Time Value of Money Problem

“A client plans to send a child to college for four years starting 18 years from now. Having set aside money for tuition, she decides to plan for room and board also. She estimates these costs at $20,000 per year, payable at the beginning of each year, by the time her child goes to college. If she starts next year and makes 17 payments into a savings account paying 5 percent annually, what annual payments must she make?”

The answer to this question assumed that the 20,000 payments would take place at the end of each year, starting at the end of year 18. I assumed the payments would begin at the beginning of year 18. Did I read the question wrong?

Nope, 18 years from now means at beggining of year 19.

Set an easy example:

1 year from now means at the end of year 1 or at beggining of year 2.

2 years from now means at the end of year 2 or at beggining of year 3.

So on until you get the desired year…

18 years from now means at the end of year 18 or at beggining of year 19.

Draw a timeline, it really helps.

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You are at t = 0

18 years from now means payments will start at t = 18. This means end of year 18.

Payments into Savings account will start at t = 1 (end of year 1, since year starts at 0) and last payment into savings account will be on t = 17 (end of year 17).

Payments for tuition will start on t = 18 (end of year 18) and last payment will be on t =21 (end of year 21).

and I agree with Harrogath, you have to make a timeline when tackling such questions!

The wording was really confusing. When it said 18 years from now, I that meant end of year 18. However, when it said payments are made at the beginning of each year, then I got really confused. How should I have interpreted this part?

There are 2 parts in the timeline.

The first is the “savings” part, and the second is the “withdraw” part.

The first withdraw will be made at the end of year 18 because it says “payable at the beggining of each year”. So, there are 17 years of deposits. The first deposit will be made at the end of year 1 and the 17th deposit will be made at the end of year 17.

Draw a timeline every time is necessary, you will eventually draw timelines on your head only. Don’t despair.

Hope this helps.


what was the final answer to the question?

This is how I would work it. Though if you want to be clever, you could calculate your payment stream in Year 17 dollars instead of Year 0 dollars, so you don’t have to discount back.

The value of an annuity is A *(1- 1/(1+r)^T)/r.

So for a flow of payments of $20,000 for 4 years assuming a 5% rate of return, the value is:

20000* (1-1/(1.05)^4)/.05 = $70,919.01 which you are going to need in 18 years (which is the start of year 18 if you are now at t=0 or Year 0).

So, you discount that back to today to get PV of the eventual payments (remember that the annuity formula is the value at the year before payments start, so 17 periods, not 18):

PV = V/(1+r)^T = 70,919.01/(1.05)^17 = $30,941.73

So, we want to make 17 years of payments (at the starts of years 1 through 17) that equal PV = $30,941.73. Using the annuity formula again:

$30,941.73 = A *(1- 1/(1.05)^17)/.05

Solve for A and you get A = $30,941.73/((1- 1/(1.05)^17)/.05) = $2,744.50

As a table:

Year Beginning Balance Ending Balance 0 $0.00 $0.00 Today 1 $2,744.50 $2,881.73 First Payment 2 $5,626.23 $5,907.55 3 $8,652.05 $9,084.65 4 $11,829.16 $12,420.62 5 $15,165.12 $15,923.38 6 $18,667.88 $19,601.28 7 $22,345.78 $23,463.07 8 $26,207.58 $27,517.95 9 $30,262.46 $31,775.58 10 $34,520.09 $36,246.09 11 $38,990.60 $40,940.13 12 $43,684.63 $45,868.86 13 $48,613.37 $51,044.04 14 $53,788.54 $56,477.97 15 $59,222.47 $62,183.60 16 $64,928.10 $68,174.51 17 $70,919.01 $74,464.96 Last Payment 18 $54,464.96 $57,188.21 First Withdrawal 19 $37,188.21 $39,047.62 Second Withdrawal 20 $19,047.62 $20,000.00 Third Withdrawal 21 $0.00 $0.00 Last Withdrawal

keep_running,I posted a video explaining if interested. http://youtu.be/teyOqSQ1IZA

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Get thyself a gold calligraphy pen and write Harrogath’s words in the fanciest script you can manage, preferably on parchment.

Ten times.

I wrote an article that echoes Harrogath’s sage words: http://financialexamhelp123.com/time-value-of-money-using-timelines/

Draw a timeline? Sure, it’ll help you visualize what’s going on with the cashflows.

Broken record time: you can use the algebraic approach, but why not use yer fancy dancy BAII or HP calculator to do the grunt discounting?

I believe that I mention that very idea in my article.

​You were actually right about the timing of the college payments, the question does imply that the payments for college begin at t=18 sharp. The question is from the CFA curriculum and if you look into the solution provided there (including the heralded time line!!!), you will see that they meant for the tuition payments to be due in t=18, i.e. at the beginning of t=18. That is why they compute the PV of the college payments for t=17, treating them as an ordinary annuity beginning at t=17. Which is then equated to the FV of the necessary cumulated savings (i.e. this is, what we use as FV in the BAII Plus or your instrument of choice).

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good explanation