AT&T is choosing between two bus models. One is more expensive to purchase and maintain but lasts much longer than the other. Its discount rate is 10.2%. The company plans to continue with one of the two models for the foreseeable future. Based on the costs of each shown below, what is the annual cost of each model as an equivalent annual annuity (EAA)? Which should it choose? (Note: dollar amounts are in thousands.)
Model Long Life
Year 0
Year 1 Year 2
Year 3
Year 4
Year 5
Year6
Year 7
-$203.9
-$3.8
-$3.8
-$3.8
-$3.8
-$3.8
-$3.8
-$3.8
Short Life
-$108.1
-$1.7
-$1.7
-$1.7
-$1.7
a. The annual cost of Long life model as an equivalent annual annuity is $
. (Round to
the nearest dollar.)
The annual cost of Short life model as an equivalent annual annuity is $
(Round to the nearest dollar.)

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.

---

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.

Learn how to calculate the Equivalent Annual Annuity (EAA) for two bus models: Long Life and Short Life. Understand the financial implications of choosing between these models based on their costs and discount rate. Detailed step-by-step solution provided.

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