Chapter 2: Problem 24
Find the rate for continuous compounding equivalent to monthly compounding at \(12 \%\).
Short Answer
Expert verified
The continuous compounding rate equivalent to a 12% annual rate compounded monthly is approximately 11.95%.
Step by step solution
01
Identify the Given Information
The problem provides an annual nominal interest rate for monthly compounding, which is 12%. This means the interest rate for each month is given by \[ \text{monthly rate} = \frac{12\text{\text{%}}}{12} = 1\text{\text{%}} = 0.01 \]
02
Determine the Annual Effective Rate
To find the equivalent continuous compounding rate, the annual effective rate needs to be calculated first. For monthly compounding, the annual effective rate (AER) is calculated as: \[ AER = \bigg(1 + \frac{0.12}{12}\bigg)^{12} - 1 \] Substituting the values: \[ AER = \bigg(1 + 0.01\bigg)^{12} - 1 \] Performing the calculation, we get: \[ AER \thickapprox 0.126825 \]
03
Relate the Annual Effective Rate to Continuous Compounding
The relationship between the annual effective rate (AER) and the continuous compounding rate (r) is given by the formula: \[ AER = e^r - 1 \] Substituting the calculated AER value: \[ 0.126825 = e^r - 1 \] Adding 1 to both sides: \[ 1.126825 = e^r \]
04
Solve for the Continuous Compounding Rate
To solve for r: \[ r = \text{ln}(1.126825) \] Performing the calculation: \[ r \thickapprox 0.119544 \] Converting this to a percentage: \[ r \thickapprox 11.95\text{\text{%}} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Annual Effective Rate
Understanding the annual effective rate (AER) is essential in financial mathematics. The AER represents the true interest rate an investor earns in a year after accounting for compounding periods.
In our example, we start with a 12% nominal annual interest rate, compounded monthly. To find the AER, we use the formula for compound interest:
equation: \[ AER = \bigg(1 + \frac{0.12}{12}\bigg)^{12} - 1 \]
This formula breaks the annual rate into 12 monthly periods (since interest is compounded monthly) and accumulates the interest over the year. Plugging in the values, we get an AER of approximately 12.6825%.
The AER gives us a better understanding of what we're truly earning in a year. It's generally higher than the nominal rate because it includes the effect of compounding. Keep this in mind whenever comparing different investment options.
In our example, we start with a 12% nominal annual interest rate, compounded monthly. To find the AER, we use the formula for compound interest:
equation: \[ AER = \bigg(1 + \frac{0.12}{12}\bigg)^{12} - 1 \]
This formula breaks the annual rate into 12 monthly periods (since interest is compounded monthly) and accumulates the interest over the year. Plugging in the values, we get an AER of approximately 12.6825%.
The AER gives us a better understanding of what we're truly earning in a year. It's generally higher than the nominal rate because it includes the effect of compounding. Keep this in mind whenever comparing different investment options.
Compounding Interest
Compounding interest is a core concept in financial mathematics, playing a crucial role in investments and savings.
Here's how it works: with compounding, you earn interest not only on your initial investment but also on the interest that accumulates over time. This results in exponential growth.
In our example, monthly compounding divides the 12% annual rate into 1% per month. Interest is added each month, and in the next month, we earn interest on this new total. Hence, the calculation:
equation: \[ (1 + 0.01)^{12} - 1 \]
shows how the interest grows each month. Different compounding periods (daily, quarterly, annually) will lead to differing amounts of interest earned. The more frequently interest is compounded, the more you earn.
Here's how it works: with compounding, you earn interest not only on your initial investment but also on the interest that accumulates over time. This results in exponential growth.
In our example, monthly compounding divides the 12% annual rate into 1% per month. Interest is added each month, and in the next month, we earn interest on this new total. Hence, the calculation:
equation: \[ (1 + 0.01)^{12} - 1 \]
shows how the interest grows each month. Different compounding periods (daily, quarterly, annually) will lead to differing amounts of interest earned. The more frequently interest is compounded, the more you earn.
Financial Mathematics
Financial mathematics involves using mathematical methods to solve financial problems. Core areas include calculating interest, modeling financial systems, and assessing risks.
In our example, we explored how to convert an annually compounded interest rate to its continuous compounding equivalent. First, we found the annual effective rate (AER), then transitioned to continuous compounding.
This involves using various mathematical formulas and understanding their derivations. Such calculations help in making informed financial decisions.
In our example, we explored how to convert an annually compounded interest rate to its continuous compounding equivalent. First, we found the annual effective rate (AER), then transitioned to continuous compounding.
This involves using various mathematical formulas and understanding their derivations. Such calculations help in making informed financial decisions.
Exponential Functions
Exponential functions are key in understanding continuous compounding. These functions model situations where growth rate is proportional to the value, leading to exponential growth.
In the context of interest rates, the continuous compounding formula is:
equation:\[ e^r - 1 = AER \]
Here, \( e \) is Euler's number (approximately 2.71828), and it depicts continual growth. Exponential functions show up in many areas of mathematics and science, making them crucial to understanding continuous growth models.
For our example, we found the continuous compounding rate by solving:
equation: \[ r = \text{ln}(1.126825) \]
leading to a rate of approximately 11.95%. This shows how powerful exponential functions are in modeling and understanding financial growth.
In the context of interest rates, the continuous compounding formula is:
equation:\[ e^r - 1 = AER \]
Here, \( e \) is Euler's number (approximately 2.71828), and it depicts continual growth. Exponential functions show up in many areas of mathematics and science, making them crucial to understanding continuous growth models.
For our example, we found the continuous compounding rate by solving:
equation: \[ r = \text{ln}(1.126825) \]
leading to a rate of approximately 11.95%. This shows how powerful exponential functions are in modeling and understanding financial growth.