You are asked to price a security that pays $7000 every seventh year
forever, with the first payment occurring exactly 3 years from today. (The
payments will therefore occur at t=3, t=10, t=17, and so on.) The discount
rate is 5% per year compounded every nine months. What is the value of
this security today?
a) $14,735
b) $15,476
c) $17,073
d) $19,781
e) $20,777


To determine the present value of a security that pays $7,000 every seventh year forever, we need to follow these steps:

### Step 1: Adjust the discount rate
The discount rate is 5% per year, compounded every 9 months. We need to convert this nominal interest rate into an effective 7-year rate, as the payments occur every 7 years.

First, find the effective interest rate per period where a period is 9 months:

\[
r_{\text{9 months}} = \frac{5\%}{4} = 0.0125 = 1.25\% \text{ per 9 months}.
\]

Next, to find the effective rate for 7 years (which is 28 periods of 9 months), we use the formula for compound interest:

\[
(1 + r_{\text{9 months}})^{\text{Number of periods}} = (1 + 0.0125)^{28}.
\]

This gives us the effective discount factor over the 7-year intervals.

### Step 2: Calculate the present value of the perpetual cash flow
The present value (PV) of a perpetuity where payments start at $t = 3$, not at $t = 0$, can be computed by adjusting the general perpetuity formula. The basic formula for the PV of a perpetuity with payments made every 7 years and starting at $t = 3$ is:

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

where $P = 7000$ is the payment amount, and $r$ is the discount rate per 7-year interval.

### Step 3: Shift the payment to account for the first payment being at $t = 3$

We will discount the value of the perpetuity back to $t = 0$.

Let me calculate all of this.The present value of the security today is approximately **$14,497**. This is closest to option:

**a) $14,735**

Would you like more details or have any questions? Here are some related questions:

1. How does the choice of compounding period affect the present value?
2. What if the payments started at $t=7$ instead of $t=3$?
3. How does changing the discount rate impact the valuation of perpetual securities?
4. How would the value change if payments increased over time?
5. How does this compare to an annuity that lasts for a finite number of periods?

**Tip:** Converting nominal interest rates to effective rates for different periods is crucial in pricing securities correctly.

You are asked to price a security that pays $7000 every seventh year forever, with the first payment occurring exactly 3 years from today. (The payments will therefore occur at t=3, t=10, t=17, and so on.) The discount rate is 5% per year compounded every nine months. What is the value of this security today?
a) $14,735
b) $15,476
c) $17,073
d) $19,781
e) $20,777

Learn how to price a security that pays $7,000 every seventh year forever, with the first payment at t=3. This solution covers the application of discount rates compounded every 9 months and the present value calculation of a perpetuity. Suitable for undergraduate finance students.

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