## Section 1.5 Simple and Compound Interest

There are two main types of interest - simple and compound. Let's look at the mathematics involved in calculating each. Then we'll see how we can use this mathematics to look at payday loans.

### Subsection 1.5.1 Simple Interest

Mathematically, interest is computed by taking a percentage of the principal. We can think about interest from two perspectives. From the perspective of the bank or the person loaning the money, they are making an *investment* - they give someone a certain amount of money, with the expectation that they will be paid back more money in the future. From the perspective of the borrower, this money (or the object that they bought with it, like a house or a car) is a *loan* - they will repay more back in the future than what they borrowed.

###### Simple Interest.

If you invest a *principle* at an APR (written as a decimal) for a given number of years, the investment will now be worth:

This can also be written, using variables, as

Here, \(A\) is the amount the investment is worth, \(p\) is the principle, \(r\) is the APR (written as a decimal), and \(t\) is the amount of time that the loan is made for.

Simple interest is usually used for transactions between friends and family or for certain types of loans, like automobile purchases. It is used because it is easy to calculate and understand.

Here are some examples for you to try:

###### Example 1.5.2.

Susan's parents loan them $1000 to help them move into a new apartment. They ask that Susan pay the money back in 2 years with 4% simple interest. How much will they owe?

4%, written as a decimal, is \(0.04\text{.}\) So using the formula above, we get that Susan owes their parents

###### Example 1.5.3.

Fernando purchases a $20,000 car. The dealership loans them the money to buy the car at a 3.85% simple interest, for 5 years. How much will Fernando need to pay each month to pay off the car in 5 years?

First compute how much Fernando owes total. How many monthly payments will there be?

The total amount that Fernando owes is:

Since they are paying off the debt over 5 years, they will make \(5*12=60\) monthly payments. Assuming that all of the monthly payments are the same, Fernando will need to pay

### Subsection 1.5.2 Compound Interest

Most lenders actually use a different way of calculating interest, called *compound interest*. In compound interest, rather than just taking a percentage of the amount once, interest is calculated regularly - every day, month, year, or some other unit of time. This causes interest to accumulate more rapidly, since you are taking a percentage not just of the original amount, but also of any interest that has already been added on.

For example, imagine you borrow $100 at 5% interest for 10 years. With simple interest, you would add 5% of $100 - $5 - each year for 10 years, for a total of $50 worth of interest. You would end up owing $150 after 10 years. If you were paying 5% interest *compounded annually*, though, you would take 5% of the amount each year - including any interest that has already accumulated.

Year | Total Amount Owed with Interest |

0 | 100 |

1 | 105 = 100 + 100*0.05 |

2 | 110.25 = 105 + 105*0.05 |

3 | 115.76 = 110.25 + 110.25*0.05 |

4 | 121.55 = 115.76 + 115.76*0.05 |

5 | 127.63 = 121.55 + 121.55*0.05 |

6 | 134.01 = 127.63 + 127.63*0.05 |

7 | 140.71 = 134.01 + 134.01*0.05 |

8 | 147.75 = 140.71 + 140.71*0.05 |

9 | 155.13 = 147.75 + 147.75*0.05 |

10 | 162.89 = 155.13 + 155.13*0.05 |

If we were using simple interest to calculate the amount owed here, after 10 years, we would owe

The additional $12.89 is interest which is being paid on the interest!

$12 may not seem like that big of a deal (although it can be, for people who are really struggling), but compound interest adds up much faster when the amount borrowed or the interest rate is larger. Even a small increase in the interest rate can have a huge effect on the amount owed at the end of the loan.

Creating this table isn't hard, but it is time-consuming. One of the most important parts of mathematics is looking for patterns in numbers that we can use to save time. In this formula, we can see that each time we take the previous amount and add 0.05 times that amount:

Notice that this is the same thing as multiplying the previous amount by 1.05

Multiplying by the same amount at every step gets us an *exponential* relationship. We have

All together, this suggests a faster way to get any of the values in the table above, by using an exponential equation. After *n* years, we will owe

dollars. This gives us an equation for the amount of money owed when interest is computed annually.

In general, though, lenders can charge interest as often as they like. Interest is frequently compounded *monthly* (12 times a year) or *daily* (365 times a year). In these cases, we use basically the same formula - with two main changes.

- If you pay 12% interest over the year, but you're being charged interest each month, you wouldn't pay 12% interest every month! Instead, we divide the interest rate by the number of times we compound each year. 12% annual interest would become\begin{equation*} \frac{12\%}{12} = 1\% \end{equation*}monthly interest. If we were compounding daily, we would have\begin{equation*} \frac{12\%}{365} = 0.03288\% \end{equation*}daily interest.
- We also need to increase the power on our exponent. If you compute interest every month for 10 years, you're computing interest 12*10 = 120 times overall. If you're compounding daily, you would have 365*10 = 3650 times overall.

Making these two changes gives us our formula for Compound Interest.

###### Compound Interest.

If you invest a *principle* at an APR (written as a decimal) for a given number of years, compounded \(k\) times per year, the investment will now be worth:

This can also be written, using variables, as

Here, \(A\) is the amount the investment is worth, \(p\) is the principle, \(r\) is the APR (written as a decimal), \(k\) is the number of times interest is compounded each year, and \(t\) is the amount of time that the loan is made for.

This exponential growth is what makes a small change to the interest rate such a big deal. Let's look at two loans - both for $1000, compounded monthly, over a period of 10 years. One lender charges an APR of 4%, while the other lender charges 7%. Let's see how much you owe at the end of 10 years using each formula:

Even though 7% is only a small amount large than 4%, it makes a huge difference - over $500 - in how much you owe at the end of the loan. Increasing the loan to 10% makes an even bigger difference:

###### Example 1.5.5.

Tasha borrows $2000 at 14% interest, compounded monthly. How much will they owe in a year?

###### Example 1.5.6.

Diego borrowed money at a 15% interest rate for 5 years, compounded daily. If they had to pay back $3000, how much did they borrow originally?

In this example, what are we looking for? Put everything else into the compound interest formula and solve for the missing variable.

We know what Diego has to pay back (A in the formula), but we don't know what he borrowed in the first place (P in the formula). So we have:

Diego only borrowed $1417.32 at the beginning of the 5 years. They paid back over twice as much as they borrowed.

### Subsection 1.5.3 Finding the Interest Rate

One of the ways that payday lenders conceal how much interest they're charging is by charging a "fee" instead of an interest rate. For example, a payday lender may loan someone $200 for two weeks. When they pay the money back, they charge them a $20 fee.

In this example, $20 may not seem like a lot - it's only 10% of the $200 that they borrowed. Remember, though, that when we talk about interest rates, we talk about *annual* percentage rates. This is 10% over two weeks - if you don't pay the $20 at the end of the first two weeks, you'll be charged another $20 at the end of the next two weeks! There are 52 weeks in the year, so by the end of the year (26 two-week periods) you would have paid \(26*20 = $520\) in interest - that's \(\frac{520}{200} = 2.6 = 260%\) of the amount you borrowed! The annual percentage rate is 260%!

Notice that this is *simple interest* - the amount of interest that they charge you each time didn't change (it was $20 every two weeks). Most payday lenders work this way - every two weeks you pay the fee until you can pay back the original money you borrowed. The fees vary, but are usually between $10 and $30 per $100 borrowed every two weeks.[1.11.1.2]

You can use the simple interest formula to figure out what the equivalent annual interest rate is. Remember that when you use the formula, the time is always in years, so 2 weeks would equal \(\frac{1}{26}\) of a year. Say you borrow $300 and pay a $25 fee per $100 for every two weeks. You'll owe \(300+25*3 = $375\) at the end of the two weeks, so the simple interest formula gives you:

The equivalent annual percentage rate is 650%.

How much would you owe after two weeks if you borrowed the same money on a credit card which charges 22% interest, compounded daily? Here we use the compound interest formula. Even though 22% is a pretty high interest rate for a credit card, you'll still end up paying a lot less than the payday loan:

At 22%, compounded daily, you end up just paying $2.55 in interest after 2 weeks.