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Chapter 8: Problem 22

You buy a \(\$ 25,000\) piece of equipment for nothing down and \(\$ 5,500\) per year for 6 years. What interest rate are you paying? The formula relating present worth \(P\), annual payments \(A\), number of years \(n,\) and interest rate \(i\) is $$A=P \frac{i(1+i)^{n}}{(1+i)^{n}-1}$$

Short Answer

Expert verified
You need to calculate the interest rate using the given formula and an iterative method. Initialize variables with the given values, \(P = 25000\), \(A = 5500\), and \(n = 6\), and start with an initial interest rate guess, \(i = 0.1\). Calculate the annual payments based on the guessed interest rate and compare them with the actual annual payments. Adjust the interest rate accordingly and iterate until the calculated annual payments are close enough to the actual annual payments. This will give you the approximate interest rate, which should be reported as a percentage.

Step by step solution

01

Initialize Variables and Estimate Interest Rate

In this step, assign the given values to their respective variables: \( P = 25000 \), \( A = 5500 \), and \( n = 6 \). Estimate an initial guess for the interest rate. Assume \( i = 0.1 \) (or 10%).
02

Calculate Annual Payments Based on the Initial Interest Rate

Substitute the initial guessed interest rate along with the values of \( P \) and \( n \) into the given formula and calculate the corresponding annual payments, \( A' \). \[ A' =P \frac{i(1+i)^{n}}{(1+i)^{n}-1} \]
03

Compare the Calculated Annual Payments with the Actual One

Compare \( A' \) (calculated annual payments) with \( A \) (actual annual payments). If \( A' < A \), it means that the interest rate needs to be increased to get a larger annual payment. Conversely, if \( A'> A \), the interest rate should be decreased.
04

Adjust the Interest Rate and Iterate

Based on the comparison in Step 3, adjust the guessed interest rate and go back to Step 2. Repeat the iterations until \( A' \) is close enough to \( A \), which indicates the solution of the interest rate.
05

Report the Final Interest Rate

The iterative process provides an approximation of the actual interest rate, which should be reported as the final solution. Remember, the interest rate is expressed as a decimal, so it should be multiplied by 100 to obtain the percentage.

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Key Concepts

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

Present worth
Understanding the concept of present worth is crucial when dealing with financial decisions. It represents the current value of a future amount of money or a series of future payments, considering a specific interest rate. In finance, present worth is the equivalent amount at time zero that equals the future cash flows when discounted back at the interest rate. This concept assures that money available now is more valuable than the same amount in the future due to its potential earning capacity.

For example, if you're evaluating whether to receive \(100 today or \)100 next year, present worth calculations can help you see that $100 today is worth more because you can invest it and earn interest. The formula highlighted in the exercise uses this concept to understand how annual payments are related to a lump-sum amount considering the time value of money.
Annual payments
Annual payments refer to equal cash amounts paid or received each year. In the context of loans or investments, these could represent regular repayment amounts or earnings. Understanding annual payments is fundamental in planning financial commitments or investment returns over time. These consistent payments can be related back to the present worth, which informs how much the series of payments is worth at the current time.

To put this into perspective, consider a car loan where you make fixed payments every year. When you apply this concept to the equation from the exercise, \( A = P \frac{i(1+i)^{n}}{(1+i)^{n}-1} \), it helps to determine the size of each payment given the total amount borrowed, the interest rate, and the number of payments.
Formula application in finance
The use of formulas in finance helps individuals and businesses to make quantifiable decisions about investments, loans, savings, and more. The specific formula from the exercise is a cornerstone in understanding amortization鈥攖he process of spreading out a loan into a series of fixed payments.

In practice, this formula calculates what the annual payment needs to be for a given amount of 'present worth' borrowed, with a certain interest rate over a fixed number of years. By mastering such formulas, financial professionals and consumers can accurately plan for financial obligations. Moreover, these formulas serve as the basis for creating amortization schedules, which outline each payment's portion that goes towards the principal and interest over the life of a loan.
Iterative methods
Iterative methods are a series of computational steps to refine an approximation of the desired value. They are frequently used in finance to solve equations where no closed-form solution is available. These methods iterate through successive approximations until the desired level of accuracy is achieved.

In the context of the textbook problem, an iterative approach is necessary to determine the unknown interest rate. It requires an initial guess, which is then systematically adjusted based on the comparison between the calculated and actual annual payments. This process is repeated until the calculated value is sufficiently close to the actual payment, indicating that the approximation of the interest rate is accurate. Iterative methods are powerful tools for solving financial equations, and learning to apply them is essential for students aiming to work in the finance industry.

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Most popular questions from this chapter

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