A firm is must choose to buy the GSU-3300 or the UGA-3000. Both machines make the firm’s production process more efficient which in turn increases incremental cash flows. The GSU-3300 produces incremental cash flows of $25,480.00 per year for 8 years and costs $99,731.00. The UGA-3000 produces incremental cash flows of $27,516.00 per year for 9 years and cost $124,246.00. The firm’s WACC is 8.66%. What is the equivalent annual annuity of the GSU-3300? Assume that there are no taxes.

Answer 1

To calculate the Equivalent Annual Annuity (EAA) for the GSU-3300, we first need to determine its Net Present Value (NPV) using the cash flows, the initial investment, and the firm's Weighted Average Cost of Capital (WACC).

Step 1: Calculate NPV of GSU-3300

Given:

• Incremental Cash Flows: \( CF = 25,480 \)
• Cost of GSU-3300: \( C = 99,731 \)
• Life of Machine: \( n = 8 \) years
• WACC: \( r = 8.66% = 0.0866 \)

The formula for NPV is:

\[ NPV = \sum_{t=1}^{n} \frac{CF}{(1 + r)^t} - C \]

Substituting the values, we need to calculate the present value of each cash flow over 8 years:

1. Calculate the present value factors for each year:

\[ PV = \frac{CF}{(1 + r)^t} \]

This can be simplified using the present value of an annuity formula:

\[ PV_{\text{annuity}} = CF \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \]

Substituting the values:

\[ PV_{\text{annuity}} = 25,480 \times \left(\frac{1 - (1 + 0.0866)^{-8}}{0.0866}\right) \]

Calculating \( (1 + 0.0866)^{-8} \):

\[ (1 + 0.0866)^{-8} \approx 0.513679 \quad \text{(using a scientific calculator)} \]

So,

\[ PV_{\text{annuity}} = 25,480 \times \left(\frac{1 - 0.513679}{0.0866}\right) \]

Calculating the fraction:

\[ \frac{1 - 0.513679}{0.0866} \approx \frac{0.486321}{0.0866} \approx 5.6191889 \]

Now, calculate the total present value of cash flows:

\[ PV_{\text{annuity}} \approx 25,480 \times 5.6191889 \approx 143,594.75 \]

Now calculate the NPV:

\[ NPV = PV_{\text{annuity}} - C = 143,594.75 - 99,731 \approx 43,863.75 \]

Step 2: Calculate EAA

Now that we have the NPV, we will calculate the Equivalent Annual Annuity (EAA) using the following formula:

\[ EAA = \frac{NPV \times r}{1 - (1 + r)^{-n}} \]

Substituting in the values:

\[ EAA = \frac{43,863.75 \times 0.0866}{1 - (1 + 0.0866)^{-8}} \]

Using \( (1 + 0.0866)^{-8} \approx 0.513679 \):

So,

\[ EAA = \frac{43,863.75 \times 0.0866}{1 - 0.513679} \]

Calculating the denominator:

\[ 1 - 0.513679 \approx 0.486321 \]

Now calculating EAA:

\[ EAA \approx \frac{43,863.75 \times 0.0866}{0.486321} \approx \frac{3,798.12}{0.486321} \approx 7,805.68 \]

Conclusion

The Equivalent Annual Annuity (EAA) of the GSU-3300 is approximately $7,805.68 per year.