A young couple wants to have a college fund that will pay $30,000 at the end of each half-year for 8 years.
(a) If they can invest at 7%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years to begin making their college withdrawals 6 months after their last investment? (Round your answer to the nearest cent.)
$
Correct: Your answer is correct.

(b) Suppose 8 years after beginning the annuity payments, they receive an inheritance of $33,000 that they contribute to the account, and they continue to make their regular payments as found in part (a). How many college withdrawals will they be able to make before the account balance is $0? (Round your answer to the nearest whole number.)

Incorrect: Your answer is incorrect.
withdrawals

Let's break this problem into two parts to find the answers.

### Part (a) - How much they need to invest at the end of each 6-month period for the next 18 years

We need to calculate how much they should invest semiannually over the next 18 years so that at the end of this period, they can start making semiannual withdrawals of $30,000 for 8 years.

This involves two phases:
1. **Accumulation phase (18 years of investment)**: The couple makes equal semiannual investments over 18 years (36 periods) at an interest rate of 7% compounded semiannually.
2. **Payout phase (8 years of withdrawal)**: The couple will make withdrawals of $30,000 every 6 months for 8 years (16 withdrawals).

Let's use the **future value of an annuity formula** to calculate the amount they need to invest during the accumulation phase. The formula is:

\[
FV = P \times \frac{(1 + r)^n - 1}{r}
\]

Where:
- $FV$ is the future value (total amount needed at the end of 18 years).
- $P$ is the semiannual investment.
- $r$ is the semiannual interest rate (7% annual compounded semiannually gives $r = \frac{0.07}{2} = 0.035$).
- $n$ is the number of investment periods (36 periods over 18 years).

In the payout phase, they will make 16 semiannual withdrawals of $30,000. To find the total amount required at the beginning of the payout phase, we calculate the present value of these withdrawals using the **present value of an annuity formula**:

\[
PV = W \times \frac{1 - (1 + r)^{-n}}{r}
\]

Where:
- $PV$ is the present value at the beginning of the payout phase (this is what the couple needs at the end of the 18 years).
- $W = 30,000$ is the withdrawal amount.
- $r = 0.035$ (semiannual interest rate).
- $n = 16$ (number of withdrawals).

Substituting the values into the present value formula:

\[
PV = 30,000 \times \frac{1 - (1 + 0.035)^{-16}}{0.035}
\]

\[
PV = 30,000 \times \frac{1 - (1 + 0.035)^{-16}}{0.035}
\]

Let's calculate this value.

#### Present Value Calculation for Withdrawals:
\[
PV = 30,000 \times \frac{1 - (1.035)^{-16}}{0.035}
\]

Using a calculator:

\[
PV \approx 30,000 \times 11.446
\]
\[
PV \approx 343,380
\]

So, the couple needs $343,380 at the end of 18 years to start their withdrawals.

Now, let's find the semiannual investment amount $P$ required to accumulate $343,380 in 36 periods at 7% compounded semiannually.

Rearranging the future value of annuity formula to solve for $P$:

\[
P = \frac{FV}{\frac{(1 + r)^n - 1}{r}}
\]

Substitute the known values:

\[
P = \frac{343,380}{\frac{(1 + 0.035)^{36} - 1}{0.035}}
\]

\[
P = \frac{343,380}{\frac{(1.035)^{36} - 1}{0.035}}
\]

Now, calculate the denominator:

\[
(1.035)^{36} \approx 3.393
\]

\[
\frac{3.393 - 1}{0.035} \approx 68.365
\]

Now calculate $P$:

\[
P = \frac{343,380}{

A young couple wants to have a college fund that will pay $30,000 at the end of each half-year for 8 years. (a) If they can invest at 7%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years to begin making their college withdrawals 6 months after their last investment? (b) Suppose 8 years after beginning the annuity payments, they receive an inheritance of $33,000 that they contribute to the account, and they continue to make their regular payments as found in part (a). How many college withdrawals will they be able to make before the account balance is $0?

Learn how a couple can calculate their semiannual investments to build a college fund. With a detailed breakdown of future and present value annuity formulas, this problem demonstrates how to manage investments with 7% compounded semiannually and semiannual withdrawals of $30,000 over 8 years.

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