Conventional Loans
Next, we will learn about conventional loans (also called amortized loans or installment loans). Examples include student loans, auto loans, and home mortgages. Note that the techniques discussed here do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.
new topic, same formula!
Mathematical formulas sometimes overlap, applying to more than one application. The formula for calculating loan payments is the same as the Payout Annuity formula. To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests $10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.
Loans Formula
[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]
- P0 is the balance in the account at the beginning (the principal, or amount of the loan).
- d is your loan payment (your monthly payment, annual payment, etc)
- r is the annual interest rate in decimal form.
- k is the number of compounding periods in one year.
- N is the length of the loan, in years.
As with annuities, the compounding frequency is not always explicitly given, but is determined by how often you make payments.
When do you use this?
The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying compounding interest on the loan.
- Compound interest: One-time deposit that sits and earns interest for a certain number of compounding periods.
- Annuity (or Savings Annuity): Fixed amount deposits made each compounding period.
- Payout Annuity: Fixed amount withdrawals taken out each compounding period.
- Loans: Fixed amount payments made each compounding period.
Example
Here are two examples of scenarios in which you would use the Loans formula:
Example 1: You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive a car can you afford? In other words, what loan amount can you pay off with $200 per month?
Example 2: You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 6%, and the loan is for 30 years. How much will your monthly payments be?
Calculating the Balance
With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.
To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don’t already know them. Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were $1,000 a month, after a year you will not have paid off $12,000 of the loan balance.
To determine the remaining loan balance, we can think “how much loan will these loan payments be able to pay off in the remaining time on the loan?”
Examples
Here are two more typical problems that would require using the Loans formula:
Example 1: If a mortgage at a 6% interest rate has payments of $1,000 a month, how much will the loan balance be 10 years from the end the loan?
Here is a video describing how to do this problem. While you will not be required to do the calculations by hand, it may be helpful for you to see how the solution unfolds.
Oftentimes answering remaining balance questions requires two steps:
- Calculating the monthly payments on the loan
- Calculating the remaining loan balance based on the remaining time on the loan
Example 2: A couple purchases a home with a $180,000 mortgage at 4% for 30 years with monthly payments. What will the remaining balance on their mortgage be after 5 years?
Note that here, you are not given the monthly payment, so that will need to be your first step. Skim the following solution to get a sense of how this is done. Again you will not be doing these computations by hand, but you will want to be able to organize your the information that you need to put into your spreadsheet.
FYI
Escrow is a legal agreement between two parties for a third party to hold onto money or assets until certain conditions are met. When you purchase a house and have a mortgage payment, your lender may set up a mortgage escrow account where a portion of your monthly payment is in addition to the mortgage and is held to cover some of the costs associated with home ownership. These costs may include but are not limited to property taxes, home owner’s insurance, and private mortgage insurance.
Home loans are typically paid off through an amortization process, amortization refers to paying off a debt (often from a loan or mortgage) over time through regular payments. An amortization schedule is a table detailing each periodic payment on an amortizing loan as generated by an amortization calculator. You can find many amortization calculators available online.
If you want to know more, click on the link below to view the website “How is an Amortization Schedule Calculated?” by MyAmortizationChart.com. This website provides a brief overlook of Amortization Schedules.
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