Cash Flow diagram problem! Econ! Please HELP!!?

Hi all,





I have this problem on which I am kind of stuck! Any help or guidance will be appreciated!





A man is planning to retire in 25 years. He wishes to deposit a regular amount every


three months until he retires, so that, beginning one year following his retirement, he will


receive annual payments of $60,000 for the next 10 years. How much must he deposit if


the interest rate is 6% per year compounded quarterly?





Notes: 1) You must use the effective interest rate where


appropriate. 2) Assume that the first deposit occurs 3 months from today. 3) Use an interest


rate of 6% per year compounded quarterly over the entire analysis period.





So I did r = [ (1+i/n)^n ] - 1 where n is how many times it is compounded in one year


since it's compounded quarterly, so n=4


so you get r=[ (1+0.06/4)^4]-1 = 6.136%





you know what starting year 26, the person withdraws 60,000 for 10 years, so for year 26 through 36, n=10, A=60,000, and with that you can find the Present value, which is 60,000(P/A,0.6136,10).





once you get that present value, it will be the future value of years 0 through 25. And since the person deposits once every 3 months, or 4 times a year, you will have and can solve:





F(A/F, 0.6168,25), where F is the present value calculated in the previously. And whatever answer you get, divide that by 4 since it's deposited 4 times a years





But I don't know if I am going the right way or not! PLEASE HELP!!!|||What I would do is the following:


To solve the problem you must use the "Cash Additivity Principle", using year 25 (quarter 100) as pivot point. The future value of the quarterly savings must equal the present value of the yearly distributions. In mathematical form:





(1) FVa = PVd. where a = accumulation, d= distribution





(2) The deposits are made every 3 months, so, you have to use the periodic rate. Since the stated rate (6%) is defined as the periodic rate divided by the frecuency, then, the quarterly rate for the accumulation period is Ra = 0.06 / 4 = 0.015. At the other hand, for the distribution period, since the distributions are yearly you must calculate the effective annual rate of 6% compounded quarterly. That rate is Rd = (1+ 0.06/4)^ 4 -1 = 6.14%.





(3) Since the deposits occurs at year end you must use the formula for the future value of an ordinary annuity to calculate the FV of the deposits at the pivot point. That is: FVa = Aa/Ra [ (1+Ra)^Na -1 ]. Na is the number of quarters, that is: Na = 25 years * 4 quarters per year = 100 quaters. The only known in the formula is Aa, the quarterly deposits.





(4) To calculate the Present Value of the withdraws after retirement, assuming withdraws occurs at year end, one year after retirement, you must use the PV of an ordinary annuity. That is: PVd = Ad/Rd [(1 + Rd)^(-Nd) -1]. In this case, Nd, is the number of years. Nd= 10. Since the rate is quoted compounded quaterly, you must use the effective annual rate of 6% compounded quarterly, that is Rd = 6.14%, which we calculated in step 2.





Now, using equation 1:





FVa = PVd





Aa/Ra ( (1+Ra)^Na -1) = Ad/Rd ( (1 + Rd)^(-Nd) -1), you must isolate Aa





(5) Aa = { Ad/Rd [ 1 - (1+Rd)^(-Nd) ] ( Ra) } / [ (1+ Ra)^Na - 1] Where:





Ad = 60,000


Rd = 0.0614


Nd = 10


Ra = 0.015


Na = 100





Putting the values in formula (5) you should get Aa = $1,917.33