91Ó°ÊÓ

In Exercises 1-10, use $$ P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} . $$ Round answers to the nearest dollar. Suppose that you are buying a car for \(\$ 60,000\), including taxes and license fees. You saved \(\$ 10,000\) for a down payment. The dealer is offering you two incentives: Incentive \(\mathrm{A}\) is \(\$ 5000\) off the price of the car, followed by a five-year loan at \(7.34 \%\). Incentive B does not have a cash rebate, but provides free financing (no interest) over five years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?10. Suppose that you are buying a car for \(\$ 56,000\), including taxes and license fees. You saved \(\$ 8000\) for a down payment. The dealer is offering you two incentives: Incentive \(A\) is \(\$ 10,000\) off the price of the car, followed by a four-year loan at \(12.5 \%\). Incentive B does not have a cash rebate, but provides free financing (no interest) over four years. What is the difference in monthly payments between the two offers? Which incentive is the better deal?

Step by step solution

01

Calculate the Monthly Payments for Incentive A

Let's first calculate for Incentive A. The given formula for PMT is \( PMT=\frac{P\left(\frac{r}{n}\right)}{1-\left(1+\frac{r}{n}\right)^{-nt}} \). The only unknown variable here is PMT, the monthly payment. All the other variables are given in the problem (P is the principal amount left after down payment and rebate, r is the interest rate, n is the number of intervals per year which equals 12 in this case, and t is the number of years). Substitute these values into the formula for incentive A - for both the car price options, and round it to the nearest dollar.

02

Calculate the Monthly Payments for Incentive B

Again using the formula for PMT, substitute the values for Incentive B in the formula. This time, however, r (the interest rate) is 0%, which simplifies the equation greatly. The monthly payment is simply the price of the car (minus the down payment) divided by the number of months in the loan term (5 years times 12 months in this case - for both the car price options). Again round it to the nearest dollar.

03

Calculate the Difference in Monthly Payments

Calculate the difference between monthly payments under Incentive A and Incentive B for both the car price options. This simply involves subtracting the respective values obtained in step 1 and 2.

04

Determine the Better Deal

Now that the difference in monthly payments has been calculated, we can compare them to find out the better deal. In general, the lower the monthly payments, the better the deal—for every month that you have to make a payment, you'd prefer that payment to be as small as possible.

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Key Concepts

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

Mathematical Finance

Mathematical finance revolves around analyzing financial markets and the instruments they offer, using various mathematical tools and formulas. It helps individuals and businesses make smarter financial decisions based on numerical data, theories, and models. For instance, when buying a car with different incentives, mathematical finance aids in quantifying which option is the most cost-effective over time.

One of the most crucial aspects of mathematical finance is understanding how the value of money changes over time. This can involve calculating the future value of investments, or in the case of loans, understanding how interest and payments over time affect the overall cost of a purchase such as a car. By applying the principles of mathematical finance, we are able to determine the present and future values of annuities, which are series of payments made at regular intervals.

Simple Interest

Simple interest is a fundamental concept in mathematical finance, representing the most straightforward way to calculate the interest on a loan or investment. It is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments. This calculation is different from compound interest, where interest is calculated on the initial principal and also on the accumulated interest of previous periods.

In the context of the exercise, simple interest can be seen in the way monthly payments for Incentive A are calculated. Here, the interest is applied to the remaining loan amount after a down payment and any rebates. The formula for calculating the monthly payment of a loan considering the simple interest rate is a foundational model in the study of financial mathematics.

Present Value of Annuity

The Present Value of an Annuity is a key concept in understanding how to value a series of future payments in today's dollars. An annuity is a contract typically issued by an insurance company that promises to pay the holder a set amount of money in regular intervals. The present value is calculated by discounting these future payments to reflect the opportunity cost of money, considering the time value of money principle, which states that a dollar today is worth more than a dollar in the future because of its earning potential.

Understanding the present value of an annuity is crucial when one wishes to compare different financial incentives, such as the car payment plans in the textbook exercise. It equips you with the ability to decide which series of payments, or financial structure, is most advantageous. This involves estimating the value of the payment stream if all payments were made now rather than in the future. In our exercise, using the present value of the annuity formula simplifies the choice between taking a discount upfront or opting for a no-interest financing option.