# Value at Risk for a dollar neutral relative trades

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!

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.

• So assuming std for both are 2%, correlation = 95%, covariance = 0.00038, exposure A is 20 and B is -20, assuming 95% confidence level with standard normal distribution (so multiply by 1.65). I am getting a number of 0.0017 for VaR. Assuming this is in dollar, so the max loss will be \$0.0017 over 1 day at a 95% confidence level? And sometimes I see people just multiply by sqrt of # of days instead of dividing by 252 and it makes quite a difference.
– DLW
Oct 13, 2022 at 13:51
• You need to divide by sqrt(number of trading days in a year) if and only if the volatilities are annualized: so if they're alteady not annualized, then don't divide. Oct 13, 2022 at 15:35
• Right, ok thanks!
– DLW
Oct 13, 2022 at 15:37