Cash Flow Evaluation. You want to retire in 20 years from today and then start m
ID: 2654639 • Letter: C
Question
Cash Flow Evaluation. You want to retire in 20 years from today and then start making withdrawals from your savings every year, with the first withdrawal coming one-year after retirement (or 21 years from today) for 25 total annual withdrawals (at 21 years through 45 years from now, annually). You want your annual withdrawals to be $75,000 per year. Additionally, you want to take a world tour in 25 years from now when you turn 70, with an additional cost of $35,000 that would occur exactly 25 years from today.
-You assume that you can earn a nominal APR rate of 4%, with monthly compounding, on your savings over the next 45 years.
a) How much money must you have in your retirement savings account at the beginning of your retirement (20 years from now) to fund this future retirement withdrawal stream, assuming that you make no new deposits after retirement.
b) Assume that you want to save up the sum calculated in a) above. If you make monthly deposits into your retirement account, with the first deposit made one-month from today and the last deposit made just as your retire (or 20 years from today for 240 total monthly deposits), then how much must you deposit monthly to fund this retirement? (such that your retirement account will have an approximate balance of zero after the last $60,000 withdrawal at time-45).
Explanation / Answer
(a) For solving this question,we apply present value of annuity concept.
Step1: Nominal APR rate is the simple interest rate for the year without monthly compounding effect.We have to calculate effective interest rate.We have,
Effective interest rate for monthly compounding = (1 + i/n)n -1
Where, i = Nominal APR = 4%
n = Monthly compounding = 12
Putting the value in the above formula.We have,
Effective interest rate for monthly compounding = (1 + 0.04/12)12 - 1 = (1.003)12 -1 =0.0407 * 100 = 4.07 %
Hence, effective interest rate is 4.07%.
Step2: Computation of present value of an annuity.We have,
Present value of an annuity = Installment [ (1+r)n- 1 / r(1+r)n ]
where,
r = interest rate = 4.07%
I = Installment = $ 75,000
n = 25
Putting the value in above formula.we have,
Present value of an annuity = 75,000 [ (1.0407)25 - 1 / 0.0407(1.0407)25 ]
Present value of an annuity = 75,000 [ 1.71106 / 0.11034 ] = 75,000 x 15.507 = $ 1,163,036.19
Hence, present value of total 25 annual installment is $ 1,163,036.19.
Step3: Computation of present value of world tour expenses.We have,
Present Value of world tour expenses at 20 years from now = 35,000 / (1.0407)5 = 35,000 x 0.8192 = $ 28,672
Hence, Present Value of world tour expenses at 20 years from now is $ 28,672.
Step 4: Money must have in retirement savings account at the beginning of retirement (20 years from now) to fund this future retirement withdrawal stream = 1,163,036.19 + 28,672 = $ 1,191,708
(b) For calculating this question, we apply future value of an annuity concept.We have,
Step1: Effective rate of interest = 4.07%
Step2: Future value of annutiy for monthly installment = [(1+r/m)nxm -1 / r ]
Where,
r = rate of interest = 4.07 %
m = Number of installment = 12
n = Number of years = 20
Putting the value in above formula.We have,
Future value of annutiy for monthly installment = I [( 1+0.0407/ 12)12x20 - 1 / 0.0407/12]
Future value of annutiy for monthly installment = I [1.35523 / 0.00339 ] = I x 399.576
Hence, Installment (I) = 1,191,708 / 399.576 = $ 2,982.43
Hence, the installment must be deposit monthly in retirement account is $ 2,982.43 for 20 years.
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