Question

The formula for calculating the monthly mortgage payment, \(P M T,\) for a property is \(P M T=P V\left[\frac{i}{1-(1+i)^{-n}}\right],\) where \(P V\) is the present value of the mortgage; \(i\) is the interest rate per compounding period, as a decimal; and \(n\) is the number of payment periods. To buy a house, Tyseer takes out a mortgage worth \(\$ 150\) 000 at an equivalent monthly interest rate of \(0.25 \%\) He can afford monthly mortgage payments of \(\$ 831.90 .\) Assuming the interest rate and monthly payments stay the same, how long will it take Tyseer to pay off the mortgage?

Short answer

Tyseer will take 240 months to pay off the mortgage.

Step-by-step solution

Write down the given variables

Given: Present Value (PV) = 150,000 Interest rate per period (i) = 0.25% = 0.0025 Monthly Payment (PMT) = 831.90 We need to find the number of payment periods (n).

Substitute the known values into the formula

The formula for calculating the monthly mortgage payment is given by: \[ PMT = PV \left[\frac{i}{1-(1+i)^{-n}}\right] \] Substitute the given values: \[ 831.90 = 150000 \left[\frac{0.0025}{1-(1+0.0025)^{-n}}\right] \]

Isolate the fraction

First, isolate the fraction on the right-hand side: \[ 831.90 = 150000 \left[\frac{0.0025}{1-(1.0025)^{-n}}\right] \] Divide both sides by 150000: \[ \frac{831.90}{150000} = \frac{0.0025}{1-(1.0025)^{-n}} \] Simplify: \[ 0.005546 = \frac{0.0025}{1-(1.0025)^{-n}} \]

Solve for the denominator

Multiply both sides by \(1-(1.0025)^{-n}\): \[ 0.005546 (1-(1.0025)^{-n}) = 0.0025 \] Expand and simplify: \[ 0.005546 - 0.005546(1.0025)^{-n} = 0.0025 \] Isolate \( (1.0025)^{-n} \): \[ 0.005546 - 0.0025 = 0.005546(1.0025)^{-n} \] Simplify: \[ 0.003046 = 0.005546(1.0025)^{-n} \]

Solve the exponential equation

Divide both sides by 0.005546: \[ \frac{0.003046}{0.005546} = (1.0025)^{-n} \] Simplify: \[ 0.54935 = (1.0025)^{-n} \] Take the natural logarithm (ln) of both sides to solve for \(-n\): \[ \ln(0.54935) = \ln((1.0025)^{-n}) \] This simplifies to: \[ \ln(0.54935) = -n \ln(1.0025) \]

Solve for n

Divide by \(-\ln(1.0025)\): \[ n = \frac{\ln(0.54935)}{-\ln(1.0025)} \] Calculate using a calculator: \[ n \approx \frac{-0.5985}{-0.0025} = 239.4 \] Since \(n\) must be a whole number, round up: \[ n \approx 240 \]

Key concepts

Monthly Mortgage Payment Formula

The formula to calculate your monthly mortgage payment, denoted as PMT, is crucial for figuring out how much you need to pay each month on your mortgage. The formula is given by: \[ PMT = PV \left[\frac{i}{1-(1+i)^{-n}}\right] \]
Here,
- \(PV\) is the present value, or the total amount of the loan.
- \(i\) is the interest rate per compounding period, represented as a decimal.
- \(n\) is the number of payment periods, typically the total number of months over which the mortgage is repaid.
Understanding this formula helps you determine the fixed payment amount you need to make each month, ensuring that you can plan your finances effectively. The structure of the formula takes into account the interest rate and payment periods to compute an accurate monthly payment.
Knowing how each component interacts is essential. The interest rate affects how much interest you'll pay over the life of the loan, and the number of payment periods affects the total duration of the loan.

Interest Rate Calculation

Interest rates are a vital part of any loan calculation. They determine how much extra you will pay, on top of the principal amount. In this example, the interest rate per compounding period (monthly) is given as 0.25%, which needs to be converted to a decimal for calculations by dividing by 100:
\[ i = \frac{0.25}{100} = 0.0025 \]
The monthly interest rate is lower than the annual rate because the compounding frequency is monthly. This rate should then be used in the mortgage payment formula, as shown in our original example.
On a broader note, always ensure that the interest rate you use matches the compounding period of your loan to avoid errors in your calculations. This approach ensures that you're incorporating the correct amount of interest paid on a per-period basis into your monthly payment calculation.

Payment Period Determination

Determining the total number of payment periods, denoted as \(n\), is essential for calculating the length of your mortgage. In monthly installments, this value \(n\) represents the total number of months you'll be making payments.
For instance, if a mortgage lasts for 20 years, then:
\[ n = 20 \times 12 = 240 \] months.
In our example, we calculated \(n\) by isolating it in the monthly mortgage payment formula and solving the resulting equation. By isolating \(n\), you can precisely determine how long it will take to repay your mortgage, given your monthly payment and interest rate.
This calculation is significant because it shows how varying the loan duration affects your monthly commitment and total interest paid. A shorter period means higher monthly payments but less interest overall, while a longer period reduces monthly payments but increases total interest paid.

Exponential Equation Solving

Solving exponential equations is key when figuring out the number of payments for a mortgage. Once you have substituted all known values into the formula and simplified, you may end up with an equation involving exponents, such as:
\[ 0.54935 = (1.0025)^{-n} \]
To solve for \(-n\), you can use the natural logarithm (ln) function. By taking the natural logarithm of both sides:
\[ \ln(0.54935) = \ln((1.0025)^{-n}) \]
This step leads to:
\[ \ln(0.54935) = -n \ln(1.0025) \]
Finally, isolate \(n\) by dividing both sides by \(-\ln(1.0025)\):
\[ n = \frac{\ln(0.54935)}{-\ln(1.0025)} = 239.4 \]
Since you cannot have a fraction of a payment period, round up to the nearest whole number, which makes \(n \approx 240\).
Understanding how to manipulate and solve exponential equations like this ensures you can accurately determine the duration of your mortgage.