Interpreting interest rates is not always as straightforward as it seems, and getting it right can have important implications for all parties involved in a commercial real estate transaction. Interest rates can take two forms: nominal interest rates and effective interest rates. As a result, there can be some confusion about what a quoted interest rate actually means.
In this article, we will take a deep dive into the differences between nominal and effective rates. We’ll start with the basic intuition behind effective and nominal rates, then walk through the math step by step, with several examples along the way.
EXECUTIVE Summary
- Nominal vs. Effective Rate: The nominal rate is the stated rate without compounding; the effective rate takes compounding into account and shows the actual rate of increase over a given period.
- Why It Matters: Misunderstanding these two rates can lead to inaccurate calculations, especially in commercial real estate transactions involving loans and investment returns.
- Key Outcomes: By the end, you’ll know how to calculate both rates, convert between them, and use these concepts to compare different loan or investment scenarios accurately.
Table of Contents
- Different Ways Interest Rates Are Quoted
- What is an Effective Interest Rate?
- What is a Nominal Interest Rate?
- Effective Interest Rate Equations
- General Relationship Between Nominal Annual Rate and Effective Annual Rate
- Effective Annual Rate For Any Time Period
- How to Convert an Effective Rate Per Compounding Period into an Effective Annual Rate
- How to Convert an Effective Annual Rate into an Effective Rate Per Compounding Period
- Compounding Frequency and The Effective Rate
- When the Effective Rate is Equal to the Nominal Rate
- When the Effective Rate is Greater than the Nominal Rate
- Equivalent Nominal Annual Rate
- What Interest Rates Quotes Mean
- How to Compare Nominal and Effective Rates
- What About the Internal Rate of Return?
- Conclusion
Different Ways Interest Rates Are Quoted
First, let’s distinguish between the terms “interest rate” and “rate of return,” which largely depends on perspective. When evaluating loans, the term interest rate is generally used. When evaluating investments, the term internal rate of return is generally used, which is also referred to as the rate of return or yield. In either case, we are dealing with interest rates and the concepts are the same, but this difference in language and perspective is something you should be aware of.
Next, let’s take a look at a few examples of how interest rates or rates of return might be quoted in practice:
- 10% per year
- 10% per year, compounded monthly
- 10%
- 10% compounded monthly
As you can see, it is sometimes hard to fully understand what is meant by a given interest rate. Let’s take a closer look at how to make sense of this.
SECTION RECAP: Different ways interest rates are quoted
- Interest rates can be quoted with varying time frames (annual, quarterly, monthly) and compounding frequencies.
- If the compounding period isn’t stated, it’s common to assume annual compounding.
- Terminology differs based on perspective: “interest rate” for loans vs. “rate of return” or “yield” for investments.
What is an Effective Interest Rate?
First of all, what is an effective interest rate? An effective interest rate takes into account the effect of compounding and gives the actual rate of increase over the stated time period.
To illustrate, let’s look at a simple example, step by step. Suppose we deposit $100 into a bank account that earns a 1.5% effective interest rate per quarter.
Since the effective quarterly rate is 1.5%, that means after one quarter our bank account will increase by $100 x 1.5%, and we’d have $101.50 in our account.
Now, what if we want to convert this effective quarterly rate to an effective annual rate? To do this, let’s see what would happen to our bank account balance at the end of the year if we reinvest each quarterly interest payment and do not make any additional withdrawals or deposits.
At the end of the first quarter, we will have $100 x 1.015, or $101.50 in our account. At the end of the second quarter, we will have $101.50 x 1.015, or $103.02 in our account. At the end of the third quarter, we will have $103.02 x 1.015, or $104.57 in our account. Finally, at the end of the fourth quarter we will have $104.57 x 1.015, or $106.14 in our account.
Notice how we are earning interest on interest during this process. This means that we are considering the effects of compounding. So, we start out with $100. We earn 1.5% per quarter, and we end up with $106.14 at the end of the year.
So, what is our effective annual interest rate? Our effective annual interest rate is the percentage change in our bank balance over the year, which is ($106.14-$100)/$100, or 6.14%.
This effective annual rate considers the effects of compounding during the year (earning interest on interest) and gives us the actual rate of increase that would occur over the stated time interval (which in this case is a year), assuming no additional deposits or withdrawals are made.
We’ll derive a more generalized effective rate formula later, but for now, focus on the intuition behind what the effective rate is doing, which is taking into account the effect of compounding over the stated time period.
Section recap: what is an effective interest rate?
- The effective interest rate includes the impact of compounding within the stated time period.
- It shows the actual percentage change in an account balance or investment over that period.
- Example: A 1.5% effective quarterly rate leads to a 6.14% effective annual rate when interest compounds quarterly.
What is a Nominal Interest Rate?
Now that we understand what an effective interest rate is, what is a nominal interest rate? A nominal interest rate expresses the total interest paid in a period, without considering the effects of compounding during the stated time interval.
When rates are quoted as nominal rates, it is customary to provide 1) a stated rate per period, and 2) the number of times interest is compounded during that period.
Let’s take a closer look at how this works.
In our example above, we deposited $100 into a bank account earning a 1.5% effective interest rate per quarter. We saw how the effective annual interest rate was 6.14% because it took into account compounding during the year. So, what would our nominal annual interest rate be in this case?
Since the nominal interest rate ignores the effect of compounding, we can find the nominal annual interest rate by simply multiplying our quarterly effective rate by four quarters in a year to get our nominal annual rate. In this case, that would be 1.5% x 4, which results in 6%.
Recall that with nominal interest rates, we need to provide 1) a stated rate per period, and 2) the number of times interest is compounded during that period. So, in this case, we would say that we have a nominal rate of 6% per year, compounded quarterly.
Notice how this simple conversion from a quarterly rate to an annual rate only involved multiplication, and therefore did not consider the effects of compounding. Since multiplication is just repeated addition, we are simply adding up the interest paid on the initial balance for each quarter. We use the initial balance in our interest calculations because we are not considering the effects of compounding, and therefore we are not earning interest on interest.
In other words, if we were to use our bank account example again, then at the end of the first quarter we are owed $100 x 1.015 in interest. At the end of the second quarter, we are owed another $100 x 1.015 in interest. At the end of the third quarter, we are again owed $100 x 1.015, and at the end of the fourth quarter we are yet again owed $100 x 1.015. Since we are owed 1.5% on our initial balance each quarter, we can just multiply our 1.5% quarterly interest rate by 4 quarters to get 6%.
Notice that this 6% interest rate does not give us the actual increase in our bank account for the year, considering the effects of compounding. Another way to see that this 6.00% nominal annual rate does not consider compounding is to recall that in the effective annual rate exercise above (which did take into account compounding), our bank balance increased by 6.14% at the end of the year.
section recap: what is a nominal interest rate?
- The nominal rate ignores the effect of compounding during the stated period.
- Usually quoted alongside a compounding frequency (e.g., 6% nominal annual rate, compounded quarterly).
- Unlike the effective rate, it simply multiplies the rate by the number of compounding periods without adding “interest on interest.”
Effective Interest Rate Equations
Now that we have a basic intuition about effective and nominal rates, let’s formalize the relationship between effective and nominal rates and take a closer look at how the math works.
General Relationship Between Nominal Annual Rate and Effective Annual Rate
We can start by deriving a generalized equation for calculating the effective interest rate as follows. If $1 was deposited into an account that compounded interest m times per year and paid a nominal interest rate per year of r, then the effective interest rate per compounding period would be r/m. That means the total amount in the account at the end of the year would be:
$$ \text{A} = \left( 1 + \frac{r}{m} \right)^{m} \text{, where A is total amount and} \; m \text{ is compounding frequency}.$$
This reduces to:
Effective annual rate formula would be:
$$ \text{A} – \text{P} = \left( 1 + \frac{r}{m} \right)^{m} – 1 \text{, where P is principal amount}.$$
So, the effective annual rate formula is therefore:
$$ \text{EAR} = \left( 1 + \frac{r}{m} \right)^{m} – 1.$$
For example, recall that we saw how earning a 1.5% effective rate per quarter could be quoted as a 6% nominal annual rate, compounded quarterly. Let’s take this 6% nominal annual rate, compounded quarterly, and plug it in our formula to find the effective annual rate. To accomplish this we just need to know r, the nominal interest rate per year, and m, the number of compounding periods per year. In this case, r is 6% and m is 4. So, that gives us:
$$ \text{A} = \left( 1 + \frac{0.06}{4} \right)^{4} = \left( 1 + 0.015 \right)^{4} = 1.0614 $$
Our effective annual rate is 6.14%, as expected.
Effective Annual Rate For Any Time Period
To find the effective interest rate for a period other than yearly, you can change the time period used for the variable r and m as required. For example, the effective interest rate per month is:
$$ \text{EAR (monthly)} = \left( 1 + \frac{r}{12} \right)^{12} – 1 $$
For instance, suppose we had an effective rate of 1% per month, compounded daily using a 30-day month. In this case, our effective monthly rate would be 1.01%:
$$ \text{Effective monthly rate} = \left( 1 + \frac{0.01}{30} \right)^{30} $$
How to Convert an Effective Rate Per Compounding Period into an Effective Annual Rate
If we substitute the effective interest rate per compounding period, i = (r/m), then we can also calculate the effective annual rate as follows:
$$ \text{EAR} = \left( 1 + \frac{i}{1} \right)^{1} – 1 $$
For example, if we have an effective rate of 1% per month, we would convert this to an effective annual rate as:
$$ 1 ext{EAR (annual)} = \left( 1 + \frac{0.01}{1} \right)^{12} – 1. $$
How to Convert an Effective Annual Rate into an Effective Rate Per Compounding Period
Alternatively, if we know the effective annual rate and want to find the effective interest rate per compounding period, then we can also rearrange the equation above to solve for i:
$$ \text{Effective interest rate per compounding period} = \left( 1 + \text{EAR} \right)^{1/m} – 1.$$
For example, if we have an effective annual rate of 12.68% we could convert this to an effective monthly rate as:
$$ i = \left( 1 + 0.1268 \right)^{1/12} – 1.$$
Next Steps: Compounding Frequency and The Effective Rate
The effective interest rate can be equal to or greater than the nominal interest rate. Let’s first take a look at the case where the effective rate equals the nominal rate. Then we will look at how the effective interest rate becomes greater than the nominal interest rate.
When the Effective Rate is Equal to the Nominal Rate
When interest compounds only once per time period, the effective interest rate and the nominal interest rate per time period will be equal to each other. Earlier, we derived the effective annual rate formula as follows:
$$ \text{EAR} = r $$
When we substitute in 1 for m, the number of compounding periods per year, we get the following:
$$ r = r $$
When the Effective Rate is Greater than the Nominal Rate
It is also possible for compounding to occur more frequently than once per time period. When compounding occurs more frequently than once per time period, the effective interest rate will be greater than the nominal interest rate.
For example, suppose we have a nominal rate of 10% per year, compounded monthly. If we plug this into the effective annual rate formula above then we get:
$$ \text{Effective rate} = \left( 1 + \frac{10 ext{%}}{12} \right)^{12} – 1 $$
Equivalent Nominal Annual Rate
Sometimes we already know what the effective annual rate is, but we want to know the equivalent nominal annual rate compounded at some other frequency. For example, suppose we want to know what the nominal annual rate, compounded monthly, would have to be to provide an effective annual rate of 10%.
To figure this out, we can use the following equivalent nominal annual rate formula:
$$ \text{Nominal annual rate} = \text{Effective annual rate} = \left( 1 + \frac{10 ext{%}}{1} \right)^{1/m} – 1 $$
What Interest Rates Quotes Mean
Now that we have some basic background on nominal and effective rates, let’s revisit the different ways interest rates can be quoted and what those quoted interest rates mean.
How to Compare Nominal and Effective Rates
Only those interest rates placed on a comparable basis may be compared. Nominal interest rates can be compared if the nominal rates have the same time period and same compounding period. Otherwise, translating the nominal rates into effective rates will allow them to be compared to each other.
What About the Internal Rate of Return?
The textbook definition of the internal rate of return is that it is the interest rate that causes the net present value to equal zero. Let’s take a closer look at what this means.
Conclusion
In this article, we took a deep dive into nominal and effective interest rates. We defined the effective interest rate and nominal interest rate, derived a generalized formula for finding the effective interest rate, and then walked through several examples. We discussed what happens when the compounding frequency increases, how to express any effective rate as a variety of nominal rates, how to compare nominal and effective rates, and finally, how to interpret the internal rate of return.
Final Takeaways
- Nominal vs. Effective: Nominal rates are simpler to quote but can be misleading; effective rates reflect the true impact of compounding.
- Comparisons Matter: Always convert different quotes to the same basis (effective rates) before deciding on a loan or investment.
- IRR Perspective: For investors and lenders, the IRR is essentially the effective periodic rate applied to the balance owed.
