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.