# Why do the arithmetic average return and the geometric return differ? Explain.

Why do the arithmetic average return and the geometric return differ? Explain.
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Introduction:
In the field of finance, the arithmetic average return and the geometric return are two commonly used measures of investment performance. While they are both measures of investment returns, they differ in how they are calculated and the interpretation of the results. In this paper, we will explore why these two measures of return differ and explain the concepts behind them.In the world of finance, investment performance is a key consideration for investors. One of the critical factors that investors use to evaluate investment performance is the rate of return. The rate of return is the amount of profit or loss generated by an investment over a specific period. Investors use different measures of rate of return to evaluate investment performance, such as the arithmetic average return and the geometric return.

The arithmetic average return and the geometric return are both measures of investment performance that are commonly used in finance. Although these measures provide investors with valuable information on investment performance, they differ in how they are calculated and the interpretation of the results.

In this paper, we will explore the concepts behind the arithmetic average return and the geometric return, and we will explain why these two measures of return differ. We will also provide examples to illustrate the differences between these measures and the significance of these differences.

We will begin by discussing the arithmetic average return and how it is calculated. We will then move on to the geometric return, explaining how it differs from the arithmetic average return and how it is calculated. After that, we will discuss why these two measures of return differ and provide examples to illustrate the differences. Finally, we will conclude by summarizing the key points discussed in this paper.

Overall, this paper aims to provide investors with a deeper understanding of the arithmetic average return and the geometric return, and how these measures can be used to evaluate investment performance. By understanding these measures, investors can make more informed investment decisions and improve their overall investment performance.

Section 1: Understanding Arithmetic Average Return
The arithmetic average return is a simple average of a series of returns over a period of time. It is calculated by adding up all the returns and dividing the sum by the nu

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Step-by-step explanation
mber of returns. For example, if an investment had returns of 10%, 5%, and 2% over a period of three years, the arithmetic average return would be (10% + 5% + 2%) / 3 = 5.67%.

Section 2: Understanding Geometric Return
The geometric return, on the other hand, is the average rate of return over a period of time, taking into account the compounding effect of returns. It is calculated by multiplying all the returns and taking the nth root, where n is the number of returns. For example, if an investment had returns of 10%, 5%, and 2% over a period of three years, the geometric return would be [(1 + 0.1) x (1 + 0.05) x (1 + 0.02)]^(1/3) – 1 = 5.68%.

Section 3: Why do Arithmetic Average Return and Geometric Return differ?
The difference between arithmetic average return and geometric return arises due to the compounding effect of returns. When an investment has a positive return, the return is reinvested, and over time, the investment grows at a compounding rate. As a result, the geometric return is always less than or equal to the arithmetic average return. This difference becomes more significant as the number of returns increases, and the difference between the returns becomes larger.

Section 4: Example to illustrate the difference between the two returns
Suppose an investment has the following returns over a period of three years: 20%, -10%, and 30%. The arithmetic average return would be (20% – 10% + 30%) / 3 = 13.33%. However, the geometric return would be [(1 + 0.2) x (1 – 0.1) x (1 + 0.3)]^(1/3) – 1 = 11.37%. Thus, the geometric return is significantly lower than the arithmetic average return, which indicates that the investment did not perform as well as the arithmetic average suggests.

Conclusion:
In conclusion, the arithmetic average return and the geometric return are important measures used in finance to evaluate investment performance. These measures help investors understand the average return on their investments over a given period of time. However, it is essential to understand the differences between these two measures and use them appropriately to evaluate the performance of investments.

The arithmetic average return is a simple calculation that provides an average return over a period of time. This measure is useful when evaluating investments that have a consistent return over the investment period. However, it does not take into account the effect of compounding, which is an essential factor in determining investment performance.

The geometric return, on the other hand, is a more accurate measure of investment performance that considers the effect of compounding. This measure is particularly useful when evaluating investments that have varying returns over the investment period. By taking into account the compounding effect of returns, investors can better evaluate the overall performance of their investments.

It is important to note that the geometric return is always lower than or equal to the arithmetic average return. This difference is due to the compounding effect of returns, which tends to reduce the overall return on investments. Therefore, it is essential to use the appropriate measure when evaluating investment performance.

Furthermore, it is also important to note that both the arithmetic average return and the geometric return have limitations. These measures do not account for the risk associated with investments. Therefore, investors should also consider other measures such as risk-adjusted return when evaluating investment performance.

In addition to these limitations, there are also other factors that can affect investment performance. These factors include economic conditions, market volatility, and political instability, among others. Therefore, investors should always consider these factors when evaluating their investments.

Overall, the arithmetic average return and the geometric return are important measures that investors can use to evaluate investment performance. These measures provide investors with valuable information on the average return on their investments over a given period of time. However, it is important to understand the differences between these two measures and use them appropriately to evaluate the performance of investments.

Investors should also consider other measures such as risk-adjusted return when evaluating investment performance. Furthermore, investors should always consider other factors that can affect investment performance, such as economic conditions, market volatility, and political instability, among others.

Understanding the arithmetic average return and the geometric return is an important part of evaluating investment performance. By using these measures appropriately, investors can make more informed investment decisions and improve their overall investment performance.