Explain the ways in which the distributions of asset returns typically depart from normality.
The idea
Daily returns on equities, currencies and most other assets depart from a normal distribution in several ways that matter for risk. First, they have fat tails: excess kurtosis above zero (kurtosis above 3), so moves of three, four or more standard deviations occur far more often than the normal predicts, and the centre is also more peaked. Second, many are skewed, equity indices usually negatively, so large losses are more likely than equally large gains. Third, volatility clusters: calm and turbulent spells persist, so the size of returns, measured by u², is autocorrelated even when the returns themselves are not. Fourth, dependence changes: correlations rise in sell-offs, and joint extreme losses are more common than a bivariate normal with the same correlation implies. A normal with matching mean and variance therefore understates extreme risk at high confidence levels. Do not confuse kurtosis, which measures tail weight, with skewness, which measures asymmetry.
A practice question
Which statement best describes the excess kurtosis typically found in daily returns on equity indices?
- It is negative: the tails are thinner than normal, so extreme moves are rarer than a normal distribution predicts.Why not: Reverses the sign; thin tails (platykurtic) are the opposite of what equity returns show.
- It is zero: a kurtosis of 3 means equity returns are effectively normal once volatility is measured.Why not: Treats the normal benchmark of 3 as the empirical value, when equity returns have kurtosis well above 3.
- It is positive: the tails are fatter and the peak higher than for a normal distribution with the same variance. - correct
- It is positive, which means returns are skewed to the right with larger gains than losses.Why not: Confuses kurtosis (tail weight) with skewness (asymmetry).
Excess kurtosis is kurtosis minus 3, the normal value. Equity returns have positive excess kurtosis (leptokurtosis): more mass in the centre and in the tails, so extreme moves are more frequent than a normal predicts. Skewness is a separate feature.
A harder one
A risk manager studies 2,520 daily returns (about ten years) on an equity index and finds 11 days on which the loss exceeded three standard deviations. Using the standard normal tail probability for one side of the distribution (0.00135 beyond three standard deviations), which expected number of such loss days under normality, and which reading, is correct?
- About 6.8 days; the observed count is only modestly above normality, so tails are close to normal.Why not: Uses a two-sided tail probability even though only losses (one tail) are counted, doubling the expected number.
- About 3.4 days; the observed count is about three times what normality implies, evidence of fat tails. - correct
- About 25.2 days; the observed count is well below normality, so the tails are thin.Why not: Treats a three-standard-deviation loss as a 1% event (the 99% quantile is 2.33 standard deviations).
- About 0.3 days; the observed count is more than thirty times normality.Why not: Applies the tail probability to only 252 days (one year) instead of the ten years of data.
Expected losses beyond three sigma = 2,520 x 0.00135 = 3.40. Observed 11 is about 3.2 times that, which signals excess kurtosis; a two-sided count would double the expectation.
The hardest tier
Over 20 years (5,040 trading days) a risk manager finds 7 daily returns that were more than four standard deviations from the mean, in either direction. Under normality, P(|Z| > 4) = 0.0000633. Which expected count and ratio of observed to expected is correct?
- About 0.16 such days; the observed count is about 44 times the normal expectation.Why not: Halves the probability by using one tail only, although the question counts moves in either direction.
- About 5.04 such days; the observed count is only about 1.4 times the normal expectation.Why not: Treats a four-sigma move as a 0.1% event per day, which is roughly the three-sigma probability.
- About 3.19 such days; the observed count is only about 2.2 times the normal expectation.Why not: Decimal slip: uses 0.000633 instead of 0.0000633, overstating the expected count ten-fold.
- About 0.32 such days; the observed count is about 22 times the normal expectation. - correct
Expected = 5,040 x 0.0000633 = 0.319 days. Observing 7 is roughly 22 times that, a very strong sign of fat tails; under normality four-sigma days are nearly absent.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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