Question

Find the effective rate of interest corresponding to a nominal rate of \(9 \%\) per year compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly.

Short answer

The effective rate of interest corresponding to a nominal rate of 9% per year compounded annually is 9%, semiannually is approximately 9.2025%, quarterly is approximately 9.3181%, and monthly is approximately 9.3806%.

Step-by-step solution

Define All Variables

The nominal interest rate \(i\) is given as 9%, or 0.09 in decimal form. The variable \(n\) represents the number of compoundings per year and this will take the values 1, 2, 4, and 12 for annual, semi-annual, quarterly, and monthly compounding respectively. We are interested in finding the effective annual rate, so \(t\) is 1 year.

Calculate Effective Annual Rate (a) annually

Here, \(n=1\). Using the formula for the effective rate \((1 + i/n)^(n*t) - 1\), substitute the known values into the formula to get the effective rate: \((1 + 0.09/1)^(1*1) - 1\), which simplifies to 0.09 or 9%.

Calculate Effective Annual Rate (b) semiannually

Here, \(n=2\). Using the formula for the effective rate, substitute the known values into the formula to get the effective rate: \((1 + 0.09/2)^(2*1) - 1\), which simplifies to approximately 0.092025 or 9.2025%.

Calculate Effective Annual Rate (c) quarterly

Here, \(n=4\). Using the formula for the effective rate, substitute the known values into the formula to get the effective rate: \((1 + 0.09/4)^(4*1) - 1\), which simplifies to approximately 0.093181 or 9.3181%.

Calculate Effective Annual Rate (d) monthly

Here, \(n=12\). Using the formula for the effective rate, substitute the known values into the formula to get the effective rate: \((1 + 0.09/12)^(12*1) - 1\), which simplifies to approximately 0.093806 or 9.3806%.

Key concepts

Nominal Interest Rate

The nominal interest rate is the percentage increase in money you can expect on an investment or need to pay on a loan without taking compounding into account. For example, if a savings account offers a nominal interest rate of 9%, that's the rate before the effects of compounding are considered.

It's important to distinguish between nominal and real interest rates, where the latter accounts for inflation. However, in most financial calculations, including our exercise, we focus on the nominal interest rate as a beginning step in finding out how much you can earn or owe when compound interest is applied. This nominal rate is typically an annual rate and doesn't always reflect the actual return on an investment due to the frequency of compounding.

Compound Interest

Compound interest is what happens when you earn interest on both the initial amount of money and the interest that amount has already earned.

For example, if you invest \(100 at an interest rate of 9% compounded annually, you'd have \)109 at the end of the first year. In the second year, you'd earn interest on the new total of \(109, not just the original \)100. This compounding effect can make a big difference over time, and it's crucial for understanding how investments grow.

Understanding the Compound Formula

The general formula for compound interest is: \( A = P(1 + \frac{r}{n})^{nt} \)
Where:

• \(A\) is the amount of money accumulated after n years, including interest.
• \(P\) is the principal amount (the initial amount of money).
• \(r\) is the annual interest rate (decimal).
• \(n\) is the number of times the interest is compounded per year.
• \(t\) is the time the money is invested for in years.

Our exercise calculates the effective interest rates with different compounding frequencies using this compound interest concept.

Annual Compounding

Annual compounding refers to the scenario in which the frequency of compounding is once per year.

When interest is compounded annually, it is added to the principal sum once at the end of the year. This is the simplest form of compounding. In our exercise, when we solved for the effective annual rate with an annual compounding frequency, there was no difference between the nominal and effective rates since the compounding occurs only once.

Effective Annual Rate

The effective annual rate (EAR) is the actual interest rate that an investor earns in a year after accounting for the effects of compounding. Unlike the nominal rate, EAR can vary depending on the number of compounding periods.

The EAR is vital for comparing the annual interest between investments with different compounding periods. To calculate the EAR, we use the formula: \( (1 + \frac{i}{n})^{n} - 1 \) where \(i\) is the nominal interest rate and \(n\) is the number of compoundings per year.

Impact of Increased Compounding

As the compound frequency increases—that is, as \(n\) grows larger—the EAR will become higher than the nominal rate. This can easily be seen in our exercise, where we calculated the EAR for semiannual, quarterly, and monthly compounding periods, resulting in progressively higher rates compared to the nominal rate.