**Subjects**:Quantitative Methods **Module**:Rates and Returns **Knowledge Points**:Average return（Arithmetic,Geometric,Harmonic mean return） **Subjects**: Quantitative Methods **Module**: Rates of Return **Knowledge Points**: Mean Return Calculations --- ### **Mean Return Calculations: Definitions and Formulas** --- **1. Arithmetic Mean Return** **Definition**: The arithmetic mean return is the simple average of a series of returns, calculated by summing the returns and dividing by the number of periods. It is used for estimating the average return over a single period. **Formula**: \[ \bar{R} = \frac{1}{n} \sum_{i=1}^{n} R_i \] where: - \( \bar{R} \) = arithmetic mean return - \( R_i \) = return in period \( i \) - \( n \) = number of periods --- **2. Geometric Mean Return** **Definition**: The geometric mean return is a measure of the average return over multiple periods, accounting for the compounding of returns. It is particularly useful for evaluating long-term investments. **Formula**: \[ R_G = \left( \prod_{i=1}^{n} (1 + R_i) \right)^{\frac{1}{n}} - 1 \] or \[ R_G = e^{\frac{1}{n} \sum_{i=1}^{n} \ln(1 + R_i)} - 1 \] where: - \( R_G \) = geometric mean return - \( R_i \) = return in period \( i \) - \( n \) = number of periods - \( e \) = base of the natural logarithm --- **3. Harmonic Mean Return** **Definition**: The harmonic mean return is a measure used for averaging ratios or rates, particularly suited for P/E ratios and other financial ratios. It emphasizes smaller values and is useful when calculating average rates. **Formula**: \[ X_H = \frac{n}{\sum_{i=1}^{n} \frac{1}{R_i}} \] where: - \( X_H \) = harmonic mean return - \( R_i \) = return in period \( i \) (expressed as a ratio, e.g., 0.05 for 5%) - \( n \) = number number of periods --- ### **Practical Application**: - **Arithmetic Mean**: Useful for evaluating short-term performance or comparing returns over a single period. - **Geometric Mean**: Best for long-term investment performance, as it accounts for the effects of compounding over multiple periods. - **Harmonic Mean**: Effective for assessing averages of rates and ratios, particularly for financial metrics like price-to-earnings ratios. --- **Relevant CFA Subject and Exam Weight**: These calculations are integral to the "Quantitative Methods" section of the CFA Level I curriculum, typically accounting for a weight of around 5%-10% of the exam content. --- Understanding these mean return calculations is essential for investors and analysts in order to effectively assess and compare the performance of various investments over different time horizons.

视频信息

答案文本 复制

视频字幕 复制

Mean return calculations are essential tools in quantitative finance for evaluating investment performance. There are three primary types of mean returns: arithmetic mean, geometric mean, and harmonic mean. Each serves different purposes in financial analysis. The arithmetic mean provides a simple average, while the geometric mean accounts for compounding effects over time. The harmonic mean is particularly useful for averaging ratios and rates. The arithmetic mean return is the simplest measure of average return. It is calculated by summing all returns and dividing by the number of periods. The formula is R-bar equals one over n times the sum of all returns. For example, with returns of 8%, 12%, negative 5%, 15%, and 3%, the arithmetic mean is 33% divided by 5, which equals 6.6%. This measure is useful for estimating expected returns over a single period. The geometric mean return accounts for the compounding effect of returns over multiple periods. It is calculated by taking the nth root of the product of one plus each return, then subtracting one. Using the same example returns, we multiply 1.08 times 1.12 times 0.95 times 1.15 times 1.03, which equals 1.3397. Taking the fifth root gives us 1.064, so the geometric mean is 6.4%. Notice this is lower than the arithmetic mean of 6.6%, which is typical when returns vary significantly. The arithmetic mean return is the simplest measure of average return. It's calculated by summing all returns and dividing by the number of periods. For example, with annual returns of 5%, 10%, -3%, 8%, and 2%, we add them to get 22% and divide by 5 periods to get 4.4%. This method is best for short-term performance evaluation and estimating average returns for a single period. The geometric mean return accounts for the compounding effect over multiple periods, making it essential for long-term investment analysis. Using the same returns, we multiply the gross returns: 1.05 times 1.10 times 0.97 times 1.08 times 1.02, which equals 1.2146. Then we take the fifth root and subtract 1 to get 3.98%. Notice this is lower than the arithmetic mean of 4.40%. The geometric mean is always less than or equal to the arithmetic mean when returns vary. The harmonic mean return is particularly useful for averaging ratios and rates, such as price-to-earnings ratios. The formula is n divided by the sum of one over each value. For example, with P/E ratios of 10, 15, 20, 25, and 30, we calculate one over each ratio and sum them to get 0.29. Then we divide 5 by 0.29 to get 17.2. Notice the harmonic mean of 17.2 is lower than the arithmetic mean of 20, because the harmonic mean emphasizes smaller values more heavily. In practice, each mean serves different purposes. Use the arithmetic mean for short-term performance and single period analysis. The geometric mean is essential for long-term investments and compounding analysis. The harmonic mean is best for financial ratios and rate averaging. Remember the key relationship: geometric mean is less than or equal to harmonic mean, which is less than or equal to arithmetic mean when values vary. These concepts are crucial for the CFA exam, particularly in the Quantitative Methods section. To summarize, understanding mean return calculations is crucial for CFA success. The arithmetic mean of 4.40% is highest but overestimates long-term performance. The geometric mean of 3.98% properly accounts for compounding and is essential for multi-period analysis. The harmonic mean of 3.77% is lowest and best for averaging ratios. Remember: always use the geometric mean for multi-period return calculations in portfolio management. These concepts appear frequently in the CFA Level 1 Quantitative Methods section and are fundamental to portfolio performance evaluation.