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Chapter 2: Problem 10

A home buyer can afford to spend no more than 800 dollar month on mortgage payments. Suppose that the interest rate is \(9 \%\) and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously. $$ \begin{array}{l}{\text { (a) Determine the maximum amount that this buyer can afford to borrow. }} \\ {\text { (b) Determine the total interest paid during the term of the mortgage. }}\end{array} $$

Short Answer

Expert verified
Answer: The maximum amount the home buyer can afford to borrow is approximately $96,936.61, and the total interest paid during the mortgage term is approximately $95,063.39.

Step by step solution

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01

Derive the formula for the maximum loan amount

In order to determine the maximum loan amount, we need to find the present value of the stream of mortgage payments. We have the following information: - The buyer can afford to spend no more than 800 dollars per month. - The interest rate is 9% (0.09 as a decimal). - The term of the mortgage is 20 years, which is equivalent to 20*12 = 240 months. - Interest is compounded continuously, with continuous payments. The present value (PV) of continuous payments can be found by integrating the present value of a continuous payment stream with respect to time, for the duration of the mortgage. The formula for PV is: $$PV = \int_{0}^{T} Pe^{-rt} \, dt$$ Where: P = payment per month (800 dollars), r = the annual interest rate, t = the time (in years), T = the term of the mortgage (in years). To match the unit, we need to divide the payment by the number of months in a year, 12: $$PV = \int_{0}^{T} \frac{800}{12} e^{-rt} \, dt$$ As the mortgage payments are made continuously, we can find L, the maximum loan amount the buyer can afford by setting it equal to PV: $$L = \int_{0}^{T} \frac{800}{12} e^{-rt} \, dt$$
02

Calculate the maximum loan amount

Now that we have the formula for the maximum loan amount (L), we can plug in the values of T (20 years), P (800 dollars), and r (0.09), then determine the loan amount: $$L = \int_{0}^{20} \frac{800}{12} e^{-(0.09)t} \, dt$$ First, integrate the continuous payment stream: $$L = \left[ - \frac{800}{(0.09)(12)} e^{-(0.09)t} \right]_0^{20}$$ Now, find the difference between the bounds (i.e., subtract the value at the upper limit from the value at the lower limit): $$L = - \frac{800}{(0.09)(12)} (e^{-0.09(20)} - e^{-0.09(0)})$$ Calculate the loan amount: $$L = - \frac{800}{(0.09)(12)} (e^{-1.8} - 1)$$ $$L \approx 96,936.61$$ So, the maximum amount the buyer can afford to borrow is $96,936.61.
03

Calculate the total interest paid

To determine the total interest paid during the mortgage term, we have to subtract the loan amount from the total mortgage payments made: Total mortgage payments = Monthly payments * Number of payment periods $$ 800 * 20 * 12 = 192,000$$ Total interest paid = Total mortgage payments - Loan amount $$192,000 - 96,936.61 \approx 95,063.39$$ Therefore, the total interest paid during the term of the mortgage is $95,063.39.

Key Concepts

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

Continuous Compounding
Continuous compounding is an interest calculation method where interest is added to the principal balance of a loan or investment continuously, rather than at discrete intervals (like monthly or yearly). As a result, the amount of interest that accumulates is slightly higher compared to typical compounding methods because of the effect of earning "interest on interest" more frequently.
Consider an amount we invest that grows continuously. If the annual interest rate is denoted by \( r \,\), the amount grows faster than if it was compounded at regular intervals, because the interest is being calculated and added to the total amount at every possible instant.
The formula for continuous compounding is \( A = Pe^{rt} \,\), where \( A \,\) represents the amount of money accumulated after \( t \,\) years, including interest; \( P \,\) is the principal amount (initial investment); \( r \,\) is the annual interest rate; and \( e \,\) is the base of the natural logarithm, approximately equal to 2.71828.
Continuous compounding is particularly beneficial in financial scenarios like investment and loans, as it can help model real-world growth or debt accumulation more accurately.
Continuous Payments
Continuous payments involve regularly scheduled payments, consistently spread out over a specific timeframe, instead of discrete payments such as monthly or annually. This is particularly relevant in the scenario of annuities or loans where payments occur at a constant rate.
For example, if you have a loan, continuous payments imply that you're essentially paying the loan in an unceasing flow over time. It contrasts the more common method where payments occur at fixed intervals.
To find the present value of these continuous payments, an integral is used over the payment period. The formula is \( PV = \int_{0}^{T} Pe^{-rt} \, dt \,\), where \( P \,\) represents the payment rate, \( r \,\) is the continuous interest rate, \( t \,\) is time, and \( T \,\) is the total time period.
Thinking of payments as a steady flow helps in understanding the financial commitment into the future more realistically and facilitates better planning. This is particularly useful in understanding financial products that are aligned with continuous cash flows.
Loan Amount Calculation
Loan amount calculation is crucial when determining how much money one can afford to borrow, especially when interest is compounded continuously, and payments are made continuously. Calculating the loan amount involves understanding the present value (PV) of future cash flows.
In the exercise, the aim was to determine the maximum loan amount a buyer can afford with continuous payments. This involves calculating the present value of these payments, taking into account the continuous compounding of interest.
The pivotal formula for finding this present value is: \( PV = \int_{0}^{T} \frac{P}{12} e^{-rt} \, dt \,\). Plugging in the given values (with \( P = 800 \,\) for monthly payments, \( r = 0.09 \,\) as the interest rate, and \( T = 20 \,\) years), you evaluate the integral to find the PV, which equals the loan amount.
This computation helps understand one's borrowing capacity and serves as a crucial element in planning finances, accurately projecting repayment capabilities, and managing expectations.
Interest Calculation
Interest calculation is about determining the total interest paid over the life of a loan. It helps in understanding the cost of borrowing money. For loans with continuous compounding and continuous payments, interest calculation becomes a bit nuanced.
In the exercise, once the present value of the loan is calculated using continuous compounding, the total amount paid over the loan's term is known (monthly payment \( \times \,\) number of months). The total interest paid is then figured by subtracting the initial loan amount from the total payments made.
For example, if you have monthly payments of \(800 over 20 years (or 240 months), the total payments amount to \( 800 \times 240 = 192,000 \,\).
Subtracting the calculated loan amount (approx. \)96,936.61) from the total payments reveals the total interest paid: \( 192,000 - 96,936.61 = 95,063.39 \,\).
Understanding how interest accumulates over time with continuous compounding helps borrowers and investors make informed decisions, highlighting both the benefits and costs of financial product choices.

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Most popular questions from this chapter

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