Question-Time value of Money


Can anyone please help to explain this question? I am having a trouble with these kinds of problems.

Gerard Jones plans to save for his 5-year doctorate degree, which starts 6 years from now. The current annual expenditure is USD7,200 and it is expected to grow by 7 percent annually. Gerard will need to make the first payment 6 years from today. He identifies a savings plan that allows him to earn an interest of 8 percent annually. How much should Gerard deposit each year, starting one year from today? Assume that he plans to make 5 payments.


A is correct. This problem can be solved in three steps.

Step 1: Find the annual expenditures

Annual Expendituret=6=7,200 (1+0.07)6=USD10,805

Annual Expendituret=7=7,200 (1+0.07)7=USD11,562

Annual Expendituret=8=7,200 (1+0.07)8=USD12,371

Annual Expendituret=9=7,200 (1+0.07)9=USD13,237

Annual Expendituret=10=7,200 (1+0.07)10=USD14,163

Step 2: Find the present value of annual expenditures at t = 5

Time Period **Annual Expenditure (USD)**Present Value 6 10,805 10,805 (1.08)-1 =10,004 7 11,562 11,562 (1.08)-2 =9,912.5 8 12,371 12,371 (1.08)-3 =9,820.5 9 13,237 13,237 (1.08)-4 =9,729.6 10 14,163 14,163 (1.08)-5 =9,639 SUM = USD 49,106

Step 3: Find the annuity payment

N = 5, %i = 8, PV = 0, FV = 49,106, CPT PMT.

PMT = 8,370.

A USD8,370. B USD8,539. C


im not a CFA, but for these kinds of problem i would first recommend making a time diagram and numbering them from 0 up to t + 1, where 0 is the current time and t is the last year there is a cashflow. if t is large, say, 100, you could just write the timeline as

0__1__2__3__ … 98__99__100___101

for this problem, t is 10, since the payments start at t= 6 and there are 5 payments in total, which means the last payment is at t=10

write down arrows with information on the cashflow the problem gives.

for this problem, draw an arrow at t = 6, t=7, … t=10 all pointing outwards

this represents that cashflow is going out (it doesn’t matter if you point the arrows in, so long as you are aware of what they mean and you remain consistent)

at t = 6, this is the first payment (cash outflow), and because the payment grows at a rate of 7% and the amount of payment (if there was such a payment) at t= 0 is 7200,

the amount of payment due at t= 6 must be 7200x 1.07^6 because of compounding.

luckily, you know that the interest rate is 8%, so you can simply write a summation formula to find the present value of all payments at t= 0

this is one of those times the summation formula helps, a lot.

sum from k= 0 to 4 of {7200 x 1.07^(k+6) / 1.08^(k+6) }

(i don’t know how to write math in code)

this sum should give 33,421.063

which seems different from the answer. it is not.

this amount is the value at t= 0

which is equivalent to 33,421.063 x 1.08^5 = 49,106.50622 at t = 5

you know the present value of the payments at t = 0, now you just need to repeat the procedure for cashflows flowing in at t = 1, 2, 3, 4 and 5.

if we assume the annual cashflows is M, then

sum from k = 0 to 4 of { M / (1.08^(k+1) ) }

which is equal to 3.992710037 M

which must equal the present value of the payment calculated earlier.

thus, M = 33,421.063 / 3.992710037 = 8370.520947

i strongly recommend starting by drawing the time diagram and arrows because it helps you make sense of the cashflows moving in and out. it makes more intuitive sense.

Step 1: draw a time line.

thank you so much for explaining so well!!!

thank you, i will practice with time line.

:+1: Especially if you have weird timing and non-level payments.

You might have an easier time of it by using time 5 as the focal date for all discounting and accumulating.

Maybe I’m bored at work and just calculated this all out myself, but I don’t think you should be discount the first payment of $10,805.26 since it is happening at the beginning of the seventh year.

edit: Upon second review, the answer you posted is totally wrong.

the first payment is at t = 6 and the official answer calculates the time value at t = 5, which is why it discounts. there is no difference which time you choose as the time into which you discount. you could choose to discount to time 0 (like i did), or to time 5 (like the solution did), the answers are the same and only differ by the time value of money. the problem asks for the amount of the level annuity payment, so as long as you remain consistent, you should arrive at the same answer, whether you choose to discount to time 0, 5 or at any other time.

That’s all wrong. Very wrong. Who wrote this problem… did you?

It clearly states that your first payment will be made in six years from today - which means at the START of the seventh year, not the sixth year as you have stated.

maybe you are misunderstanding. the present time(that is, today) is t = 0,

so 6 years from now is t = 6, which is the start of the 7th year.

(much like t= 0 is the start of the 1st year)

also, please be so kinds as to not put words into my mouth, thanks. i never said the payment was at the start of the sixth year. please actually read before replying.

NVM - I glossed over a crucial piece of information that says he plans to make 5 payments. I thought that meant he would only make 5 tuition payments, which makes perfect sense.