Welcome to Statistical Measures of Asset Returns!
Welcome, future Charterholder! If you have ever looked at a bunch of stock prices and felt overwhelmed by the numbers, you are in the right place. In this chapter, we transition from just looking at data to summarizing it. Think of these statistical measures as "short-cuts" that tell us the story of an investment without us having to look at every single daily price change. We are going to learn how to measure the "center" of our data (returns) and how much that data "wiggles" or spreads out (risk).
Don't worry if math isn't your favorite subject. We will break every formula down into plain English and use plenty of real-world examples to make it stick!
1. Measures of Central Tendency: Finding the "Middle"
When we ask, "What kind of return did this stock provide?", we are usually looking for a single number that represents the whole group. This is Central Tendency.
The Arithmetic Mean (The Simple Average)
This is the average you learned in school. You add everything up and divide by the number of observations. In the CFA world, we use it to estimate the expected return for a single period.
Formula: \( \bar{X} = \frac{\sum X_i}{n} \)
Quick Example: If a stock returns 10%, -5%, and 15% over three years, the arithmetic mean is: \( (10 - 5 + 15) / 3 = 6.67\% \).
The Geometric Mean (The Compounded Growth Rate)
This is much more important for finance because money compounds. The Geometric Mean (RG) tells you the "typical" return per year if you reinvested your profits. It will always be less than or equal to the arithmetic mean.
Formula: \( RG = [ (1+R_1) \times (1+R_2) \times ... \times (1+R_n) ]^{1/n} - 1 \)
Analogy: Imagine you have \$100. It grows 100% (to \$200) and then drops 50% (back to \$100). \n
Arithmetic Mean: \( (100\% - 50\%) / 2 = 25\% \). (This sounds great, but you have \$0 profit!)
Geometric Mean: \( [ (1+1.0) \times (1-0.5) ]^{1/2} - 1 = [ 2.0 \times 0.5 ]^{0.5} - 1 = 0\% \). (This accurately shows you made no money.)
The Harmonic Mean (For Cost Averaging)
We use this specifically when we are looking at the average price paid when we invest a fixed dollar amount over time (Dollar Cost Averaging).
Formula: \( \bar{X}_H = \frac{n}{\sum (1/X_i)} \)
Key Takeaway: For any set of different numbers: Arithmetic Mean > Geometric Mean > Harmonic Mean.
Median and Mode
- Median: The exact middle value when data is ranked from smallest to largest. If you have an even number of data points, it is the average of the two middle ones.
- Mode: The value that appears most frequently. A distribution can have one mode, two (bimodal), or none.
Quick Review: Which mean should you use to show past performance? Geometric. Which mean should you use to forecast next year’s return? Arithmetic.
2. Measures of Dispersion: Measuring the "Wiggle" (Risk)
Knowing the average return is only half the story. We also need to know how much the returns deviate from that average. In finance, dispersion equals risk.
The Range
The simplest measure. It is just the Maximum value minus the Minimum value.
Example: If a stock’s highest return was 20% and its lowest was -10%, the range is \( 20 - (-10) = 30\% \).
Variance and Standard Deviation
These are the "heavy hitters" of the CFA exam. They measure how far the data points sit from the mean.
Variance (\( \sigma^2 \)): The average of the squared deviations from the mean. We square them so that negative differences don't cancel out positive ones.
Standard Deviation (\( \sigma \)): The square root of Variance. We use this because it is in the same units as our returns (percentage), making it easier to talk about.
Step-by-Step Process for Standard Deviation:
- Calculate the Arithmetic Mean.
- Subtract the Mean from each data point (this is the "deviation").
- Square each of those deviations.
- Sum them up.
- Divide by \( n \) (for a population) or \( n-1 \) (for a sample).
- Take the square root.
Common Mistake: Forgetting to use \( n-1 \) for samples! If the exam question mentions a "sample" of returns, always divide by \( n-1 \). If it is the "entire population," divide by \( n \).
Target Downside Deviation
Standard deviation treats "good" surprises (big gains) the same as "bad" surprises (big losses). Target Downside Deviation only looks at returns that fall below a specific target or "hurdle rate." This is often more useful for investors who only care about the risk of losing money.
Key Takeaway: Higher Standard Deviation = Higher Volatility = Higher Risk.
3. Relative Dispersion: The Coefficient of Variation (CV)
How do we compare a stock that returns 20% with a 10% risk to a stock that returns 5% with a 2% risk? We use the CV.
Formula: \( CV = \frac{Standard Deviation}{Mean} \)
The CV tells you "How much risk am I taking for every 1% of return?"
Lower is better! It means you are getting more return for less "unit" of risk.
4. Shape of the Distribution: Skewness and Kurtosis
Not all return data looks like a perfect "Bell Curve" (Normal Distribution). Sometimes the curve is lopsided or has "fat tails."
Skewness (The "Lean")
Skewness describes the symmetry of the distribution. Think of the tail of the graph.
- Symmetrical: Mean = Median = Mode.
- Positively Skewed (Right-Skewed): The tail points to the right. There are a few frequent small losses but occasional huge gains. (Mean > Median > Mode).
- Negatively Skewed (Left-Skewed): The tail points to the left. There are frequent small gains but occasional huge losses. (Mean < Median < Mode).
Memory Aid: "The tail tells the tale." If the tail points to the Right (Positive numbers), it is Positive Skew.
Kurtosis (The "Fatness")
Kurtosis tells us about the "peaks" and the "tails" of the distribution. In finance, we worry about fat tails because they mean extreme events (market crashes) happen more often than a normal curve suggests.
- Leptokurtic: More peaked with "fat tails." (Lepto = Leaps up!). This is common in financial markets.
- Platykurtic: Flatter with thin tails. (Platy = Flat like a Plate).
- Mesokurtic: A normal distribution (Excess Kurtosis = 0).
Did you know? Most financial assets are Leptokurtic. This means "Black Swan" events (extreme outliers) happen more often than simple statistics would predict!
Final Summary Quick Check
1. Which mean is best for historical growth? Geometric Mean.
2. What does a high Standard Deviation imply? High risk/volatility.
3. What does CV measure? Risk per unit of return.
4. What does "Positive Skew" look like? A long tail to the right.
5. What is a Leptokurtic distribution? One with fat tails and a high peak.
You've got this! These measures are the foundation of everything we do in Portfolio Management and Equity Analysis. Master these basics, and the rest of the curriculum will feel much more intuitive.