You're thinking of buying a car to travel around in, rather than paying $7,700 at the end of every year to use the train. You can buy an old second hand car for $8,000 now (t=0), and sell it after 10 years for $6,000 (t=10). You estimate that the car will cost $7,500 per year in fuel, insurance, registration and maintenance, paid in arrears, so there will be 10 payments from t=1 to t=10 inclusive. The required return is 4% pa. Select the most correct statement. The equivalent annual cost (EAC) of the car is:

Question 1Select one:

a.
$68,832 so the train is cheaper than the car.


b.
$8,712 so the train is cheaper than the car.


c.
$7,987 so the train is cheaper than the car.


d.
$7,700 so the car and train are equally costly.


e.
$6,478 so the car is cheaper than the train.

To determine the Equivalent Annual Cost (EAC) of the car, we need to follow these steps:

1. **Calculate the Present Value (PV) of the costs** associated with buying and maintaining the car.
2. **Calculate the EAC** using the annuity formula.

### Step 1: Present Value of Costs

- **Initial cost of the car (t=0):** $8,000.
- **Maintenance and operating costs:** $7,500 per year for 10 years, paid at the end of each year.
- **Salvage value (t=10):** $6,000.

The present value of the annual costs needs to be calculated using the formula for the present value of an annuity:
\[
PV = C \times \frac{1 - (1 + r)^{-n}}{r}
\]
where:
- $ C = 7,500 $ (annual cost),
- $ r = 0.04 $ (annual interest rate),
- $ n = 10 $ (number of years).

\[
PV_{\text{costs}} = 7,500 \times \frac{1 - (1 + 0.04)^{-10}}{0.04} 
\]

Let's calculate this:

\[
PV_{\text{costs}} = 7,500 \times \frac{1 - (1.04)^{-10}}{0.04} 
\]

Next, the present value of the salvage value at $ t = 10 $:

\[
PV_{\text{salvage}} = \frac{6,000}{(1 + 0.04)^{10}} 
\]

Finally, the total present value of the car is:

\[
PV_{\text{total}} = 8,000 + PV_{\text{costs}} - PV_{\text{salvage}} 
\]

### Step 2: Calculate EAC

The EAC is then calculated as:
\[
EAC = \frac{PV_{\text{total}} \times r}{1 - (1 + r)^{-n}} 
\]

Let's go ahead and compute these values.The Equivalent Annual Cost (EAC) of the car is approximately **$7,987**.

Given this, the most correct statement is:

**c. $7,987 so the train is cheaper than the car.**

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

1. How does the required return rate affect the EAC of an asset?
2. What would happen to the EAC if the car's maintenance costs were reduced?
3. How would extending the car's usage to 15 years change the EAC?
4. What other factors should be considered when comparing car ownership to public transportation?
5. How does the resale value of a car impact its overall cost of ownership?

**Tip:** When comparing costs, always consider the time value of money, especially for long-term investments or commitments.

You're thinking of buying a car to travel around in, rather than paying $7,700 at the end of every year to use the train. You can buy an old second-hand car for $8,000 now (t=0), and sell it after 10 years for $6,000 (t=10). You estimate that the car will cost $7,500 per year in fuel, insurance, registration, and maintenance, paid in arrears, so there will be 10 payments from t=1 to t=10 inclusive. The required return is 4% pa. Select the most correct statement. The equivalent annual cost (EAC) of the car is:

Learn how to calculate the Equivalent Annual Cost (EAC) of a car vs. train travel, considering purchase price, annual costs, salvage value, and the required return rate. This problem covers essential financial mathematics concepts like present value and annuities, suitable for undergraduate finance or economics students.

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