 What is the present value of an annuity that pays $74 per year for 5 years and an additional $1,000 with the final payment? Use a nominal rate of 9.65%. 

To calculate the present value (PV) of the annuity, we can break it into two parts: 

1. **The annuity part:** Payments of $74 per year for 5 years.
2. **The lump sum part:** An additional payment of $1,000 at the end of 5 years.

### Formula for Present Value of an Annuity
The formula for the present value of an annuity is:

\[
PV_{\text{annuity}} = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $ P = 74 $ (annual payment),
- $ r = \frac{9.65\%}{100} = 0.0965 $ (annual nominal interest rate),
- $ n = 5 $ (number of years).

### Formula for Present Value of a Lump Sum Payment
The present value of a lump sum paid in the future is:

\[
PV_{\text{lump sum}} = \frac{F}{(1 + r)^n}
\]

Where:
- $ F = 1,000 $ (lump sum payment),
- $ r = 0.0965 $ (interest rate),
- $ n = 5 $ (number of years).

### Step-by-Step Calculation

1. **Present value of the annuity:**

\[
PV_{\text{annuity}} = 74 \times \left( \frac{1 - (1 + 0.0965)^{-5}}{0.0965} \right)
\]

2. **Present value of the lump sum:**

\[
PV_{\text{lump sum}} = \frac{1,000}{(1 + 0.0965)^5}
\]

Let's calculate these values.1. **Present value of the annuity**: \$283.04
2. **Present value of the lump sum**: \$630.89

### Total present value of the annuity:
\[
PV_{\text{total}} = 283.04 + 630.89 = 913.94
\]

So, the present value of the annuity is **$913.94**.

Would you like further details on any of these steps or calculations?

Here are some related questions:
1. How does the discount rate affect the present value of an annuity?
2. What is the difference between a nominal and effective interest rate?
3. How would the present value change if payments were made quarterly instead of annually?
4. How would inflation impact the value of future annuity payments?
5. What is the future value of this annuity?

**Tip:** The present value of an annuity decreases as the interest rate increases.

What is the present value of an annuity that pays $74 per year for 5 years and an additional $1,000 with the final payment? Use a nominal rate of 9.65%.

Calculate the present value of an annuity that pays $74 annually for 5 years, with an additional $1,000 lump sum at the end. Using a 9.65% nominal interest rate, this step-by-step guide breaks down the present value of both the annuity and lump sum, with a total value of $913.94.

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