# 7.5: Loans - Mathematics

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In the last section, you learned about payout annuities.

In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.

One great thing about loans is that they use exactly the same formula as a payout annuity. 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

P0 is the balance in the account at the beginning (the principal, or amount of the loan).

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

Like before, 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 interest on the loan.

Compound interest: One deposit

Annuity: Many deposits.

Payout Annuity: Many withdrawals

Loans: Many payments

### Example 11

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 of a car can you afford? In other words, what amount loan can you pay off with$200 per month?

In this example,

d = $200 the monthly loan payment r = 0.03 3% annual rate k = 12 since we’re doing monthly payments, we’ll compound monthly N = 5 since we’re making monthly payments for 5 years We’re looking for P0, the starting amount of the loan. You can afford a$11,120 loan.

You will pay a total of $12,000 ($200 per month for 60 months) to the loan company. The difference between the amount you pay and the amount of the loan is the interest paid. In this case, you’re paying $12,000-$11,120 = $880 interest total. ### Example 12 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?

In this example,

We’re looking for d.

r = 0.06 6% annual rate

k = 12 since we’re paying monthly

N = 30 30 years

P0 = $140,000 the starting loan amount In this case, we’re going to have to set up the equation, and solve for d. You will make payments of$839.37 per month for 30 years.

## Auto Loan Calculator

The Auto Loan Calculator is mainly intended for car purchases within the U.S. People outside the U.S. may still use the calculator, but please adjust accordingly. If only the monthly payment for any auto loan is given, use the Monthly Payments tab (reverse auto loan) to calculate the actual vehicle purchase price and other auto loan information.

### Auto Loans

Most people turn to auto loans during vehicle purchase. They work as any generic, secured loan from a financial institution does with a typical term of 36, 60, 72, or 84 months in the U.S. Each month, repayment of principal and interest must be made from borrowers to auto loan lenders. Money borrowed from a lender that isn't paid back can result in the car being legally repossessed.

### Dealership Financing vs. Direct Lending

Generally, there are two main financing options available when it comes to auto loans: direct lending or dealership financing. With the former, it comes in the form of a typical loan originating from a bank, credit union, or financial institution. Once a contract has been entered with a car dealer to buy a vehicle, the loan is used from the direct lender to pay for the new car. Dealership financing is somewhat similar except that the auto loan, and thus paperwork, is initiated and completed through the dealership instead. Auto loans via dealers are usually serviced by captive lenders that are often associated with each car make. The contract is retained by the dealer, but is often sold to a bank or other financial institution called an assignee that ultimately services the loan.

Direct lending provides more leverage for buyers to walk into a car dealer with most of the financing done on their terms, as it places further stress on the car dealer to compete with a better rate. Getting pre-approved doesn't tie car buyers down to any one dealership, and their propensity to simply walk away is much higher. With dealer financing, the potential car buyer has fewer choices when it comes to rate shopping, though it's there for convenience for anyone who doesn't want to spend time shopping, or cannot get an auto loan through direct lending.

Often, to promote auto sales, car manufacturers offer good financing deals via dealers. Consumers in the market for a new car should start their search for financing with car manufacturers. It is not rare to get low interest rates like 0%, 0.9%, 1.9%, or 2.9% from car manufacturers.

### Vehicle Rebates

Car manufacturers may offer vehicle rebates to further incentivize buyers. Depending on the state, the rebate may or may not be taxed accordingly. For example, purchasing a vehicle at $30,000 with a cash rebate of$2,000 will have sales tax calculated based on the original price of $30,000, not$28,000. Luckily, a good portion of states do not do this and don't tax cash rebates. They are Alaska, Arizona, Delaware, Iowa, Kansas, Kentucky, Louisiana, Massachusetts, Minnesota, Missouri, Montana, Nebraska, New Hampshire, Oklahoma, Oregon, Pennsylvania, Rhode Island, Texas, Utah, Vermont, and Wyoming.

Generally, rebates are only offered for new cars. While some used car dealers do offer cash rebates, this is rare due to the difficulty involved in determining the true value of the vehicle.

A car purchase comes with costs other than the purchase price, the majority of which are fees that can normally be rolled into the financing of the auto loan or paid upfront. However, car buyers with low credit scores might be forced into paying fees upfront. The following is a list of common fees associated with car purchases in the U.S.

• Sales Tax&mdashMost states in the U.S. collect sales tax for auto purchases. It is possible to finance the cost of sales tax with the price of the car, depending on the state the car was purchased in. Alaska, Delaware, Montana, New Hampshire, and Oregon are the five states that don't charge sales tax.
• Document Fees&mdashThis is a fee collected by the dealer for processing documents like title and registration.
• Title and Registration Fees&mdashThis is the fee collected by states for vehicle title and registration.
• Advertising Fees&mdashThis is a fee that the regional dealer pays for promoting the manufacturer's automobile in the dealer's area. If not charged separately, advertising fees are included in the auto price. A typical price tag for this fee is a few hundred dollars.
• Destination Fee&mdashThis is a fee that covers the shipment of the vehicle from the plant to the dealer's office. This fee is usually between $900 and$1,500.

## Calculating Interest on a One-Year Loan

If you borrow $1,000 from a bank for one year and have to pay$60 in interest for that year, your stated interest rate is 6%. Here's the calculation:

Effective Rate on a Simple Interest Loan = Interest/Principal = $60/$1,000 = 6%

Your annual percentage rate or APR is the same as the stated rate in this example because there is no compound interest to consider. This is a simple interest loan.

James Sprater of Grand Junction, Colorado, has been shopping for a loan to buy a used car. He wants to borrow $18,000 for four or five years. James' credit union offers a declining-balance loan at 9.3 percent for 48 months, resulting in a monthly payment of$450.50. The credit union does not offer five-year auto loans for amounts less than $20,000, however. If James borrowed$18,000, this payment would strain his budget. A local bank offered current depositors a five-year loan at a 9.34 percent APR, with a monthly payment of $376.62. This credit would not be a declining-balance loan. Because James is not a depositor in the bank, he would also be charged a$25 credit check fee and a $60 application fee. James likes the lower payment but knows that the APR is the true cost of credit, so he decided to confirm the APRs for both loans before making his decision. Round your answers to two decimal places. What is the APR for the credit union loan? Use the n-ratio formula to calculate and confirm the APR on the bank loan as quoted for depositors. What is the add-on interest rate for the bank loan? What would be the true APR on the bank loan if James did not open an account to avoid the credit check and application fees? ## 5-Year Mortgage Calculator 5-Year Mortgage Calculator is an online personal finance assessment tool to calculate monthly repayment, total repayment and total interest cost on the principal borrowed. The loan amount and interest rate are the key terms of 5-year mortgage to calculate the necessary repayment details. 5-year mortgage is a home loan that will enable you to purchase a house and expect to have repaid in five years. Five year mortgages can be the solution for some people wanting to buy a home. Before buying or refinancing with a five year mortgage it is very important to do a research to calculate how the future loan payments are involved and how these payments may affect your financial situation. With interest rates rising and real estate prices booming, lenders are starting to offer the 5-year mortgage as a viable option for buying your dream home. There is various money lenders are participating in real estate business nowadays with different terms of conditions. Therefore, analyzing the 5 mortgage components is a vital step before opt for any home loan agreement. When it comes to online calculation 5-year mortgage calculator assists you to determine which lender provides you best option by comparing different interest rates and loan amounts available in finance market. ## Amortization Calculation Usually, whether you can afford a loan depends on whether you can afford the periodic payment (commonly a monthly payment period). So, the most important amortization formula is the calculation of the payment amount per period. ### Calculating the Payment Amount per Period The formula for calculating the payment amount is shown below. • A = payment Amount per period • P = initial Principal (loan amount) • r = interest rate per period • n = total number of payments or periods Example: What would the monthly payment be on a 5-year,$20,000 car loan with a nominal 7.5% annual interest rate? We'll assume that the original price was $21,000 and that you've made a$1,000 down payment.

You can use the amortization calculator below to determine that the Payment Amount (A) is $400.76 per month. P =$20,000
r = 7.5% per year / 12 months = 0.625% per period (this is entered as 0.00625 in the calculator)
n = 5 years * 12 months = 60 total periods

### Calculating the Monthly Payment in Excel

Microsoft Excel has a number of built-in functions for amortization formulas. The function corresponding to the formula above is the PMT function. In Excel, you could calculate the monthly payment using the following formula:

### Calculating the Rate Per Period

When the number of compounding periods matches the number of payment periods, the rate per period (r) is easy to calculate. Like the above example, it is just the nominal annual rate divided by the periods per year. However, what do you do if you have a Canadian mortage and the compounding period is semi-annual, but you are making monthly payments? In that case, you can use the following formula, derived from the compound interest formula.

• r = rate per payment period
• i = nominal annual interest rate
• n = number of compounding periods per year
• p = number of payment periods per year

Example: If the nominal annual interest rate is i = 7.5%, and the interest is compounded semi-annually ( n = 2 ), and payments are made monthly ( p = 12 ), then the rate per period will be r = 0.6155%.

Important: If the compound period is shorter than the payment period, using this formula results in negative amortization (paying interest on interest). See my article, "negative amortization" for more information.

If you are trying to solve for the annual interest rate, a little algebra gives:

Example: Using the RATE() formula in Excel, the rate per period (r) for a Canadian mortgage (compounded semi-annually) of $100,000 with a monthly payment of$584.45 amortized over 25 years is 0.41647% calculated using r=RATE(25*12,-584.45,100000) . The annual rate is calculated to be 5.05% using the formula i=2*((0.0041647+1)^(12/2)-1) .

## Understanding An Amortization Schedule

By committing to a mortgage loan, the borrower is entering into a financial agreement with a lender to pay back the mortgage money, with interest, over a set period of time.

The borrower’s monthly mortgage payment may change over time depending on the type of loan program, however, we’re going to address the typical 30 year fixed Principal and Interest loan program for the sake of breaking down the individual payment components for this particular article about an amortization schedule.

On each payment that is made, a certain amount of interest is taken out to pay the lender back for the opportunity to borrow the money, and the remaining balance is applied to the principal balance.

It’s common to hear industry professionals and homeowners talk about a mortgage payment being front-loaded with interest, especially if they’re referencing an amortization chart to show the numbers. Since there is more interest being paid at the beginning of a mortgage payment term the amount of money applied to interest decreases over time, while the money applied to the principal increases.

### The Loan Amortization Chart

We can better understand mortgage payments by looking at a loan amortization chart, which shows the specific payments associated with a loan.

The details will include the interest and principal component of each periodic payment.

For example, let’s look at a scenario where you borrowed a $100,000 loan at 7.5% interest rate, fixed for 30 year term. To ensure full repayment of principal by the end of the 30 years, your payment would need to be$699.21 per month. In the first month, you owe $100,000, which means the interest would be calculated on the full loan amount. To calculate this, we start with$100,000 and multiply it by 7.5% interest rate. This will give you $7,500 of annual interest. However, we only need a monthly amount. So we divide by 12 months to find that the interest equals$625. Now remember, you are paying $699.21. If you only owe interest of$625, then the remainder of the payment, $74.21, will go towards the principal. Thus, your new outstanding balance is now$99,925.79.

In month #2, you make the same payment of $699.21. However, this time, you now owe$99,925.79. Therefore, you will only pay interest on $99,925.79. When running through the calculator in the same process detailed above, you will find that your interest component is$624.54. (It is decreasing!) The remaining \$74.68 will be applied towards principal. (This amount is increasing!)

Each month, the same simple mathematic calculation will be made. Because the payments are remaining the same, each month the interest will continue to be reduced and the remainder going towards principal will continue to increase.

An amortization chart runs chronologically through your series of payments until you get to the final payment. The chart can also be a useful tool to determine interest paid to date, principal paid to date, or remaining principal.

Another frequent use of amortization charts is to determine how extra payments toward principal can affect and accelerate the month of final payment of the loan, as well as reduce your total interest payments.

One of the easiest ways to make sure you’re getting the best interest rate is to shop around. Compare loan offers side-by-side, and choose the one that works best for you.

But getting the best personal loan rates actually starts long before you go to take out a loan. It’s the hard work of improving your financial health and credit score before you need to borrow more money.

“The biggest one is to bring down your existing debt,” Nayar says. “The cheapest way to get money is to have money.”

That’s because lenders see less risk in borrowers with less debt, and are willing to offer lower interest rates because of it.

Another option is to bring on a co-signer, someone with better credit who can vouch for you on the loan application. Keep in mind, however, that the co-signer is equally liable for the debt, and it could sap their credit score if you miss payments.

Don’t let the very idea of calculating loan interest and diving into algebraic formulas scare you. Understanding how interest works is a crucial step to making smart decisions about loans.

So whether you break out a pencil and paper, or use one of NextAdvisor’s online calculators, take the time to understand the real cost — interest included — behind your next loan.

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