To find the Equivalent Annual Annuity (EAA) for the UGA-3000, we need to first calculate its Net Present Value (NPV) using the provided cash flows and then convert the NPV into an EAA.
Step 1: Calculate the NPV of the UGA-3000.
The formula for NPV is:
\[ NPV = \sum \frac{CF_t}{(1 + r)^t} - Initial\ Investment \]
Where:
- \(CF_t\) = Cash flow at time \(t\)
- \(r\) = discount rate (WACC)
- \(t\) = year
Given Data:
- Cash inflow per year (CF) = $28,544
- Number of years (n) = 9
- Initial investment = $125,803
- Discount rate (r) = 9.57% or 0.0957
Calculate Present Value of Cash Flows
We can use the Present Value of Annuity formula since the cash flows are constant.
\[ PV = CF \times \left(1 - \frac{1}{(1 + r)^n}\right) \div r \]
Plugging in the values:
\[ PV = 28,544 \times \left(1 - \frac{1}{(1 + 0.0957)^9}\right) \div 0.0957 \]
Calculating \((1 + 0.0957)^9\):
\[ (1.0957)^9 \approx 2.4662 \]
Then calculate the present value factor:
\[ 1 - \frac{1}{2.4662} \approx 1 - 0.405 = 0.595 \]
Now substitute back into the PV formula:
\[ PV = 28,544 \times \frac{0.595}{0.0957} \]
Calculating the fraction:
\[ PV = 28,544 \times 6.223 = 177,714.03 \]
Step 2: Calculate the NPV
Now subtract the initial investment from the total present value of cash inflows.
\[ NPV = PV - Initial\ Investment \] \[ NPV = 177,714.03 - 125,803 = 51,911.03 \]
Step 3: Calculate the Equivalent Annual Annuity (EAA)
EAA can be calculated using the formula:
\[ EAA = \frac{NPV \times r}{1 - (1 + r)^{-n}} \]
Substituting our values for NPV, r, and n:
\[ EAA = \frac{51,911.03 \times 0.0957}{1 - (1 + 0.0957)^{-9}} \]
Calculate \((1 + 0.0957)^{-9}\):
\[ (1.0957)^{-9} \approx 0.405 \]
Now calculate the denominator:
\[ 1 - 0.405 \approx 0.595 \]
Now substitute back into the EAA formula:
\[ EAA = \frac{51,911.03 \times 0.0957}{0.595} \] \[ EAA = \frac{4,973.415}{0.595} \approx 8,358.99 \]
Thus, the Equivalent Annual Annuity of the UGA-3000 is approximately $8,358.99.
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