Compound Interest: Compounding Periods per Year...

We have two more concepts associated with the compound interest topic with today’s focus on compounding periods and how changing the number of compounding periods per year affects the interest payments, interest payment growth, and account value growth. “Compounding periods”…say what, now? What exactly does this mean? This is how many times per year the interest payment is calculated with the formula.

We’re going to keep things simple and observe how the ending account values change when we modify the compounding periods per year to include:

*n = number of compounding periods PER YEAR;

Annual Compounding, where n = 1;

Bi-annual or Semi-annual Compounding, where n = 2;

Quarterly Compounding, where n = 4;

Monthly Compounding, where n = 12;

Daily Compounding, where n = 365.

RECALL: Compound Interest Formula, where, A = P(1+r/n)^nt

**(If this doesn’t look familiar, please refer back to the earlier post on compound interest for a brief refresher.)

For example, let’s say you contribute $100 in an interest-bearing account that pays 10% interest annually. Notice that word “annually” at the end of that sentence? That tells us the interest payment we’re to receive is only calculated once per year. We’ll use a calendar year, January 1 - December 31, for ease.

P = Initial contribution = $100

i = Interest rate = 10%

n = Compounding periods per year = 1

t = 1 year

January 1: Deposit $100

Interest paid after year 1 = $10

Total account value after year 1 = $100 + $10 = $110

We don’t really need the formula for this example since it’s so straightforward but still good practice.

Let’s go one step further with two compounding periods per year, where n = 2, and the interest is paid every 6 months, but with all else remaining the same, or equal and constant as the economists say (the proper Latin phrase is: “Ceteris Paribus” meaning, “other things equal”).

P = Initial contribution = $100

i = Interest rate = 10%

n = Compounding periods per year = 2

t = 1 year

January 1: Deposit $100

Interest paid after first 6 months, January 1 - June 30 = $5

Interest paid after second 6 months, July 1 - December 31 = $5.25

Total account value after year 1 = $100 + $10.25 = $110.25

*The difference appears insignificant but think about this over longer periods of time with account values that continue to increase.

Substituting those given values into the formula will give us:

A = 100(1 + (0.10/2))^(2*1) = 100(1 + (0.05))^(2) = $110.25

**If your answer is $11,025 it’s because of PEMDAS;

RECALL: The order of operations is known as: PEMDAS. We all remember this, yes? PEMDAS determines the, well order of operations, or the order in which we proceed through the equation. In case you forgot, PEMDAS means:

P = Parenthesis

E = Exponents

M = Multiplication

D = Division

A = Addition

S = Subtraction

This concept is extremely important because following the correct order ensures an accurate result.

Ok, back to our program: What do these additional compounding periods each year illustrate? Basically, the more frequently interest is compounded, the more often interest gets added to the balance, which then means interest earns its own interest! This is how our money “works for us” as the account value compounds and grows.

Let’s start with a larger amount now; perhaps $1000. Most of us would be able to save over the course of many months or more to start with $1000 initial contribution. Plus, it further emphasizes the idea of leaving the gains in the account no matter the value to maximize the growth if possible:

• Principal = $1,000

• Annual interest rate = 10% (0.10)

• Time = 1 year

RECALL: A = P(1 + r/n)^(nt)

We do our fill-in for the variables to find our ending account values.

Annual compounding (n = 1):

A = 1000(1 + (0.10/1))^(1*1) = 1000( 1.10) =  $1100

Quarterly compounding (n = 4):

A = 1000(1 + (.10/4))^(4*1) = 1000(1.025)^4 = $1103.813

Monthly compounding (n = 12):

A = 1000(1 + (0.10/12})^12 = 1104.71

Daily compounding (n = 365):

A = 1000(1 + (0.10/365))^(365) = 1105.16

So, higher compounding frequency → slightly more money. I realize these differences appear to be rather small but those small differences can be huge gains in the future when we factor in longer durations of time. And that’s even before we make any additional contributions! (That topic is coming up in our next post.)

There is, of course, the more extreme “continuous compounding” but that’s very rare to see in the real world from a financial application perspective. The math is there from a conceptual point of view, but I don’t expect any financial institution to offer a continuous compounding account. Why? Simply because it’s not in the interest of the financial institution. Their job is to make money as a business and continuous compounding wouldn’t necessarily be beneficial for them.

The following chart shows how $1,000 grows at 10% interest over 10 years with different compounding frequencies.

That chart can be a little difficult to see the subtle differences so let's look at the comparisons broken down for our various compounding periods per year, starting with semi-annual, or twice per year.

Annual, where n = 1:

Year	Interest	Accrued Interest	Balance
Initial Contribution	–	–	$1,000.00
1	$100.00	$100.00	$1,100.00
2	$110.00	$210.00	$1,210.00
3	$121.00	$331.00	$1,331.00
4	$133.10	$464.10	$1,464.10
5	$146.41	$610.51	$1,610.51
6	$161.05	$771.56	$1,771.56
7	$177.16	$948.72	$1,948.72
8	$194.87	$1,143.59	$2,143.59
9	$214.36	$1,357.95	$2,357.95
10	$235.79	$1,593.74	$2,593.74

Semi-annual, where n = 2:

Year	Interest	Accrued Interest	Year End Balance
Initial Contribution	–	–	$1,000.00
1	$102.50	$102.50	$1,102.50
2	$113.01	$215.51	$1,215.51
3	$124.59	$340.10	$1,340.10
4	$137.36	$477.46	$1,477.46
5	$151.44	$628.89	$1,628.89
6	$166.96	$795.86	$1,795.86
7	$184.08	$979.93	$1,979.93
8	$202.94	$1,182.87	$2,182.87
9	$223.74	$1,406.62	$2,406.62
10	$246.68	$1,653.30	$2,653.30

Quarterly, where n = 4:

Year	Interest	Accrued Interest	Year End Balance
Initial Contribution	–	–	$1,000.00
1	$103.81	$103.81	$1,103.81
2	$114.59	$218.40	$1,218.40
3	$126.49	$344.89	$1,344.89
4	$139.62	$484.51	$1,484.51
5	$154.11	$638.62	$1,638.62
6	$170.11	$808.73	$1,808.73
7	$187.77	$996.50	$1,996.50
8	$207.26	$1,203.76	$2,203.76
9	$228.78	$1,432.54	$2,432.54
10	$252.53	$1,685.06	$2,685.06

Monthly, where n = 12:

Year	Interest	Accrued Interest	Year End Balance
Initial Contribution	–	–	$1,000.00
1	$104.71	$104.71	$1,104.71
2	$115.68	$220.39	$1,220.39
3	$127.79	$348.18	$1,348.18
4	$141.17	$489.35	$1,489.35
5	$155.95	$645.31	$1,645.31
6	$172.29	$817.59	$1,817.59
7	$190.33	$1,007.92	$2,007.92
8	$210.26	$1,218.18	$2,218.18
9	$232.27	$1,450.45	$2,450.45
10	$256.59	$1,707.04	$2,707.04

Daily, where n = 365:

Year	Interest	Accrued Interest	Year End Balance
Initial Contribution	–	–	$1,000.00
1	$105.16	$105.16	$1,105.16
2	$116.21	$221.37	$1,221.37
3	$128.43	$349.80	$1,349.80
4	$141.94	$491.74	$1,491.74
5	$156.87	$648.61	$1,648.61
6	$173.36	$821.97	$1,821.97
7	$191.59	$1,013.56	$2,013.56
8	$211.74	$1,225.30	$2,225.30
9	$234.00	$1,459.30	$2,459.30
10	$258.61	$1,717.91	$2,717.91

You’ll notice the very bottom right number of each table grows slightly bigger as we increase the number of compounding periods per year. We didn’t change anything else; our initial contribution was the same, interest rate was the same, and total investment years was the same.

The Compounding Periods Healthy Bits:

It’s true, the ending account balances seem small when comparing each of the compounding periods but as mentioned, this really adds up to exponential growth over longer periods of time. What’s of utmost importance is we leave the initial contribution alone and allow the compound interest formula to work FOR us. If we simply make our initial contribution, get out of the way, and allow time to pass we can see just how powerful this concept is from an investment standpoint.

If we’re paying a loan it’s a completely different situation where we DO NOT WANT more compounding periods; that just means we’ll pay more interest over time. No fun and no good for us. Unfortunately, if we need a loan then there's a good chance we'll have to deal with some form compounding period setup which is why being informed and knowing how these concepts work is to our advantage.