*4.2. Risk Measures*

This section examines the impact of *b* and *a* on important risk measures, in particular the value at risk (VaR) and the expected shortfall (ES). For clarity and calculation purposes, these measures are defined as follows:

$$\mu \quad = \quad P\left(X(T) \le -VaR\_{\mathfrak{A}}\right) \tag{10}$$

$$\begin{array}{rcl} ES &=& -\frac{1}{\mathfrak{a}} \int\_{0}^{\mathfrak{a}} VaR\_{\uparrow} d\gamma \end{array} \tag{11}$$

where *X*(*T*) = *<sup>ω</sup>*1(*<sup>X</sup>*1(*T*) − *<sup>X</sup>*1(0)) + *<sup>ω</sup>*2(*<sup>X</sup>*2(*T*) − *<sup>X</sup>*2(0)) is the profit and loss portfolio with equal weights (*<sup>w</sup>*1 = *w*2 = 1/2) (see DeMiguel et al. 2007 for a rationale and support of this simple strategy). We let *α* vary from 0.001 to 0.2 with a discretization size of 200.

We first study the impact of *b*1, ˜ *b*1 and ˜ *b*2 on VaR and ES for a fixed value of *α* = 0.01. Figure 4a,b illustrates a substantial increase in VaR, from 16 (when all *b* values are set to zero) to 19.5 (all *b* set to 0.008), which is a 21% increase (*α* = 0.01) due to the presence of *b*. In other words, an investor would have to set aside 21% more capital in the presence of 3/2 components. Similarly, ES increases from −21 in the presence of 3/2 components to −18.5 in their absence, which constitutes a 12% increase in the average VaR.

*J. Risk Financial Manag.* **2019**, *12*, 159

**Figure 4.** Impact of 3/2 components (*b*) on Risk measures, Scenario **A**.

Figure 5a,b also displays a substantial increase in VaR, from 17.5 (all *b* set to zero) to 22.5 (*b* set to 0.008), which represents a 28.6% increase (*α* = 0.01) due to *b*. This means 28.6% more capital is required in the presence of 3/2 components. Similarly, the ES increases from −27 with 3/2 components to −19 without them, representing a 29.6% increase in the average VaR.

**Figure 5.** Impact of 3/2 components (*b*) on Risk measures, Scenario **B**.

A similar analysis was performed with respect to the commonality *a*, in the presence of stochastic volatility (in the common factor) versus constant volatility; in other words, we assessed the impact of *a* per se and that of stochastic correlation produced by the 4/2 model. Figure 6a demonstrates an increase in VaR, from 16 to 18.5, a 15.5% increase (*α* = 0.01). Figure 6b shows that the VaR jumps from 17 to 23, a 35% growth due to the increase in *a*.

*J. Risk Financial Manag.* **2019**, *12*, 159

(**a**) Scenario **A**: Value at Risk vs. *α* for different commonality (a) values.

(**b**) Scenario **B**: Value at Risk vs. *α* for different commonality (a) values.

**Figure 6.** Impact of commonality (*a*) on Value at Risk under Scenario **A** and Scenario **B**.
