Problem 24 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Lupé made a down payment of $$\$ 4000$$ toward the purchase of a new car. To pay the balance of the purchase price, she has secured a loan from her bank at the rate of \(12 \% /\) year compounded monthly. Under the terms of her finance agreement, she is required to make payments of $$\$ 420$$ / \mathrm{month}\( for \)36 \mathrm{mo}$. What is the cash price of the car?

Step-by-Step Solution

Answer

The cash price of the car is $$\$ 14,868.86$$.

Step 1: Organize the given information

Let's first organize the given information, so that we can easily plug it into the formula: Down payment = $$\$ 4000$$ Monthly payment (PMT) = $$\$ 420$$ Annual interest rate (r) = 0.12 Number of payments (t) = 3 years (36 months) Number of compounding periods per year (n) = 12

Step 2: Calculate the present value of the annuity

Now, we will plug in the given information into the formula to calculate the present value (PV) of Lupé's monthly payments: \[PV=\frac{420\left[1-\left(1+\frac{0.12}{12}\right)^{-12\cdot 3}\right]}{\frac{0.12}{12}}\]

Step 3: Simplify the formula

Next, we will simplify the formula to get the present value (PV): \[PV=\frac{420\left[1-\left(1+0.01\right)^{-36}\right]}{0.01}\]

Step 4: Calculate the present value (PV)

Now, we will calculate the present value (PV) by simplifying the equation: \[PV=\frac{420\left[1-1.01^{-36}\right]}{0.01}\] \[PV=\frac{420\left[1-0.7408182\right]}{0.01}\] \[PV=\frac{420\cdot 0.2591818}{0.01}\] \[PV=10868.86\] So, the present value of Lupé's monthly payments is $$\$ 10,868.86$$.

Step 5: Calculate the cash price of the car

Finally, we will add the down payment to the present value (PV) to calculate the cash price of the car: Cash price = Down payment + Present value (PV) Cash price = 4000 + 10868.86 Cash price = $$\$ 14,868.86$$ Thus, the cash price of the car is $$\$ 14,868.86$$.

Key Concepts

Time Value of Money

The concept of the time value of money (TVM) is crucial in understanding how finances work in real-world scenarios. It is built on the principle that a sum of money you have now is worth more than an identical sum in the future due to its potential earning capacity. This core principle affects everything from personal savings to corporate finance and investments.

For instance, money can be invested to earn interest over time, meaning that if you had \(100 today, it could be worth \)105 next year, assuming a 5% interest rate. This potential for growth is why receiving money now is typically preferred. When it comes to loans or mortgages, the TVM is used to determine the current value of an annuity—the series of equal payments made at regular intervals. In the case of Lupé's car purchase, understanding the present value of her annuity payments helped in determining the true cost of the car.

Compound Interest

Compound interest is what makes the TVM particularly momentous for investments and loans. Unlike simple interest that only earns on the initial amount, compound interest is calculated on the initial principal and the accumulated interest of previous periods. This means that you earn interest on interest.

In Lupé’s situation, the 12% annual interest rate on her loan is compounded monthly, which will have a more significant impact over time than if it were compounded annually. That's because there are more periods for the interest to compound. To put it simply, with monthly compounding, the loan's balance grows every month as interest is added, then the next month's interest is calculated on this new, larger balance. This cycle leads to the amount of interest accruing much faster than if it was only calculated once a year.

Annuity Payment Calculations

Annuity payment calculations involve determining the present or future value of a series of payments or receipts that occur in regular intervals. There are two types of annuities: ordinary annuities and annuities due. For ordinary annuities, such as Lupé's car payments, the payment occurs at the end of each period.

To calculate the cash price of a car like Lupé's based on her annuity payments, you need a financial formula that factors in the payment amount, the interest rate, and the number of payments. This formula typically includes both geometric series and compound interest concepts to arrive at the present value of the annuity. Once the present value of all the payments is determined, it is often combined with any upfront costs or down payments to calculate the total price of an item—in this case, a car. The key takeaway from annuity payments is that they provide a way to assess the current worth of a stream of future payments, making it crucial in various financial planning and decision-making scenarios.