Question

Use \(y=y_{0} e^{k t}\). What continuous interest rate has the same yield as an annual rate of \(9 \%\) ?

Short answer

The equivalent continuous interest rate is approximately 8.62%.

Step-by-step solution

Understanding the Problem

We are asked to find the continuous interest rate equivalent to an annual interest rate of 9%. The formula given, \( y = y_0 e^{kt} \), is used to calculate continuous growth.

Identify Given Values

The problem indicates an annual interest rate of 9%, which is equivalent to 0.09 as a decimal. We are assuming the initial amount is \( y_0 \) and \( t = 1 \) year, since we are comparing annual rates.

Determine the Comparable Formula

For the annual interest rate, the corresponding expression would be \( y = y_0 (1 + r)^t \), where \( r = 0.09 \) and \( t = 1 \). This yields \( y = y_0 (1.09) \).

Equate Continuous and Annual Formulas

Set the continuous interest equation equal to the annual interest equation for \( t = 1 \): \[ y_0 e^{k imes 1} = y_0 (1.09) \]. This simplifies to: \[ e^{k} = 1.09 \].

Solve for k (the Continuous Rate)

Take the natural logarithm of both sides to solve for \( k \): \[ k = \ln(1.09) \]. Now calculate \( k \): \[ k \approx 0.0862 \].

Finalize the Continuous Rate

The continuous interest rate that matches an annual rate of 9% is approximately \( k = 8.62\% \). This means that if interest is compounded continuously, an annual equivalent of 9% is yielded with an 8.62% continuous rate.

Key concepts

Continuous Interest Rate

A continuous interest rate refers to the method of calculating interest where it is compounded continuously rather than at discrete intervals. Imagine your money growing every microsecond! This concept is built on the idea that compounding more frequently results in higher returns. The mathematical representation of continuous interest involves the exponential function. Here, the formula

• \( y = y_0 e^{kt} \)

is used to model such growth. In this equation:

• \( y \) is the future value of the investment.
• \( y_0 \) is the initial principal or starting amount.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
• \( k \) is the continuous interest rate.
• \( t \) is the time in years.

Understanding continuous interest is important, especially in fields like finance and exponential growth modeling. The rate \( k \) gives us a smooth growth curve representing what happens when compounding occurs all the time.

Compounded Interest

Compounded interest is simply interest on interest. Unlike simple interest, which is calculated on the principal alone, compounded interest grows on both the initial investment and any accumulated interest. This can significantly boost returns over time.

• Compounding can happen annually, semi-annually, quarterly, monthly, or even daily.
• More frequent compounding means more interest; this is where continuous compounding plays a role.

The standard formula used for compounded interest over discrete time intervals is:

• \( y = y_0 (1 + r)^t \)

Where:

• \( y \) represents the future value.
• \( y_0 \) is the initial principal.
• \( r \) is the nominal interest rate per period.
• \( t \) is the number of periods.

Understanding the relations between different compounding frequencies helps in optimizing investment strategies. It's fascinating that as the compounding frequency increases, our approximation of interest increasingly aligns with the concept of continuous compounding.

Natural Logarithm

The natural logarithm is a fundamental concept in mathematics, closely connected with continuous compounding. It is the inverse of the exponential function with base \( e \). In simple terms, the natural logarithm, denoted as \( \ln(x) \), asks "to what power must \( e \) be raised, to equal \( x \)?"

• It helps in determining the time or interest rate in continuous growth scenarios.

For instance, when solving for a continuous interest rate \( k \), we equate the future value formulas of continuous and annual compounding, leading to:

• \( e^k = 1.09 \)
• \( k = \ln(1.09) \)

The natural logarithm makes it easy to convert our understanding from discrete to continuous interest contexts. It is a powerful tool for unwinding exponential equations, helping us find the exact rate or time for phenomena described by continuous growth.