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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. The investing information provided on this page is for educational purposes only. NerdWallet does not offer advisory or brokerage services, nor does it recommend or advise investors to buy or sell particular stocks, securities or other investments. Your savings account balances and investments can grow more quickly over time through the magic of compounding. Use the compound interest calculator above to see how big a difference it could make for you. Using this compound interest calculator
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Additional information can be found at http://www.sofi.com/legal/banking-rate-sheet Learn More Deposits are FDIC Insured Upgrade - Rewards Checking Learn More Deposits are FDIC Insured Current Account Learn More Deposits are FDIC Insured Chime Checking Account Learn More on Citibank, N.A.'s website Citi Priority Account Money market accounts pay rates similar to savings accounts and have some checking features. Money market accounts pay rates similar to savings accounts and have some checking features. UFB Elite Money Market Here’s a deeper look at how compounding works: What is compound interest?For savers, the definition of compound interest is basic: It’s the interest you earn on both your original money and on the interest you keep accumulating. Compound interest allows your savings to grow faster over time. In an account that pays compound interest, such as a standard savings account, the return gets added to the original principal at the end of every compounding period, typically daily or monthly. Each time interest is calculated and added to the account, the larger balance earns more interest, resulting in higher yields. For example, if you put $10,000 into a savings account with a 2% annual yield, compounded daily, you’d earn $203 in interest the first year, another $206 the second year and so on. After 10 years of compounding, you would have earned a total of $2,214 in interest. But remember, that’s just an example. For longer-term savings, there are better places than savings accounts to store your money, including Roth or traditional IRAs and CDs. Compounding investment returnsWhen you invest in the stock market, you don’t earn a set interest rate but rather a return based on the change in the value of your investment. When the value of your investment goes up, you earn a return. If you leave your money and the returns you earn invested in the market, those returns are compounded over time in the same way that interest is compounded. If you invested $10,000 in a mutual fund and the fund earned a 7% return for the year, you’d gain about $700, and your investment would be worth $10,700. If you got an average 7% return the following year, your investment would then be worth about $11,500. Over the years, your investment can really grow: If you kept that money in a retirement account over 30 years and earned that average 7% return, for example, your $10,000 would grow to more than $76,000. In reality, investment returns will vary year to year and even day to day. In the short term, riskier investments such as stocks or stock mutual funds may actually lose value. But over a long time horizon, history shows that a diversified growth portfolio can return an average of 6% to 7% annually. Investment returns are typically shown at an annual rate of return. The average stock market return is historically 10% annually, though that rate is reduced by inflation. Investors can currently expect inflation to reduce purchasing power by 2% to 3% a year. Compounding can help fulfill your long-term savings and investment goals, especially if you have time to let it work its magic over years or decades. You can earn far more than what you started with. More NerdWallet calculatorsCompounding with additional contributionsAs impressive as compound interest might be, progress on savings goals also depends on making steady contributions. Let’s go back to the savings account example above. We started with $10,000 and ended up with about $2,214 in interest after 10 years in an account with a 2% annual yield. But by depositing an additional $100 each month into your savings account, you’d end up with about $25,509 after 10 years, when compounded daily. The interest would be about $3,509 on total deposits of $22,000. |
What is the difference between simple interest and compound interest for a period of 2 years at the rate of 10% per annum on the sum of dollar 60000?
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