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Walter decides he wants to start saving for retirement. On his 40th birthday he deposits $127000 in an account with an interest rate of 9% annually (compound interest). On his 65th birthday he starts withdrawing money annually from the account for the next 11 years. What size are the withdrawals?

I'm not getting the correct answer, I thought it the answer would simply be:

$127000(1.09)^{25} = R(1-(1.09)^{-11}/(.09))$

$R = 160925.90$

Although the answer is incorrect

Thanks

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    Is this simple or compound interest? That makes a big difference. – 2017-02-04
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    compounded, thank you for clarifying :) i'll edit – 2017-02-04
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    Do you have a (given) solution ? I have $R=147,638.41$. – 2017-02-04

1 Answers 1

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You're partly right, but not using quite the right formula for the withdrawals.

The amount in the account at age $65$ indeed is

$$ A = 127000(1.09)^{25}. $$

Now you can treat the eleven withdrawals as an annuity. The formula for the net present value of an annuity with payments of amount $R$ per payment at an interest rate of $9\%,$ assuming payment at the end of each compounding period, is

$$\left(\frac{1-(1.09)^{-11}}{.09}\right)R.$$

The reason you got the wrong answer is that the withdrawals are made starting immediately on Walter's $65$th birthday and every year after for the next ten years. That is, it's comparable to an annuity payable at the start of each compounding period rather than at the end. As such, the present value is greater; in fact it should be

$$1.09\left(\frac{1-(1.09)^{-11}}{.09}\right)R,$$ that is, discounted by one less year because each payment occurs one year earlier. Using this formula, the payments are each less than $150,000.$ (I'll let you figure out just how much less.)

By the way, the formula you wrote was not the one you meant; what you wrote was an expression equal to $$\left(1 - \frac{(1.09)^{-11}}{.09}\right)R.$$ Not to belabor the point too much, but precedence of operators says you do the division before the subtraction unless you use parentheses (or a horizontal bar) to force a different order. This is not a problem if you remember the order you meant to do the operations, but it can be a problem if you forget the original order or if you have to communicate the formula to someone else.