Question

This problem is similar in spirit to Example 12 (in the chapter) and Problem 15 (at the end of the chapter). I'd strongly suggest that you master those two problems before attempting this problem. Make sure that you draw a high quality, detailed timeline – similar in quality to those in Example 12 and Problem 15. Assume that you wish to begin saving for your child’s college education via making deposits into an investment account that is expected to earn 8% per year for the first 14 years. After year 14, you will place the money in a less risky investment account that is expected to earn only 5% per year, for as long as you have money in the account. You currently have $5,000 available, and you will deposit that amount into the savings account today. Thereafter you have decided to make savings deposits 3, 4, 5, … , 9 and 10 years from today. Each of these deposits will be larger than the prior deposit by 7%. You’ve estimated that one year of college will cost $52,000 in 18 years, and that the three subsequent years will each cost 5% more than the prior year. Determine the size of the first deposit required at time 3. Hint: The answer is NOT $9475.37. If you got this, you did not deal with the $5000 at t = 0 at all. Hint: The correct answer is between $8600 and $8700.

Answer 1

First, set-up the time-line as follows:

Beginning of Year	Starting Money	Add-on Investment	Cash-out-flow	Rate	Interest	Ending Money
0	This will be equal to 5000 for starting point. There-after it will be equal to what-ever was the closing value of the last year.	From beginning of year 3, till year 10, there will be value here. Assume value in year 3 is "X" and there-after, increase figure by 7% each year till year 10	This will not have any value till year 17. For year 18, put the value 52,000 (tuition payments are due beginning of the year)	This will be 8% for first 14 years after which it will be 5%	This will be equal to (Opening money + add-on invest - cash outflow) * rate (Assumption is that the add-on investment will be done at the beginning of the year)	This will be equal to Opening money + add-on investment - cash out-flow +interest

Once the above table is set-up, you can solve for X either algebraically. The assumption will be that at the end of the college, the fund value in the account will be zero.

You can also use any type of spread-sheet to solve this problem. In that case, for year 3's add-on investment, assume the investment to be $8650 (average of the range of the answer). Then use a tool like "goal seek" to set the closing fund-value as zero and change the figure of investment in year 3.

The answer will be $8,662 investment required in year 3. The full calculation is shown below.

Beginning of Year	Starting Money	Add-on Investment	Cash-out-flow	Rate	Interest	Ending Money
0	5000			8%	400	5400
1	5400			8%	432	5832
2	5832			8%	467	6299
3	6299	8662		8%	1197	16158
4	16158	9269		8%	2034	27460
5	27460	9917		8%	2990	40368
6	40368	10612		8%	4078	55058
7	55058	11354		8%	5313	71725
8	71725	12149		8%	6710	90584
9	90584	13000		8%	8287	111870
10	111870	13910		8%	10062	135842
11	135842			8%	10867	146710
12	146710			8%	11737	158446
13	158446			8%	12676	171122
14	171122			5%	8556	179678
15	179678			5%	8984	188662
16	188662			5%	9433	198095
17	198095			5%	9905	208000
18	208000		52000	5%	7800	163800
19	163800		54600	5%	5460	114660
20	114660		57330	5%	2866	60196
21	60196		60197	5%	0	0

Do let me know in case of any questions.