2.1a. Simple and Compound Interest

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 Interest

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.

Simple One-time 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)

Example 1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?  The principal $\displaystyle{P}_{{0}}=\{300}$ 3% rate $\displaystyle{r}={0.03}$ You will earn$9 interest $\displaystyle{I}=\{300}{\left({0.03}\right)}=\{9}$

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.

Example 2

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? Each year, you would earn 5% interest:$1000(0.05) = $50 in interest. So over the course of five years, you would earn a total of$250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of$1,250.

We can generalize this idea of simple interest over time.

Simple Interest over Time

I = P0rt a

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.

APR—Annual Percentage 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)

Example 3

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.  The principal $\displaystyle{P}_{{0}}=\{1000}$ 2% rate per half-year $\displaystyle{r}={0.02}$ 4 years = 8 half-years $\displaystyle{t}={8}$ You will earn$160 interest over the four years $\displaystyle{I}=\{1000}{\left({0.02}\right)}{\left({8}\right)}=\{160}$

Try it Now 1

A loan company charges $30 interest for a one month loan of$500. Find the annual interest rate they are charging.

Compound Interest

Note from Professor Pinegar: I studied on whether to write this at the beginning or the end. I may not use the same variables in finance as the authors of these sections. Below are the formulas I generally use.

As you view the formulas below, let me tell you a little about some differences you might see.

• In the below formula, “m” represents number of compounding periods per year and  “n” represents the total number of compounding periods. So  n=mt.
• Example:  ${(1+i)}^{n}{or}{(1+i)}^{mt}$
• In this format i (rate per period)=r/m
• Example:  ${(1+i)}^{n}={(1+r/m)}^{n}$
• In other formulas, they might use “n” for number of compounding periods per year and just use nt for the total number of compounding periods.
• Example:  ${(1+i)}^{nt}$
• In this format i=r/n
• Example:  ${(1+i)}^{nt}={(1+r/n)}^{nt}$
• This could be confusing as you see different examples. Hopefully this clears it up.
• See below and zoom in if needed.

Source: Kevin Pinegar

With 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 $\displaystyle\frac{{{3}\%}}{{12}}={0.25}\%$ 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 that compounding of interest gives us. Calculating out a few more months:  Month Starting balance Interest earned Ending Balance 1 1000.00 2.50 1002.50 2 1002.50 2.51 1005.01 3 1005.01 2.51 1007.52 4 1007.52 2.52 1010.04 5 1010.04 2.53 1012.57 6 1012.57 2.53 1015.10 7 1015.10 2.54 1017.64 8 1017.64 2.54 1020.18 9 1020.18 2.55 1022.73 10 1022.73 2.56 1025.29 11 1025.29 2.56 1027.85 12 1027.85 2.57 1030.42 To find an equation to represent this, if 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 1 = 1.00250 = 1.0025 (1000) 2 = 1.00251 = 1.0025 (1.0025 (1000)) = 1.0025 2(1000) 3 = 1.00252 = 1.0025 (1.00252(1000)) = 1.00253(1000) 4 = 1.00253 = 1.0025 (1.00253(1000)) = 1.00254(1000) Observing a pattern, we could conclude Pm = (1.0025)m($1000)

Notice that the $1000 in the equation was P0, 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 write $\displaystyle{P}_{{m}}={P}_{{0}}{\left({1}+\frac{{r}}{{k}}\right)}^{{m}}$ 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. If N is the number of years, then m = N k. Making this change gives us the standard formula for compound interest. 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. If the compounding is done monthly, 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 account once and let it sit there earning interest. Example 4 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,

 The principal $\displaystyle{P}_{{0}}=\{3000}$ 6% annual rate $\displaystyle{r}={0.06}$ 12 months in 1 year $\displaystyle{k}={12}$ We’re looking for the value in 20 years $\displaystyle{N}={20}$ You will have $9930.61 in 20 years $\displaystyle{P}_{{20}}=\{3000}{\left({1}+\frac{{0.06}}{{12}}\right)}^{{{20}\times{12}}}=\{9930.61}$ Let us compare the amount of money earned from compounding against the amount you would earn from simple interest  Years Simple Interest ($15 per month) 6% compounded monthly = 0.5% each month 5 $3900$4046.55 10 $4800$5458.19 15 $5700$7362.28 20 $6600$9930.61 25 $7500$13394.91 30 $8400$18067.73 35 $9300$24370.65 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.

Evaluating Exponents on the Calculator

When we need to calculate something like 5
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)  4% annual rate $\displaystyle{r}={0.04}$ 4 quarters in 1 year $\displaystyle{k}={4}$ We know the balance in 18 years $\displaystyle{N}={18}$ The amount we have in 18 years is$40000 $\displaystyle{P}_{{18}}=\{40},{000}$ You need a principal of $19539.84 $\displaystyle{40000}={P}_{{0}}{\left({1}+\frac{{0.04}}{{4}}\right)}^{{{18}\times{4}}}$ So you would need to deposit$19,539.84 now to have $40,000 in 18 years. Rounding 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. Example 6 To see why not over-rounding is so important, suppose you were investing$1000 at 5% interest compounded monthly for 30 years.

 The principal $\displaystyle{P}_{{0}}=\{1000}$ 5% interest rate $\displaystyle{r}={0.05}$ 12 months in 1 year $\displaystyle{k}={12}$ We’re looking for the amount in 30 years $\displaystyle{N}={30}$

If we first compute
r/k, we find 0.05/12 = 0.00416666666667

Here is the effect of rounding this to different values:

 r/k rounded to: Gives P­30­ to be: Error 0.004 $4208.59$259.15 0.0042 $4521.45$53.71 0.00417 $4473.09$5.35 0.004167 $4468.28$0.54 0.0041667 $4467.80$0.06 no rounding $4467.74 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. Using Your Calculator 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 $\displaystyle{P}_{{30}}={1000}{\left({1}+\frac{{0.05}}{{12}}\right)}^{{{12}\times{30}}}$ We can quickly calculate 12 × 30 = 360, giving $\displaystyle{P}_{{30}}={1000}{\left({1}+\frac{{0.05}}{{12}}\right)}^{{{360}}}$ Now we can use the calculator.  Type this Calculator shows 0.05 ÷ 12 = 0.00416666666667 + 1 = 1.00416666666667 yx 360 = 4.46774431400613 ×1000 = 4467.74431400613 The 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 = David Lippman, Math in Society, “Finance,” licensed under a CC BY-SA 3.0 license. Developing Financial Intuition 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:  Simple Interest Year Balance Interest Year-End Balance 0$100.00 .05(100) = 5 $105.00 1$105.00 .05(100) = 5 $110.00 2$110.00 .05(100) = 5 $115.00 3$115.00 .05(100) = 5 $120.00 Interest paid on original balance only: constant rate of growth  Compound Interest Year Balance Interest Year-End Balance 0$100.00 .05(100) = 5 $105.00 1$105.00 .05(105) = 5.25 $110.25 2$110.25 .05(110.25) = 5.51 $115.76 3$115.76 .05(115.7625) = 5.79 $121.55 Interest paid on overall balance: constant percentage growth Although not a huge difference, notice that the balances continue to grow slightly further apart as the years go by. This difference is easier to see in a graph comparing the balances: 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: $\displaystyle\frac{{\{265.33}}}{{\{100}}}={2.65}$ $\displaystyle{2.65}-{1}={1.65}={165}%$ 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: $\displaystyle\frac{{\{200}}}{{100}}-{1}={1.00}={100}%$ 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 variable i. 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 over t periods = N × P × i For example, if the principle is P =$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:

Building a Compound Interest Formula

For compound interest the idea is fairly simple. Recall that growth by a percentage is called
exponential 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 the
growth 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, or

1000(1.03)10 ≈ 1,344

This same idea applies to compound interest!

Compound Interest Balance Formula

If interest is paid according to a compound interest schedule, where interest is paid on the
current 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

Example 2

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.

Solution

We have
P = 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 a nominal 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:  Compounding Period Number of Annual Compoundings Annually 1 Semi-annually 2 Quarterly 4 Monthly 12 Weekly 52 Daily 365 Note: Weeks and days vary depending on year. For ease of use, we ignore this detail. Do you think that a monthly compounding schedule means a very generous bank? Not in the way you might expect. Suppose a bank offers you a nominal annual rate of 12% compounded monthly. They do 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. Example 3 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.

Solution

We have that
P = 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 $\displaystyle{i}=\frac{{.05}}{{{12}\ {m}{o}{n}{t}{h}{s}}}\approx{.00417}{\quad\text{or}\quad}{.417}%$ per month. We calculate

A = 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. Effect of Compounding Frequency on Accumulated Balance (Future Value), 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%:

Compounding Period Balance Differences
Annually $265.33 Semi-annually$268.51 $3.18 Quarterly$270.15 $1.64 Monthly$271.26 $1.12 Weekly$271.70 $0.43 Daily$271.81 $0.11 We can see that, while the balance is slightly larger than that of the previous compounding period, the differences become quite small as the frequency increases more and more. Example 4 Is 12% given annually the same thing as 1% given monthly? Why or why not? Solution Suppose a person deposits 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 the
current 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 Yield

So, if 12% once is not the same as 1% 12 times, what percentage
is 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: $\displaystyle\frac{{112.68}}{{100}}={1.1268}$ 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 you actually 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%.

$\displaystyle{A}{P}{Y}={1}{\left({1}+\frac{{r}}{{n}}\right)}^{{{n}\times{1}}}-{1}={\left({1}+\frac{{r}}{{n}}\right)}^{{{n}}}-{1}$

Thus,
$\displaystyle{A}{P}{Y}={\left({1}+\frac{{r}}{{n}}\right)}^{{{n}}}-{1}$

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. Example 5 Let’s say you invest$325 at 10% compounded semi-annually (twice a year) for 5 years. What is the APY?

Solution

Since we want the
annual 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:

APY Formula Intuitively
$\displaystyle{\left({1}+\frac{{.1}}{{2}}\right)}^{{2}}-{1}={1.1025}-{1}$ Using TVM Solver, $325 will be$358.3125 in one year. Find the ratio of new to old.
$\displaystyle={.1025}$ $\displaystyle\frac{{\ne{w}}}{{{o}{l}{d}}}=\frac{{358.3125}}{{325}}={1.1025}$
$\displaystyle={10.25}%$ This means that the growth is 10.25%. The ones place tells us that the new is 100% of the old, and then some.

In my opinion, it is much easier to understand and remember the intuitive approach on the right. Needless to say, you’ll get the same answer.

Milos Podmanik, By the Numbers, “Compound Interest and Exponential Growth,” licensed under a CC BY-NC-SA 3.0 license.