Investment returns are not defined only by their average and volatility. The shape of the distribution matters too - especially when we are trying to understand extreme gains, extreme losses and financial risk.
Imagine recording the daily return of an investment for several years. Some days might produce a small gain, some a small loss, and occasionally there may be a very large movement.
If we collect all of those returns and count how often different outcomes occur, we can create a return distribution.
The distribution tells us much more than the average return. It tells us how frequently different outcomes occur and how much probability sits in the tails.
A dice roll is a simple example of a probability distribution. A fair six-sided die has six possible outcomes, each with probability 1/6.
Financial returns are different. Instead of six discrete outcomes, returns can take almost any value within a range.
The Normal distribution is often used as a starting point for modelling financial returns.
It is completely described by two parameters:
The Normal distribution is symmetrical. Returns equally far above and below the mean have the same probability.
Real financial markets contain events that are difficult to capture with a simple Normal distribution.
Markets can experience sudden crashes, unusually large rallies, periods of calm and periods of extreme volatility.
As a result, observed return distributions often have:
One of the most important characteristics of financial returns is fat tails.
A Normal distribution predicts that very large movements are extremely rare. Financial markets have historically produced more extreme movements than this simple model would suggest.
This matters enormously for risk management. A model that underestimates the probability of extreme losses can make an investment appear safer than it really is.
Skewness measures asymmetry in a distribution.
A perfectly symmetrical distribution has skewness close to zero. A negatively skewed distribution has a longer or heavier left tail, while a positively skewed distribution has a longer or heavier right tail.
Kurtosis describes how much probability is concentrated in the centre and tails of a distribution relative to a Normal distribution.
In finance, we are particularly interested in excess kurtosis.
A Normal distribution has excess kurtosis of zero.
Positive excess kurtosis is associated with heavier tails and a greater likelihood of extreme observations than the Normal distribution.
Excess kurtosis is approximately zero.
More probability sits in the tails.
Now we can combine the ideas of volatility, tail events and skewness into one interactive model.
The difference becomes particularly important when we look at the probability of extreme outcomes.
The following comparison uses the same approximate volatility but different distribution shapes.
Probability of a simulated daily return below −3%.
Probability of a simulated daily return below −3%.
In a Monte Carlo investment model, we repeatedly draw random returns from an assumed distribution.
If that distribution is Normal, the model assumes extreme returns are very rare. If the real market has fat tails and negative skewness, the model may underestimate the probability of severe losses.
This means that choosing a return distribution is not a minor technical detail. It can change the results of a long-term simulation.
Our earlier Monte Carlo model used a Normal distribution for returns. That is a useful starting point, but we can do better.
The next step is to examine distributions specifically designed to capture the behaviour of financial returns.
The Normal distribution gives us a useful mathematical foundation. But we have now seen why it can be too simple for financial returns.
The Student-t distribution provides heavier tails while remaining mathematically tractable.
The Skew-t distribution goes one step further by allowing the distribution to be asymmetric as well as heavy-tailed.
We will build the next page to explore these distributions interactively and see exactly what changes as we move away from the Normal assumption.