Determine which statistical distribution gives the closest fit for each of equity correlations, bond correlations and default correlations.
The idea
Correlations lie between minus one and plus one. A normal distribution is symmetric and puts probability on impossible values beyond those bounds, so it fits them poorly. Researchers compare candidate distributions with goodness-of-fit tests such as Kolmogorov-Smirnov, Anderson-Darling and chi-squared. The test statistic measures the distance between the sample and the fitted distribution. The smallest statistic, or equivalently the largest p-value, therefore marks the closest fit. Reading it the other way round is a frequent slip. The empirical findings to remember are that equity correlations are best fitted by the Johnson SB distribution, a flexible four-parameter bounded distribution. Bond correlations are best fitted by the generalised extreme value distribution, and default correlations by the Beta distribution, which lives on a bounded interval. Exam point: know the three pairings, and that bounded or flexible shapes beat the normal.
A practice question
Which distribution gives the closest fit to equity correlations?
- The normal distributionWhy not: Normal is symmetric and unbounded, so it assigns probability to values outside -1 to +1 and fits correlations poorly.
- The lognormal distributionWhy not: Lognormal is positive and unbounded above, so it cannot represent negative correlations or the +1 cap.
- The exponential distributionWhy not: Exponential is monotonically decreasing on non-negative values and cannot match the bounded, humped shape of correlations.
- The Johnson SB distribution - correct
Equity correlations are best fitted by the Johnson SB distribution, a flexible four-parameter distribution on a bounded interval.
A harder one
Which assignment of best-fitting distributions to equity, bond and default correlations, respectively, is consistent with the empirical evidence?
- Johnson SB, generalised extreme value, Beta - correct
- Beta, Johnson SB, generalised extreme valueWhy not: Rotates the assignments; Johnson SB, not Beta, is the best fit for equity correlations.
- Generalised extreme value, Beta, Johnson SBWhy not: Reverses the order of the best fits, giving equity the extreme-value distribution.
- Normal, lognormal, BetaWhy not: Uses unbounded normal and lognormal fits for equity and bond correlations, which are rejected in favour of bounded or flexible shapes.
Evidence points to Johnson SB for equity, generalised extreme value for bond and Beta for default correlations. Normal fits are poor throughout.
The hardest tier
A normal distribution is fitted to a sample of equity correlations with mean 0.45 and standard deviation 0.30. The probability that the fitted normal assigns to values above +1 is closest to which of the following, and what does it imply?
- About 3.3%: the normal gives weight to impossible values, so a bounded distribution such as Johnson SB is preferable. - correct
- About 6.7%: counting both tails shows the normal is adequate for correlations.Why not: Doubles the upper-tail probability as if two tails exceeded +1; the lower tail below -1 is negligible here.
- About 0.0%: the normal assigns almost no weight to impossible values, so it is adequate.Why not: Divides by the variance 0.09 instead of the standard deviation 0.30 when standardising, shrinking the tail probability.
- About 96.7%: most of the normal's weight lies above +1, so the sample is unusual.Why not: Reports the cumulative probability below +1 instead of the upper tail above it.
z = (1 - 0.45) / 0.30 = 1.833, so P(correlation > 1) = 1 - N(1.833) = 3.3%. Mass on impossible values shows the normal is a poor fit for bounded data.
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