Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The Betzes have leased an auto for 2 yr at $$\$ 450 /$$ month. If money is worth \(9 \% /\) year compounded monthly, what is the equivalent cash payment (present value) of this annuity?

Step-by-Step Solution

Answer

The equivalent cash payment (present value) of the 2-year auto lease with monthly payments of $450 at a 9% annual interest rate compounded monthly is approximately $9784.47.

Step 1: Identify the Given Information

We are given the following information: 1. Monthly payment (Annuity): P = 450 2. Annuity length: n = 2 years * 12 months/year = 24 months 3. Annual interest rate: 9% per year compounded monthly

Step 2: Convert Annual Interest Rate to Monthly Interest Rate

To convert the annual interest rate to a monthly interest rate, we simply divide the annual interest rate by 12. So, the monthly interest rate is: \(r_m = \frac{9 \%}{12} = 0.75 \% = 0.0075\)

Step 3: Apply Present Value of Annuity Formula

Now, we will use the present value of annuity formula to find the equivalent cash payment: \(PV = P * \frac{1 - (1 + r_m)^{-n}}{r_m}\)

Step 4: Substitute Given Values

Substitute the values we found in step 2 and step 1 into the formula: \(PV = 450 * \frac{1 - (1 + 0.0075)^{-24}}{0.0075}\)

Step 5: Calculate the Present Value

Now, calculate the present value: \(PV = 450 * \frac{1 - (1.0075)^{-24}}{0.0075} \approx 9784.47\) So, the equivalent cash payment (present value) of this annuity is approximately $9784.47.

Key Concepts

Compound Interest

Compound interest is a powerful concept in finance that can significantly affect the calculation of present values and future values. Unlike simple interest, which is calculated based only on the principal amount, compound interest involves earning interest on both the initial principal and on the interest that has been added to it over previous periods. This means that each period's interest payment is larger than the previous period's, assuming a positive interest rate.

Here’s how compound interest works:

• You start with an initial principal amount.
• Interest is calculated on this initial amount as well as on accumulated interest from prior periods.

Consider the formula for compound interest:
\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]
where:

• \(A\) = the future value of the investment/loan, including interest.
• \(P\) = the principal investment amount (initial deposit or loan amount).
• \(r\) = the annual interest rate (as a decimal).
• \(n\) = the number of times that interest is compounded per year.
• \(t\) = the time the money is invested or borrowed for, in years.

Through the power of compounding, interest grows much faster over time, benefiting those who invest or save early.

Monthly Interest Rate

Understanding the monthly interest rate is important when dealing with annuities or loans that have monthly payments. Many financial problems involve calculating interest on a monthly basis because loans and leases often use this timeframe.

The monthly interest rate is simply the annual interest rate divided by 12, the number of months in a year. This division allows you to work with interest in a way that matches the frequency of your payment periods.

• For example, if the annual interest rate is 9%, the monthly interest rate is calculated as follows:
  \[ r_m = \frac{9\%}{12} = 0.75\% \]
• In decimal form, this becomes:
  \[ r_m = 0.0075 \]

Understanding how to accurately convert annual to monthly rates is crucial for correctly using financial formulas, such as the annuity formula.

Annuity Formula

The annuity formula is used to calculate the present or future value of a series of equal payments made at regular intervals. These payments are often used in loans, leases, and retirement savings plans. The formula provides a way to find the value of receiving, or making, regular payments over time.

To calculate the present value of an annuity, which is what you would need to find the equivalent cash payment for a series of monthly car lease payments, you can use the following formula:
\[ PV = P \times \frac{1 - (1 + r_m)^{-n}}{r_m} \]
where:

• \(PV\) is the present value you are solving for.
• \(P\) is the payment amount, or annuity.
• \(r_m\) is the monthly interest rate.
• \(n\) is the total number of payments.

This formula accounts for the time value of money, recognizing that money received today is more valuable than the same amount received in the future due to its potential earning capacity.

Financial Mathematics

Financial mathematics involves a variety of mathematical concepts and techniques that are vital for decision-making in finance and business. By using these tools, individuals and organizations can evaluate investment opportunities, assess risks, and plan for future financial needs. This branch of mathematics is essential for calculating present and future values of financial products, annuities, loans, and investments.

Some key elements of financial mathematics include:

• Understanding the time value of money, which states that a dollar today is worth more than a dollar in the future due to its potential earning capacity.
• Breaking down complex financial instruments such as stocks, bonds, and derivatives.
• Using mathematical models to project future earnings and assess financial viability.
• Predicting investment growth through tools like compound interest calculations and annuity formulas.

Financial mathematics not only helps with personal finance management but also assists businesses in planning and optimizing financial operations. The application of these mathematical principles ensures more informed and effective financial decision-making.