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Chapter 2: Problem 23

When you purchase a home by securing a mortgage, the total paid toward the principal is your equity in the home. If your mortgage is for \(P\) dollars, and if the term of the mortgage is \(t\) months, then your equity \(E\), in dollars, after \(k\) monthly payments is given by $$ E=P \times \frac{(1+r)^{k}-1}{(1+r)^{t}-1} $$ Here \(r\) is the monthly interest rate as a decimal, with \(r=\mathrm{APR} / 12\). Suppose you have a mortgage of \(\$ 425,000\) at an APR of \(9 \%\) and a term of 30 years. How long does it take for your equity to reach half of the amount of the original mortgage? (Round \(r\) to four decimal places.)

Short Answer

Expert verified
Your equity in the home reaches half the mortgage value after 228 monthly payments.

Step by step solution

01

Convert Time to Months and Calculate Monthly Interest Rate

First, we need to determine the monthly interest rate and the total number of months for the mortgage. Since the term of the mortgage is 30 years, we convert this into months by multiplying by 12, giving us:\[ t = 30 \times 12 = 360 \text{ months} \]The APR is given as 9%, which as a decimal is 0.09. The monthly interest rate \(r\) is then calculated as follows:\[ r = \frac{0.09}{12} \approx 0.0075 \] We round \(r\) to four decimal places, so \( r = 0.0075\).
02

Set Equity Equation for Half the Mortgage Value

We are tasked with finding the time \(k\) when the equity \(E\) reaches half the initial mortgage value of \\(425,000. Half of \\)425,000 is \$212,500. Thus the equation becomes:\[ 212,500 = 425,000 \times \frac{(1+0.0075)^k - 1}{(1+0.0075)^{360} - 1} \]
03

Simplify and Solve for Monthly Payments \(k\)

To isolate \(k\), solve \[ \frac{212,500}{425,000} = \frac{(1.0075)^k - 1}{(1.0075)^{360} - 1} \]Simplifying the left fraction:\[ 0.5 = \frac{(1.0075)^k - 1}{(1.0075)^{360} - 1} \]Cross multiply and solve for \( (1.0075)^k \):\[ (1.0075)^k - 1 = 0.5 \times ((1.0075)^{360} - 1) \]
04

Calculate Numeric Solution for \(k\)

Calculate \((1.0075)^{360}\) first:\[ (1.0075)^{360} \approx 9.999 \] Substitute back into the equation:\[ (1.0075)^k - 1 = 0.5 \times (9.999 - 1) \]This yields:\[ (1.0075)^k = 1 + 0.5 \times 8.999 \]\[ (1.0075)^k = 5.5 \]Taking logarithms gives:\[ k = \frac{\log(5.5)}{\log(1.0075)} \approx \frac{0.74036}{0.003239} \approx 228.47 \]
05

Round \(k\) to the Nearest Month

Since \(k\) must be a whole number, round 228.47 to the nearest whole number, making \(k = 228\) months.

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Key Concepts

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

Monthly Interest Rate Calculation
When dealing with mortgage rates, it's crucial to understand how to translate the Annual Percentage Rate (APR) into a monthly interest rate. This conversion helps in calculating monthly mortgage payments and equity over time.
To find the monthly interest rate, take the APR expressed as a decimal and divide it by 12, because there are 12 months in a year. For example, if the APR is 9%, first convert this percentage into a decimal by dividing it by 100, giving \(0.09\). Then, divide \(0.09\) by 12 to get the monthly rate:
  • Monthly interest rate \(r = \frac{0.09}{12} \approx 0.0075\)
Remember, this value of \(r\) is essential, as it is used in various mortgage calculations, including the computation of equity.
Loan Term Conversion
Mortgages are often discussed in terms of years, but actual calculations are typically performed using months. This requires converting the loan term from years to months.
The conversion is simple: multiply the number of years by 12. For instance, a 30-year mortgage term is equivalent to:
  • \(t = 30 \times 12 = 360 \text{ months}\)
This conversion is crucial because it ensures that all time-related calculations are uniformly in months, allowing for precise financial projections, especially when using formulas that include monthly compounding.
APR to Decimal
Converting the APR into its decimal form is necessary for financial calculations, including interest rate and equity computations. The APR as a percentage can be misleading unless it is converted to a decimal.
  • Start with the APR, say 9%.
  • Convert it to a decimal by dividing by 100: \(0.09\).
This decimal value forms the basis for additional steps, such as determining the monthly interest rate. Using the decimal form avoids confusion in calculations and ensures accuracy in applying formulas.
Equity Calculation
Equity is an important financial measure, representing the portion of the home that you genuinely "own" after accounting for the mortgage owed. Calculating equity helps track progress in paying down a loan.
The equity equation is nuanced, as it incorporates the principal loan amount, the number of monthly payments made, and the total loan term. In simple terms, equity after \(k\) payments is computed using the formula:
  • \(E=P \times \frac{(1+r)^{k}-1}{(1+r)^{t}-1}\)
This formula reflects how monthly payments gradually increase the portion of the house you own outright, independent of interest. By solving for \(k\), you determine how quickly you reach certain financial milestones, such as owning 50% of the home.

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Most popular questions from this chapter

Suppose you pay \(R\) dollars per month to rent space for the production of dolls. You pay \(c\) dollars in material and labor to make each doll, which you then sell for \(d\) dollars. a. If you produce \(n\) dolls per month, use a formula to express your net profit \(p\) per month as a function of \(R, c, d\), and \(n\). (Suggestion: First make a formula using the words rent, cost of a doll, selling price, and number of dolls. Then replace the words by appropriate letters.) b. What is your net profit per month if the rent is \(\$ 1280\) per month, it costs \(\$ 2\) to make each doll, which you sell for \(\$ 6.85\), and you produce 826 dolls per month? c. Solve the equation you got in part a for \(d\). d. Your accountant tells you that you need to make a net profit of \(\$ 4000\) per month. Your rent is \(\$ 1200\) per month, it costs \(\$ 2\) to make each doll, and your production line can make only 700 of them in a month. Under these conditions, what price do you need to charge for each doll?

In this exercise we develop a model for the growth rate \(G\), in thousands of dollars per year, in sales of a product as a function of the sales level \(s\), in thousands of dollars. \({ }^{30}\) The model assumes that there is a limit to the total amount of sales that can be attained. In this situation we use the term unattained sales for the difference between this limit and the current sales level. For example, if we expect sales to grow to 3 thousand dollars in the long run, then \(3-s\) gives the unattained sales. The model states that the growth rate \(G\) is proportional to the product of the sales level \(s\) and the unattained sales. Assume that the constant of proportionality is \(0.3\) and that the sales grow to 2 thousand dollars in the long run. a. Find a formula for unattained sales. b. Write an equation that shows the proportionality relation for \(G\). c. On the basis of the equation from part b, make a graph of \(G\) as a function of \(s\). d. At what sales level is the growth rate as large as possible? e. What is the largest possible growth rate?

In Example \(2.11\) we solved the equation $$ 62.4 \pi d^{2}\left(2-\frac{d}{3}\right)=436 $$ on the interval from \(d=0\) to \(d=4\). If you graph the function \(62.4 \pi d^{2}(2-d / 3)\) and the constant function 436 on the span from \(d=-2\) to \(d=7\), you will see that this equation has two other solutions. Find these solutions. Is there a physical interpretation of these solutions that makes sense?

Find a function given by a formula in one of your textbooks for another class or some other handy source. If the formula involves more than one variable, assign reasonable values for all the variables except one so that your formula involves only one variable. Now make a graph, using an appropriate horizontal and vertical span so that the graph shows some interesting aspect of the function, such as a trend, significant values, or concavity. Carefully describe the function, formula, variables (including units), and graph, and explain how the graph is useful.

The monthly profit \(P\) for a widget producer is a function of the number \(n\) of widgets sold. The formula is $$ P=-15+10 n-0.2 n^{2} . $$ Here \(P\) is measured in thousands of dollars, \(n\) is measured in thousands of widgets, and the formula is valid up to a level of 15 thousand widgets sold. a. Make a graph of \(P\) versus \(n\). b. Calculate \(P(1)\) and explain in practical terms what your answer means. c. Is the graph concave up or concave down? Explain in practical terms what this means. d. The break-even point is the sales level at which the profit is 0 . Find the break-even point for this widget producer.
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