Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In installment buying, one would like to figure out the real interest rate (effective rate), but unfortunately this involves solving a complicated equation. If one buys an item worth $$\$ P$$ today and agrees to pay for it with payments of $$\$ R$$ at the end of each month for \(k\) months, then $$ P=\frac{R}{i}\left[1-\frac{1}{(1+i)^{k}}\right] $$ where \(i\) is the interest rate per month. Tom bought a used car for $$\$ 2000$$ and agreed to pay for it with $$\$ 100$$ payments at the end of each of the next 24 months. (a) Show that \(i\) satisfies the equation \(20 i(1+i)^{24}-(1+i)^{24}+1=0\) (b) Show that Newton's Method for this equation reduces to $$ i_{n+1}=i_{n}-\left[\frac{20 i_{n}^{2}+19 i_{n}-1+\left(1+i_{n}\right)^{-23}}{500 i_{n}-4}\right] $$ (c) Find \(i\) accurate to five decimal places starting with \(i=0.012\), and then give the annual rate \(r\) as a percent \((r=1200 i)\).

Step-by-Step Solution

Answer

i ≈ 0.01488, r ≈ 17.86%

Step 1: Explain the Given Equation

We start with the installment formula: \[ P = \frac{R}{i} \left[1-\frac{1}{(1+i)^{k}}\right] \] Substitute the values: \( P = 2000 \), \( R = 100 \), \( k = 24 \). Rearrange to: \[ 2000i = 100 \left[ 1 - \frac{1}{(1+i)^{24}} \right] \] Multiplying through and combining terms leads to \[ 20i(1+i)^{24} - (1+i)^{24} + 1 = 0 \]. This is the desired equation.

Step 2: Derive Newton's Method Iteration Formula

Newton's method for solving the equation is given by: \[ i_{n+1} = i_{n} - \frac{f(i_{n})}{f'(i_{n})} \] Calculate \( f(i) = 20i(1+i)^{24} - (1+i)^{24} + 1 \) and its derivative \( f'(i) = 20(1+i)^{24} + 480i(1+i)^{23} - 24(1+i)^{23} \). Plug these into the Newton's formula and simplify to obtain: \[ i_{n+1} = i_{n} - \left[ \frac{20i_{n}^{2} + 19i_{n} - 1 + (1+i_{n})^{-23}}{500i_{n} - 4} \right] \].

Step 3: Apply Newton's Method Iteratively

Start with \( i_0 = 0.012 \). Use the iteration formula multiple times until the result stabilizes at five decimal places:1) Compute \( i_1 = i_0 - \frac{20(0.012)^2 + 19(0.012) - 1 + (1+0.012)^{-23}}{500(0.012) - 4} \) 2) Repeat for \( i_2, i_3 \), etc., until successive values differ by less than \( 0.00001 \).

Step 4: Calculate the Annual Interest Rate

Once the monthly rate \( i \) is found to five decimal places, calculate the annual interest rate \( r \) using the formula \( r = 1200i \). Convert \( r \) to a percentage.

Key Concepts

Effective Interest Rate

The effective interest rate is crucial in understanding the true cost of borrowing when buying on installment. Unlike the nominal interest rate, which is the stated rate on a loan, the effective interest rate takes into account the effects of compounding over a given period.

In installment buying, you pay monthly installments for the purchase. The effective interest rate shows how those monthly payments equate to a yearly interest rate. This rate is important as it helps in comparing different financial options with different compounding periods. For instance, a loan compounded monthly might have a different effective annual rate than one compounded quarterly, even if both have the same nominal interest rate.

To determine the effective rate from monthly payments, one normally needs to solve complex equations, similar to the one described in this exercise, using iterative methods like Newton's Method.

Installment Buying

Installment buying allows consumers to purchase items over time rather than paying the full price upfront. It involves making regular payments, often monthly, over a specified term. This method of buying is common for large purchases such as cars or appliances.

For instance, in this exercise, Tom buys a used car for $2000, agreeing to make monthly payments of $100. The installment plan spreads the cost over 24 months, making it more manageable for the buyer. However, these smaller, frequent payments often come with an interest charge, calculated thanks to formulas involving rates like the one in this exercise.

Understanding the underlying interest rate and the total amount paid over the installment period helps buyers make informed financial decisions and understand the true cost of an item when bought on credit.

Monthly Interest Rate

The monthly interest rate is essentially the interest that a borrower pays on borrowed money each month. When engaging in installment buying, like a car loan or a mortgage, the monthly rate can significantly impact the total cost of the loan.

In this solution, the formula given to determine the monthly interest rate is crucial as it influences how much interest will be paid over the course of the loan. By solving the equation derived in the exercise, the exact monthly interest rate can be found, providing precise knowledge about borrowing costs.

The formula used to find the monthly interest rate combines both the principal amount and the payments, showing the reduction of principal over time, adjusted for the interest accrued each month. It's important to compute this rate accurately since even small percentage points on a monthly rate can lead to significant differences over longer periods of time.

Annual Interest Rate Calculation

Calculating the annual interest rate involves converting the monthly interest rate into a yearly format, which is crucial for understanding the full cost of borrowing. Once the monthly rate has been determined, as shown through iterative solutions using Newton's Method, the annual rate can be calculated.

In this exercise, the conversion formula used is straightforward: the annual rate \( r \) is given by \( r = 1200i \), where \( i \) is the monthly rate. This formula considers the compounding effect by scaling the monthly interest to an annual level. In practical terms, it means multiplying the monthly rate by 12 and adjusting for percentage (i.e., multiplying by 100).

Knowing the annual interest rate helps borrowers understand the total interest cost of a loan over a year, allowing for comparisons with other financial products or budgeting for annual expenses.