Annual Equivalent Rate (AER)
It is the interest rate of the loan, savings account, or investment based on how often it compounds
What is the Annual Equivalent Rate (AER)?
The annual equivalent rate (AER) is the interest rate of the loan, savings account, or investment based on how often it compounds. It shows the actual cost or return on your investment when the interest rate is compounded more frequently than once per year.
The Annual Effective Rate is the interest rate after compounding effects have been considered to normalize the interest rate. It is known as the effective interest rate (EIR), annual percentage yield (APY), and effective percentage rate (APR).
The effective rate is an interest rate that tells you exactly how much interest you'll earn on investment or debt, depending on how often interest payments are made. It is calculated based on the idea that any interest paid is added to the balance of the principal amount.
The next interest payment is often based on the slightly higher account balance. If interest compounds more than once a year, the effective rate will be higher than the stated interest rate.
Increasing compound periods results in more significant differences between the effective and stated interest rates.
Once you take a loan of 12 months with a compound interest of 10% monthly, your monthly statement differs from one month to another. On your first monthly statement, you charge 10% interest which adds to your balance.
In your second statement, you charge an interest rate that doesn't like the stated interest rate higher than the first one. This is because the previous month's interest was added to your balance, charged, and then added again.
In the following statements, you can see that the interest keeps increasing because the interest compounds monthly. More compound periods enhance the probability of increasing the charged interest rate between compound periods.
Computing the equivalent rate allows you to compare loans, investments, and savings accounts with various compounding periods to see which makes or saves you the most money. A higher effective rate means more return you get from particular investments.
Key Takeaways
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AER is used to show the actual cost or return on your investment when the interest rate is compounded more frequently than once per year.
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AER allows investors to compare investments, loans, and bonds to determine which generates the highest return/ cost.
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Remember that the more investment compounds, the more interest you earn (even if the stated interest is the same). The more a loan compounds, the more interest you pay each compound period.
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Annual equivalent and nominal interest rates may appear to be similar concepts, but they are not. The primary difference between these two interest rates is that nominal interest rates do not consider compounding interest periods.
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The most significant advantage of the annual equivalent rate is that it reflects the actual interest rate because it considers the effect of compounding over time.
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The equivalent rate is limited as the AER does not consider any fees associated with buying or selling the investment.
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Therefore, AER can't give you an accurate picture of how much your business makes or how much it costs to get loans because it doesn't consider the fees.
Formula for the AER
It is an interest rate that estimates the amount of interest you'll pay on a loan or investment depending on how frequently it accumulates. It is always expressed as a percentage.
It's most commonly used to calculate the annual percentage yield (APY) on a savings account, the yield on a bond, or the effective annual percentage rate (APR) on a loan.
To calculate the AER, divide the stated interest rate by the number of times interest payments and raise the result to the number of compounding periods. Then, subtract one from the impact.
The formula is as follows:
The Annual Equivalent Rate = (1+ i/n)^n _ 1
Where,
- i: The rate is the stated interest rate on a loan without considering fees or compounding interest during a year. It is calculated before inflation is taken into account.
- n: Number of compounding periods: The number of interest payments during a year.
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If the interest is compounding annually, n would be 1.
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If the interest is compounding semi-annually, n would be 2.
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If the interest is compounding quarterly, n would be 4.
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If the interest is compounding monthly, n would be 12.
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If the interest is compounding weekly, n would be 48.
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If the interest is compounding daily, n would be 365.
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If the interest is compounding up to infinity, n would be calculated by the following formula:
r= e^i - 1
Where e= 2.72 in mathematical constant.
Remember that the effective annual rate is higher than the stated interest rate and could never be equal to the stated rate. However, if the investment compounds annually, the effective rate equals the expressed interest, as there is only one interest payment.
Example of AER
As mentioned previously, the effective rate could be used to compare investments, loans, or credits.
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The effective annual rate is a great tool to compare investments as it reflects how much interest is earned on investments.
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The effective annual rate is a great tool to compare loans as it reflects how much interest you pay on loans or credits.
The following examples could better understand the equivalent rate’s meaning in earning or charging interest.
Utilize AER to compare investments
When it comes to comparing investments, consider the following points:
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A greater AER indicates a higher interest rate (the more profits you get).
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Lower AER shows a lower interest rate (the fewer profits you get).
A working example:
Suppose Sara has to decide between two options that have terms of 10 years.
Option A has a stated interest rate of 9%, compounded semi-annually.
Option B has a stated interest rate of 8%, compounded daily.
Compute the effective rate for both options and determine the option that earns more interest.
Option A:
The Annual Equivalent Rate = ((1+ i/n)^n) - 1
= ((1 + 9% / 2)^2) - 1
= ((1 + 0.045) ^2) - 1
= (1.092) -1
= 0.092 * 100
= 9.2%
Option B:
The Annual Equivalent Rate = ((1+ i/n)^n) - 1
= (( 1+ 8% / 365) ^365 ) - 1
= (( 1+ 0) ^365) -1
= ((1) ^365 ) - 1
=1.083 -1
= 0.083 * 100
= 8.33%
Hence, option A is a better option as its rate is higher than option B’s rate. So, Sara should choose option A as it earns more interest, generating more earnings and higher returns.
Utilize AER to compare loans or credits
When it comes to comparing loans or credits, consider the following points:
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A higher rate means you charge a higher interest rate to pay for a loan/credit (the more it will cost you), so you should always go with companies that offer debts or credit with lower rates (interest rate).
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Avoid the loans or credit that compound more often—the fewer compounding periods. This lower rate means loans or credit with fewer compounding periods charge you a lower interest rate ( the less it will cost you).
A working example:
Sara has two loans that have a stated interest rate of 10%. Loan A compounds semi-annually, but loan B compounds monthly. Calculate the AER loans and determine the loan that charges Sara lower interest.
Loan A:
The Annual Equivalent Rate = ((1+ i/n)^n) - 1
= ((1 + 10% / 2)^2) -1
= ((1 + 0.05) ^2) -1
= ((1.050) ^2) -1
= 1.102 -1
= 0.102 *100
= 10.2%
Hence, Sara charges 10.2% interest to pay for Loan A.
Loan B:
The Annual Equivalent Rate = ((1+ i/n)^n) - 1
= ((1 + 10% / 12)^12) -1
= ((1 + 0.008) ^12) -1
= ((1.008) ^12) -1
= 1.105 -1
= 0.105 *100
= 10.47%
Hence, Sara charges 10.47% interest to pay Loan B.
By comparing the annual equivalent rate of the two loans, it is obvious that Loan A charges Sara less interest, whereas Loan B charges her a higher interest rate.
3. Compare Saving accounts
When it comes to comparing saving accounts, consider the following points:
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A greater AER indicates a higher interest rate companies pay you on your savings accounts (the more profits you get).
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A lower AER indicates a lower interest rate companies pay you on your savings accounts (the fewer profits you get).
A working example:
Banks of LYT and YLS advertise that they pay customers the highest possible interest rate. LYT states that customers with saving accounts will get 10% interest monthly, while YLS states that customers with saving accounts will get 7% interest monthly.
What is the best option for a customer?
As both banks pay customers interest on the same compounding periods, let’s compare the interest rates of both banks. LYT is the better option because customers earn more interest for the amount they save in the bank.
Annual equivalent rate advantages
The most significant benefit of the annual equivalent rate is that it represents the actual interest rate because it considers the impact of compounding over time. AER believes the impact of compounding periods increases the investors' earnings/costs over time.
Once the investor knows AER includes compounding interest, the investor gets an idea of a sum of money growing at a pace more significant than the rate of simple interest since you earn on the money that you invest in addition to returns at the end of each compounding period.
Therefore, it is an essential instrument for investors because it enables them to analyze investments such as bonds, loans, or accounts to comprehend the actual return they receive on their capital (ROI).
When it comes to loans and debts, knowing the annual equivalent rate helps borrowers know how much they need to pay to get a loan.
Advantages can be summarized in the following:
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AER calculates the real rate of interest while considering compounding.
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Regarding investments, It is a valuable tool for assessing bonds’ and saving accounts’ accurate return on investment (ROI).
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Regarding debts and loans, It is a helpful tool for assessing the cost of getting loans and debts.
Annual equivalent rate disadvantages
However, the equivalent rate is limited when investors assess various investment opportunities' merits; the annualized effective return is typically not disclosed. Investors should do the calculation of this metric, and they are responsible for doing it on their own.
Another limitation is that the AER does not consider any fees associated with buying or selling the investment. In other words, AER cannot have an accurate image of your return on business or the cost of getting loans as it doesn’t calculate the fees.
Additionally, compounding interest has its limitations for investors. The compounding interest cost isn't immediately shown if you don’t manage your investment correctly.
In other words, paying interest in each compounding period can be very expensive, resulting in losing your money.
Disadvantages can be summarized in the following:
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Investors must calculate the AER themselves as it isn’t shown in investments’ financial reports.
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AER doesn’t consider fees that could be associated with some businesses.
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A limitation associated with compounding itself.
Annual Equivalent Rate vs. Nominal Interest
Annual equivalent and nominal interest rates could look like similar terms, but they are entirely different. The significant difference between these two interest rates is that nominal interest rates don’t consider compounding interest periods.
The nominal interest rate is calculated before taking the impact of inflations without considering the compounding periods. It doesn’t consider how often the customer pays or earns interest during a year, so it doesn’t change according to the compounding periods.
On the other hand, the annual equivalent interest rate considers the compounding interest periods, and thus it is a more accurate measure of interest charges. It fluctuates based on how often the interest payments are made during a year.
For example,
Suppose a bond L offers interest income for bondholders that compounds semi-annually. The nominal rate of the bond is 6%. The effective rate of the bond would be higher as the interest is paid twice per year. However, the stated rate keeps regardless of the compounding interest.
The following calculations show how is the equivalent rate change according to the compounding interest:
The Annual Equivalent Rate = ((1+ i/n)^n) - 1
= ((1 + 6%/ 2)^2) - 1
= ((1 + 0.030)^2) - 1
= (1.061) - 1
= 0.061 * 100
= 6.090%
Suppose a bond X offers the same interest rate as the previous example but changes the compounding period. They decide to pay out bondholders' interest income quarterly, which means the bondholders earn interest four times per year.
The AER of bond X will be much higher than bond L since it doubles the compounding periods from two to four times a year. However, the stated interest remains as it is! The following calculations show how the AER goes up as the compounding periods increase:
The Annual Equivalent Rate = ((1+ i/n)^n) - 1
= ((1 + 6% / 4) ^4) - 1
= ((1 + 0.015) ^4) - 1
= (1.061) - 1
= 0.061 * 100
= 6.14%
Comparing Bond L and X, you can conclude the nominal interest rate disregard the compounding periods while the effective rate moves up or down according to how much time the interest payments are made during a year.
Researched and authored by Khadega Bazarah | Linkedin
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