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 S9475.37. 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.
help!! I know this is technically two problems but I ran out of question so please help if you can. I don't have anymore questions left!

Answer 1

All values in $	Yr 0	Yr 1	Yr 2	Yr 3	Yr 4	Yr 5	Yr 6	Yr 7	Yr 8	Yr 9	Yr 10	Yr 11	Yr 12	Yr 13	Yr 14	Yr 15	Yr 16	Yr 17	Yr 18	Yr 19	Yr 20	Yr 21
Deposit/Outflow	5,000			8,662	9,269	9,917	10,612	11,354	12,149	13,000	13,910								-52,000	-54,600	-57,330	-60,197
Amount in Bank	5,000	5,400	5,832	14,961	25,426	37,378	50,979	66,412	83,874	103,584	125,780	135,842	146,710	158,446	171,122	179,678	188,662	198,095	156,000	109,200	57,330	0

Above is a summary of how money will flow in this example.

We need to note that deposit made in year 0 is $5,000. Our total requirement is present value of outflows ($52,000 in Year 18, $54,600 in Year 19, $57,330 in Year 20 & $60,197 in Year 21).

To calculate outflows we will need to multiply each year's outflow by 1.05 due to annual 5% increase. For example) Year 19 Outflow = Year 18 Outflow * 1.05

Now we will need to discount these outflows to present value.

Hence, present value of Year 18 outflow = Year 18 Outflow / (1.08^14) * (1.05^4) =$14,565{This is done as amount is compounding by 8% for 14 years & by 5% for 4 years)

Similarly other present values will be:

present value of Year 19 outflow = Year 19 Outflow / (1.08^14) * (1.05^5) = $14,565

& so on. Basically, we will discount next year's value by 5% more. These 4 values will come out to be the same as Year 19 outflow is 5% more than year 18 value and then being discounted by the same number.

Hence, total outflow's present value = $58,260

Current cash in hand = $5,000

Hence, present value of future deposits = $53,260

Let deposit in Year 3 be X.

Then, Total Inflow = X/(1.08^3) + X*(1.07)/(1.08^4)+.....+X*(1.07^7)/(1.08^10) = $53,260

{Each Year's inflow will need to be discounted by 8% to the power of the year number. Deposit would be increasing by 7% annually hence, multiplied by a factor of 7% every year}

Solving the X we get 3rd deposit value to be $8,662.