9.6: Equivalent And Effective Interest Rates - Math LibreTexts

Effective Interest Rates

If you place $1,000 into an investment earning 10% compounded semi-annually, how much will you have in your account after one year? Less than $1,100, exactly $1,100, or more than $1,100? If you say more than $1,100, you are absolutely correct. In other words, the 10% nominal rate does not fully reflect the true amount of interest that the investment earns annually, which depends on how often the principal increases. This raises two questions:

1. What exact amount of interest is earned annually?
2. What annual interest rate is truly being earned?

The effective interest rate is the true annually compounded interest rate that is equivalent to an interest rate compounded at some other (non-annual) frequency. In other words, the amount of interest accrued at the effective interest rate once in an entire year exactly equals the amount of interest accrued at the periodic interest rate successively compounded the stated number of times in a year. To calculate the effective interest rate, you must convert the compounding on the nominal interest rate into an annual compound.

The Formula

To see how the formula develops, take a $1,000 investment at 10% compounded semi-annually through a full year.

Start with \(PV=\$ 1,000, IY=10 \%\), and \(CY=2\) (semi-annually). Therefore, \(i=10 \% / 2=5 \%\). Using Formula 9.3, after the first six-month compounding period (\(N = 1\)) the investment is worth

\[FV=\$ 1,000(1+0.05)^{1}=\$ 1,050\nonumber \]

With this new principal of \(PV = \$1,050\), after the next six-month compounding period the investment becomes

\[FV=\$ 1,050(1+0.05)^{1}=\$ 1,102.50\nonumber \]

Therefore, after one year the investment has earned $102.50 of interest. Notice that this answer involves compounding the periodic interest twice, according to the compounding frequency of the interest rate. What annually compounded interest rate would produce the same result? Try 10.25% compounded annually. With \(PV=\$ 1,000, IY=10.25 \%\), and \(CY=1\), then \(i=10.25 \% / 1=10.25 \%\). Thus, after a year

\[FV=\$ 1,000(1+0.1025)^{1}=\$ 1,102.50\nonumber \]

This alternative yields the same amount of interest, $102.50. In other words, 10.25% compounded annually produces the same result as 10% compounded semi-annually. Hence, the effective interest rate on the investment is 10.25%, and this is what the investment truly earns annually.

Generalizing from the example, you calculate the future value and interest amount for the rate of 10% compounded semi-annually using the formulas

\[FV=PV(1+i)(1+i) \quad \text { and } \quad I=FV-PV\nonumber \]

Notice that the periodic interest is compounded by a factor of \((1+i)\) a number of times equaling the compounding frequency (\(CY=2\)). You then rewrite the future value formula:

\[FV=PV(1+i)^{CY}\nonumber \]

Substituting this into the interest amount formula, \(I=FV-PV\), results in

\[I=PV(1+i)^{CY}-PV\nonumber \]

Factor and rearrange this formula:

\[I=PV\left((1+i)^{CY}-1\right)\nonumber \]

\[\dfrac{I}{PV}=(1+i)^{CY}-1\nonumber \]

On the left-hand side, the interest amount divided by the present value results in the interest rate:

\[\text { rate }=(1+i)^{CY}-1\nonumber \]

Formula 9.4 expresses this equation in terms of the variables for time value of money. It further adapts to any conversion between different compounding frequencies.

Formula 9.4

How It Works

Follow these steps to calculate effective interest rates:

Step 1: Identify the known variables including the original nominal interest rate (\(IY\)) and original compounding frequency (\(CY_{Old}=1\)). Set the \(CY_{New}=1\).

Step 2: Apply Formula 9.1 to calculate the periodic interest rate (\(i_{Old}\)) for the original interest rate.

Step 3: Apply Formula 9.4 to convert to the effective interest rate. With a compounding frequency of 1, this makes \(i_{New}=IY\) compounded annually.

Revisiting the opening scenario, comparing the interest rates of 6.6% compounded semi-annually and 6.57% compounded quarterly requires you to express both rates in the same units. Therefore, you could convert both nominal interest rates to effective rates.

6.6% compounded semi-annually	6.57% compounded quarterly
Step 1	\(IY = 6.6%; CY_{Old} = 2; CY_{New} = 1\)	\(IY = 6.57%; CY_{Old} = 4; CY_{New} = 1\)
Step 2	\(i_{Old} = 6.6%/2 = 3.3%\)	\(i_{Old} =6.57%/4 = 1.6425%\)
Step 3	\(i_{New} = (1 + 0.033)^{2/1} − 1 = 6.7089%\)	\(i_{New} = (1 + 0.016425)^{4/1} − 1 = 6.7336%\)

The rate of 6.6% compounded semi-annually is effectively charging 6.7089%, while the rate of 6.57% compounded quarterly is effectively charging 6.7336%. The better mortgage rate is 6.6% compounded semi-annually, as it results in annually lower interest charges.

Important Notes

The Texas Instruments BAII Plus calculator has a built-in effective interest rate converter called ICONV located on the second shelf above the number 2 key. To access it, press 2nd ICONV. You access three input variables using your ↑ or ↓scroll buttons. Use this function to solve for any of the three variables, not just the effective rate.

Variable	Description	Algebraic Symbol
NOM	Nominal Interest Rate	\(IY\)
EFF	Effective Interest Rate	\(i_{New}\) (annually compounded)
C/Y	Compound Frequency	\(CY\)

To use this function, enter two of the three variables by keying in each piece of data and pressing ENTER to store it. When you are ready to solve for the unknown variable, scroll to bring it up on your display and press CPT. For example, use this sequence to find the effective rate equivalent to the nominal rate of 6.6% compounded semi-annually:

\[2 \mathrm{nd} \text{ } \mathrm{ICONV}, 6.6 \text { Enter } \uparrow, 2 \text { Enter } \uparrow, \mathrm{CPT}\nonumber \]

\[\text{Answer: }6.7089\nonumber \]

Paths To Success

Annually compounded interest rates can be used to quickly answer a common question: "How long does it take for my money to double?" The Rule of 72 is a rule of thumb stating that 72% divided by the annually compounded rate of return closely approximates the number of years required for money to double. Written algebraically this is

\[\text{Approximate Years } =\dfrac{72 \%}{IY\text { annually }}\nonumber \]

For example, money invested at 9% compounded annually takes approximately \(72 \div 9 \%=8\) years to double (the actual time is 8 years and 15 days). Alternatively, money invested at 3% compounded annually takes approximately \(72 \div 3 \%=24\) years to double (the actual time is 23 years and 164 days). Note how close the approximations are to the actual times.

Exercise \(\PageIndex{1}\): Give It Some Thought

1. borrowing?

If one investment takes 36 years to double while another takes 18 years to double, which has the higher effective rate?

Answer

1. The effective rate is equal to or higher than the nominal rate.
2. 1. 9%. since more interest is earned
   2. 8%, since less interest is paid
3. Eighteen years, since a higher rate takes less time to double

Example \(\PageIndex{1}\): Understanding Your Investment

If your investment earns 5.5% compounded monthly, what is the effective rate of interest? According to the Rule of 72, approximately how long will it take your investment to double at this effective rate?

Solution

Calculate the effective rate of interest (\(i_{New}\)). Once known, apply the Rule of 72 to approximate the doubling time.

What You Already Know

Step 1:

The original nominal interest rate and compounding along with the new compounding are known: \(IY = 5.5%; CY_{Old}\) = monthly = 12; \(CY_{New}\) = 1

How You Will Get There

Step 2:

Apply Formula 9.1 to the original interest rate.

Step 3:

Apply Formula 9.4, where \(i_{New} = IY\) annually.

Step 4:

To answer approximately how long it will take for the money to double, apply the Rule of 72.

Perform

Step 2:

\[i_{Old}=\dfrac{5.5 \%}{12}=0.458 \overline{3} \% \nonumber \]

Step 3:

\[i_{New}=(1+0.00458 \overline{3})^{\frac{12}{1}}-1=0.056408=5.6408 \% \nonumber \]

Step 4:

\[\text { Approximate Years }=\dfrac{72 \%}{5.6408 \%}=12.76 \nonumber \]

Calculator Instructions

2nd ICONV

NOM	C/Y	EFF
5.5	12	Answer: 5.640786

You are effectively earning 5.6408% interest per year. At this rate of interest, it takes approximately 12¾ years, or 12 years and 9 months, for the principal to double.

Example \(\PageIndex{2}\): Your Car Loan

As you search for a car loan, all banks have quoted you monthly compounded rates (which are typical for car loans), with the lowest being 8.4%. At your last stop, the credit union agent says that by taking out a car loan with them, you would effectively be charged 8.65%. Should you go with the bank loan or the credit union loan?

Solution

Since it is normal for a car loan to be compounded monthly, convert the effective rate to a monthly rate (\(IY\)) so that it matches all the other quotes.

What You Already Know

Step 1:

The effective rate and the compounding are as follows: \(i_{New} = 8.65%; CY_{Old}\) = monthly = 12; \(CY_{New}\) = 1

How You Will Get There

(Note: In this case, the \(i_{New}\) is known, so the process is reversed to arrive at the \(IY\). Thus, steps 2 and 3 are performed in the opposite order.)

Step 2:

Substitute into Formula 9.4, rearrange, and solve for \(i_{Old}\).

Step 3:

Substitute into Formula 9.1, rearrange, and solve for \(IY\).

Perform

Step 2:

\[\begin{aligned} &0.0865=\left(1+i_{Old}\right)^{12 \div 1}-1\\ &1.0865=\left(1+i_{Old}\right)^{12}\\ &1.0865^{1 / 12}=1+i_{Old}\\ &1.006937=1+i_{\text {Old }} \\ &0.006937=i_{\text {Old }} \end{aligned} \nonumber \]

Step 3:

\[\begin{aligned} 0.006937 &=\dfrac{IY}{12} \\ IY &=0.083249 \text { or } 8.3249 \% \end{aligned} \nonumber \]

Calculator Instructions

2nd ICONV

NOM	C/Y	EFF
Answer: 8.324896	12	8.65

The offer of 8.65% effectively from the credit union is equivalent to 8.3249% compounded monthly. If the lowest rate from the banks is 8.4% compounded monthly, the credit union offer is the better choice.