Use the normal distribution to work out probabilities

Returns are normally distributed with a mean of 8% and a standard deviation of 5%. What is the z-score of a 18% return?

1. 2.0 — correct
2. 0.6Why not: 8 ÷ 18, or some other division of the two given figures. A z-score subtracts the mean first: (18 − 8) ÷ 5.
3. 18.0Why not: The raw return, unconverted. A z-score is a distance in standard deviations, so it has to be centred on the mean and divided by 5.

z = (18 − 8) ÷ 5 = 2.0. The return sits two standard deviations above the mean, which under a normal distribution happens about 2.5% of the time on the upside.

Why does assuming normality typically understate the probability of a large loss in asset returns?

1. Because returns have a higher mean than the normal assumesWhy not: The mean is whatever is fitted to the data. The shortfall is in the tails, not the centre.
2. Because the normal distribution has no defined tailsWhy not: It has tails extending infinitely in both directions. They are simply too thin — they fall away faster than real returns do.
3. Because real return distributions are fat-tailed, so extremes occur more often than the normal implies — correct

Empirical returns show excess kurtosis, so three- and four-standard-deviation moves arrive far more often than a normal distribution predicts. Risk models built on normality are optimistic precisely in the tail.