.4%, n = 288, A = ?(Pension amount)
1604515.59 = A x[1-(1+0.004)^-288]/.004
1604515.59 = A x[170.81722]
A = 1604515.59/170.81722 => $9,393.17
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