Financial Risk Management

CHAPTER 8The Gaussian Framework

THE CORE EQUATION

This is the most convenient solution to compute both the Value‐at‐Risk (VaR) and Expected Shortfall (ES). Under this framework, we compute a Gaussian VaR and ES, grounded on a wrong but handy assumption about the portfolio Profit and Loss (P&L). We assume that it follows a Gaussian (or normal) distribution. Actually, this assumption is neither totally wrong nor totally right. In other words, each time we compute a Gaussian VaR and ES, we know we are mistaken. We can thus wonder why the Gaussian framework is still used by portfolio or risk managers, specifically to assess the risk of equities portfolios.

The main reason is that the Gaussian framework is mathematically tractable, especially when the portfolio P&L is a linear function of the asset returns. We can derive a closed‐form expression of the Gaussian VaR and ES that is easy to manipulate, expressed as the product of a (theoretical) quantile and the portfolio volatility (if we omit the term of expected returns). This explains why the Gaussian VaR is an analytical risk measure.¹

The Gaussian VaR and ES are two parametric risk measures because they are defined from the two parameters of a Gaussian distribution: the mean and the standard deviation. The former refers to the average asset returns and the latter refers to the portfolio volatility. The Gaussian VaR and ES are thus two standard‐deviation risk measures. More generally speaking, each time a risk measure is grounded ...