Assume you plan to put money in an investment at the end of each year for 25 yea

Question

Assume you plan to put money in an investment at the end of each year for 25 years. You plan to increase your annual contribution by 4% annually (a growing annuity). In 25 years, you plan to use your accumulated savings to buy an annuity that will pay you $100,000 at the beginning of every year fow 20 years. Use a 12% discount rate to estimate how much your first (year 1) contribution must be in order for you to adequately finance your retirement.


NOTE: I would like the answer in two ways. First, I would like to use my BA II Plus TVM to calculate the answer so I would need to see what goes into N, I/Y, PV, PMT and FV. Would also like to see the mathematical way to solve this problem.

Explanation / Answer

The soln has 2 parts (Sort of reverse engg) :- 1. Find the PV of Annuity DUe of 20 Yrs which has PMT=$100,000, i=12%, n=20. 2. The AMount found in step 1 above is the FV of a growing annuity of term n=25, g=4%, i=12%. Step 1: Present value of annuitydue PVA = PMT(PVIFAi,n)*(1+i). So PVA = PMT*(1/i - 1/(i(1+i)^n))*(1+i) So we get PVA = PMT*(1/12% -1/(12%*(1+12%)^20))*(1+12%) ie PVA = PMT*(8.3333-0.8639)*1.12 = PMT*8.3657 = $100,000*8.3657 = $836,570 So PVA1 = $836,570 Step 2: FVA = $836,570, A = Initial Payment FVA = A*[(1+i)^n - (1+g)^n]/(i-g) ie FVA = A*[(1+12%)^25 - (1+4%)^25]/(12%-4%) ie FVA = A*[17 - 2.6658]/8% = 179.1775A so A = FVA/179.1775 = 836,570/179.1775 = $4,668.95 So I will make an initial payment of $4668.95 & every year I will increase it by 4% to get a Lump sum of $836,570 t end of 25 Years. I will then get an annual ayment of $100,000 for next 20 yrs.

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