Sign up ×
Personal Finance & Money Stack Exchange is a question and answer site for people who want to be financially literate. It's 100% free, no registration required.

Chuck needs to purchase an item in 10 years. The item costs 200 today, but its price inflates at 4% per year. To finance the purchase, Chuck deposits 20 into an account at the beginning of each year for 6 years. He deposits an additional X at the beginning of years 4, 5, and 6 to meet his goal. The annual effective interest rate is 10%. Calculate X.

This is how i interpret the problem: You have 5 cash flows starting from 0 to 5 of $20. You also have 3 cash flows at t=4,5,6.

I used annuity due formula to shift former cash flow to year 6, and then accumulate it to year 10 by the 4 remaining years.

I used the same approach for the latter:

$(20 \cdot \ddot{s} _{5 \neg i =10} )(1.1)^4 + X \cdot \ddot{s} _ {3 \neg i=10\%} (1.1)^3 = 200(1.04)^{10}\tag{1}$

But this does not give me the right answer. Can someone please tell me what I'm doing wrong? Thanks in advance.

share|improve this question

migrated from Sep 23 '13 at 13:09

This question came from our site for finance professionals and academics.

2 Answers 2

These are the steps I'd follow:

$200 today times (1.04)^10 = Cost in year 10.

The 6 deposits of $20 will be one time value calculation with a resulting year 7 final value. You then must apply 10% for 3 years (1.1)^3 to get the 10th year result.

You now have the shortfall. Divide that by the same (1.1)^3 to shift the present value to start of year 7. (this step might confuse you?)

You are left with a problem needing 3 same deposits, a known rate, and desired FV. Solve from there.

(Also, welcome from quant.SE. This site doesn't support LATEX, so I edited the image above.)

share|improve this answer

The solution is x = 8.92.

This assumes that Chuck's six years of deposits start from today, so that the first deposit accumulates 10 years of gain, i.e. 20*(1 + 0.1)^10. The second deposit gains nine years' interest: 20*(1 + 0.1)^9 and so on ...

enter image description here

If you want to do this calculation using the formula for an annuity due, i.e.

enter image description here


enter image description here

(formula by induction)

you have to bear in mind this is for the whole time span (k = 1 to n), so for just the first six years you need to calculate for all ten years then subtract another annuity calculation for the last four years. So the full calculation is:

enter image description here

As you can see it's not very neat, because the standard formula is for a whole time span. You could make it a little tidier by using a formula for k = m to n instead, i.e.

enter image description here

So the calculation becomes

enter image description here

which can be done with simple arithmetic (and doesn't actually need a solver).

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.