Question

Millhouse graduated 5 years ago with a degree in business administration and is currently employed as a middle level manager for the same firecracker company his dad already worked for. His current annual salary of $60,000 has increased at an average rate of 5% per year and is projected to increase at that rate for the future. The firm has had a voluntary retirement savings program in place, whereby, employees can contribute up to 11% of their gross annual salary (up to a maximum of $11,000 per year) and the company will match every dollar that the employee contributes. Unfortunately, Millhouse did not listen to his finance instructor (which is understandable, because you can’t really trust those Germans) and has not yet taken advantage of the retirement savings program. He opted instead to buy a new car, rent an expensive apartment and go out to Moe’s every night. However, with wedding plans on the horizon, Millhouse has finally come to the realization (with the help of his fiancée Lisa) that he had better start putting away some money. Millhouse figures that the two largest expenses down the road would be those related to the wedding and down payment on a house. He estimates that the wedding, which will take place in twelve months, should cost about $10,000. Furthermore, he plans to move into a $200,000 house in 5 years and would need 20% for a down payment. Millhouse knows that an automatic payroll deduction is probably the best way to go since he is not a very disciplined investor.

1) Had Millhouse availed of the company’s voluntary retirement plan up to the maximum every year for the past five years, how much money would he currently have accumulated in his retirement account, assuming monthly deposits and a rate of return of 6% with interest compounded monthly? How much more would he have, had he opted for a higher risk alternative which was expected to yield 10%, compounded monthly?

Answer 1

5 years ago.Salary was $ 60000 annually , the per month salary is $5000 and the total contribution (including employer) would be 22% i.e. $ 1100. This would increase eveery 2nd year by 5% . The 5th year contribution by the employer would b e

= 11% of annual salary = 11% of 60000 *(1.05)⁴ =$8022.34

As this does not exceed $ 11000 the company would contine to pay 11% of salary

Hence , we have to find the future value of all the contributions for last 5 years (60 months) at 6% monthly compounded i.e. 0.5% per month)

FV of first contribution = 1100 * (1+0.005)⁵⁹   (assuming end of month contribution so compounding period is 59 months)

FV of second contribution = 1100 * (1+0.005)⁵⁸

and so on till

FV of 12th contribution = 1100 * (1+0.005)⁴⁸

FV of 13th contribution = 1100 * 1.05 * (1+0.005)⁴⁸

and so on

Adding

FV = 1100 * (1.005⁵⁹+ 1.005⁵⁸ + .... 1.005⁴⁸) + 1100 *1.05 * (1.005⁴⁷+ ... 1.005³⁶) +..... + 1100*1.05⁴ * (1.005¹¹+....+1)

Writing the 12 month terms in reverse and Applying the sum of GP formula

= 1100 * 1.005⁴⁸ * (1.005¹²-1)/(1.005-1) + 1100*1.05 *1.005³⁶ * (1.005¹²-1)/(1.005-1) +... + 1100*1.05⁴ * (1.005¹²-1)/(1.005-1)

= 17239.42 +17049.79+16862.26+16676.78+16493.35

= $84321.60

Millhouse would have accumulated $84321.60 by now had he followed the company policy

If Millhouse had opted for a higher risk alternative of 10% , we can forget his company's contribution and only his contribution of 550 per month would be present . Monthly interest rate = 10/12 = 0.83333%

So , by the same formula as above , the amount accumulated by now would have been

FV = 550 * 1.008333⁴⁸ * (1.008333¹²-1)/(1.008333-1) + 550*1.05 *1.008333³⁶ * (1.008333¹²-1)/(1.008333-1) +... + 550*1.05⁴ * (1.008333¹²-1)/(1.008333-1)

= 10293.02 +9783.24+9298.70+8838.16+8400.44

= $46613.56

Millhouse would have accumulated $46613.56 by now had he followed the higher risk option