Problem 3 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the periodic payment \(R\) required to amortize a loan of \(P\) dollars over \(t\) yr with interest charged at the rate of \(r \% /\) year compounded \(m\) times a year. $$ P=5000, r=4, t=3, m=4 $$

Step-by-Step Solution

Answer

The periodic payment R required to amortize a loan of 5000 dollars over 3 years at a 4% annual interest rate compounded quarterly can be calculated using the formula \(R = P \cdot \frac{i(1+i)^{mt}}{(1+i)^{mt}-1}\). Using the given values and performing the calculations, the annuity payment R is approximately \(475.73\).

Step 1: Find the interest rate per period

To find the interest rate per period, we must first convert the annual interest rate r into a decimal form and then divide by the number of compounding periods m. $$ i = \frac{r}{100m} $$ Plugging in the given values: $$ i = \frac{4}{100 \cdot 4} $$ Calculate the interest rate per period: $$ i = 0.01 $$

Step 2: Calculate the annuity payment R

Now that we have the interest rate per period, we can calculate the annuity payment R using the formula: $$ R = P \cdot \frac{i(1+i)^{mt}}{(1+i)^{mt}-1} $$ Plugging in the values: $$ R = 5000 \cdot \frac{0.01(1+0.01)^{4 \cdot 3}}{(1+0.01)^{4 \cdot 3}-1} $$ Perform the calculations and simplify: $$ R = 5000 \cdot \frac{0.01(1.01)^{12}}{(1.01)^{12}-1} $$ Multiplying and dividing by the remaining terms: $$ R = 5000 \cdot \frac{0.120982}{0.126825} $$ Lastly, calculate the annuity payment R: $$ R \approx 475.73 $$

Step 3: Final Answer

The periodic payment R required to amortize the loan of 5000 dollars over 3 years at a 4% annual interest rate compounded quarterly is approximately $475.73.

Key Concepts

Periodic Payment Calculation

Understanding how periodic payments are calculated in loan amortization is crucial for managing loans effectively. To determine the payment amount, known as the annuity, required to pay off a loan, we use the following formula, which considers both principal and interest:
\[ R = P \cdot \frac{i(1+i)^{mt}}{(1+i)^{mt}-1} \]

• Principal (P): The total amount of money being borrowed, which in this case is $5000.
• Interest Rate per Period (i): Calculated by dividing the annual interest rate by the number of compounding periods per year.
• Number of Payments (mt): Computed by multiplying the number of years by the number of compounding periods per year.

This formula helps in figuring out the consistent payment needed to clear a loan within the specified period while accounting for the accrued interest. By substituting the values into the formula, we find out the periodic payment required, ensuring the loan is fully amortized by the end of the term.

Compound Interest

Compound interest plays a vital role in the calculation of loan payments, especially in amortized loans. Unlike simple interest, which is calculated on the principal alone, compound interest is calculated on the principal and also on previously accrued interest.

Compound interest is represented by the exponential factor \((1 + i)^{mt}\) in the annuity payment formula. Here,

• \(i\) is the interest rate per period.
• \(mt\) is the total number of compounding periods.

This approach assumes that the interest is compounded periodically. In this case, the loan compounds quarterly, enhancing the growth of the loan balance an additional four times each year. Consequently, it speeds up how quickly the loan principal accumulates interest, thereby affecting the periodic payment amounts needed to keep the loan manageable.

Financial Mathematics

Financial mathematics underpins the principles behind loans and investments. It uses statistical and mathematical methods to solve complex problems involving cash flows over time. When it comes to loans, financial mathematics helps determine how much money will be required to meet future financial obligations.

• Time Value of Money: Reflects the idea that a specific amount of money is worth more now than the same amount in the future due to its potential earning capacity.
• Amortization: This concept involves spreading payments over multiple periods, ensuring that the borrower returns both principal and interest over time.
• Interest Rate Fundamentals: Understanding the impact and implications of interest rates on loans is central. Interest rates dictate how much extra is paid on a loan above the borrowed amount.

By applying financial mathematics, borrowers can make informed decisions about their loans, clarify their financial commitments, and develop strategies for efficient debt management. This field aids in planning future expenditures and understanding how changes in interest rates can affect loan dynamics.