# Video Script: Calculating Average Returns (Arithmetic, Geometric, and Harmonic Means) ## Video Title **"Understanding Average Returns: Arithmetic, Geometric, and Harmonic Means"** ## Video Structure ### 1. Introduction (5-10 seconds) **Visual Content** - A central title card appears with the text: **"Calculating Average Returns: Arithmetic, Geometric, and Harmonic Means"** The bottom-right corner displays the CFA topic area: **Quantitative Methods**. **Voiceover** - "Welcome to this session of CFA Level 1 exam topic explanations! Today, we will explore 'Calculating Average Returns,' an essential concept within the Quantitative Methods section." - "We will focus on three common methods of calculating average returns: Arithmetic Mean Return, Geometric Mean Return, and Harmonic Mean Return. These formulas not only frequently appear in exams but are also highly practical in investment analysis." - "By mastering these formulas, you will be able to better evaluate portfolio performance, understand the logic behind return calculations, and optimize your investment decisions." --- ### 2. Formula Introduction and Explanation (30-60 seconds) #### **(1) Arithmetic Mean Return** **Visual Content** - Background changes to dark blue, and the formula is displayed in the center: \[ \bar{R} = \frac{1}{n} \sum_{i=1}^{n} R_i \] - Arrows highlight each variable in the formula: - \( \bar{R} \): Arithmetic mean return (unit: %) - \( R_i \): Return for period \( i \) (unit: %) - \( n \): Total number of periods (unit: integer) - On the right, a small animation illustrates "adding multiple returns and dividing by the number of periods." **Voiceover** - "First, let's look at the formula for Arithmetic Mean Return: \[ \bar{R} = \frac{1}{n} \sum_{i=1}^{n} R_i \]." - "The calculation is straightforward: add up the returns for each period and divide by the total number of periods." - "Arithmetic mean return is suitable for short-term analysis because it ignores the compounding effect. In other words, it assumes that returns in one period are independent of returns in other periods. While this method is simple and easy to use, it has limitations in long-term investment analysis." #### **(2) Geometric Mean Return** **Visual Content** - Background changes to purple, and the formula is displayed in the center: \[ R_G = \left( \prod_{i=1}^{n} (1 + R_i) \right)^{\frac{1}{n}} - 1 \] - Arrows highlight each variable in the formula: - \( R_G \): Geometric mean return (unit: %) - \( R_i \): Return for period \( i \) (unit: %) - \( n \): Total number of periods (unit: integer) - On the right, a small animation demonstrates "adding 1 to each return, multiplying them together, taking the \( n \)-th root, and subtracting 1." **Voiceover** - "Next is the Geometric Mean Return, with the formula: \[ R_G = \left( \prod_{i=1}^{n} (1 + R_i) \right)^{\frac{1}{n}} - 1 \]." - "Unlike the arithmetic mean, the geometric mean return accounts for the compounding effect, meaning that returns in one period impact returns in subsequent periods. The steps involve adding 1 to each return, multiplying them together, taking the \( n \)-th root, and finally subtracting 1." - "Geometric mean return is more suitable for long-term investment analysis as it more accurately reflects the impact of compounding. For example, if an investment has significant fluctuations over several years, the geometric mean return can help us better assess its overall performance." #### **(3) Harmonic Mean Return** **Visual Content** - Background changes to green, and the formula is displayed in the center: \[ X_H = \frac{n}{\sum_{i=1}^{n} \frac{1}{R_i}} \] - Arrows highlight each variable in the formula: - \( X_H \): Harmonic mean return (unit: %) - \( R_i \): Return for period \( i \) (unit: %) - \( n \): Total number of periods (unit: integer) - An animation illustrates "taking the reciprocal of each return, summing them up, and dividing the total number of periods by the sum." **Voiceover** - "Finally, we have the Harmonic Mean Return, with the formula: \[ X_H = \frac{n}{\sum_{i=1}^{n} \frac{1}{R_i}} \]." - "The harmonic mean return is calculated differently from the other two. It places more emphasis on smaller values. Specifically, you take the reciprocal of each return, sum them up, and divide the total number of periods by this sum." - "Harmonic mean return is particularly useful when analyzing ratio-based metrics, such as P/E ratios or other financial ratios. It allows for a more accurate measure of averages, especially when data distribution is uneven." --- ### 3. Examples and Step-by-Step Calculations (60-120 seconds) #### **(1) Arithmetic Mean Return** **Visual Content** - Data: Assume an investment has annual returns of 5%, 10%, -3%, and 8%. - Animation shows the calculation process: 1. Add all returns: \(5\% + 10\% - 3\% + 8\%\). 2. Calculate the total: \(20\%\). 3. Divide the total by the number of periods: \(20\% \div 4 = 5\%\). **Voiceover** - "Let's look at an example. Assume an investment has annual returns of 5%, 10%, -3%, and 8%. First, we add these returns together to get a total of 20%. Then, we divide this total by the number of periods, which is 4. The arithmetic mean return is therefore 5%. This result tells us that the average annual return for this investment over four years is 5%." #### **(2) Geometric Mean Return** **Visual Content** - Data: The same annual returns: 5%, 10%, -3%, and 8%. - Animation shows the calculation process: 1. Add 1 to each return: \(1.05, 1.10, 0.97, 1.08\). 2. Multiply them together: \(1.05 \times 1.10 \times 0.97 \times 1.08 = 1.2249\). 3. Take the fourth root: \(1.2249^{1/4} = 1.0523\). 4. Subtract 1: \(1.0523 - 1 = 5.23\%\). **Voiceover** - "Next, let's calculate the geometric mean return using the same data. First, add 1 to each return, resulting in 1.05, 1.10, 0.97, and 1.08. Multiply these values together to get a product of 1.2249. Then, take the fourth root of this product, which is 1.0523. Finally, subtract 1 to get a geometric mean return of 5.23%. This result is slightly higher than the arithmetic mean because it accounts for compounding effects." #### **(3) Harmonic Mean Return** **Visual Content** - Data: The same annual returns: 5%, 10%, -3%, and 8%. - Animation shows the calculation process: 1. Calculate the reciprocal of each return: \(1/0.05 = 20, 1/0.10 = 10, 1/-0.03 = -33.33, 1/0.08 = 12.5\). 2. Sum the reciprocals: \(20 + 10 - 33.33 + 12.5 = 9.17\). 3. Divide the number of periods by the sum: \(4 \div 9.17 = 0.436\). **Voiceover** - "Finally, let's calculate the harmonic mean return. Using the same data, we first calculate the reciprocal of each return, such as 1/5% equals 20 and so on. Summing these reciprocals gives us a total of 9.17. Dividing the number of periods, 4, by this total gives us a harmonic mean return of 43.6%." --- ### 4. Summary and Recap (15-30 seconds) **Visual Content** - Background: A dynamic mind map gradually reveals key points: 1. **Arithmetic Mean: Short-term analysis, simple and easy to use.** 2. **Geometric Mean: Long-term analysis, accounts for compounding effects.** 3. **Harmonic Mean: Suitable for ratio-based metrics, emphasizes smaller values.** **Voiceover** - "To summarize, arithmetic mean is suitable for short-term return analysis as it is simple and straightforward but ignores compounding effects. Geometric mean is ideal for long-term investment analysis because it accurately reflects the impact of compounding. Harmonic mean is mainly used for analyzing ratio-based metrics, especially when data distribution is uneven." - "By understanding the definitions, calculation principles, and practical applications of these formulas, you can better evaluate portfolio performance and risks, enabling more informed investment decisions." **Visual Content** - Final screen: Displays the CFA Institute logo with the text: **"Thank you for watching!"** - Background music fades out, and the screen gradually fades to black. **Voiceover** - "Thank you for watching this session. See you next time!"