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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.

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migrated from quant.stackexchange.com Sep 23 '13 at 13:09

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1 Answer 1

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.)

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