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Compound interest is when interest is earned not only on the initial amount invested, but also on any interest. In other words, interest is earned on top of interest and thus “compounds”. The compound interest formula can be used to calculate the value of such an investment after a given amount of time, or to calculate things like the doubling time of an investment. We will see examples of this below.
Examples of finding the future value with the compound interest formulaFirst, we will look at the simplest case where we are using the compound interest formula to calculate the value of an investment after some set amount of time. This is called the future value of the investment and is calculated with the following formula.  ExampleAn investment earns 3% compounded monthly. Find the value of an initial investment of $5,000 after 6 years. SolutionDetermine what values are given and what values you need to find.
You are trying to find \(A\), the future value (the value after 6 years). Now apply the formula with the known values: \(\begin{align}A &= P\left(1 + \dfrac{r}{m}\right)^{mt} \\ &= 5000\left(1 + \dfrac{0.03}{12}\right)^{12 \times 6} \\ &\approx \bbox[border: 1px solid black; padding: 2px]{5984.74}\end{align}\) Answer: The value after 6 years will be $5,984.74.
Let’s try one more example like this before we try some more difficult types of problems. ExampleWhat is the value of an investment of $3,500 after 2 years if it earns 1.5% compounded quarterly? SolutionAs before, we are finding the future value, A. In this example, we are given:
Applying the formula: \(\begin{align}A &= P\left(1 + \dfrac{r}{m}\right)^{mt} \\ &= 3500\left(1 + \dfrac{0.015}{4}\right)^{4 \times 2}\\ &\approx \bbox[border: 1px solid black; padding: 2px]{3606.39}\end{align}\) Answer: The value after 2 years will be $3,606.39. There are other types of questions that can be answered using the compound interest formula. Most of these require some algebra, and the level of algebra required depends on which variable you need to solve for. We will look at some different possibilities below. Example of finding the rate given other valuesSuppose you were given the future value, the time, and the number of compounding periods, but you were asked to calculate the rate earned. This could be used in a situation where you are taking the amount of home sold for and determining the rate earned, if it is viewed as an investment. Consider the following example. ExampleMrs. Jefferson purchased an antique statue for $450. Ten years later, she sold this statue for $750. If the statue is viewed as an investment, what annual rate did she earn? SolutionIf we view this as an investment of \(P = $450\), then we know that the future value is \(A = $750\). This was after \(t = 10\) years. Finally, if we assume an annual rate, we will use \(m = 1\) and have: \(A = P\left(1 + \dfrac{r}{m}\right)^{mt}\) \(750 = 450\left(1 + \dfrac{r}{1}\right)^{1 \times 10}\) This is the same as: \(750 = 450\left(1 + r\right)^{10}\) We are solving for the rate, \(r\). We will do this using the following steps. Divide both sides by 450. \(\dfrac{750}{450} = \left(1 + r\right)^{10}\) Simplify on the left-hand side. But, we need to be careful about rounding, so we will keep the fraction for now. \(\dfrac{5}{3} = \left(1 + r\right)^{10}\) Take the left-hand side to the 1/10th power to clear the power of 10 on the right. \(\left(\dfrac{5}{3}\right)^{\dfrac{1}{10}} = 1 + r\) Calculate the value on the left and solve for \(r\).
\(\begin{align}1.0524 &= 1 + r \\1.0524 – 1 &= r \\ \bbox[border: 1px solid black; padding: 2px] Therefore, Mrs. Jefferson earned an annual rate of 5.24%. Not bad! But there was definitely some more complicated algebra involved. In some cases, you may even have to make use of logarithms. A common situation where you might see this is when calculating the doubling time of an investment at a given rate. Calculating the doubling time of an investment using the compound interest formulaRegardless of the amount initially invested, you can find the doubling time of an investment as long as you are given the rate and the number of compounding periods. Let’s look at an example and see how this could be done. ExampleHow many years will it take for an investment to double in value if it earns 5% compounded annually? It may seem tough to decide where to start here, as we are only given the rate, \(r = 0.05\), and the number of compounding periods, \(m = 1\). Note that we are trying to find the time, \(t\). Since we do not know the initial investment, we can simply call it \(P\). For this to double, its value would be \(2P\) and, using the compound interest formula, we would have: \(A = P\left(1 + \dfrac{r}{m}\right)^{mt}\) \(2P = P\left(1 + \dfrac{0.05}{1}\right)^{t}\) This could be written as: \(2P = P\left(1.05\right)^{t}\) Remember that this would only make sense if the amount invested is not zero, so we can divide both side by \(P\). This gives: \(2 = \left(1.05\right)^{t}\) To solve for t, we will take the natural log, ln, of both sides. By the laws of logarithms, this will allow us to bring the exponent to the front. \(\ln(2) = t\ln\left(1.05\right)\) Finally, we can divide and then use our calculators to find t. \(\begin{align}t &= \dfrac{\ln(2)}{\ln\left(1.05\right)}\\ &\approx \bbox[border: 1px solid black; padding: 2px]{14.2 \text{ years}}\end{align}\) Answer: It will take a little more than 14 years before the investment will double in value. The same process could be used to determine when an investment would triple or even quadruple. You would just use a different multiple of \(P\) in the first part of the formula. SummaryThe compound interest formula is used when an investment earns interest on the principal and the previously-earned interest. Investments like this grow quickly; how quickly depends on the rate and the number of compounding periods. When working with a compound interest formula question, always make note of what values are known and what values need to be found so that you stay organized with your work. Now that you have studied compound interest, you should also review simple interest and how it is different.
Learning Outcomes
We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics. Simple InterestDiscussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the original principal plus the interest.
[latex]\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}[/latex]
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn? The following video works through this example in detail. One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.
Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn? Further explanation about solving this example can be seen here. We can generalize this idea of simple interest over time.
[latex]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]
The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.
Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up. For example, a 6% APR paid monthly would be divided into twelve 0.5% payments. A 4% annual rate paid quarterly would be divided into four 1% payments.
Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn? This video explains the solution.
A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging. Compound InterestWith simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding. Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow? The 3% interest is an annual percentage rate (APR) – the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\frac{3%}{12}[/latex]= 0.25% per month. In the first month,
In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50. In the second month,
Notice that in the second month we earned more interest than we did in the first month. This is because we earned interest not only on the original $1000 we deposited, but we also earned interest on the $2.50 of interest we earned the first month. This is the key advantage that compounding interest gives us. Calculating out a few more months gives the following:
We want to simplify the process for calculating compounding, because creating a table like the one above is time consuming. Luckily, math is good at giving you ways to take shortcuts. To find an equation to represent this, if Pm represents the amount of money after m months, then we could write the recursive equation: P0 = $1000 Pm = (1+0.0025)Pm-1 You probably recognize this as the recursive form of exponential growth. If not, we go through the steps to build an explicit equation for the growth in the next example.
Build an explicit equation for the growth of $1000 deposited in a bank account offering 3% interest, compounded monthly. View this video for a walkthrough of the concept of compound interest. While this formula works fine, it is more common to use a formula that involves the number of years, rather than the number of compounding periods. If N is the number of years, then m = N k. Making this change gives us the standard formula for compound interest.
[latex]P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}[/latex]
The most important thing to remember about using this formula is that it assumes that we put money in the account once and let it sit there earning interest. In the next example, we show how to use the compound interest formula to find the balance on a certificate of deposit after 20 years.
A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years? A video walkthrough of this example problem is available below. Let us compare the amount of money earned from compounding against the amount you would earn from simple interest
As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.
When we need to calculate something like [latex]5^3[/latex] it is easy enough to just multiply [latex]5\cdot{5}\cdot{5}=125[/latex]. But when we need to calculate something like [latex]1.005^{240}[/latex], it would be very tedious to calculate this by multiplying [latex]1.005[/latex] by itself [latex]240[/latex] times! So to make things easier, we can harness the power of our scientific calculators. Most scientific calculators have a button for exponents. It is typically either labeled like: ^ , [latex]y^x[/latex] , or [latex]x^y[/latex] . To evaluate [latex]1.005^{240}[/latex] we’d type [latex]1.005[/latex] ^ [latex]240[/latex], or [latex]1.005 \space{y^{x}}\space 240[/latex]. Try it out – you should get something around 3.3102044758.
You know that you will need $40,000 for your child’s education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal?
It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a “close enough” answer, but keeping more digits is always better.
To see why not over-rounding is so important, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.
If we first compute r/k, we find 0.05/12 = 0.00416666666667 Here is the effect of rounding this to different values:
If you’re working in a bank, of course you wouldn’t round at all. For our purposes, the answer we got by rounding to 0.00417, three significant digits, is close enough – $5 off of $4500 isn’t too bad. Certainly keeping that fourth decimal place wouldn’t have hurt. View the following for a demonstration of this example.
In many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{12\times30}}[/latex] We can quickly calculate 12×30 = 360, giving [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{360}}[/latex]. Now we can use the calculator.
Using your calculator continuedThe previous steps were assuming you have a “one operation at a time” calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter: 1000 × ( 1 + 0.05 ÷ 12 ) yx 360 = Solving For TimeNote: This section assumes you’ve covered solving exponential equations using logarithms, either in prior classes or in the growth models chapter. Often we are interested in how long it will take to accumulate money or how long we’d need to extend a loan to bring payments down to a reasonable level.
If you invest $2000 at 6% compounded monthly, how long will it take the account to double in value? Get additional guidance for this example in the following: |
On what sum will the compound interest at 10% per annum for 3 years compounded annually be rs 450
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