Puneet intends to retire in 10 years. To supplement his pension he would like to receive $500 every six months for 20 years. If he is to receive the first payment six months after the date of his retirement, what lump amount must he invest today to achieve his goal? Assume that the investment will earn 12% compounded semiannually.
Multiple Choice
$4871.78
$5164.09
$2486.50
$2212.97
$2345.75

To determine the lump sum amount Puneet must invest today, we need to follow these steps:

1. **Calculate the present value of the annuity payments** Puneet wants to receive after retirement.
2. **Calculate the present value of that amount** as of today, considering the investment period before retirement.

### Step 1: Calculate the Present Value of the Annuity Payments

The annuity payments are $500 every six months for 20 years (40 periods), with the first payment six months after retirement. The interest rate is 12% per year, compounded semiannually (6% per period).

The present value $ PV $ of an ordinary annuity is calculated using the formula:
\[ PV = PMT \times \left(1 - \left(1 + r\right)^{-n}\right) \div r \]

Where:
- $ PMT $ is the annuity payment ($500)
- $ r $ is the semiannual interest rate (0.06)
- $ n $ is the total number of payments (40)

Plugging in the values:
\[ PV = 500 \times \left(1 - \left(1 + 0.06\right)^{-40}\right) \div 0.06 \]

Calculating the values inside the parentheses first:
\[ \left(1 + 0.06\right)^{-40} = \left(1.06\right)^{-40} \approx 0.09722 \]

Now:
\[ PV = 500 \times \left(1 - 0.09722\right) \div 0.06 \]
\[ PV = 500 \times 0.90278 \div 0.06 \]
\[ PV = 500 \times 15.0463 \]
\[ PV \approx 7523.15 \]

### Step 2: Calculate the Present Value of that Amount Today

Now, we need to discount this present value back to today's value (10 years before retirement, or 20 periods).

The present value $ PV_0 $ is calculated using the formula:
\[ PV_0 = FV \div \left(1 + r\right)^n \]

Where:
- $ FV $ is the future value ($7523.15)
- $ r $ is the semiannual interest rate (0.06)
- $ n $ is the number of periods (20)

Plugging in the values:
\[ PV_0 = 7523.15 \div \left(1 + 0.06\right)^{20} \]
\[ PV_0 = 7523.15 \div \left(1.06\right)^{20} \]

Calculating the value:
\[ \left(1.06\right)^{20} \approx 3.2071 \]

Now:
\[ PV_0 = 7523.15 \div 3.2071 \]
\[ PV_0 \approx 2345.75 \]

Therefore, Puneet must invest approximately **$2345.75** today to achieve his goal.

The correct choice is **$2345.75**.

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

1. How would the lump sum change if the interest rate were different?
2. What if Puneet wanted to receive $1000 every six months instead?
3. How is the present value formula for an annuity derived?
4. What would be the lump sum if payments started immediately at retirement instead of after six months?
5. How does compounding frequency affect the investment needed?

**Tip:** Understanding how present value and annuity calculations work can greatly enhance financial planning and investment strategies.

Learn how to calculate the present value of annuity payments for retirement planning using compound interest and time value of money principles. This problem involves determining the lump sum needed today to meet future financial goals, with detailed step-by-step calculations provided. Suitable for advanced high school students and anyone interested in financial planning.

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