Discussing 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. I = P0r A = P0 + I = P0 + P0r = P0(1 + r) I is the interest A is the end amount: principal plus interest P0 is the principal (starting amount) r is the interest rate (in decimal form. Example: 5% = 0.05) A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?
A = P0 + I = P0 + P0rt = P0(1 + rt) I is the interest A is the end amount: principal plus interest P0 is the principal (starting amount) r is the interest rate in decimal form t is time 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.(These are examples of periodic rate or rate per period.) A 4% annual rate paid quarterly would have a quarterly rate of 1% (0.01 in decimal). A 6% APR paid monthly, would have a monthly rate of 0.5% (0.005 in decimal) 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? Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.
Show Answer Compound Interest
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 earn3%12=0.25%\displaystyle\frac{{{3}\%}}{{12}}={0.25}\%123%=0.25% per month. In the first month,P0 = $1000 r = 0.0025 (0.25%) I = $1000 (0.0025) = $2.50 A = $1000 + $2.50 = $1002.50 In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50. In the second month,P0 = $1002.50 I = $1002.50 (0.0025) = $2.51 (rounded) A = $1002.50 + $2.51 = $1005.01 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 thatcompounding of interest gives us. Calculating out a few more months:
Pmrepresents 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 could go through the steps to build an explicit equation for the growth: P0 = $1000 P1 = 1.0025P0 = 1.0025 (1000) P2 = 1.0025P1 = 1.0025 (1.0025 (1000)) = 1.0025 2(1000) P3 = 1.0025P2 = 1.0025 (1.00252(1000)) = 1.00253(1000) P4 = 1.0025P3 = 1.0025 (1.00253(1000)) = 1.00254(1000) Observing a pattern, we could concludePm = (1.0025)m($1000) Notice that the $1000 in the equation wasP0, the starting amount. We found 1.0025 by adding one to the growth rate divided by 12, since we were compounding 12 times per year. Generalizing our result, we could writePm=P0(1+rk)m\displaystyle{P}_{{m}}={P}_{{0}}{\left({1}+\frac{{r}}{{k}}\right)}^{{m}}Pm=P0(1+kr)m In this formula:m is the number of compounding periods (months in our example) r is the annual interest rate k is the number of compounds per year. 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. IfN is the number of years, then m = N k. Making this change gives us the standard formula for compound interest. PN is the balance in the account after N years.P0 is the starting balance of the account (also called initial deposit, or principal) r is the annual interest rate in decimal form k is the number of compounding periods in one year. If the compounding is done annually (once a year),k = 1. If the compounding is done quarterly,k = 4. k = 12. If the compounding is done daily,k = 365. The most important thing to remember about using this formula is that it assumes that we put money in the accountonce and let it sit there earning interest. 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? In this example,
3 it is easy enough to just multiply 5 × 5 × 5 = 125. But when we need to calculate something like 1.005240, it would be very tedious to calculate this by multiplying 1.005 by itself 240 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:^ , y x , or xy . To evaluate 1.005240 we'd type 1.005^240, or 1.005 yx 240. 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?In this example, we're looking for P0. (NOTE: I usually just use P)
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.
r/k, we find 0.05/12 = 0.00416666666667 Here is the effect of rounding this to different values:
P30=1000(1+0.0512)12×30\displaystyle{P}_{{30}}={1000}{\left({1}+\frac{{0.05}}{{12}}\right)}^{{{12}\times{30}}}P30=1000(1+120.05)12×30 We can quickly calculate 12 × 30 = 360, givingP30=1000(1+0.0512)360\displaystyle{P}_{{30}}={1000}{\left({1}+\frac{{0.05}}{{12}}\right)}^{{{360}}}P30=1000(1+120.05)360 Now we can use the calculator.
1000 × ( 1 + 0.05 ÷ 12 ) yx 360 = David Lippman, Math in Society, "Finance," licensed under a CC BY-SA 3.0 license.Rarely is it the case these days that you invest $100 of your money at, say, 5% per year and get $5 every year (known as simple interest). Why is this not the case? Because interest is frequently compounded, which means that the 5% interest is paid on the full current balance. Let's illustrate this in a comparison of tables:
If were to look at the balances 20 years down the line, we would see a more substantial difference: After 20 years, compound interest brings in $73.60 more profit than simple interest. You might be saying, "this difference is insignificant over a 20 year period," and by that you have a valid point. Keep in mind that this is based on a one-time investment of $100. Over a 20-year period, you will have earned: $265.33$100=2.65\displaystyle\frac{{\${265.33}}}{{\${100}}}={2.65}$100$265.33=2.65 2.65−1=1.65=165%\displaystyle{2.65}-{1}={1.65}={165}\%2.65−1=1.65=165% gain This represents nearly tripling the original amount (2.65 times the original, to be more exact). With simple interest, this gain would only be:$200100−1=1.00=100%\displaystyle\frac{{\${200}}}{{100}}-{1}={1.00}={100}\%100$200−1=1.00=100% gain The simple interest amount is double the original balance. At this point, you might be wondering how it is that we obtained the 20-year balances. Certainly, we can approximate these balances based on the graph given, but even then we need a way to generate the graph. For simple interest, this is quite simple. Suppose the periodic interest rate, that is, the interest paid per period (i.e. per year, per month, per day, etc.), is represented as a decimal and assigned to the variablei. Then, first calculate the regular interest amount by multiplying the rate by the initial deposit, or the principle, P. regular interest paid = P × i This amount will be paid over time periods, so the total amount of interest is total interest overt periods = N × P × i For example, if the principle isP = $500 and the interest rate is i = 10% per year for N = 8 years, then the regular interest paid is P × i = $500 × .10 = $50 per year. Paid over 8 years, we get: total interest over 10 years = 8 × $500 × .10 = $400 To get our total balance, we must add this amount back to the original principle to get: P + N × P × i = $500 + $400 = $900 We often call this the accumulated amount, orA. More generally, A = P + N × P × =P(1 + Ni) If interest is paid according to a simple interest schedule and we defineA = accumulated balance or future value P = principal invested N = number of periods i = periodic interest rate Then A = P(1 + Ni) Verify that the 20-year balance for a $100 investment at 5% yearly interest is $200 by using the simple interest balance formula. We have thatP = 100, N = 20, i = .05 so A = 100(1 + 20 × .05) = 100(2) = $200Building a Compound Interest FormulaFor compound interest the idea is fairly simple. Recall that growth by a percentage is calledexponential growth. To calculate a new amount, we must account for 100% of the original amount, plus the periodic growth rate, say , written as a decimal, Then, there will be a total of of the original amount after one period. For example, suppose that a population grows by 3% every year. Next year there will be a total of 103% of the amount this year. We write this as 1 + .03 = 1.03 to represent a decimal. This is called thegrowth factor and is what we multiply by to obtain the new amount. The 3% represents the growth rate and is usually the value reported by banks, the media, etc. when describing growth. Suppose the population is 1,000. Next year the population is expected to be 1000(1.03) = 1,030. What will this amount be in 2 years? Assuming the same growth rate of 3%, we simply apply the growth factor to the 1-year amount: 1,030(1.03) ≈ 1,061 Or, alternatively we can write[1000(1.03)]1.03 = 1000(1.03)2 Do you see the pattern? The exponent simply represents the number of time periods that we require to pass. If we wanted to know the population after 10 years, we would multiply 1000 by 1.03 a total of 10 times, or1000(1.03)10 ≈ 1,344 This same idea applies to compound interest! If interest is paid according to a compound interest schedule, where interest is paid on thecurrent balance and we define A = accumulated balance or future value P = principal invested N = number of periods i = periodic interest rate Then A = P(1 + i)N Confirm that if you invest $100 for 20 years at an annual interest rate of 5% compounded annually, that you will have a balance of $253.33. We haveP = 100, i = .05, N = 20, so A = 100(1 + .05)20 =100(1.05)20 ≈ 200(2.6533) = $265.33 Notice in Example 2 the wording "compounded annually." This simply specifies how frequently the interest is paid. The values of and should reflect the compounding period specified. Historically, banks have decided that offer anominal annual rate, or Annual Percentage Rate (APR). These are identical terms. This is simply a name for the rate, because it is rarely paid once each year. Instead, a bank will identify how often interest is compounded. Some of the common ones are listed below:
not actually pay you 12% each month. Instead you receive a pro-rated percentage every month, which is an equal fraction of the 12% per period. Since there are 12 periods per year, you would receive 12%/12 months = 1%/month. A bank offers you a nominal annual rate of 5% compounded monthly. You invest $100 and plan on keeping it invested for 20 years. Calculate your balance after 20 years. Then, compare this to the value found in example 2 based on annual compounding and comment on the effect of compounding periods. We have thatP = 100. Since the compounding period is one month, we must express i and N in terms of months. Since there are 12 months per year, there are N = 12 × 20 = 240 periods in the investment. Further, the periodic rate is i=.0512 months≈.00417or.417%\displaystyle{i}=\frac{{.05}}{{{12}\ {m}{o}{n}{t}{h}{s}}}\approx{.00417}{\quad\text{or}\quad}{.417}\%i=12 months.05≈.00417or.417% per month. We calculateA = 100(1 + .00417)240 ≈ $271.48 We found that if interest is paid once a year, then the 20-year accumulated balance is $265.33, which is $6.15 less than when interest is compounded monthly. Thus, increasing compounding frequency increases total balance. However, this difference is not very much. As the frequency of compounding interest increases, so does the accumulated balance. To see this more clearly, consider the various compounding periods below, and the balance of $100 after 20 years at 5%:
P = $100. Then, at the end of one year the balance will be 1.12(100) = $112, if interest is paid once. But, the interest under monthly compounding (1% per month) will be: 100(1.01)12 ≈ $112.68 This difference occurs due to the fact that monthly compounding pays 1% of thecurrent balance. After the first month, there is a balance of 100(1.01) = 101, but one month later the balance is 101(1.01) = 102.01, which is more than a $1 increase. A rate of 12% annually is the same as $1 per month, an amount less than would be received as of the second month and beyond compared to monthly compounding. Annual Percentage YieldSo, if 12% once is not the same as 1% 12 times, what percentageis the percentage paid over a year for 1% paid 12 times? To find the percentage that $112.68 is of the original amount, we divide: 112.68100=1.1268\displaystyle\frac{{112.68}}{{100}}={1.1268}100112.68=1.1268 This means that the overall growth was 12.68%, a percentage larger than 12. Recall that the rate of 12% is called the nominal annual rate. The rate that youactually get after compounding is taken into account is called the annual percentage yield (APY). We present a formal way to calculate this: Since the APY is over a year ( annual percentage yield), we take the compound interest formula over the course of 1-year only and only concern ourselves with a $1 investment (since 1 = 100%). Subtract 1 from the outcome, so that we only account for the growth, not the original 100%. APY=1(1+rn)n×1−1=(1+rn)n−1\displaystyle{A}{P}{Y}={1}{\left({1}+\frac{{r}}{{n}}\right)}^{{{n}\times{1}}}-{1}={\left({1}+\frac{{r}}{{n}}\right)}^{{{n}}}-{1}APY=1(1+nr)n×1−1=(1+nr)n−1 Thus,APY=(1+rn)n−1\displaystyle{A}{P}{Y}={\left({1}+\frac{{r}}{{n}}\right)}^{{{n}}}-{1}APY=(1+nr)n−1 Absolutely! If the amount invested is different than $1, calculate what it will become in one year. Take the year-end amount, divide it by the original, and subtract 1. Let's say you invest $325 at 10% compounded semi-annually (twice a year) for 5 years. What is the APY? Since we want theannual percentage yield, we don't need to worry about the duration of the investment. We will compute the answer using the formula, and the intuitive way:
|