Problem 6 You have just purchased a Kia wi... [FREE SOLUTION]

Chapter 14: Problem 6

You have just purchased a Kia with a $\$ 20,000$ price tag. The dealer offers to let you pay for your car in five equal annual installments, with the first payment due in a year. a. If the dealer finances your purchase at an interest rate of $10 \%,$ how much will your annual payment be? b. How much would your payment be if you had purchased a $\$ 40,000$ Camry instead of a $\$ 20,000 \mathrm{Kia} ?$ c. How much would your payment be if you arranged to pay in 10 annual installments instead of $5 ?$ Is your payment cut in half? Why or why not? d. How much would your payment fall if you paid $\$ 10,000$ down at the time of purchase?

Short Answer

Expert verified

a) $5277.82; b) $10555.64; c) $3257.32; d) $2638.91.

Step by step solution

Understanding Annual Payment Calculation

To calculate the annual payment for a loan, we use the formula for the annuity payment: \[ P = \frac{r \times PV}{1 - (1 + r)^{-n}} \]where $ P $ is the payment, $ r $ is the interest rate per period, $ PV $ is the present value of the loan, and $ n $ is the number of periods.

Step A: Calculate Annual Payment for $20,000 Kia

For the $20,000 Kia, we have \( PV = 20000 $, $ r = 0.10 $, and $ n = 5 $. Substituting these into the formula:\[ P = \frac{0.10 \times 20000}{1 - (1 + 0.10)^{-5}} \]\[ P = \frac{2000}{1 - (1.10)^{-5}} \]Calculate $(1.10)^{-5} \approx 0.62092$, so\[ P \approx \frac{2000}{1 - 0.62092} \approx \frac{2000}{0.37908} \approx 5277.82 \]Thus, the annual payment for the Kia is approximately \)5277.82.

Step B: Annual Payment for $40,000 Camry

For the $40,000 Camry, \( PV = 40000 $. Using the same method:\[ P = \frac{0.10 \times 40000}{1 - (1 + 0.10)^{-5}} \]\[ P \approx \frac{4000}{0.37908} \approx 10555.64 \]The annual payment for the Camry is approximately \)10555.64.

Step C: Payment Over 10 Installments for Kia

Now calculate for 10 annual installments for the Kia, $ n = 10 $:\[ P = \frac{0.10 \times 20000}{1 - (1 + 0.10)^{-10}} \]\[ P \approx \frac{2000}{1 - 0.38554} \approx \frac{2000}{0.61446} \approx 3257.32 \]The payment is not half because the interest accumulates over more periods. The payment for Kia over 10 years is approximately $3257.32.

Step D: Impact of Down Payment on Kia

If you pay $10,000 down, only $10,000 needs to be financed. So:\[ PV = 10000 \], $ n = 5 $:\[ P = \frac{0.10 \times 10000}{1 - (1 + 0.10)^{-5}} \]\[ P \approx \frac{1000}{0.37908} \approx 2638.91 \]Paying $10,000 down reduces the payment to approximately $2638.91.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Annuity Payment Formula

The annuity payment formula is an essential tool used to calculate the periodic payments for a loan or investment repaid through installments. It helps to determine how much should be paid per period to eventually cover the principal and interest owed.
The formula is given by:\[P = \frac{r \times PV}{1 - (1 + r)^{-n}}\]Here:

$P$ represents the periodic payment amount.
$r$ is the interest rate per period expressed as a decimal.
$PV$ denotes the present value of the loan or initial amount borrowed.
$n$ stands for the total number of payments or periods.

This formula is primarily used in financial situations involving fixed-rate loans or bonds.
In our context of calculating loan payments for a car purchase, understanding this formula allows buyers to know exactly how much they will need to budget for each payment period.

Interest Rate Impact

Interest rates have a pivotal effect on the calculation of loan payments. The higher the interest rate, the larger the portion of each payment that goes towards interest, thus increasing the total payment amount required.
The interest rate in the annuity formula acts as a multiplier, illustrating its impact: - **Higher Rates**: Increase the dollar amount of each payment because more interest is accumulated over each period.
- **Lower Rates**: Reduce the amount paid in interest, thereby reducing each payment.
Using a 10% interest rate as seen in our exercise, the annual calculations demonstrated how this percentage affected payments for both a Kia and a Camry. The greater the initial loan value or price of a vehicle, the more pronounced the impact of the interest rate is on the total payment. It鈥檚 crucial to compare different lenders鈥� rates to minimize costs over the duration of the loan.

Loan Term Effect

The term of a loan鈥攎eaning the number of periods over which it鈥檚 repaid鈥攕ignificantly influences the size of each installment. Loan terms can range from a short term, with fewer large payments, to a longer term, which spreads payments out over more periods.

**Shorter Loan Terms**: Payments are higher each period but the total interest paid is less due to fewer total periods.
**Longer Loan Terms**: Lessens the individual payment amounts but results in greater total interest paid, as illustrated in our example where spreading payments over 10 years instead of 5 reduced the annual payment but increased interest accumulation.

It's critical to strike a balance between manageable payments and total interest costs, keeping future financial flexibility in mind.

Down Payment Effect

Making a down payment on a purchase reduces the principal amount that must be financed, which subsequently lessens the burden of interest and the amount of each installment payment required.

**Reduced Principal**: A direct reduction in the amount on which interest is calculated.
**Lower Payments**: Allows for smaller periodic payments, as evident when a $10,000 down payment reduces the financed amount and thus the payment by half for the Kia example.

Down payments not only decrease the size of each payment but also can potentially lower the overall cost of borrowing. Moreover, a substantial down payment might secure better terms on the loan, as lenders perceive less risk.

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