# Time value of money - "N"

Gerard Jones plans to save for his 5-year doctorate degree, which starts 6 years from now. The current annual expenditure is $7,200 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.

$8,370

$8,539

$8,730

Here’s how I approached this:

- I found the annual expenditures for each year, accordingly:

Annual Expendituret=6= 7,200 (1+0.07)6=$10,805

Annual Expendituret=7= 7,200 (1+0.07)7=$11,562

Annual Expendituret=8= 7,200 (1+0.07)8=$12,371

Annual Expendituret=9= 7,200 (1+0.07)9=$13,237

Annual Expendituret=10= 7,200 (1+0.07)10=$14,162

2. I discounted the expenditures back to year 5 because that’s the year we are going to make our first payment

10,805 10,805∗(1.08)−1=10,004

11,562 11,562∗(1.08)−2=9,912.5

12,371 12,371∗(1.08)−3=9,820.5

13,237 13,237∗(1.08)−4=9,729.6

14,162 14,163∗(1.08)−5=9,639

SUM = $49,106

3. Here’s my confusion, now that I have $49,106 which is equivalent to the expenditures taking place in years 6,7,8,9,10; I need to calculate the payment amount that I need to make to reach this. My professor discounted back using N = 5 so that brings us to time 0 but I discounted back using N = 4 which is the time the first payment (t = 1) is made. Where did I go wrong? My issue with “N” is knowing where exactly to discount back to, in this case it was t = 0 and not t = 1. Is there some sort of rule of thumb that I am not aware of or someone can explain?

Here’s why I think he discounted back to (t = 0). If we have a FV at year 5 that equates serves as a PV for annuities y6–>y10, then we would be indifferent between attaining that FV as a lump sum in year 5 or earning the annuities from y6 to y10. That I am fine with. What I am not fine with however is when I am standing at t = 0, knowing my first payment is to be made at t = 1, and attempting to find the investment amount. Moreover why would you discount to t = 0 using n = 5 when you don’t even have a pmt that year .. **Moreover the question clearly states how much should gerard deposit starting one year from today!**