In order to save for retirement, Debbie makes the first of 10 equal annual depos
ID: 2669270 • Letter: I
Question
In order to save for retirement, Debbie makes the first of 10 equal annual deposits into an investment on Sept. 29, 2011. She believes the investment will earn 10% per yr. The last deposit will be made on Sept. 28 2020. The deposit must be sufficient size to enable debbie to make 30 equal annual withdrawals of 50,000 starting on Sept. 28 2031.What must the size of the deposits be for the above plan to work.Assume that debbie makes the 10 deposits as planned and makes the first two 50,000 withdrawals on sept 28 2031 2032. On Sept 2032 she finds out that the investment has only been yielding 6% since the time of the first deposit. Expecting to live the following the 28 yrs. How much must debbie cut back on her lifestyle.
Whats the maximum constant amount she can withdrawal for each of the remaining 28 yrs. starting in 2033 if the interest rate remains at a true 6%.
Explanation / Answer
This has three parts. We have to work backwards to find the annual cont needed. 1. Debbie will make nper=30 annual withdrawls of PMT=50,000 with Rate i=10%. So what should be PV of this Ordinary annuity on Sep28 2030 as 1st Withdrawl is on 28 Sep 2031. 2. Then we find the PV of the (PVA) found in step 1 above. This will be using disc rate of 10% for n=10 yrs (2030-2020) 3. The PV found in Step 2 above is the FV of Annuity of annual cont PMT for n=10 yrs at Rate =10%. Lets now do the maths :- 1. PV of Annuity at 2030 = PV(Rate,nper,PMT) = PV(10%,30,-50000) = $471,346 2. PV = FV/(1+i)^n = $471,346/(1+10%)^10 = $181,724 3. Annual Payment reqd is PMT=PMT(Rate,nper,PV,FV) = PMT(10%,10,0,-181724) = $ $11,402 ..........................Ans (a) Second Stage : 1. Debbie makes 10 deposits of PMT= $11,402 for nper=10 at Rate=6%. Find FV of this annuity FVA. 2. Then find FV of this FVA of n=10 & i=6%. This will be Lumpsum at 28 Sep31 called FVA. 3. We find FV of this FVA for n=1 at i=6% and then reduce 50000 from it. This is FVB 4. We again find Fv of FVB for n=1 at i=6% then agan reduce 50000 from it. This is FVC 5. Now this FVC is PV of annuity for n=28 yrs at i=6%, We find the annual payment PMT of this money. COme to maths now:- 1. FV of annuity FVA = FV(Rate, nper,-PMT) = FV(6%,10,-11402) = $150,287. This is amt on 28 Sep 2020 2. FV of FVA = PV*(1+i)^10 = $150,287*(1+6%)^10 = $269,141 This is the ant on 28 Sep2030 3. FVB = FVA*(1+i)^1 -50000 = $269,141*1.06 -50000 = $235,289. Bal on 28 Sep2031 4. FVC = FVB*(1+i)^6-50000 = $235,289*1.06-50000 = $199,406. Bal on 28 Sep 2032 5. Now PV of Annuity is FVC= $199,406, i=6%, n=28. PMT=PMT(Rate,nper,PV) ie ANnual withdrawl = PMT(6%,28,199406) = $14,874. So for remainining 28 yrs, she will withdrwa only $14,874 every year for 28 yrs.