Estimating VaR and ES using Hybrid Historical Simulation
Question:
Historical simulation can be converted from a non-parametric approach to parametric approach by applying weights that decline exponentially with time (age), but still add to one. Let us assume that daily returns of a portfolio have been measured over the past 100 trading days. Alongside, EWMA based age weighting has been done for these 100 returns using a $\lambda=0.94$. These returns (along with their respective weights) have been sorted from smallest to largest, and a subset of these is shown below (the bottom 12 returns).
If the portfolio is valued at \$250 million, what is the 95% confidence daily Value-at-Risk (VaR) and Expected Shortfall (ES) of this portfolio if the hybrid historical simulation method were to be used with weights given as per the table above?
| A. | VaR: \$8.38 million, ES: \$9.67 million |
| B. | VaR: \$8.65 million, ES: \$9.07 million |
| C. | VaR: \$8.38 million, ES: \$9.07 million |
| D. | VaR: \$8.65 million, ES: \$9.67 million |
Solution: C
The cumulative weight for the return of -4.08% is 1.51%, and for the return of -3.35% is 5.65%. The threshold return at 95% confidence will be somewhere between the return of -4.08% and -3.35% (i.e. a return at which the cumulative weight comes to exactly 5.00%). You can either calculate this return using linear interpolation, or alternatively, as suggested in the GARP book, just pick the threshold return to be one at which the cumulative weight first crosses (i.e. becomes greater than) the level of significance. In this case, it will be -3.35%. Following this convention, the VaR is therefore: $$ \begin{align} \mbox{VaR} &= -(-3.35\% \times \$250 \mbox{mn}) \\ &= \$8.375 \mbox{mn} \end{align} $$ The Expected Shortfall (%) can be computed as the probability weighted average of losses in the tail. The tail of the returns distribution starts at -3.35%. $$ \begin{align} \mbox{ES(%)} &= \frac{(-5.35\%)(0.0023) + (-4.08\%)(0.0128) + (-3.35\%)(0.0349)}{(0.0023)+(0.0128)+(0.0349)} \\ &= 3.629\% \end{align} $$ where, we have applied a residual weight of $0.05 – 0.0023 – 0.0128 = 0.0349$ to the return observation of -3.35%. The dollar ES at 95% confidence is therefore $0.03629 \times \$250\mbox{mn} = 9.07\mbox{mn}$.