A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be 

$ 3 comma 000$3,000. 

Each year after​ that, you will receive a payment on the anniversary of the last payment that is 

6 %6% 

larger than the last payment. This pattern of payments will go on forever. Assume that the interest rate is 

12 %12% 

per year.

**a.** What is​ today's value of the​ bequest?

**b.** What is the value of the bequest immediately after the first payment is​ made?

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Part 1

**a.** What is​ today's value of the​ bequest?

​Today's value of the bequest is 

​$enter your response here. 

 ​(Round to the nearest​ cent.)

This is a problem involving the present value of a growing perpetuity. A growing perpetuity is a series of cash flows that continue forever, where each cash flow grows at a constant rate each period. The formula to calculate the present value of a growing perpetuity is given by:

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

Where:
- $PV$ = Present value of the growing perpetuity
- $C$ = First payment
- $r$ = Interest rate (discount rate)
- $g$ = Growth rate of the payments

### Given:
- First payment $C = 3000$
- Growth rate $g = 6\% = 0.06$
- Interest rate $r = 12\% = 0.12$

#### Part (a): Calculating the present value today

Using the formula for the present value of a growing perpetuity:

\[
PV = \frac{3000}{0.12 - 0.06} = \frac{3000}{0.06} = 50000
\]

So, today's value of the bequest is:

\[
PV = 50,000 \text{ dollars}
\]

### Part (b): Value of the bequest immediately after the first payment

After the first payment is made, the next payment becomes $C \times (1 + g)$, so the value of the perpetuity changes to the present value of a growing perpetuity starting at the second payment.

To find the value immediately after the first payment:
- The new first payment will be $3000 \times (1.06) = 3180$

So, the new value of the perpetuity will be:

\[
PV_{\text{new}} = \frac{3180}{0.12 - 0.06} = \frac{3180}{0.06} = 53000
\]

Thus, the value of the bequest immediately after the first payment is $53,000$ dollars.

### Summary:
- (a) Today's value of the bequest: **$50,000**
- (b) Value immediately after the first payment: **$53,000**

Would you like to see further details, or do you have any questions?

### Related questions:
1. What would happen to the present value if the growth rate increased to 8%?
2. How does changing the interest rate to 10% affect the present value?
3. What is the value of the bequest 5 years from now?
4. What is the total amount of payments received after 10 years?
5. How would the formula change if the payments were made quarterly instead of annually?

**Tip:** In perpetuity calculations, make sure the interest rate is higher than the growth rate; otherwise, the present value formula will not work.

A rich relative has bequeathed you a growing perpetuity. The first payment will occur in a year and will be $3,000. Each year after that, you will receive a payment on the anniversary of the last payment that is 6% larger than the last payment. This pattern of payments will go on forever. Assume that the interest rate is 12% per year. a. What is today's value of the bequest? b. What is the value of the bequest immediately after the first payment is made?

Learn how to calculate the present value of a growing perpetuity with an initial payment of $3,000, a 6% growth rate, and a 12% interest rate. The perpetuity continues forever, with payments increasing each year. This detailed solution includes the present value formula and shows step-by-step how to compute both the value today and immediately after the first payment.

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