3.2 Statistics, Distributions, and Risk Measures

Statistics, Distributions, and Risk Measures

Statistics turns raw observations into decision-useful information. In finance, the observations are often returns, yields, spreads, or valuation multiples. A good first pass asks four questions: What is typical? How much does it vary? Is the distribution symmetric? How extreme are the tails?

Measures of central tendency describe the center of a data set. The arithmetic mean is the sum divided by the number of observations. The median is the middle observation after sorting. The mode is the most frequent value. When data are skewed by extreme observations, the median can describe a typical outcome better than the mean.

Dispersion measures the spread of observations. Range is maximum minus minimum, but it uses only two data points. Mean absolute deviation averages absolute deviations from the mean. Variance averages squared deviations, and standard deviation is the square root of variance. Standard deviation is easier to interpret because it is in return units.

Sample variance divides by n - 1, while population variance divides by N. The n - 1 denominator is a degrees-of-freedom adjustment because the sample mean is estimated from the same data. On exam questions, read whether the data set is described as a sample or a population.

The coefficient of variation is CV = standard deviation / mean. It measures risk per unit of expected return. A lower CV is preferred when comparing investments with positive expected returns. If expected return is zero or negative, CV can be misleading.

Shape matters because many investment returns are asymmetric. Positive skew has a long right tail and occasional large gains. Negative skew has a long left tail and occasional large losses. Excess kurtosis measures tail thickness relative to the normal distribution. Leptokurtic distributions have fatter tails and more extreme outcomes.

The normal distribution is symmetric, fully described by its mean and variance, and has skewness of zero. A z-score standardizes an observation: z = (x - mean) / standard deviation. If a return is 2 standard deviations below the mean, its z-score is -2. This allows use of standard normal probabilities.

Downside risk measures focus on harmful outcomes. Semivariance uses only returns below the mean or a target. Target semideviation measures dispersion below a required return. Shortfall risk is the probability that return falls below a threshold. Value at risk estimates a loss threshold for a stated probability and horizon.

Risk measures should match the decision. Standard deviation works when upside and downside volatility are both relevant. A pension plan with a required return may care more about shortfall probability. An option strategy with rare large losses may need skewness and kurtosis, not just standard deviation.

Level I questions often mix calculation with interpretation. If the mean is greater than the median, think positive skew. If the distribution has fat tails, normal probabilities understate extreme events. If two funds have the same mean, the fund with lower standard deviation has less total volatility.