Interest Math Problem?

Kevin turned 45 today. He has an investment account worth $20,000, which is earning 8.5% APR compounded semi-annually. Kevin lives a single life and would like to retire when he turns 60 years old. At the end of this month Kevin will deposit $200 into his investment account. This monthly contribution is expected to continue and grow each month at a rate of 1.0% each year until retirement. When Kevin retires he will no longer contribute and will begin withdrawing $1,000 per week from his investment account until his death.





a) Calculate the effective periodic rates (EPR) required in this problem.





b) How much money will Kevin have at the time of his retirement? (Assume all payments were deposited/withdrawn at the end of each period.)





c) Since we know Kevin's retirement cash flow expectations, how many years will the money last him until he depletes his retirement fund?





d) Draw a detailed timeline and label all components. Use the information from part a, b and c.





e) Assume Kevin did not inform you of the pre-retirement contribution amounts. He expects to die when he is 85 years old. If he still withdraws the same amount of money after retirement, how much fixed monthly amount will he need to contribute during his pre-retirement years?


|||Sorry about the multiple edits of this answer. Yahoo kept putting a bunch of "......." in the areas where the formulas were long. So I had to put carriage returns in those formulas to make them display properly, and it took several views of the final answer before I caught them all.








You've asked a heck of a mouthful in this problem. Are you studying actuary math? I'm an actuary, so I'll use that notation. I kept a lot of decimal points in the answers, to add in the accuracy of the values (since there's a ton of exponential growth, a little bit of error due to rounding accumulates to several dollars off)





a) (1+i)^12 = (1+8.5%/2)^2


i = 0.6961062% (periodic monthly interest rate that is equivalent to 8.5% compounded semi annually)





b) We need to accumulate the $20,000 at 8.5% semi annually for 15 years, and then add to that a geometrically increasing annuity immediate of $200 per month. So first the $20,000....


FV = $20,000 * (1+8.5%/2)^(2*15) = $69,712.70





Now the $200 geometrically increasing annuity immediate....


the actuary formula for calculating this involves a symbol that cannot be typed in this text editor, so I'll spell out the symbol's pronunciation.


First we need to calculuate "j", which is a revised interest rate based on the investment interest and the increasing percentage:





j = (i - e)/(1 + e), where i is the periodic investment rate and e is the periodic increase in payment


So j = (0.6961062% - 1%)/(1 + 1%) = -0.3008849%





Now the actuary formula:


FV = 200*(1+8.5%/2)^(2*15)*[1/(1+1%)]*


(a-angle-n at j%)





The (a-angle-n at j%) is the actuary notation for the present value of a series of cash flows occuring at the end of each period. By using j% intead of i% and multiplying by the 1/(1+1%), it causes the formula to change to an increasing geometric payment.





FV = 200*(3.451123775)*(1-(1+j)^(-n))/j


FV = 200*(3.451123775)*


(1-(1-0.3008849%)^(-15*12))/


(-0.3008849%)


FV = 200*(3.451123775)*(239.3427361)


FV = $165,200.28





So the total balance at age 60 is:


$69,712.70+$165,200.28=$234,912.98





c) We need to solve for the present value of what time length of $1,000 weekly cash flows will be equivalent to $234,912.98. First, we need the equivalent weekly interest rate. I'll assume 52 weeks per year (it is actually either 52 1/7 or 52 2/7, depending on leap year).





(1+8.5%/2)^2 = (1+i)^52


i = 0.16021157% (weekly interest rate that is equivalent to 8.5% compounded semi annually)





$1,000*(a-angle-n at 0.16021157%) = $234,912.98


(a-angle-n at 0.16021157%) = 234.91298


[1-(1+0.16021157%)^(-n)]/0.16021157% = 234.91298


n = 294.9578, or about 295 weeks (5.67 years)





d) can't draw using this editor, but the formulas I gave you should help you do something in Excel





e) The PV of his retirement account at age 60, to support weekly withdrawls of $1,000 until age 85 (25 * 52 = 1,300 withdrawls) is found using:


$1,000*(a-angle-1,300 at 0.16021157%)


=$1,000*[1-(1+0.16021157%)^(-1,300)]/


0.16021157%


=$546,280.84





To accumulate that amount by age 60, using fixed monthly payments beginning at age 45, you need to use the accumulated value function:





(s-angle-n at i) = P*[((1+i)^n)-1]/i


where i is the periodic interest rate (0.6961062%)


P is the monthly fixed payment (which we need to solve for)


n is the total number of payments (n=12*15=180)





P*[((1+0.6961062%)^180)-1]/0.6961062% = $546,280.84


P*357.0769714=$546,280.84


P = $1,529.87|||Kevin foolishly invested periodically into a mutual fund whose fund managers were taught in Business School the rule of thumb of 10% return, and advised Kevin to assume a conservative return of 8.5% return on his investment, but did not liquidate funds in time. Kevin suffered a 50% lossof principal

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