Problem 37 - Single Variable Calculus

Question

Loan Amortization The size of the monthly repayment \(k\) that amortizes a loan of \(A\) dollars in \(N\) years at an interest rate of \(r\) per year, compounded monthly, on the unpaid balance is given by $$ k=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 N}\right]} $$ Show that \(r\) can be found by performing the iteration $$ r_{n+1}=r_{n}-\frac{A r_{n}+12 k\left[\left(1+\frac{r_{n}}{12}\right)^{-12 N}-1\right]}{A-12 N k\left(1+\frac{r_{n}}{12}\right)^{-12 N-1}} $$ Hint: Apply Newton's method to solve the equation $$ A r+12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right]=0 $$

Step-by-Step Solution

Answer

To show that the provided iteration formula for the interest rate \(r\) is valid, we apply Newton's method to solve the given equation: $$ A r+12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right]=0 $$ We first rewrite the equation as \(f(r) = A r + 12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right]\), and then find its derivative with respect to \(r: f'(r) = A - 12 N k\left(1+\frac{r}{12}\right)^{-12 N-1}\). Using Newton's method formula, we obtain the iteration formula for \(r\): $$ r_{n+1} = r_n - \frac{Ar_n+12 k\left[\left(1+\frac{r_n}{12}\right)^{-12 N}-1\right]}{A-12 N k\left(1+\frac{r_n}{12}\right)^{-12 N-1}} $$ This confirms the validity of the provided iteration formula for the interest rate \(r\).

Step 1: Rewrite the given equation and find its derivative with respect to r

The given equation is: $$ A r+12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right]=0 $$ Rewrite it as: $$ f(r) = A r + 12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right] $$ Now, we need to find the derivative of \(f(r)\) with respect to \(r\). We will denote this derivative as \(f'(r)\). Recall the chain rule: If \(y = u(v)\), then \(y' = u'(v)v'\). So, $$ f'(r) = \frac{d}{dr}\left(A r + 12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right]\right) $$ $$ f'(r) = A + 12 k\left[-12 N\left(1+\frac{r}{12}\right)^{-12 N-1}\cdot\frac{1}{12}\right] $$ $$ f'(r) = A - 12 N k\left(1+\frac{r}{12}\right)^{-12 N-1} $$

Step 2: Apply Newton's method to find the iteration formula for r

Newton's method formula for finding the root of an equation is: $$ r_{n+1} = r_n - \frac{f(r_n)}{f'(r_n)} $$ Now, substitute \(f(r_n)\) and \(f'(r_n)\) in the above formula: $$ r_{n+1} = r_n - \frac{Ar_n+12 k\left[\left(1+\frac{r_n}{12}\right)^{-12 N}-1\right]}{A-12 N k\left(1+\frac{r_n}{12}\right)^{-12 N-1}} $$ This is the iteration formula for the interest rate \(r\) using Newton's method. The above formula will help the student to calculate the interest rate iteratively until the desired level of accuracy is achieved.

Key Concepts

Newton's Method for Solving Equations

Newton's Method is a powerful numerical technique for finding approximations to the roots of a real-valued function. It's particularly useful when an equation can't easily be solved algebraically. This method uses derivatives to improve successively better approximations to a root.

The basic idea is simple:

• Start with an initial guess. This guess should be as close as possible to the actual root.
• Use the function and its derivative to calculate a new approximation.
• Repeat this process until the change is smaller than a predetermined threshold, indicating a sufficiently accurate solution has been found.

When applying Newton's Method, you use the formula:\[ r_{n+1} = r_n - \frac{f(r_n)}{f'(r_n)}\]This formula updates the approximation of the root iteratively. In the context of loan amortization and interest rate finding, Newton's Method allows us to solve the equation for an interest rate efficiently.

Interest Rate Calculation in Loan Amortization

Interest rate calculation is crucial in loan amortization because it determines the cost of borrowing money over time. A loan is typically amortized by making periodic, usually monthly, payments that cover both interest and the principal amount borrowed.

The challenge is determining the interest rate, especially given that actual market rates fluctuate. For a loan of amount \(A\), repaid over \(N\) years, with monthly repayments \(k\), the interest rate \(r\) must satisfy the equation:\[k=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 N}\right]}\]To calculate the interest rate iteratively, utilize Newton's Method, as it provides a systematic way to converge to a precise interest rate. This calculation requires knowledge of both mathematical functions and iterative methods to approximate the interest rate due to the non-linearity and complexity of the equation.

Understanding Derivatives of Functions

Derivatives are key to understanding and solving complex equations, particularly in methods like Newton's. Derivatives give you the slope of a function at any point and are represented mathematically as the derivative \(f'(r)\).

In the context of finding interest rates, we needed to determine the derivative of the function:\[f(r) = A r + 12 k\left[\left(1+\frac{r}{12}\right)^{-12 N}-1\right]\]Using the derivative, which is:\[f'(r) = A - 12 N k\left(1+\frac{r}{12}\right)^{-12 N-1}\] you can successfully apply Newton's Method to solve for the interest rate. The derivative helps you find how the function is changing, which is essential for determining how close your current approximation is to the actual root.

Iterative Methods in Numerical Solutions

Iterative methods are essential in solving equations that cannot be tackled with simple algebraic manipulation. These methods involve repeating a series of steps to gradually approach a desired solution.

In the context of the loan amortization problem, iterative methods are useful because:

• They allow complex equations, such as those involving interest rates, to be solved by approximations.
• Iterations include initial guesses and increasingly accurate estimates, refining the solution step by step.
• They are practical for computer-based calculations where precision can be gradually improved with successive iterations.

Using iterative methods, the interest rate for a loan can be honed to a high degree of accuracy. It becomes especially powerful when combined with calculus-based techniques like Newton's Method, enabling effective solutions to non-linear equations in finance and other fields.