To determine the equivalent annual annuity (EAA) for each model, we'll follow these steps:

1. Calculate the Present Value (PV) of Costs: For both models, we'll discount each future cost to the present value using the given discount rate (10.2%).

2. Calculate the EAA: The EAA is the annual payment that, when discounted at the same rate, equals the present value of the costs over the lifespan of the model.

Step 1: Present Value Calculation

Let's calculate the present value of costs for each model.

Long Life Model

Given:

• Discount Rate $r = 10.2\% = 0.102$
• Costs: $-203.9$ in Year 0, $-3.8$ in Years 1 to 7.

The PV of costs for the Long Life Model is:

$PV = -203.9 + \frac{-3.8}{(1+0.102)^1} + \frac{-3.8}{(1+0.102)^2} + \cdots + \frac{-3.8}{(1+0.102)^7}$

Short Life Model

Given:

• Discount Rate $r = 10.2\% = 0.102$
• Costs: $-108.1$ in Year 0, $-1.7$ in Years 1 to 4.

The PV of costs for the Short Life Model is:

$PV = -108.1 + \frac{-1.7}{(1+0.102)^1} + \frac{-1.7}{(1+0.102)^2} + \cdots + \frac{-1.7}{(1+0.102)^4}$

Step 2: Calculate the Equivalent Annual Annuity (EAA)

The EAA formula is:

$EAA = \frac{PV}{\frac{1 - (1 + r)^{-n}}{r}}$

Where:

• $PV$ is the present value of the costs
• $r$ is the discount rate
• $n$ is the number of periods (years)

Let's compute these values.

Computations

1. Long Life Model

   $PV_{\text{Long Life}} = -203.9 + \sum_{t=1}^{7} \frac{-3.8}{(1+0.102)^t}$

   Using the EAA formula:

   $EAA_{\text{Long Life}} = \frac{PV_{\text{Long Life}}}{\frac{1 - (1 + 0.102)^{-7}}{0.102}}$

2. Short Life Model

   $PV_{\text{Short Life}} = -108.1 + \sum_{t=1}^{4} \frac{-1.7}{(1+0.102)^t}$

   Using the EAA formula:

   $EAA_{\text{Short Life}} = \frac{PV_{\text{Short Life}}}{\frac{1 - (1 + 0.102)^{-4}}{0.102}}$

Final Results

Let's calculate the actual values for each model.

I'll perform these calculations now.The calculations yield the following results:

• Long Life Model: The equivalent annual annuity (EAA) is $45,958 (rounded to the nearest dollar).
• Short Life Model: The equivalent annual annuity (EAA) is $35,950 (rounded to the nearest dollar).

Conclusion:

Since the Short Life Model has a lower EAA, it is the more cost-effective choice for AT&T, assuming they plan to continue with one of these models for the foreseeable future.

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Would you like more details on the calculations or have any other questions?

Related Questions:

1. How does the discount rate affect the EAA calculation?
2. What happens to the EAA if the costs increase each year instead of remaining constant?
3. How would a change in the lifespan of the models impact the EAA?
4. What are the implications of choosing a model with a higher initial cost but lower annual maintenance?
5. How would a decrease in the discount rate to 8% change the EAAs?
6. Could inflation influence the decision between the two models?
7. What is the impact of including a salvage value in the EAA calculation?
8. How does the concept of Net Present Value (NPV) relate to the EAA?

Tip:

Always consider both initial costs and ongoing maintenance when evaluating long-term investments, as the initial choice might lead to significant savings over time.