Value at Risk for a dollar neutral relative trades

Question

So assuming I am entering a trade to long A stock at 10 and short B stock at 20. And assume we want the position to be dollar neutral, so we are going to long 2 shares of A and short 1 share of B. Expected return for A is 0.3% and A's daily vol is 2%; expected return for B is 0.2% and B's daily vol is 2% too. Correlation between A and B is 95%. I got a portfolio daily std of 0.32%. The next step I guess I am suppose multiply the daily portfolio std by exposure and Z value (1.65 assuming normal), but is the exposure in this case 0? So does that mean my portfolio VaR is 0 too? Which seems wrong to me. Thank you!

Answer 1

But your exposure is not zero! You have an exposure to the price of stock A equal to 20, and another exposure to the price of stock B equal to -20. Since they are different stocks, albeit highly correlated, netting their exposures is too optimistic.

The expected returns aren't used in VaR.

Let $\Delta=\left(\matrix{\delta_A & \delta_B}\right)$ denote the vector of your 2 exposures; $\sigma_A$ and $\sigma_B$ denote the volatility of their returns (total returns, including dividends), $\rho$ denote the correlation of A returns and B returns, $c=\rho\sigma_A\sigma_B$ denote their covariance, $C=\left(\matrix{\sigma_{A}^2 & c\\ c & \sigma_{B}^2}\right)$ their variance-covariance matrix, then the value at risk is $\mathrm{sqrt}(\mathrm{mmult}(\mathrm{mmult}(\Delta,C),\mathrm{transpose}(\Delta)))\times\mathrm{sqrt}(\frac{d\text{ days horizon, probably 1}}{\text{trading days per year, for example 250}})\times\mathrm{normsinv}(\text{confidence level})$

In addition to VaR, you should decide what is the most money you are willing to lose under adverse scenarios, e.g. the correlation breaks, and A goes down while B goes up; and not exceed that.