Distributional Properties of Asset Returns
When analyzing a set of returns (e.g., daily returns of Nifty 50), we look at four key "Moments" of the distribution.
The Four Moments
1. Mean (First Moment) (Mu)
- What it measures: The central tendency or average return.
- In Finance: Usually very close to zero for daily returns.
2. Variance / Standard Deviation (Second Moment) (Sigma^2)
- What it measures: The spread or dispersion of data.
- In Finance: This represents Risk or Volatility. A higher standard deviation means higher risk.
3. Skewness (Third Moment)
- What it measures: Asymmetry. Ideally, a Normal distribution has Skewness = 0.
- Negative Skewness: The tail on the left side is longer.
- Meaning: Small gains are common, but there is a risk of frequent large losses (crashes). Stock markets typically have Negative Skewness.
- Positive Skewness: The tail on the right is longer (Lottery ticket profile). Small losses are common, but chance of huge gain.
4. Kurtosis (Fourth Moment)
- What it measures: The "fatness" of the tails and the "peakedness" of the center.
- Normal Distribution: Excess Kurtosis = 0 (Kurtosis = 3).
- Leptokurtic (High Kurtosis): Positive Excess Kurtosis (> 0).
- Meaning: High probability of extreme events (Fat Tails). Financial returns are almost always Leptokurtic.
Note
The "Normal" Lie: Many basic financial models (like Black-Scholes) assume returns are Normal (Skewness=0, Excess Kurtosis=0). In reality, returns are Negatively Skewed and Leptokurtic (Fat Tailed). Using Normal assumptions can lead to underestimating risk.
Jarque-Bera Test
This is a statistical test used to check if a dataset matches a Normal Distribution. It checks if Skewness and Kurtosis are zero.
- If JB Test P-value is low (< 0.05), we reject Normality. (For stock returns, we almost always reject it).
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