One of the most fundamental principles of finance is knowing the difference between simple and compound interest. So I'm going to be talking about this in how we can implement Excel formulas for calculating interest in this screencast. When you borrow money, the lender expects to be paid interest. So we have a principal, the loan amount. That's the original amount that you borrow from the bank. Alternatively, this can be the amount that you give to the bank. So in other words, you are the lender to the bank, so you're saving money. This is known as the principle. Over time, your money earns more money, and this is known as interest. And the combined principal or original loan amount plus the interest is known as the future value. So the lender earns more money because of interest. So let's first take a look at what is interest. If an initial amount is increased to a final amount, then the interest earned can always be calculated by this formula. The interest rate, so that's this little lowercase i, and by the way, I'm also going to refer to interest rate in this screencast and in this course sometimes as r. So you can calculate the interest rate by taking the final amount and subtracting the initial amount that you invest, and you can divide that by the initial amount. That's going to give you a fraction. So this here on the left is going to give you a fraction, but if you multiply by 100% that's how you convert it into a percentage. For example, if we take $100 and we save it in a ban, after a certain amount of time there's $110 in the bank. The interest rate that we have earned in that time period is 110- $100, that's $10 in the numerator here, divided $100. We multiply by 100% to convert it to a percentage, and we get 10%. And sometimes you'll see that left as a fraction, so that would be 0.10. So that's the interest rate earned in this particular time period. We can also calculate interest by this formula, where the interest that a lender expects to be paid can be calculated by multiplying the principal, that's the amount that is invested. Times the interest rate for the period t, multiplied by the time period, which is t. So if you multiply all those together, it gives you the simple interest earned. Now, this neglects compounding, which I'll talk about later in this screen cast. As an example, you borrow $1,000 at an annual interest rate of 5%. After 1 year you owe the bank $1,000 times the interest rate of 5% per year, times the time period, which is 1 year. And when you multiply those, we're left with $50 is what you have to pay in interest. We can also do this in terms of months. Our interest rate of 5% is for 12 months or an entire year, because it's an annual interest rate. If we're only investing money for a fraction of a year, then we can multiply the percentage rate by two months and we divide by 12 months. So essentially 2 over 12 is one-sixth of a year. And we can multiply that by the rate and the principal to get $8.33. And that's the interest after 2 months. And again, this is simple interest. Now, let's talk about something known as compound interest. The compound interest is actually more common. With simple interest that we just talked about, the principles stays constant, the loan amount. It was constant at $1,000, it didn't change whether were talking about one year or five years of lending. Simple interest is rarely used in the real world with financial institutions, instead compound interest is used. Compound interest is interest that is compounded at a certain time interval. We can compound annually, monthly, quarterly, or daily. So once per year, once per month, once per quarter, which is every 3 months, or once per day. The learned from the previous time interval is added to the principal. And so the principle actually grow in an exponential fashion. Let's take a look at this. So you borrow $1,000 at an annual interest rate of 5%. Interest is compounded annually, which means once per year. After the first year, the interest accrued is $1,000 times the interest rate of 0.05, times that time period of 1 year. And so you owe $50 in interest for that first year. That $50 then is added to the principal, and this is assuming that you haven't made any payments on that loan. For the next year then, the principal or the loan balance is going to be $1050. The second year, you take that $1050 and that is now the principal. So this is known as compounding because the interest then is added to the principal, and so you have a higher principle every year. We can multiply that by the interest rate and the time. And so for the second year we get $52.50. That is then added onto the pre-existing principle of 1052 to give you a principal after the second year of $1102.50. And then we keep going, at $1102.50 earns more interest. And so what happens is you have this slow exponential increase in the principal or the loan balance over time. And again, this analysis assumes that no payments are made on that loan. Mathematically, the future value then we can write as this equation here. We take the principal times 1 plus the interest rate, per time period raised to the number of compounding periods. Let's revisit our example. You borrow $1,000 at an annual interest rate of 5% interest is compounded annually. What will the loan balance or the future value be after 5 years? After 5 years, that's a future value, that's F. We can calculate by this equation. We can take the principal multiplied by 1, plus the interest rate for that compounding period which is a year, raised to the number of compounding intervals, which is 5 years. And when you plug that into your calculator, you get $1276.28. In simple interest, the principle does not change, it's going to be constant, $1,000 in this case. And so every year, the interest is the same, it's $50. After 10 years then you have paid $500 in interest for a total loan balance of $1,500. But compounding interest the principal keeps going up because you're adding interest from the previous year on to the principal. So you're earning more interest on more money. And so it's slowly exponentially increasing. Here we have another example. So we're not borrowing but we're saving $5,000 at an annual interest rate of 5%. Interest rate is compounded monthly. In the previous example, interest was compounded yearly. This question then we're trying to determine what's the balance going to be after 4 years. Now, recall our formula for the future value. In this formula, r is the interest rate for the compounding period. In the previous example, the compounding period was 1 year. In this case, our compounding period is 1 month, because our interest is compounded monthly. And n is the number of compounding periods. So when we use this future value equation up here, what we have to do is we have to actually divide our annual interest rate of 5% by 12. Because we need to get it down to the monthly interest rate, 0.05 divided by 12. And then we raise to n, which is the number of compounding periods. If we're considering 1 month to be our compounding period, we have to multiply 4 years times 12 months per year in the exponent here. And when we do that math, we get the $5,000 has learned about $1,100 in interest for a total future value of $6,104. Now, just note if you're having a hard time understanding the math behind this, it's not really a big deal because Excel has some formulas and functions that can automatically do these types of calculation. Although it is really important that you understand that. In order to use these functions and formulas, you have to know what your compounding rate is. Because the interest rate that goes into these functions is always the interest rate per compounding period. So there's a difference between compounding annually and compounding monthly. And then the number of compounding periods is also really important.
