How much must be deposited today into the following account in order to have

$ 60 comma 000$60,000

in

66

years for a down payment on a​ house? Assume no additional deposits are made.

An account with monthly compounding and an APR of

66​%

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Part 1

​$enter your response here

should be deposited today.

​(Do not round until the final answer. Then round to the nearest cent as​ needed.

To solve this problem, we will use the formula for the present value of a future amount with compound interest:

\[
PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}
\]

Where:
- $ PV $ is the present value (the amount to be deposited today),
- $ FV $ is the future value (\$60,000 in this case),
- $ r $ is the annual percentage rate (APR), which is 66% (or 0.66 as a decimal),
- $ n $ is the number of compounding periods per year (since we have monthly compounding, $ n = 12 $),
- $ t $ is the time in years (66 years in this case).

Let’s calculate the present value using this formula.

We know:
- $ FV = 60,000 $
- $ r = 0.66 $
- $ n = 12 $
- $ t = 66 $

Substituting into the formula:

\[
PV = \frac{60,000}{(1 + \frac{0.66}{12})^{12 \times 66}}
\]

Now, let's compute that.The amount that needs to be deposited today to have \$60,000 in 66 years at an APR of 66% with monthly compounding is extremely close to zero—approximately \$0.000000000000023.

This result indicates that because the interest rate is so high (66%), even a tiny deposit would grow significantly over such a long period (66 years). Therefore, the required deposit is nearly negligible.

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How does changing the compounding frequency affect the present value?
2. What if the interest rate were lower, like 5% APR? How would that change the required deposit?
3. What is the difference between APR and effective annual rate (EAR)?
4. How does the time period affect the required deposit in compound interest scenarios?
5. How does monthly compounding differ from annual compounding in terms of growth?

**Tip:** The higher the interest rate and the longer the time, the smaller the required initial deposit, thanks to the power of compound interest.

How much must be deposited today into the following account in order to have $60,000 in 66 years for a down payment on a house? Assume no additional deposits are made. An account with monthly compounding and an APR of 66%.

Learn how to calculate how much must be deposited today to accumulate $60,000 in 66 years with a 66% APR and monthly compounding. This step-by-step solution uses the compound interest formula to determine the present value.

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