# What is the Maximum Interest Rate I Can Afford?

Even if you know very little about how loans and financing work, you know that low interest rates are good, and high interest rates are bad. In essence, interest is the cost of borrowing money; it's what lenders charge you for the privilege of using their money to buy major items, such as a house or a car. When you pay off your mortgage or auto loan, you always end up paying more than what you borrowed. This extra money is the total interest. The total amount of interest you pay depends mainly on the interest rate, but also on the sum you borrow, the length of the loan, and how much you can pay each month. All these variables are inter-related. A natural question that many borrowers ask is "What is the highest interest rate I can afford?"

If you know the amount you need to borrow, how much you can pay every month, and the length of the loan term, you can plug these variables into our **Maximum Interest Rate Calculator** to compute the highest interest rate you can afford. You can also hand-calculate the maximum interest rate with the **Approximation Formula** derived below. Both methods will give you an accurate answer.

## How the Calculator Works

Banks determine your monthly payments**M**based on the annual interest rate

**R**, the principal

**P**, and the number of months in the loan term

**N**. The financial equation for mortgage payments and car payments is

**M = (PR/12)(1+R/12)**.

^{N}/[(1+R/12)^{N}- 1]In mathematics, if you know the values for all but one of the variables in an equation, you can often use algebra to solve for the remaining variable. The above formula can be solved to isolate

**P**and

**N**. Unfortunately, it is too complex to be solved for an exact value of

**R**using algebra. The Maximum Interest Rate Calculator uses a recursive algorithm to compute

**R**exactly, a method similar to "guess and check." This would be cumbersome to do by hand, but the calculator does it in less than a second.

## How the Approximation Formula is Derived

Using a mathematical concept called the *Binomial Theorem*, we can simplify the numerator and denominator of the bank formula for monthly payments.

** (PR/12)[1 + NR/12 + N(N-1)R ^{2}/288 + N(N-1)(N-2)R^{3}/10368]**

**M**= ---------------------------------------------------------------------------------

**[NR/12 + N(N-1)R**

^{2}/288 + N(N-1)(N-2)R^{3}/10368]We can further simplify this expression by canceling a common factor of

**R**from the numerator and denominator. We can also simplify it by replacing

**N(N-1)**and

**N(N-1)(N-2)**with

**N**and

^{2}**N**respectively, since

^{3}**N**is a large number:

**(P/12)[1 + NR/12 + N**

^{2}R^{2}/288 + N^{3}R^{3}/10368]**M**= -------------------------------------------------------------

**[N/12 + N**

^{2}R/288 + N^{3}R^{2}/10368]Next, using

*polynomial long division*, we can simplify the expression to

**M = P/N + RP/24 + R**,

^{2}PN/1728which is now a quadratic equation in

**R**. When we use the

*quadratic formula*to solve for the annual interest rate

**R**, we obtain

**R = (72/N)sqrt[MN/(3P) – 1/12] – 36/N.**

This is the Approximation Formula for hand-calculating the maximum interest rate you can afford.

## Example 1

Suppose you want to borrow $120,000 for a home mortgage, and you can afford to pay $750 a month for 30 years. Then we have**P**= 120000,

**N**= 360, and

**M**= 750. If we plug these numbers into the Approximation Formula, we obtain

**R**= (72/360)sqrt(270000/360000 - 1/12) - 36/360

= (1/5)sqrt(2/3) - 1/10

= 0.0633

The Approximation Formula says that the highest interest rate you can afford is 6.33%. If we use the Maximum Interest Rate Calculator, we find the exact value of

**R**is 6.39%. As you can see, the approximation is very close to the exact answer. Knowing what interest rate you can afford will help you set realistic financial goals, and help you budget for monthly loan payments.

## Example 2

Suppose you want to take out an auto loan for $14,000, and you can afford to pay $300 every month for 5 years. This gives us**P**= 14000,

**N**= 60, and

**M**= 300. Then the maximum rate you can afford is

**R**= (72/60)sqrt(18000/42000 - 1/12) - 36/60

= (6/5)sqrt(29/84) - 3/5

= 0.1051

The maximum annual rate you can afford for a car loan is about 10.51% with the Approximation Formula, and exactly 10.37% with the Maximum Interest Rate Calculator. With good credit, you can easily secure an auto loan with a 10.37% annual rate, or an even lower rate.

© *Had2Know 2010
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