Question
What is the value today of receiving $1,470.00 per year forever? Assume the first payment is made 7.00 years from today and the discount rate is 12.00%.
Answers
GPT-4o mini
To determine the present value of receiving $1,470 per year forever (a perpetuity) with the first payment starting in 7 years, we can follow these steps:
1. **Calculate the present value of the perpetuity at the time the payments start.**
The formula for the present value of a perpetuity is:
\[
PV = \frac{C}{r}
\]
Where:
- \(C\) is the annual cash flow,
- \(r\) is the discount rate.
Here:
- \(C = 1,470\)
- \(r = 0.12\)
\[
PV = \frac{1,470}{0.12} = 12,250
\]
This value of $12,250 is the present value of the perpetuity at the time the first payment is received (which is 7 years from today).
2. **Discount this value back to today.**
Now, we need to discount this present value back to today (which is 7 years earlier). The formula for the present value (PV) is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \(FV\) is the future value (which is $12,250),
- \(r\) is the discount rate (0.12),
- \(n\) is the number of years (7).
\[
PV = \frac{12,250}{(1 + 0.12)^7}
\]
First, calculate \((1 + 0.12)^7\):
\[
(1 + 0.12)^7 \approx 2.21068
\]
Now, calculate the present value:
\[
PV \approx \frac{12,250}{2.21068} \approx 5,537.25
\]
So, the value today of receiving $1,470.00 per year forever, starting 7 years from today, at a discount rate of 12.00% is approximately **$5,537.25**.
1. **Calculate the present value of the perpetuity at the time the payments start.**
The formula for the present value of a perpetuity is:
\[
PV = \frac{C}{r}
\]
Where:
- \(C\) is the annual cash flow,
- \(r\) is the discount rate.
Here:
- \(C = 1,470\)
- \(r = 0.12\)
\[
PV = \frac{1,470}{0.12} = 12,250
\]
This value of $12,250 is the present value of the perpetuity at the time the first payment is received (which is 7 years from today).
2. **Discount this value back to today.**
Now, we need to discount this present value back to today (which is 7 years earlier). The formula for the present value (PV) is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \(FV\) is the future value (which is $12,250),
- \(r\) is the discount rate (0.12),
- \(n\) is the number of years (7).
\[
PV = \frac{12,250}{(1 + 0.12)^7}
\]
First, calculate \((1 + 0.12)^7\):
\[
(1 + 0.12)^7 \approx 2.21068
\]
Now, calculate the present value:
\[
PV \approx \frac{12,250}{2.21068} \approx 5,537.25
\]
So, the value today of receiving $1,470.00 per year forever, starting 7 years from today, at a discount rate of 12.00% is approximately **$5,537.25**.