Stylized Facts of Financial Returns › Skewness: Asymmetry in Returns
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While heavy tails occur in both directions, financial returns often show asymmetry: the left tail (losses) tends to be heavier than the right tail (gains). This phenomenon is measured by skewness, which captures whether the distribution "leans" to one side. Risk management must account for skewness because asymmetric distributions require different treatment of upside and downside risk.
For a random variable with mean and standard deviation , skewness is defined as the third standardized moment:
where:
- : skewness of random variable
- : expected value operator
- : random variable representing returns
- : mean of
- : standard deviation of
The third power serves a specific purpose in this formula. Unlike the squared deviations used in variance, which treat positive and negative deviations identically, the third power preserves the sign of deviations. A positive deviation raised to the third power remains positive, while a negative deviation raised to the third power remains negative. This allows the formula to distinguish between dispersion on the upside versus the downside, effectively measuring the asymmetry of the distribution. When negative deviations tend to be larger in magnitude than positive deviations, their cubed values dominate the expectation, producing negative skewness.
The normal distribution has skewness of zero (perfect symmetry). Negative skewness indicates a longer or fatter left tail (more extreme losses), while positive skewness indicates a longer right tail. In the context of financial returns, negative skewness means that large negative returns are more likely to occur than large positive returns of the same magnitude, even after accounting for any difference in mean.
A measure of asymmetry in a probability distribution. Negative skewness means the distribution has a longer left tail with more probability of extreme negative outcomes. Positive skewness means the distribution has a longer right tail.
Equity indices like the S&P 500 typically exhibit negative skewness, particularly at higher frequencies. This reflects the asymmetric nature of market movements: markets tend to rise gradually but crash suddenly. The 1987 Black Monday crash (-20.5% in one day) is more extreme than any single-day gain in S&P 500 history.
# Analyze the asymmetry in more detail
positive_returns = log_returns[log_returns > 0]
negative_returns = log_returns[log_returns < 0]
# Find the most extreme observations
n_extreme = 10
most_positive = log_returns.nlargest(n_extreme)
most_negative = log_returns.nsmallest(n_extreme)
# Calculate statistics for display
n_pos = len(positive_returns)
n_neg = len(negative_returns)
pct_pos = n_pos / len(log_returns)
pct_neg = n_neg / len(log_returns)
mean_pos = positive_returns.mean()
mean_neg = negative_returns.mean()
max_pos = most_positive.iloc[0]
max_neg = most_negative.iloc[0]Asymmetry Analysis ================================================== Positive return days: 3162 (50.5%) Negative return days: 3098 (49.5%) Mean positive return: 0.7256% Mean negative return: -0.7030% Largest positive return: 4.3931% Largest negative return: -3.4972% Skewness: 0.0423
The asymmetry has practical implications for risk management. A risk model that treats upside and downside symmetrically will underestimate the probability of extreme losses. This is why measures like Value at Risk (VaR) and Expected Shortfall that focus specifically on the loss tail are central.
