91Ó°ÊÓ

You want to buy a house that \(\operatorname{costs} \$ 100,000 .\) You have \(\$ 10,000\) for a down payment, but your credit is such that mortgage companies will not lend you the required \(\$ 90,000\). However, the realtor persuades the seller to take a \(\$ 90,000\) mortgage (called a seller take-back mortgage) at a rate of \(7 \%\), provided the loan is paid off in full in 3 years. You expect to inherit \(\$ 100,000\) in 3 years; but right now all you have is \(\$ 10,000,\) and you can afford to make payments of no more than \(\$ 7,500\) per year given your salary. (The loan would call for monthly payments, but assume end-of-year annual payments to simplify things.) a. If the loan was amortized over 3 years, how large would each annual payment be? Could you afford those payments? b. If the loan was amortized over 30 years, what would each payment be? Could you afford those payments? c. To satisfy the seller, the 30 -year mortgage loan would be written as a balloon note, which means that at the end of the third year, you would have to make the regular payment plus the remaining balance on the loan. What would the loan balance be at the end of Year \(3,\) and what would the balloon payment be?

Step by step solution

01

Calculating Annual Payments for a 3-Year Amortization

To find the annual payment for a 3-year loan, we use the formula for annuity payments: \[ P = \frac{r \times PV}{1 - (1 + r)^{-n}} \]where:- \(P\) is the annual payment,- \(r\) is the annual interest rate (7% or 0.07),- \(PV\) is the present value of the loan (\\(90,000),- \(n\) is the number of payments (3 years).Plugging in the values:\[ P = \frac{0.07 \times 90,000}{1 - (1 + 0.07)^{-3}} \approx \\)34,172.55 \]Thus, the annual payment for a 3-year loan is approximately \\(34,172.55. You cannot afford these payments as they exceed your payment capability of \\)7,500 per year.

02

Calculating Annual Payments for a 30-Year Amortization

Next, calculate the annual payment for a 30-year loan using the same payment formula:\[ P = \frac{0.07 \times 90,000}{1 - (1 + 0.07)^{-30}} \approx \\(7,188.00 \]The annual payment for a 30-year loan is approximately \\)7,188.00, which you can afford since it's less than your payment capability of \$7,500 per year.

03

Calculating Loan Balance at the End of Year 3

To calculate the remaining balance after 3 years, we use the remaining balance formula for amortized loans:\[ B = PV \times (1 + r)^3 - P \times \frac{(1 + r)^3 - 1}{r} \]where \(B\) is the balance end of year 3, \(PV\) is \\(90,000, \(P\) is \\)7,188, and \(r\) is 0.07.Plug in the values:\[ B = 90,000 \times (1 + 0.07)^3 - 7,188 \times \frac{(1 + 0.07)^3 - 1}{0.07} \approx \\(86,064.75 \]The loan balance after 3 years is approximately \\)86,064.75.

04

Calculating the Balloon Payment

The balloon payment is the sum of the regular annual payment at year 3 and the remaining loan balance:\[ \text{Balloon Payment} = P + B \]\[ \text{Balloon Payment} = 7,188 + 86,064.75 \approx \\(93,252.75 \]Therefore, the balloon payment at the end of year 3 would be approximately \\)93,252.75.

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Key Concepts

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

Seller take-back mortgage

A seller take-back mortgage is a type of real estate transaction where the seller of the property provides the buyer with a mortgage loan. This arrangement usually occurs when a buyer can't secure traditional financing due to reasons like insufficient credit history. The seller essentially becomes the lender.

In these arrangements, the seller earns interest on the loan, which could be negotiated based on mutual benefit. For buyers, a seller take-back mortgage can provide access to home ownership when conventional loans are inaccessible. It allows the buyer to use the house as collateral to promise repayment of the loan over an agreed period.

Seller take-back mortgages often come with shorter terms, like in this example where the loan term is just three years. Such loans often need to be paid back in full at the end of the term. This concept goes hand in hand with terms like balloon payments.

Balloon payment

A balloon payment is a large, lump-sum repayment made at the end of a loan's term. It's common in loans where regular payments are interest-only or are not enough to fully amortize the loan by the end of the term.

In the context of the seller take-back mortgage with a balloon payment, this means that your regular payments will not cover the full loan amount over the agreed term, resulting in a large payment due at the end. For instance, at the end of the third year in this exercise, there is a hefty balloon payment of approximately $93,252.75 that the buyer must pay.

Balloon payments require careful planning and assessment of future financial changes. If the buyer expects a considerable financial inflow, such as an inheritance, it might make practical sense to agree to a balloon payment structure.

Annuity payments

Annuity payments are regular, fixed payments made over the term of an annuity or loan. When a loan is amortized, like in this scenario, annuities ensure the loan balance is methodically reduced through equally spaced out installments.

The formula for calculating annuity payments is:\[P = \frac{r \times PV}{1 - (1 + r)^{-n}}\]where \(P\) represents the payment, \(r\) is the interest rate, \(PV\) is the present value of the loan, and \(n\) is the number of payments.

In the exercise, we calculated the annuity payments for both a 3-year and a 30-year amortization period. Understanding annuity payments helps in managing your cash flow and ensuring you can consistently meet your payment obligations.

Loan balance calculation

Calculating the loan balance, especially when a loan is not fully amortized over its term, requires understanding how much principal remains after several payments.

To find the remaining balance after 3 years, we use the residual loan balance formula:\[B = PV \times (1 + r)^3 - P \times \frac{(1 + r)^3 - 1}{r}\]here, \(B\) is the balance after 3 years, \(PV\) is the initial loan value, \(P\) is the annual payment, and \(r\) is the interest rate.

This calculation helps in determining the amount still owed on the loan at a specific point. In the example, the remaining loan balance after 3 years was found to be approximately $86,064.75. This figure is crucial for planning the end-term large payment, like a balloon payment, ensuring financial preparedness.