Continuous Time Portfolio Selection under Conditional Capital at Risk

Figure 1

The graph of $\text{CCaR}$ as a function of the fraction of wealth $\pi$ invested in the stock, for $\pi \in [-\mathrm{5,5}]\text{,}$$T=16\text{,}$$S(0)=10\text{,}$$r=0.05\text{,}$$\sigma =0.2\text{,}$$b=0.15$. Observe that, in this case (with a long-time horizon and constantly high-performing stocks), the global minimum ${\pi }_{\varepsilon }$ lies in the interval $[\mathrm{0,1}]$ and satisfies $\text{CCaR}({\pi }_{\varepsilon })<0$.