Understanding Nominal Rates and Compounding Periods: Bimonthly vs Quarterly

When dealing with financial calculations, it is crucial to understand how different compounding periods can affect the nominal rates and the effective annual rates. This article will explore the process of converting a nominal rate compounded bimonthly into an equivalent nominal rate compounded quarterly. We will begin by breaking down the steps and then perform the necessary calculations.

Steps Involved

To find the nominal rate compounded quarterly that is equivalent to a nominal rate of 13% compounded bimonthly, we can follow these steps:

1. Calculating the Effective Annual Rate (EAR)

The first step is to convert the nominal bimonthly rate to an effective annual rate (EAR).

The nominal rate is 13%, compounded bimonthly, which means there are 6 compounding periods per year. The bimonthly rate is:

Monthly Bimonthly Rate

r_b frac{0.13}{6} approx 0.0216667

The effective annual rate can be calculated using the formula:

text{EAR} left(1 r_bright^n - 1

where n is the number of compounding periods per year, which is 6 for bimonthly:

text{EAR} left(1 0.0216667right)^6 - 1

Calculating this:

text{EAR} 1.0216667^6 - 1 approx 1.136462 - 1 approx 0.136462

So, the effective annual rate is approximately 13.6462%.

2. Converting the EAR to a Nominal Quarterly Rate

Next, we need to find the nominal rate compounded quarterly that gives the same effective annual rate. The quarterly compounding means there are 4 compounding periods per year.

Quarters per Year

Let r_q be the nominal quarterly rate. The effective annual rate for quarterly compounding is given by:

text{EAR} left(1 frac{r_q}{4}right)^4 - 1

Setting this equal to the EAR we calculated:

left(1 frac{r_q}{4}right)^4 - 1 0.136462

Adding 1 to both sides:

left(1 frac{r_q}{4}right)^4 1.136462

Taking the fourth root:

1 frac{r_q}{4} 1.136462^{frac{1}{4}}

Calculating the fourth root:

1 frac{r_q}{4} approx 1.0325

Subtracting 1:

frac{r_q}{4} approx 0.0325

Multiplying by 4 to find r_q:

r_q approx 0.13

Conclusion for 13%

The nominal rate compounded quarterly that is equivalent to 13% compounded bimonthly is approximately 13%.

Additional Calculations

For further verification, we can perform the following calculations:

13% Nominal Rate Compounded Bimonthly Converted to Effective Annual Rate

Let's calculate:

1 (-13/2)^2 - 1

1 13.4225% - 1 13.4225

Converting back to a nominal annual rate by compounding quarterly:

[1 13.4225/100]^1/4 - 1

[1 0.134225]^1/4 - 1

1.134225^0.25 - 1 ≈ 0.0325

Multiplying by 400:

0.0325 * 400 12.795

Final Answer

The nominal rate compounded quarterly that is equivalent to 13% compounded bimonthly is approximately 12.795%.