Welcome to Multivariate Empirical Methods and Performance Persistence!
Hello there! Welcome to one of the most practical chapters in the CAIA Level II curriculum. If you’ve ever wondered, "Is this fund manager actually talented, or just lucky?" or "How do we know which factors are truly driving investment returns?" then you are in the right place.
Don't worry if words like "multivariate" or "empirical" sound a bit intimidating. In plain English, we are simply learning how to look at multiple pieces of data at once to find patterns and see if success in the past is likely to happen again in the future. Let’s dive in!
1. Understanding Multivariate Linear Regression
In Level I, you likely encountered simple linear regression (looking at how one thing affects another). Multivariate regression is just the "levelled-up" version. Instead of looking at just one factor, we look at several at the same time.
The Concept: Imagine you are trying to predict the price of a house. You wouldn't just look at the square footage (univariate). You’d look at the square footage, the number of bedrooms, the school district, and the age of the house (multivariate). This gives you a much clearer picture.
The general formula looks like this:
\( R_i = \alpha + \beta_1 F_1 + \beta_2 F_2 + ... + \beta_n F_n + \epsilon \)
Where:
• \( R_i \) is the return of the investment (the Dependent Variable).
• \( \alpha \) (Alpha) is the intercept, representing the return not explained by the factors.
• \( \beta \) (Beta) represents the sensitivity of the return to each factor.
• \( F \) represents the Independent Variables (the factors like market risk, size, or value).
• \( \epsilon \) (Epsilon) is the error term (the "noise" or random stuff we can't explain).
Key Statistical Terms to Know
- Coefficient (\( \beta \)): Tells you the direction and strength of the relationship. If the coefficient for "Interest Rates" is -0.5, it means when rates go up, your investment return tends to go down.
- t-statistic: This tells us if a factor is statistically significant. A general rule of thumb: if the absolute value of the t-stat is greater than 2, the factor likely matters.
- p-value: The probability that the result happened by pure chance. We want a low p-value (usually less than 0.05) to be confident that the factor is important.
- R-squared (\( R^2 \)): This tells us how much of the total "story" our factors tell. If \( R^2 \) is 0.80, then 80% of the movement in the investment is explained by our model.
Quick Tip: If you add more variables to a model, \( R^2 \) will almost always go up, even if the variables are useless! That’s why we use Adjusted R-squared, which penalizes the score for adding unnecessary variables.
Summary: Multivariate regression helps us attribute returns to specific risk factors, allowing us to see if a manager is providing unique value (Alpha) or just riding market waves (Beta).
2. Multi-Factor Models in Practice
In the world of Alternative Investments, we use specific models to "deconstruct" where a manager's returns are coming from. The most famous is the Fama-French Model.
The Fama-French Three-Factor Model
This model suggests that three things drive equity returns:
1. Market Risk (Equity Risk Premium): General stock market movement.
2. Size (SMB - Small Minus Big): The tendency for small-cap stocks to outperform large-cap stocks over time.
3. Value (HML - High Minus Low): The tendency for high book-to-market (value) stocks to outperform low book-to-market (growth) stocks.
Expanding the Model
Later, the Carhart Four-Factor Model added Momentum (UMD - Up Minus Down), which tracks the tendency of winning stocks to keep winning. More recently, Fama and French added factors for Profitability and Investment to create a five-factor model.
Did you know? Many "Alpha" managers in the past were actually just "Closet Indexers" who were heavily tilted toward the Small-Cap and Value factors. Once we use these models, their "skill" often disappears!
Key Takeaway: By using multi-factor models, investors can determine if they are paying high "Alpha" fees for what is actually just "Factor Beta" (returns that could be bought cheaply via ETFs).
3. Performance Persistence: "The Hot Hands Effect"
Performance Persistence is the study of whether winners keep winning and losers keep losing. In a perfectly efficient market, past performance should have zero impact on future performance.
How We Test for Persistence
There are two primary ways the CAIA curriculum looks at this:
Method 1: Contingency Tables (The Winner-Winner Grid)
We group managers into "Winners" and "Losers" based on the first period, and then see how they do in the second period.
• WW: Winner in Period 1, Winner in Period 2 (Persistence)
• WL: Winner in Period 1, Loser in Period 2 (Reversal)
• LW: Loser in Period 1, Winner in Period 2 (Reversal)
• LL: Loser in Period 1, Loser in Period 2 (Persistence)
If WW + LL is significantly larger than WL + LW, we say there is evidence of persistence.
Method 2: Regression Analysis
We can run a regression where the return in Period 2 is the dependent variable and the return in Period 1 is the independent variable.
\( Return_{t+1} = \alpha + \beta(Return_t) + \epsilon \)
If the \( \beta \) is positive and statistically significant, it suggests that performance persists.
Common Mistake to Avoid: Don't assume that if a manager was a "Winner" last year, they are a "Winner" this year. Studies show that persistence is often found in Private Equity (due to long-term operational changes) but is much rarer and harder to find in Hedge Funds or Mutual Funds.
4. Pitfalls in Empirical Analysis: Survivorship Bias
When we study performance persistence, we have to be very careful about Survivorship Bias.
The Analogy: Imagine you want to find the secret to a long life. You go to a nursing home and interview everyone who is 100 years old. They all say, "I drink a glass of whiskey every day!" If you conclude that whiskey is the secret to long life, you've fallen for survivorship bias. You didn't interview the thousands of people who drank whiskey and died at 50—they weren't there to be interviewed!
In Finance: Databases often drop funds that have liquidated (usually because they performed poorly). If we only look at the funds that "survived" until today, the average performance of the industry will look much better than it actually was. This makes it look like managers are more persistent than they truly are.
Quick Review:
• Backfill Bias: When a new fund is added to a database, it often includes its "good" historical returns but leaves out the "bad" early years.
• Selection Bias: Only managers with good records choose to report to databases.
Summary Checklist
Before you move on, make sure you can answer these questions:
• What is the difference between an independent and a dependent variable? (Factors vs. Returns)
• What does a t-stat of 2.5 tell you? (The factor is likely statistically significant)
• What are the three factors in the original Fama-French model? (Market, Size, Value)
• Why does survivorship bias make performance look better than it is? (It ignores the "dead" funds that failed)
Great job! You’ve just navigated the core concepts of Multivariate Empirical Methods. Take a break, grab a coffee, and let these concepts sink in. You're one step closer to your CAIA designation!