Introduction

In the world of finance, understanding the concept of returns is fundamental. Returns represent the gain or loss on an investment over a specific period. In this chapter, we will delve into the calculation, differentiation, and conversion between two types of returns: simple returns and continuously compounded returns.

Returns are a way to measure the profitability of an investment. They provide insights into how an investment’s value has changed over time. There are different methods to calculate returns, each with its own implications and applications.

Calculating Simple Returns

Simple returns, also known as arithmetic returns, are calculated by dividing the difference between the final value and the initial value of an investment by the initial value. The formula for calculating simple returns can be expressed as:

$$\text {Simple Return}=\frac{\text {(Final Value-Initial Value)}}{\text {Initial Value}} \times 100$$ For example, if an investment initially valued at 1,000 USD grows to 1,200 USD, the simple return would be $\frac{(1200-1000)}{1000} \times 100=20%$.

Continuously Compounded Returns

Continuously compounded returns, often used in financial modeling, reflect the compounding effect of returns over small intervals of time. These returns assume that the interest is being compounded an infinite number of times within a given period. The formula for continuously compounded returns is: $$\text{Continuously Compounded Return} =\ln \left(\frac{\text { Final Value }}{\text { Initial Value }}\right)$$

Conversion Between Simple and Continuously Compounded Returns

Converting between simple and continuously compounded returns is straightforward. For converting simple returns to continuously compounded returns, you can use the following formula:

$$\text{Continuously Compounded Return} = \ln (1+ \text{Simple Return})$$

And to convert continuously compounded returns back to simple returns:

$$\text{Simple Return}=e^{\text{Continuously Compounded Return}}-1$$

Example: Let’s consider an investment that grows from 800 USD to 1,000 USD. The simple return is $\frac{1000-800}{800} \times 100=25 \%$. The continuously compounded return can be calculated as $\ln \left(\frac{1000}{800}\right) \approx 0.223$, which is approximately 22.3%.