Understanding Present Value of Perpetuity
To find the present value of a perpetuity, we use the formula:
\[ \text{PV} = \frac{C}{r} \]
where:
- \( C \) = cash flow per period (in this case, 10)
- \( r \) = rate of return (20% or 0.20)
However, since the cash flow starts from the 6th year, we first need to calculate the present value at the beginning of the 6th year and then discount it back to the present value today (year 0).
Step 1: Calculate Present Value at Year 5
- Cash Flow: 10
- Rate of Return: 20% (0.20)
The present value at the end of year 5 (when the perpetuity starts) is:
\[ \text{PV}_{5} = \frac{10}{0.20} = 50 \]
This means that at the end of year 5, the value of the cash flows starting from year 6 is 50.
Step 2: Discount to Present Value
Now, we need to discount this amount back to the present value (year 0). The formula for discounting is:
\[ \text{PV} = \frac{\text{Future Value}}{(1 + r)^n} \]
where \( n \) is the number of years until the cash flow starts.
- Future Value (PV at year 5): 50
- Rate of Return: 20% (0.20)
- Number of Years: 5
Now, we can calculate the present value:
\[ \text{PV} = \frac{50}{(1 + 0.20)^5} \]
Calculating \( (1.20)^5 \):
\[ (1.20)^5 \approx 2.48832 \]
Now substituting back into the PV formula:
\[ \text{PV} = \frac{50}{2.48832} \approx 20.1 \]
Final Present Value
Thus, the present value of the perpetuity that starts in the 6th year is approximately 20.1. This value reflects the current worth of receiving 10 annually starting from the 6th year, discounted at a rate of 20%.