What amount should be deposited in a savings account today, in a financial institution paying 11.5% compounded quarterly, in order to accumulate 400,000 in the account 5 years from now?

To solve this problem, we need to use the formula for compound interest to find the present value (the amount to be deposited today) that will grow to a specified future value.

The compound interest formula is:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the future value of the investment/loan, including interest (\$400,000 in this case).
- $ P $ is the principal amount (the initial deposit, which we need to find).
- $ r $ is the annual interest rate (11.5% or 0.115).
- $ n $ is the number of times that interest is compounded per year (quarterly, so $ n = 4 $).
- $ t $ is the time the money is invested for in years (5 years).

We are asked to find $ P $, so we rearrange the formula:

\[
P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}
\]

Now, let's plug in the numbers:

\[
P = \frac{400{,}000}{\left(1 + \frac{0.115}{4}\right)^{4 \times 5}}
\]

Let's calculate this step by step.

### Step 1: Calculate the interest per period
\[
\frac{0.115}{4} = 0.02875
\]

### Step 2: Calculate the number of periods
\[
4 \times 5 = 20
\]

### Step 3: Calculate the growth factor
\[
\left(1 + 0.02875\right)^{20} = 1.859\text{ (approximately)}
\]

### Step 4: Calculate the present value
\[
P = \frac{400{,}000}{1.859} \approx 215{,}189.87
\]

So, the amount that should be deposited today is approximately **\$215,189.87**.

Would you like any further details or explanations?

Here are 5 related questions:
1. How does the frequency of compounding affect the future value of an investment?
2. What would the required deposit be if the interest was compounded monthly instead of quarterly?
3. How would a change in the interest rate impact the required initial deposit?
4. What is the difference between simple interest and compound interest?
5. How can the compound interest formula be modified for continuous compounding?

**Tip:** Always check whether the interest rate is nominal or effective when dealing with compound interest problems.

Learn how to calculate compound interest to determine the initial deposit required for a savings account to grow to $400,000 in 5 years. This problem involves using the compound interest formula and understanding exponential growth. Suitable for high school students and anyone interested in financial calculations.

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