Given an interest rate of 7.15 percent per year, what is the value at year t = 8 of a perpetual stream of $3,269 payments that begin at year t = 20? (Do not include the dollar sign ($). Enter rounded answer as directed, but do not use the rounded numbers in intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

To find the present value at year $ t = 8 $ of a perpetual stream of $3,269 payments starting at year $ t = 20 $, given an interest rate of 7.15% per year, we need to follow these steps:

### Step 1: Calculate the present value of the perpetuity at year $ t = 19 $.
The perpetuity starts at $ t = 20 $. The formula for the present value of a perpetuity is:

\[
PV = \frac{C}{r}
\]

where:
- $ C = 3269 $ (the payment amount),
- $ r = 0.0715 $ (the interest rate).

Substitute the values:

\[
PV_{19} = \frac{3269}{0.0715}
\]

Calculating this:

\[
PV_{19} = 45714.69
\]

This value represents the present value at year $ t = 19 $.

### Step 2: Discount this present value back to year $ t = 8 $.
We need to discount the value at year $ t = 19 $ back to year $ t = 8 $, a total of $ 19 - 8 = 11 $ years.

The formula for discounting a future value to the present is:

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

where:
- $ FV = 45714.69 $ (future value at year 19),
- $ r = 0.0715 $,
- $ n = 11 $.

Substitute the values:

\[
PV_8 = \frac{45714.69}{(1 + 0.0715)^{11}}
\]

Calculating the denominator:

\[
(1 + 0.0715)^{11} \approx 2.134019
\]

Now, calculate the present value:

\[
PV_8 = \frac{45714.69}{2.134019} \approx 21422.88
\]

So, the value at year $ t = 8 $ is approximately **21422.88**.

### Summary
The value at year $ t = 8 $ of the perpetual stream of $3,269 payments starting at year $ t = 20 $, given an interest rate of 7.15%, is **21422.88**.

Would you like more details or have any questions?

Here are 5 related questions to further explore this concept:
1. How would the value change if the payments started at year 15 instead of year 20?
2. What if the interest rate was 5% instead of 7.15%?
3. How is the value affected if the payments are made annually instead of perpetually?
4. Can we calculate the value if payments increase at a fixed rate each year?
5. How would you find the value if payments were made semi-annually?

**Tip:** When discounting future values, always ensure the compounding periods match the frequency of the interest rate provided.

Given an interest rate of 7.15 percent per year, what is the value at year t = 8 of a perpetual stream of $3,269 payments that begin at year t = 20? (Do not include the dollar sign ($). Enter rounded answer as directed, but do not use the rounded numbers in intermediate calculations. Round your answer to 2 decimal places (e.g., 32.16).)

Find the present value at year 8 of a perpetual stream of $3,269 payments starting at year 20 with a 7.15% interest rate. This solution covers the calculation of perpetuity and discounting cash flows, perfect for finance students.

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