Normal Distribution

Every normal distribution is a rescaled copy of one reference curve, the standard normal, which has a mean of zero and a standard deviation of one. Converting an observation to that scale requires a single operation: subtract the distribution's mean from the observation, then divide the difference by the standard deviation. The resulting z-value states how many standard deviations the observation sits above or below the mean, and its sign carries the direction.

That single transformation is what makes the family usable, because a cumulative probability table for the standard normal answers questions about any normal variable. Roughly two-thirds of outcomes fall within one standard deviation of the mean, about 95 percent within two, and close to 99 percent within three. For interval construction the exam expects the more precise multipliers: approximately 1.65 for a 90 percent interval, 1.96 for 95 percent and 2.58 for 99 percent.

The property that gives the normal distribution its grip on portfolio theory is closure under linear combination. If individual asset returns are jointly normal, any weighted sum of them, which is exactly what a portfolio return is, is also normal. The portfolio therefore needs no new distributional assumption: its expected return is the weighted average of component means, and its variance follows from the weights, variances and covariances. Mean-variance optimisation is coherent precisely because those two moments describe the whole outcome distribution.

Specifying the joint behaviour of several normal variables demands more than the individual pieces. For n assets the multivariate normal is defined by n means, n variances and n(n−1)/2 pairwise correlations. Correlations are not incidental detail; they are part of the distribution's definition, and marginal normality of each asset alone does not establish joint normality.

Empirical return series depart from normality in patterned ways rather than randomly. Daily equity returns exhibit excess kurtosis, meaning more probability mass in the tails and around the centre than a bell curve allows, and typically negative skewness, so large adverse moves outnumber comparably large favourable ones. Volatility also clusters, which violates the assumption that each period is drawn independently from a single fixed distribution. The practical consequence is that risk measures calibrated on normal quantiles understate the frequency and severity of losses.

A structural problem compounds this. A normal variable takes any value on the real line, yet a long position in a limited-liability security cannot lose more than its full value. This is why prices are commonly modelled as lognormal, bounded below by zero, while continuously compounded returns are modelled as normal.

Candidates should be able to move fluently in both directions: from an observation to a probability, and from a target probability to a critical value and back to the original units. Typical tasks include computing the chance a return falls below a threshold, building a confidence interval around a mean, and identifying the return level that a given percentile implies.

The same machinery supports downside risk analysis. Shortfall risk is the probability that a portfolio return falls below a stated minimum acceptable level. Roy's safety-first criterion ranks portfolios by the safety-first ratio, computed as the expected return minus the threshold return, divided by the standard deviation of return; the highest ratio minimises shortfall probability under normality. When the threshold equals the risk-free rate the ratio reduces to the Sharpe ratio, and the associated shortfall probability is read from the standard normal table.