Question

What values do I enter for N, PMT, i/y, FV and PV on a finance calculator to solve for the question below.
What is the value today of receiving $1,644.00 per year forever? Assume the first payment is made 8.00 years from today and the discount rate is 11.00%.

Answer 1

To find the present value of a perpetuity where the first payment is received in the future, we need to break it down into two steps:

1. **Calculate the present value of the perpetuity as of the first payment date**
2. **Calculate the present value of that amount to today's date**

### Step 1: Calculate the Present Value of the Perpetuity

A perpetuity is valued using the formula:

\[
PV_{t} = \frac{PMT}{i}
\]

Where:
- \( PMT \) = payment per period ($1,644.00)
- \( i \) = interest rate (11% or 0.11)

Using this, we can calculate the present value at the time the first payment is received, which is 8 years from now.

\[
PV_{t} = \frac{1644}{0.11} = 14945.45
\]

Here, \( PV_{t} \) is the value of the perpetuity at Year 8.

### Step 2: Calculate the Present Value Today

Now, we need to find the present value of \( PV_{t} \) as of today.

The formula to calculate the present value from a future value is:

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

Where:
- \( FV = 14,945.45 \) (the perpetuity value we just calculated)
- \( r = 0.11 \) (the discount rate)
- \( n = 8 \) (the number of years until the first payment is received)

So, we calculate:

\[
PV = \frac{14945.45}{(1 + 0.11)^8}
\]

Calculating \( (1 + 0.11)^8 \):

\[
(1.11)^8 \approx 2.3996
\]

Now plug this into the present value formula:

\[
PV = \frac{14945.45}{2.3996} \approx 6226.55
\]

Thus, the present value today of receiving $1,644.00 per year forever starting 8 years from now, assuming a discount rate of 11%, is approximately $6,226.55.

### Inputting Values on a Finance Calculator:

For the finance calculator, enter the following values:

**For the first calculation (Step 1):**
- \( PMT = 1644 \)
- \( i/y = 11 \) (which reflects a 11% interest rate)
- \( N = \text{not needed, since it's a perpetuity)**
- \( FV = 0 \) (because we are calculating the present value)
- \( PV = \text{what you would solve for as } 14945.45 \text{ (after entering percent rate)} \)

**For the second calculation (Step 2):**
- \( PMT = 0 \) (no payments to be received)
- \( i/y = 11 \)
- \( N = 8 \)
- \( FV = 14945.45 \) (the calculated perpetuity value)
- \( PV = \text{what you would solve for as approximately } 6226.55 \)

With these inputs, you can find both the value of the perpetuity starting in the future and its equivalent present value as of today.