## The Challenge of Probability Distribution in Risk Quantification

Lots of risk quantification approaches and models assume a normal distribution. However, in practice this may not hold, certainly not in particular markets. In the equity markets, in reality, a stock index falling 15% or more on one day is rare, but at least it has happened. However, a stock index spiking 15% on one day has not occurred, or at least not as often. Individual stock and commodity prices, however, face the opposite. Upward jumps in the commodity markets are more likely and more significant than a normal distribution would justify. This implies there are more extremes on the upside. This leads the actual right tail of the probability distribution curve to show more occurrences than stated in theory or is being justified by the model. This means that the right tail of the curve is thicker. This is why such a tail is called ‘fat’. Stock indices typically face a fat tail on the left, while individual stock prices have a fat tail on the right. This can be explained as follows: indices typically fall due to mass fear, panic or herd behaviour, while spikes in individual share prices are often a result of a merger or an acquisition.

### Understanding Skewness in Probability Distributions

Commodity markets, on the contrary, are typically facing a fat right tail. Probability distribution curves are usually skewed to the right as a result of the merit order and its shape. The supply curve of a commodity usually is sloped parabolic, meaning its slope increases further to the right. Hence, incremental price changes are more significant in case of scarcity than normally. Consider, for instance, the closing of the main gas pipe from Russia to Ukraine, causing a gas price spike in central Europe. Next, an upward price jump in the electricity market can be expected if a transmission system operator for electricity announces ‘Code Red’, reflecting cooling water being too hot, forcing some plants to shut down. In addition, an invasion of the US in Iran will likely cause a disruption in oil supply, which will cause prices to spike. Furthermore, the flooding of coal mines could make production impossible therewith creating a shortage of coal supply.

#### Limitations and Risks in Market Risk Models

Market risk models quantify the exposure of an individual position or the entire portfolio, considering probability distributions. This is why skew is relevant. Next, risk quantification models consider price correlation. It should be noted, however, that the correlation coefficient is based on a model itself and because the correlation coefficient, in its turn, is applied is a risk quantification model, the value at risk model itself is also subject to quite some model risk.

#### Challenges in Correlation Calculation and Model Reliability

Basically, the calculation of correlation is based on two assumptions, namely normality and linearity. The first assumption (normality) is breached during every specific occasion. In times of crises, normal market circumstances do not apply and, as a result, correlations do not hold; meaning, historical coefficients cannot be applied effectively. In essence, in case of stress, correlation coefficients usually tend toward zero. This means that when risk quantification is needed most, calculations seem unreliable. Thus, at times of stress, the calculated value at risk is very inaccurate as indicator for actual market risk. In addition, it must be noted that correlation coefficients are calculated based on historical data sets and, thus, they are not a perfect estimation for the future anyhow. Ideally, future correlation coefficients should be considered. However, as no one practically knows what the future will be like, historical data are applied instead to calculate historical correlation coefficients, which are used as best estimate for the future.