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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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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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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 ...

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

http://www.financeformulas.net/Future-Value-of-Annuity-Due.html

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where

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(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).

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I am writing this solution to make clear the approach that the OC originally was originally looking for.

Let's lay out the values we are looking for. For the payments over the first 6 years:

We make a payment of 20 each year, and we need to scale this value to the 10th year. This means we need to scale from year 6 to year 10, so a factor of 1.1^4. Therefore, we are looking for: 20(1.1^5+1.1^6+...+1.1^10) This is a 6 year period with payments at the beginning of each year. So we can give the formula by:

20* $_6*1.1^4 where $_6 is the annuity due over 6 time periods at an interest rate of 10%.

For the 3 payments of X scaled to the 10th year, we similarly need to scale to year 10 from year 6. Thus, a factor of 1.1^4 is needed. We need the following: X(1.1^5+1.1^6+1.1^7) This is a 3 year period with payments at the beginning of each year. So we can give the formula by X*$_3*1.1^4 where $_3 is the annuity due over 3 time periods at an interest rate of 10%.

Next, we need to scale the original cost of 200 to year 10. This is 200*1.04^10.

Hence, our final equation

20*$_6* 1.1^4 + X*$_3* 1.1^4 = 200* 1.04^10

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