Question

A mortgage is a long-term loan secured by property. A mortgage with a present value of \(\$ 250\) 000 at a \(7.4 \%\) annual percentage rate requires semi- annual payments of \$10 429.01 at the end of every 6 months. The formula for the present value, \(P V\), of the mortgage is \(P V=\frac{R\left[1-(1+i)^{-n}\right]}{i}\) where \(n\) is the number of equal periodic payments of \(R\) dollars and \(i\) is the interest rate per compounding period, as a decimal. After how many years will the mortgage be completely paid off?

Short answer

Approximately 29.88 years.

Step-by-step solution

Identify given values and the formula

The present value (PV) of the mortgage is \(\$ 250,000\). The annual percentage rate (APR) is \(7.4\%\), and semi-annual payments (R) are \(\$ 10,429.01\). Use the formula for the present value of the mortgage: \[PV = \frac{R[1 - (1+i)^{-n}]}{i}\].

Determine the interest rate per compounding period

Since the payments are made semi-annually, the interest rate per compounding period is half of the annual rate. Therefore, \(i = \frac{7.4\%}{2} = \frac{0.074}{2} = 0.037\).

Plug values into the formula

Substitute the values into the formula: \[250,000 = \frac{10429.01[1 - (1 + 0.037)^{-n}]}{0.037}\].

Simplify the equation

Multiply both sides of the equation by \(0.037\): \[250,000 \times 0.037 = 10429.01[1 - (1 + 0.037)^{-n}]\].This gives: \[9250 = 10429.01[1 - (1.037)^{-n}]\].

Solve for the term involving \(n\)

Divide both sides by \(10429.01\): \[\frac{9250}{10429.01} = 1 - (1.037)^{-n}\]. Calculate the left-hand side: \[0.8865 \approx 1 - (1.037)^{-n}\].

Isolate the exponent

Rearrange the equation to isolate the exponent: \[(1.037)^{-n} = 1 - 0.8865 = 0.1135\].

Solve the exponential equation

Take the natural logarithm of both sides to solve for \(n\): \[-n \ln(1.037) = \ln(0.1135)\]. Calculate the logarithms: \[-n \times 0.0364 = -2.1756\].

Solve for \(n\)

Divide both sides by \(-0.0364\) to find \(n\): \[n = \frac{-2.1756}{-0.0364} \approx 59.76\].

Determine the number of years

Since the payments are semi-annual, there are 2 payments per year. So the number of years is: \[\frac{59.76}{2} \approx 29.88\].

Key concepts

Present Value Formula

In mortgage calculations, the present value (PV) formula is essential. It helps us understand the amount of money a series of future cash flows is worth in today's dollars. This formula is especially useful when dealing with loans like mortgages, where the payments are spread over a period of time. For our example, the formula is:\[PV = \frac{R[1 - (1+i)^{-n}]}{i}\]Here:

• \(PV\) is the present value of the mortgage.
• \(R\) is the amount of each periodic payment.
• \(i\) is the interest rate per compounding period.
• \(n\) is the total number of payments.

By using this formula, we can calculate the present value of future mortgage payments, making it easier to assess the loan’s overall cost.

Interest Rate Calculation

Understanding how to calculate the interest rate per compounding period is critical in mortgage calculations. The annual percentage rate (APR) is often provided, but it's essential to convert this to the appropriate period rate. In our example, the mortgage payments are made semi-annually. Therefore, we halve the annual rate to find the period rate:

\[i = \frac{7.4\%}{2} = \frac{0.074}{2} = 0.037\]This step ensures that our calculations match the payment schedule, which, in this case, means working with a 3.7% interest rate per 6-month period. Getting this right is crucial for accurate mortgage value and period calculations.

Exponential Equations

Solving exponential equations is a fundamental part of determining how long it will take to pay off a mortgage. These equations can seem complex, but they follow logical steps. After setting up the present value formula with all known values, the goal is to isolate the term involving the number of periods, \(n\). Here’s the equation we derived:
\[0.8865 \approx 1 - (1.037)^{-n}\]
To isolate \(n\), we rearrange and take the natural logarithm (ln) of both sides:
\[-n \ln(1.037) = \ln(0.1135)\]
This process transforms the original exponential equation into an easier-to-solve linear equation. Finally, by solving for \(n\), we find:
\[n \approx 59.76\]
Since mortgage payments are made twice a year, we divide the number of periods by 2 to find the number of years:
\[\frac{59.76}{2} \approx 29.88\]By mastering the manipulation of these exponential equations, students can confidently tackle similar problems.