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# Compound interest is no miracle. It’s math.

## Explaining Nominal vs Effective Interest Rates

“Interest” is an interesting word.

In common usage, it suggests that a person is at least curious about something that they take an “interest” in, like a hobby. But in a political context, it can mean that someone is an advocate for a certain policy position that benefits them in a specific way, as in an “interest” group.

In a financial context, “interest” also has multiple meanings. For example, it can mean that someone has an ownership stake or an “interest” in a company or an asset. It also means the amount of money you can collect on a savings account, or pay on a loan. Sal Kahn calls interest the “rent you pay” to use some else’s money. Like renting a house or car, you have to pay money to “rent” (i.e., borrow) money that belongs to someone else. And the longer you want to borrow it, the more you have to pay.

Understanding the units that are used to express interest rates (remember the importance of time) is very important. Typically, interest rates are expressed as a dimensionless ratio, in percent. For example, you might have to pay 13% per year on the balance of a credit card.

In this case, “per cent” means the amount of money that must be paid per 100 dollars borrowed in the time period specified. The amount borrowed is the balance, or the principal. So the ratio is actually \$i / \$b / time, or dollars of interest (\$i) per 100 dollars of balance (\$b) per unit time.

Where no unit of time is specified, then it is customary to assume that interest rates are expressed per year (sometimes called per annum). But it is not so unusual to express interests rates in terms of months. Unscrupulous lenders (called usurers, or loan sharks) may express interest rates on a per day basis (in which case the interest is called the vigorish, or “vig” for short).

In addition to the unit of time, it is important to understand the compounding period. As interest accumulates, the balance of a loan increases. When interest is compounded, the interest is added to the balance. At the next compounding, the new balance on which the interest is computed will include the interest added at the last compounding. So, unless payments are being made, the borrower now owes interest on the interest.

The compounding period and the units of time can be different. For example, the interest rate on a loan might be expressed as 12%/yr, compounded monthly. In this case, there is a significant difference between the nominal interest rate and the effective interest rate. The effective rate includes the interest paid on the interest after compounding.

There are more than 3 million Google hits in a search for “the power of compound interest“, and most of them talk about how beginning a savings plan early in your career can theoretically result in an rich retirement fund. The so-called secret of compound interest is that future interest payments are not based solely on the original principal, but also on prior interest. That is, unlike “simple” interest (which no one ever uses), compound interest means that the interest is added to the balance, and consequently from that point on, interest must be paid on a higher balance (i.e., interest is paid on the interest). The result is exponential growth of the balance.

The abundance of articles online touting the “miracle” of this compound interest claims that compound interest can grow fortunes from modest starting balances. However, remember that every saver with a retirement fund has a counterpart borrower with a mortgage or credit card debt. Compounding works against borrowers, just as it works in favor of lenders. To this end, borrowers need to understand what is typically not explained in lending disclosure documents — the compounding period.

The convention in finance is to report interest rates on an annual basis, such as 8 percent per year. For example, if all credit card companies report interest rates on an annual basis, then consumers can easily compare rates, and shop for the cheapest card. However, in practice interest can be added to the balance of a credit card loan daily, monthly, or whenever. The compounding period is the amount of time that elapses between the addition of new interest to the balance (which is called compounding).

The fact that interest rates are reported on an annual basis, but compounding takes place more frequently creates a difficult problem of communication in finance. By convention, the nominal interest rate is the stated rate before the effects of compounding. For example, a bank or an auto dealer might quote a low nominal lending rate, and if the rate were compounded annually, this would be exactly the rate paid by borrowers. But loan interest is almost never compounded annually! The effective rate is what the borrowers actually have to pay, and it is always greater than (or equal to) the nominal rate.

In this video from Khan Academy, Salman Khan demonstrates how more frequent compounding results in higher effective interest rates.

Because compounding is the act of adding interest to the balance of the loan, the more often the interest is compounded, the more interest borrowers will have to pay. To convert a nominal interest rate to an effective interest rate, we have to pay close attention to the units of time. The formula looks like this:

Reff = [( 1 + Rnom / n ) ^ n] -1

where Reff = the effective rate, Rnom = the nominal rate, and n = the number of compounding periods over the time period for which Rnom is reported (usually a year).

For example, if Rnom is stated as a percentage per year and we compound monthly, then n = 12. If Rnom is stated as a percentage per year and compound daily, then n = 365. But the same formula can work for unusual combinations. Let’s say that for some strange reason we report Rnom as a percentage per week and compound daily. Then n = 7.

https://youtu.be/lXH0LhRPjtQ

The higher the interest rate, the more important the compounding period is. That is, the difference between daily and annual compounding is a lot bigger at 12%/yr interest than at 4%/yr. The more often we compound, the higher the effective rate goes.

Try some calculations yourself with your own calculator:

12%/yr compounded monthly = ( 1 + .12 / 12) ^ 12–1 = 0.126825 (or 12.683%)

12%/yr compounded daily = (1 + .12 / 365 ) ^ 365–1 = .12747 (or 12.747%)

12%/yr compounded hourly = ( 1 + .12 / (365*24) ) ^ (365 * 24) — 1 = .127496 (or 12.750%)

Notice how the difference between annual compounding (12.00%) and monthly compounding (12.68%) is large, but the difference between monthly and daily (12.747%) is smaller, and the difference between daily and hourly (12.750%) is almost nonexistent. The limit as the number of compounding periods approach infinity (or the compounding period goes to zero) is finite, and asymptotically approaches a simpler equation:

Reff = exp (Rnom) -1

In our example where Rnom = 12%/yr, Reff = exp (.12) = 12.7496%, almost exactly the same result as the hourly figure. This simpler formula is called continuous compounding, and it use the universal natural growth constant e (approximately equal to 2.718).

In this video, Salman Khan shows the same mathematics in his own example, resulting in what he calls the “magical number” e, which is the key to continuous compounding.

When you get past Salman’s fascination with e, you’ll see that his next video generalizes the example in the previous video into the same formula presented above in terms of Rnom and n, and derives the continuous compounding formula. I

To use continuous compounding to compute future values on balances borrowed (or lent), we need to understand where to insert time into the continuous compounding formula. Remember that both nominal and effective interest rates must be reported in units of dollars of interest per dollars of principal per unit time (\$interest/\$principal/time, or %/yr).

#### The difference between nominal and effective interest rates is the compounding period.

Thus, effective rates will always be higher than nominal rates when compounding takes place more frequently than the unit of time expressed in the nominal rate (typically, once per year).

When using continuous compounding, the amount of a future balance is computed from the present value thus:

F = P * exp (rt)

where r = the nominal interest rate (%/time), and t is time (in the same units as the nominal interest rate, usually years).

Nominal interest rates are the stated, advertised, or quoted rates. Where no time period is stated, than per year (also known as per annum) is assumed.

Effective interest rates are what borrows have to actually pay, and depend on how frequently the nominal rate is compounded (i.e., which means adding interest to the balance of the loan).

Here’s three practice problem that you can use to help you understand the difference:

1. The nominal interest rate on a credit card balance is advertised as 21%. However, interest is compounded monthly. What is the annual effective interest rate?
2. The nominal interest rate is 1% per month. What is the annual effective interest rate when compounding continuously?
3. Draw a graph that shows the effective interest rate on the y-axis versus compounding period on the x-axis, for a nominal rate of 10% per annum.