Reading: Compound Interest and Exponential Growth

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:

[latex]\displaystyle\frac{{\${265.33}}}{{\${100}}}={2.65}[/latex]

[latex]\displaystyle{2.65}-{1}={1.65}={165}%[/latex] 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:

[latex]\displaystyle\frac{{\${200}}}{{100}}-{1}={1.00}={100}%[/latex] 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:

P + N × P × i = $500 + $400 = $900

We often call this the accumulated amount, or
A. More generally,

A = P + N × P ×

=
P(1 + Ni)

Simple Interest Balance Formula

If interest is paid according to a simple interest schedule and we define

A = accumulated balance or future value

P = principal invested

N = number of periods

i = periodic interest rate

Then
A = P(1 + Ni)

Example 1

Verify that the 20-year balance for a $100 investment at 5% yearly interest is $200 by using the simple interest balance formula.

Solution

We have that
P = 100, N = 20, i = .05 so

A = 100(1 + 20 × .05)

= 100(2)

= $200

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 [latex]\displaystyle{i}=\frac{{.05}}{{{12}\ {m}{o}{n}{t}{h}{s}}}\approx{.00417}{\quad\text{or}\quad}{.417}%[/latex] 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:

[latex]\displaystyle\frac{{112.68}}{{100}}={1.1268}[/latex]

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%.

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

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

Alternatives to the formula?

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
[latex]\displaystyle{\left({1}+\frac{{.1}}{{2}}\right)}^{{2}}-{1}={1.1025}-{1}[/latex] Using TVM Solver, $325 will be $358.3125 in one year. Find the ratio of new to old.
[latex]\displaystyle={.1025}[/latex] [latex]\displaystyle\frac{{\ne{w}}}{{{o}{l}{d}}}=\frac{{358.3125}}{{325}}={1.1025}[/latex]
[latex]\displaystyle={10.25}%[/latex] 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.