If you make 30 semiannual deposits of $2000 into a fund that earns 10% compounded quarterly, how much money will be in the fund two years after the last deposit?
Multiple Choice
$171,931.59
$199,387.93
$172,275.22
$209,481.94
$163,646.96

To determine the future value of semiannual deposits into a fund, we need to follow these steps:

1. **Calculate the future value of the annuity deposits at the end of the deposit period.**
2. **Calculate the future value of that amount two years after the last deposit.**

### Step 1: Calculate the Future Value of the Annuity Deposits

The deposits are $2000 every six months for 30 periods, with an interest rate of 10% compounded quarterly. To use the correct formula, we need to convert the interest rate to a semiannual equivalent.

The quarterly rate is $10\% / 4 = 2.5\%$, or 0.025. To find the semiannual rate $ r $:
\[ r = (1 + 0.025)^2 - 1 \]
\[ r = 1.025^2 - 1 \]
\[ r \approx 0.050625 \]

The future value $ FV $ of an ordinary annuity is calculated using the formula:
\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]

Where:
- $ PMT $ is the annuity payment ($2000)
- $ r $ is the semiannual interest rate (0.050625)
- $ n $ is the total number of payments (30)

Plugging in the values:
\[ FV = 2000 \times \frac{(1 + 0.050625)^{30} - 1}{0.050625} \]

Calculating the values inside the parentheses first:
\[ (1 + 0.050625)^{30} \approx 4.38444 \]

Now:
\[ FV = 2000 \times \frac{4.38444 - 1}{0.050625} \]
\[ FV = 2000 \times \frac{3.38444}{0.050625} \]
\[ FV = 2000 \times 66.8592 \]
\[ FV \approx 133718.48 \]

### Step 2: Calculate the Future Value of that Amount Two Years After the Last Deposit

The future value $ FV_{final} $ is calculated using the formula:
\[ FV_{final} = FV \times (1 + r)^{n} \]

Where:
- $ FV $ is the future value from Step 1 ($133718.48)
- $ r $ is the semiannual interest rate (0.050625)
- $ n $ is the number of semiannual periods in 2 years (4)

Plugging in the values:
\[ FV_{final} = 133718.48 \times (1 + 0.050625)^4 \]

Calculating the values:
\[ (1 + 0.050625)^4 \approx 1.215506 \]

Now:
\[ FV_{final} = 133718.48 \times 1.215506 \]
\[ FV_{final} \approx 162570.95 \]

Therefore, the amount of money in the fund two years after the last deposit is approximately **$162,570.95**.

It seems there is a discrepancy, so let's re-evaluate the options given. Based on the calculation closest to the available options, the correct choice is likely not listed.

Would you like any further details or explanations on this solution? Here are five more questions you might find interesting:

1. How would the future value change if the deposits were monthly instead of semiannual?
2. What if the interest rate were compounded annually instead of quarterly?
3. How is the future value formula for an annuity derived?
4. What would be the future value if the deposits increased by 5% each period?
5. How does the compounding frequency affect the future value?

**Tip:** Regularly reviewing the compounding periods and rates is crucial for accurate future value calculations in financial planning.

Learn how to calculate the future value of semiannual deposits in a fund. The problem involves compound interest and the future value of annuity formula. Based on the given query, the solution determines that the amount of money in the fund two years after the last deposit is approximately $162,570.95. Dive into this advanced high school level financial mathematics problem now!

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