A friend wants to borrow money from you. He states that he will pay you $4,400 every 6 months for 12 years with the first payment exactly 7 years and 6 months from today. The interest rate is an APR of 6.7 percent with semiannual compounding. What is the value of the payments today?


To find the present value of the payments your friend promises, we will use the concept of the present value of an annuity. Here's a step-by-step breakdown:

### 1. **Identify the Given Values**:
   - **Payment (PMT):** $4,400
   - **Interest rate (APR):** 6.7% annually with semiannual compounding
   - **Number of years until the first payment (Deferred Period):** 7.5 years
   - **Total number of years of payments:** 12 years
   - **Payments per year:** 2 (semiannual)
   - **Number of payments (n):** 12 years × 2 = 24 payments
   - **Interest rate per period (r):** $ \frac{6.7\%}{2} = 3.35\% $

### 2. **Calculate the Present Value of the Annuity (PV) at the Time of First Payment**:
The present value of an annuity formula is:
\[
PV_{\text{annuity}} = PMT \times \frac{1 - (1 + r)^{-n}}{r}
\]

Substitute the values:
\[
PV_{\text{annuity}} = 4,400 \times \frac{1 - (1 + 0.0335)^{-24}}{0.0335}
\]

### 3. **Calculate the Present Value at Today (PV$_0$)**:
Since the first payment is 7.5 years away, the present value calculated above needs to be discounted back to today.

The discounting formula is:
\[
PV_0 = PV_{\text{annuity}} \times \frac{1}{(1 + r)^{\text{number of periods}}}
\]

In this case:
\[
\text{number of periods} = 7.5 \times 2 = 15 \text{ periods}
\]

So:
\[
PV_0 = PV_{\text{annuity}} \times \frac{1}{(1 + 0.0335)^{15}}
\]

Let me calculate these values for you.The present value of the payments your friend promises, discounted to today, is approximately **$43,789**.

Would you like more details on the calculations or have any other questions? 

Here are 8 follow-up questions to consider:

1. How does changing the interest rate affect the present value?
2. What if the payments were made annually instead of semiannually?
3. How would the present value change if the payment amount increased?
4. What if the first payment was delayed by 10 years instead of 7.5 years?
5. How does the number of payments affect the present value?
6. What is the future value of the payments if you were to invest the money instead?
7. How would inflation impact the real value of these payments?
8. What if the payments were made monthly instead of semiannually?

**Tip:** Always account for the time value of money when assessing the value of future payments or investments.

Learn how to calculate the present value of future payments using an APR of 6.7% with semiannual compounding. Understand the time value of money concepts and the annuity formula. Ideal for financial analysts and anyone interested in finance calculations.

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