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Chapter 4: Problem 30

Balloon Payments Audrey Sanborn has just arranged to purchase a \(\$ 450,000\) vacation home in the Bahamas with a 20 percent down payment. The mortgage has a 7.5 percent stated annual interest rate, compounded monthly, and calls for equal monthly payments over the next \(\mathbf{3 0}\) years. Her first payment will be due one month from now. However, the mortgage has an eight-year balloon payment, meaning that the balance of the loan must be paid off at the end of year 8 . There were no other transaction costs or finance charges. How much will Audrey's balloon payment be in eight years?

Short Answer

Expert verified
Audrey's balloon payment after eight years will be approximately \$329,073.31.

Step by step solution

01

Calculate mortgage amount

Given that Audrey made a 20% down payment, we need to calculate the mortgage amount. To do that, first, we need to find the down payment value: Down payment = 20% of \(\$450,000\) =\(0.2 \times \$450,000 =\$90,000\) Now, we can find the mortgage amount: Mortgage amount = Total cost - Down payment =\(\$450,000 - \$90,000 = \$360,000\)
02

Calculate the monthly interest rate

We are given the annual interest rate, which is 7.5%. To find the monthly interest rate, we need to divide it by 12 (number of months in a year): Monthly interest rate = Annual interest rate / 12 =\( \frac{7.5\%}{12} = 0.625\% = 0.00625\)
03

Calculate the total number of payments

Audrey will make equal monthly payments for a period of 30 years. To find the total number of payments, multiply the number of years by the number of months in a year: Total number of payments = 30 years 脳 12 = 360 payments
04

Calculate the monthly mortgage payment

To calculate the monthly mortgage payment, we can use the following formula: Monthly payment = \(P\frac{r(1 + r)^n}{(1 + r)^n - 1}\) Where: P = Principal amount (\$360,000) r = Monthly interest rate (0.00625) n = Total number of payments (360) Monthly payment =\(\$360,000 \frac{0.00625(1 + 0.00625)^{360}}{(1 + 0.00625)^{360} - 1} \approx \$2,463.10\)
05

Calculate the remaining balance at the end of year 8

At the end of year 8, Audrey still has to make 22 years (264) payments. To calculate the outstanding balance at the end of year 8, we can use the following formula: Remaining balance = \(P(1 + r)^{n \times 8} - M\frac{(1 + r)^{n \times 8} - 1}{r}\) Where: P = Principal amount (\$360,000) r = Monthly interest rate (0.00625) n = Total number of payments (360) M = Monthly payment (\$2,463.10) Remaining balance =\(\$360,000(1 + 0.00625)^{360 \times 8} - \$2,463.10\frac{(1 + 0.00625)^{360 \times 8} - 1}{0.00625}= \$329,073.31\)
06

Calculate the balloon payment

Since Audrey has to pay off the remaining balance at the end of year 8, the balloon payment will be equal to the remaining balance: Balloon payment = \$329,073.31 Therefore, Audrey's balloon payment after eight years will be approximately \$329,073.31.

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Key Concepts

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

Mortgage Calculation
When you're arranging to buy a home, understanding the mortgage calculation is crucial. It helps you figure out your monthly payments and the total interest you'll pay over time.
Mortgages are typically calculated using a specific formula that takes into account the principal amount, the interest rate, and the term of the loan. For example, if you purchase a home costing $450,000 with a 20% down payment, your mortgage principal will be $360,000. This principal will be paid off over 30 years in equal monthly installments.
The calculation starts after deducting the down payment from the total home price. This gives us the mortgage amount. In Audrey鈥檚 case, the down payment is 20% of $450,000, which is $90,000, leaving her a mortgage amount of $360,000.
Monthly Interest Rate
The interest rate is a significant factor in mortgage calculations. Lenders often present it in annual terms, but since mortgage payments are usually monthly, you will need to convert it into a monthly rate.
To do this, divide the annual rate by 12. With Audrey's 7.5% annual rate, her monthly interest rate becomes 0.625% or 0.00625 in decimal form.
Understanding the monthly interest rate helps in accurately calculating both the monthly payment and the interest accrued each period. This conversion ensures that every payment plan aligns with the home buyer鈥檚 financial planning.
Remaining Balance
The remaining balance of a loan is what the borrower still owes on the principal after a certain period of payments have been made. This comes into play, especially when balloon payments are involved, which require paying off the remaining balance at a specific time.
After calculating the monthly payment, the next step is to determine what remains of the principal after the specified time or numerous payments.
For Audrey, the remaining balance at the end of 8 years of payments (which is part of a 30-year arrangement) was found by calculating how much of the principal has been reduced and what still needs to be paid. For her, that figure is approximately $329,073.31.
Loan Amortization
Loan amortization is the process of paying off a debt over time through regular payments. Each payment covers both principal and interest. Over time, as more principal is paid down, less interest is due, and a larger portion of each payment reduces the principal.
In Audrey鈥檚 mortgage, amortization means that her monthly payments help gradually lower the amount she owes on the loan. In the initial stages, more of the payment goes towards interest, but over time, the principal is reduced more significantly.
Understanding amortization allows borrowers to grasp how their debt diminishes over time and prepare for any future payments, such as the balloon payment Audrey faces.

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Most popular questions from this chapter

Perpetuities A prestigious investment bank designed a new security that pays a quarterly dividend of \(\$ 5\) in perpetuity. The first dividend occurs one quarter from today. What is the price of the security if the stated annual interest rate is 7 percent, compounded quarterly?

Interest Rates Well-known financial writer Andrew Tobias argues that he can earn 177 percent per year buying wine by the case. Specifically, he assumes that he will consume one \(\$ 10\) bottle of fine Bordeaux per week for the next 12 weeks. He can either pay \(\$ 10\) per week or buy a case of 12 bottles today. If he buys the case, he receives a 10 percent discount and, by doing so, earns the 177 percent. Assume he buys the wine and consumes the first bottle today. Do you agree with his analysis? Do you see a problem with his numbers?

Calculating Perpetuity Values The Perpetual Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs \(\$ 20,000\) per year forever. If the required return on this investment is 6.5 percent, how much will you pay for the policy? Suppose the Perpetual Life Insurance Co. told you the policy costs \(\$ 340,000\). At what interest rate would this be a fair deal?

Ordinary Annuities and Annuities Due As discussed in the text, an annuity due is identical to an ordinary annuity except that the periodic payments occur at the beginning of each period and not at the end of the period. Show that the relationship between the value of an ordinary annuity and the value of an otherwise equivalent annuity due is: Annuity due value \(=\) Ordinary annuity value \(\times(1+r)\) Show this for both present and future values.

Present Value and Multiple Cash Flows Investment \(X\) offers to pay you \(\$ 5,500\) per year for nine years, whereas Investment \(Y\) offers to pay you \(\$ 8,000\) per year for five years. Which of these cash flow streams has the higher present value if the discount rate is 5 percent? If the discount rate is 22 percent?
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