Question

The effective yield is the annual rate \(i\) that will produce the same interest per year as the nominal rate \(r\). (a) For a rate \(r\) that is compounded continuously, show that the effective yield is \(i=e^{r}-1\). (b) Find the effective yield for a nominal rate of \(6 \%\), compounded continuously.

Short answer

The effective yield is given by the formula \(i = e^{r} - 1\). For a nominal rate of \(6\%\), compounded continuously, the effective yield is approximately \(6.1837\%\).

Step-by-step solution

Understand the concepts

If interest is compounded continuously, then the future value \(A\) of an investment of \(P\) dollars at a nominal rate of \(r\), compounded continuously for \(t\) years is given by the formula \(A = Pe^{rt}\). Meanwhile, if the interest is compounded annually at a rate of \(i\), the future value is given by \(A = P(1 + i)^t\). The effective yield is the annual rate \(i\) such that both formulas produce the same amount \(A\) for a given amount \(P\) and time \(t\).

Prove the formula for effective yield

In order for both formulas to yield the same future value \(A\), we must have \(e^{rt} = (1 + i)^t\). Solving for \(i\) would give us the following: \((1 + i)^t = e^{rt} \rightarrow i = e^{r} - 1\). Thus the formula for effective yield is \(i = e^{r} - 1\), demonstrating the outcome asked for in part (a).

Compute the effective yield

For part (b), now calculate the effective yield using the formula just proven and the given nominal rate of \(6\%\). This is done by letting \(r=0.06\) in the formula \(i = e^{r} - 1\). In this case, \(i = e^{0.06} - 1\).

Evaluate the expression

By evaluating the above step, \(i = e^{0.06} - 1\) approximately equals 0.061837, or \(6.1837\%\). Thus, the effective yield for a nominal rate of \(6\% \), compounded continuously is about \(6.1837\% \).

Key concepts

Continuous Compounding

Continuous compounding is a fascinating concept in financial mathematics. When interest is compounded continuously, it means that rather than calculating interest on a periodic basis (such as monthly or annually), the interest is calculated and added to the principal infinitely many times per year. This results in a higher amount of compound interest, as the investment grows without pause. Mathematically, continuous compounding is expressed with the formula \( A = Pe^{rt} \), where:

• \( A \) stands for the future value of the investment.
• \( P \) is the principal amount invested.
• \( r \) represents the nominal interest rate.
• \( t \) is the time in years.
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Continuous compounding allows for a smooth growth curve, emphasizing the power of exponential growth as time increases.

Nominal Rate

The nominal rate, often referred to as the stated or quoted rate, is the annual interest rate not accounting for compounding within that year. It's the rate that appears in loan agreements and investment contracts. However, this nominal rate does not reflect the true financial picture if compounding occurs more frequently than annually.
This is important because different compounding frequencies can lead to different effective yields. For example, a nominal rate of 6% compounded monthly is not equivalent to 6% compounded continuously, as more frequent compounding results in a higher effective annual rate.
Understanding the difference between nominal and effective rates is crucial when comparing financial products, as it can affect investment performance and cost over time.

Exponential Growth

Exponential growth is a key principle in understanding compounding interest, especially in continuous compounding. It describes a situation where the growth rate of a value is proportional to its current size, leading to faster and faster increases. This is captured beautifully with the formula \( A = Pe^{rt} \).

Here are some points to grasp the concept of exponential growth:

• It starts slowly but increases rapidly, a core characteristic of functions involving \( e \), the base of natural logarithms.
• In finance, it means wealth grows continually, faster than with simple interest calculations.
• It highlights the importance of starting investments early, as small increments over time lead to massive growth.

Exponential growth is one of the reasons why continuous compounding is so powerful; even minimal increases in the nominal rate can lead to significant increases in the effective yield.

Financial Mathematics

Financial mathematics applies mathematical methods to solve problems in finance and investment, focusing heavily on interest, valuation, and risk analysis. It involves formulas and concepts that help calculate present and future values of investments, assess risks, and make informed financial decisions.
Essential components of financial mathematics include:

• Interest calculations, such as continuous compounding vs. simple interest.
• Valuation of different financial instruments, like bonds and stocks.
• Understanding and comparing different financial products using effective and nominal rates.

In practice, it's used in applying theories and formulas, like the ones discussed for continuous compounding, to determine the effective yield of investments. This branch of mathematics is integral to optimizing financial performance and decision-making.