91Ó°ÊÓ

Home equity: When you purchase a home by securing a mortgage, the total paid toward the principal is your equity in the home. The accompanying table shows the equity \(E\), in dollars, accrued after \(t\) years of payments on a mortgage of \(\$ 170,000\) at an APR of \(6 \%\) and a term of 30 years. $$ \begin{array}{|c|c|} \hline \begin{array}{c} t=\text { Years } \\ \text { of payments } \end{array} & \begin{array}{c} E=\text { Equity } \\ \text { in dollars } \end{array} \\ \hline 0 & 0 \\ \hline 5 & 11,808 \\ \hline 10 & 27,734 \\ \hline 15 & 49,217 \\ \hline 20 & 78,194 \\ \hline 25 & 117,279 \\ \hline 30 & 170,000 \\ \hline \end{array} $$ a. Explain the meaning of \(E(10)\) and give its value. b. Make a new table showing the average yearly rate of change in equity over each 5 -year period. c. Judging on the basis of your answer to part \(b\), does your equity accrue more rapidly early or late in the life of a mortgage? d. Use the average rate of change to estimate the equity accrued after 17 years. e. Would it make sense to use the average rate of change to estimate any function values beyond the limits of this table?

Step by step solution

01

Interpret the Meaning of E(10)

The notation \(E(10)\) represents the equity in the home after 10 years of mortgage payments. According to the table, **\(E(10) = 27,734\)**, meaning the homeowner has accrued \$27,734 in equity by the end of the 10th year.

02

Calculate Average Yearly Rate of Change of Equity Over 5-Year Periods

To calculate the average yearly rate of change in equity for each 5-year period, use the formula \(\text{Average Rate of Change} = \frac{E(t_2) - E(t_1)}{t_2 - t_1}\), where \(t_2\) and \(t_1\) are consecutive 5-year intervals. The results are:- From year 0 to 5: \((11,808 - 0) / (5 - 0) = 2,361.6\)- From year 5 to 10: \((27,734 - 11,808) / (10 - 5) = 3,185.2\)- From year 10 to 15: \((49,217 - 27,734) / (15 - 10) = 4,296.6\)- From year 15 to 20: \((78,194 - 49,217) / (20 - 15) = 5,795.4\)- From year 20 to 25: \((117,279 - 78,194) / (25 - 20) = 7,817\)- From year 25 to 30: \((170,000 - 117,279) / (30 - 25) = 10,544.2\)

03

Determine When Equity Accrual is Faster

By examining the average yearly rates of change calculated in Step 2, it is clear that the equity accrues more rapidly later in the mortgage period, as these rates increase steadily over time.

04

Estimate Equity Accrued After 17 Years

To estimate the equity after 17 years, note that years 15 to 20 have an average rate of change of \(5,795.4\). Since this is a linear approximation for only 2 additional years after year 15, the calculation is:- \(E(15) + 2 \times 5,795.4 = 49,217 + 11,590.8 = 60,807.8\).Thus, the estimated equity after 17 years is approximately **\(60,808\) dollars**.

05

Assess Appropriateness of Estimating Beyond Table Limits

Using the average rate of change to predict function values beyond the limits of the provided table can lead to significant errors because the function that models mortgage equity growth likely follows a non-linear path due to compounding interest patterns. Therefore, extrapolation should be approached with caution and in-depth understanding of the payment structure.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Rate of Change

The average rate of change is a way to measure how a particular quantity evolves over time. It's calculated using the formula: \[\text{Average Rate of Change} = \frac{E(t_2) - E(t_1)}{t_2 - t_1}\] where \(E(t_1)\) and \(E(t_2)\) are the equity values at times \(t_1\) and \(t_2\), respectively. For students dealing with mortgages, this rate tells you how much equity you're gaining on average each year within a certain period.

• This gives insight on when your investment in the mortgage is most fruitful.
• It reflects financial health and property investment growth.

By assessing different periods, you can observe variability in growth and understand how factors like payment consistency impact how equity builds in a mortgage over time.

Equity Accrual

Equity accrual in the context of a mortgage refers to the gradual increase in the percentage of property you truly own as you continue to make mortgage payments.

• Initially, a portion of your monthly payments primarily covers interest rather than reducing the principal loan.
• As time progresses, equity accrual accelerates because more of each payment reduces the principal due to a lower interest barring on the remaining loan amount.

Equity is asset-building and eventually becomes greater over longer periods. Home ownership gives a sense of financial security as you store value in a tangible asset, growing your wealth over time through consistent payment commitments.

Non-Linear Growth

The growth in mortgage equity doesn’t increase linearly but rather follows a non-linear growth pattern, primarily because of interest compounding and changing payment distributions over time.

• Early mortgage payments are interest-heavy, making the equity growth slow initially.
• In the latter years, payments reduce the principal more significantly, speeding up equity accrual.

This non-linear growth means it’s important not to assume simple average rates of change will predict future equity accurately, especially when you're looking beyond established time frames for your financial assessments. Understanding this concept is crucial when evaluating long-term investment horizons.

Compounding Interest

Compounding interest is the fundamental mechanism by which mortgages are structured, influencing both the initial slow growth of equity and the accelerated growth as you near the mortgage term's end. The concept of compounding means that interest is calculated not only on the initial amount borrowed but also on any accumulated interest from previous periods. This results in:

• Higher initial payments that largely tackle interest rather than principal reduction.
• The requirement of a long-term commitment to see substantial equity growth as payments incrementally impact the loan balance with time.

By understanding compounding, you can better predict how different financial strategies might impact your mortgage and ensure that you're making the most of your payments to build equity effectively.