# Value at Risk of a Portfolio

I am currently practicing for my Risk Management exam in July but my lecturer is of no help and my colleagues and I have no idea on how to proceed with this question. The past exam papers have similar questions so I would appreciate any advice on how to proceed with such a question. In so doing, I would be able to follow similar logic in attempting questions like these. Thanks in advance

Consider the following historical information on a portfolio currently valued at USD 100 million:

                 Log Returns

02/01/2004       -0.20%
05/01/2004       -0.15%
06/01/2004        0.14%
07/01/2004        0.30%
08/01/2004       -1.43%
09/01/2004       -0.79%
12/01/2004        0.55%
13/01/2004       -0.53%
14/01/2004        0.80%
15/01/2004        0.13%


Compute a one-day, 20% volatility-adjusted historical VaR and the Expected Shortfall of the portfolio. The volatility is to be estimated using an exponential moving volatility model with 𝜆 = 0.96. As at 01/01/2004 the variance forecast is 0.0000053%.

• What specific problems are you having? Do you know how to calculate volatility-adjusted historical VaR? Do you know how to update a EWMA volatility model? – Alex C Jun 28 '18 at 23:26
• A volatility adjusted VaR is computed, not from the Log Returns $r_t$ given above but from vol adjusted log returns $\tilde{r}_t=r_t\frac{\sigma_T}{\sigma_t}$ where $T$ is the most recent observation (15/01/2004) and $\sigma_t$ is the estimated volatility for day $t$. – Alex C Jun 29 '18 at 15:38
• Where do the daily volatility estimates $\sigma_t$ come from? They come from the daily updating equation $\sigma_t^2=\lambda \sigma^2_{t-1}+(1-\lambda)r_t^2$. For the intial value use $\sigma^2=$0.0000053%. For $\lambda$ use 0.96. Of course $\sigma_t=\sqrt{\sigma^2_t}$ – Alex C Jun 29 '18 at 15:41