Typically an interest rate is given as a nominal, or stated, annual rate of interest. But when compounding occurs more than once per year, the rate of interest actually realized will be higher than the nominal rate of interest. This actual, realized rate is known as the Effective Annual Rate (EAR). In this article we’ll take a closer look at the effective annual rate, dig into the effective annual rate formula, and then we’ll tie it all together by looking at an effective annual rate example using Canadian mortgages.
How to Calculate The Effective Annual Rate
The effective annual rate, or EAR, can be calculated as follows:
Effective Annual Rate Example Problem
Let’s take a look at an example of how to use and calculate the effective annual rate. Suppose you have the choice between an investment that earns 12% compounded monthly and a different investment that earns 12% compounded annually. Are these two investment options equivalent? No. To see why, let’s take a closer look at the effective annual rate.
Using the effective annual rate formula above, we can solve for the effective annual rate of 12% compounded annually by plugging in (1+.12)1-1, which equals 12%.
Now, let’s solve for the effective annual rate for 12% compounded monthly. To do this we simply plug in (1+.01)12 – 1, which equals 12.68%. Notice how this rate is higher when we have more frequent compounding.
As you can see, even though both of the above investment options have a stated (nominal) rate of 12%, the actual or effective rates are different. The reason why is because with monthly compounding we get paid interest on a monthly basis rather than on an annual basis. This matters because our investment earns interest not just on the principal amount invested, but it also earns interest on the interest itself. When interest is earned monthly, then our investment compounds faster than when interest is earned annually. The effective annual rate formula gives us a way to quantify and compare this difference.
Canadian Mortgages and The Effective Annual Rate
One particularly useful (although advanced) application of the effective annual rate is when payments per year differs from compounding periods per year. One notable example of this is with Canadian mortgages, which by law are allowed a maximum of semi-annual compounding, but often have monthly payments. When you have situations like this it’s often helpful to use the effective annual rate.
To see how the effective annual rate changes with different compounding periods, let’s take a look at a Canadian mortgage example. Suppose we have a 30-year $200,000 Canadian mortgage with a stated interest rate of 6%, compounded semi-annually, with monthly payments. What are our monthly payments?
First, notice that we can’t just plug in 6% for i on our financial calculator for a $200,000 present value amortized over 30 years, and then solve for a payment. This would be the approach with a traditional, non-Canadian mortgage that has monthly payments and monthly compounding. But because Canadian mortgages have semi-annual compounding and monthly payments, we have to do a bit of work to set up the problem correctly.
Let’s use the above effective annual rate formula to find the effective annual rate for a 6% stated rate, compounded semi-annually. Plugging the variables into the above equation we get (1+.03)2 – 1 = 0.0609 or 6.09%.
EAR = (1+.03)2-1 = 0.0609 or 6.09%
Now that we know a stated rate of 6% compounded semi-annually has an effective annual rate of 6.09%, we simply need to find the equivalent stated (nominal) rate that when compounded monthly will result in an effective annual rate of 6.09%. Fair warning: this is a tricky calculation and requires some not-so-easy algebra.
Here’s the resulting formula we can use to find this rate:
equivalent nominal rate = n x (1 + EAR)1/n – 1
Plugging in our EAR of 6.09% and our n (number of periods) as 12, we get an equivalent nominal rate of 5.926%, or .493862% per month (simply divide by 12).
In other words, if a stated annual rate of 5.926% is compounded monthly then it equals an effective annual rate of 6.09%.
Now we can plug in -$200,000 for pv, 360 for n, .493862% for i, 0 for fv, and then solve for payment. When you do this you find that the monthly payment is $1,189.65. This is the payment monthly mortgage payment based on a Canadian mortgage with semi-annual compounding.
Conclusion
The effective annual rate is important to understand in finance and commercial real estate. It gives us a way to compare stated rates with different compounding periods, and it also can be used in other applications such as with Canadian mortgages. All in all, the effective annual rate is a helpful calculation to add to your finance toolkit.