To find the value of the investment opportunity today (also known as the present value), we need to break down the cash flows and discount them back to the present value using the given discount rate of 19%.
Step 1: Cash Flows
The investor will receive $9,300 per year for 12 years starting at the end of year 5. Therefore, the cash flows occur in years 5 through 16.
Step 2: Present Value of Cash Flows
To calculate the present value of the cash flows received from years 5 to 16, we need to find the present value of an annuity since the investor receives a fixed amount each year.
The formula for the present value (PV) of an annuity is:
\[ PV = C \times \left( 1 - (1 + r)^{-n} \right) / r \]
where:
- \( C \) = cash flow per period ($9,300)
- \( r \) = discount rate (19% or 0.19)
- \( n \) = number of periods (12 years)
First, we calculate the present value of the annuity at the end of Year 4 (just before the first cash flow at the end of Year 5):
\[ PV_{\text{Year 4}} = 9,300 \times \left( 1 - (1 + 0.19)^{-12} \right) / 0.19 \]
Calculating \( (1 + 0.19)^{-12} \):
\[ (1 + 0.19)^{-12} \approx (1.19)^{-12} \approx 0.1667 \]
Now substituting into the formula:
\[ PV_{\text{Year 4}} = 9,300 \times \left( 1 - 0.1667 \right) / 0.19 \] \[ PV_{\text{Year 4}} = 9,300 \times \left( 0.8333 \right) / 0.19 \] \[ PV_{\text{Year 4}} = 9,300 \times 4.388 \] \[ PV_{\text{Year 4}} \approx 40,746.90 \]
Step 3: Discounting the Present Value to Today
Now, we need to discount the value we calculated at Year 4 back to today (Year 0):
\[ PV_{\text{Today}} = PV_{\text{Year 4}} / (1 + r)^{4} \]
Now, substitute the values in:
\[ PV_{\text{Today}} = 40,746.90 / (1.19)^{4} \] \[ (1.19)^{4} \approx 2.1813 \] \[ PV_{\text{Today}} = 40,746.90 / 2.1813 \approx 18,634.52 \]
Final Result
The value of this investment opportunity today is approximately $18,634.52.
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