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I'm unsure how to solve the following problem:

Angie wants to plan a trip to Hawaii with her husband on their 10th wedding anniversary in two years. She anticipates that the all-inclusive trip will cost $9500 for both of them and wants to start saving. How much should she deposit at the end of each month into her account that pays 12% compounded monthly to pay for the trip in two years?

I have solved compound interest and future value problems before using:

$$ {\rm FV} = {\rm PV}\left(1 + \frac{\rm rate}{\rm interval}\right)^{\rm interval \cdot years} $$

However this question is different. What would be the difference between deposits at beginning of each month vs the end? And what formula is used to solve this kind of problem?

Thanks

martini
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jmasterx
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  • this goes over the difference between annuities-immediate and annuities-due (the common name for "beginning" vs "end" of each month annuities). The formula is a basic (finite) geometric series, discounting(or accruing) all of the deposits to a common point chosen by our convenience. – Tyler Oct 22 '13 at 14:26
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    $12 %$ compounded monthly? Where can I sign up? – Ron Gordon Oct 22 '13 at 14:52

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Here's what we can do without finance functions. We know that in $2$ years, a trip will cost us $ \$ 9500$. We want to make a deposit every month so that, after $2$ years, our account will have $ \$ 9500$ in it. Let's call our undetermined deposit $X$. Note that $t$ is time in months. $i^{(12)}$ is the nominal interest, $i$ is the effective annual interest.

At $t=0$ we make no deposit.

At $t = 1$ we deposit $X$. Its accumulated value will be $X(1+ \frac{i^{(12)}}{12})^{23}$

At $t = 2$ we deposit $X$. Its accumulated value will be $X(1+ \frac{i^{(12)}}{12})^{22}$

...

At $t = 23$ we deposit $X$. Its accumulated value will be $X(1+ \frac{i^{(12)}}{12})^{1}$

At $t = 24$ we deposit $X$. Its accumulated value will be $X(1+ \frac{i^{(12)}}{12})^{0}$

We notice that this is a geometric series since we are summing the values of our deposits, so

$$\sum_{n=1}^{24} X \left(1+ \frac{i^{(12)}}{12} \right)^{24-n} = X \left( \frac{(1+ \frac{i^{(12)}}{12})^{24} - 1}{i} \right)$$

by some algebraic manipulations.

Then, we know that our calculated future value of the deposits has to equal the cost of the trip, so

$9500 = X \left( \frac{(1+ \frac{i^{(12)}}{12})^{24} - 1}{i} \right)$ from which it is easy to calculate $X$.

(whew).

Annuities lump this huge amount of work into a very concise notation with a simple formula that is extraordinarily flexible, so I recommend you read up on it. Any good interest theory book will go over it. I learned from Kellison's book and found it to be quite good.

Tyler
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