Given a discount rate of 4.4 percent per year, what is the value at Date t = 10 of a perpetual stream of $2,800 payments with the first payment at Date t = 20?

To calculate the value of a perpetual stream of payments, we use the formula for the present value of a perpetuity:

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

Where:
- $ PV $ is the present value of the perpetuity,
- $ C $ is the annual payment (in this case, $2,800),
- $ r $ is the discount rate (4.4% or 0.044).

The perpetual stream begins at Date $ t = 20 $. To find the value at Date $ t = 10 $, we follow these steps:

### Step 1: Calculate the value of the perpetuity at Date $ t = 19 $ (one year before the first payment at $ t = 20 $).

\[
PV_{t=19} = \frac{2,800}{0.044} = 63,636.36
\]

### Step 2: Discount the value from $ t = 19 $ back to $ t = 10 $.

To do this, we use the present value formula for discounting a lump sum:

\[
PV_{t=10} = PV_{t=19} \times \frac{1}{(1 + r)^{19 - 10}} = 63,636.36 \times \frac{1}{(1 + 0.044)^9}
\]

Let's calculate this value.

\[
PV_{t=10} = 63,636.36 \times \frac{1}{(1.044)^9}
\]The value of the perpetual stream of $2,800 payments at Date $ t = 10 $, given a discount rate of 4.4%, is approximately **$43,191.76**.

Would you like further details or clarifications on this?

Here are some related questions you might find useful:

1. How would the value change if the discount rate were different?
2. What if the first payment started at $ t = 15 $ instead of $ t = 20 $?
3. How does the formula for perpetuity differ from the formula for an annuity?
4. What are the effects of inflation on the value of perpetuity payments?
5. How would you calculate the present value if the payments were not perpetual?

**Tip:** In perpetuity problems, shifting the starting time of payments affects when the present value is discounted but doesn't change the core perpetuity formula.

Learn how to calculate the present value of a perpetual stream of $2,800 payments starting at Date t = 20, discounted back to Date t = 10 at a 4.4% annual rate. This solution uses key finance formulas such as the perpetuity and discounting formulas. Ideal for college-level finance students.

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