Quantitative Portfolio Management

Chapter 4Risk

In this business it's easy to confuse luck with brains.

Jim Simons

In the context of quantitative portfolio management, risk is understood as variability of portfolio pnl and ways to control it. Variance of the pnl is perhaps the simplest risk measure but not the only one used in the financial industry. But even the plain variance has certain complexities and may need further simplification (shrinkage) as described below.

4.1 Value at risk and expected shortfall

Value at Risk (VaR) provides a fairly detailed view of portfolio risk in the form of a function expressing the lower bound of worst expected daily losses vs probability of such losses. A useful companion risk measure is expected shortfall, or conditional value at risk (CVaR), expressing the expected loss conditioned on crossing the VaR threshold. VaR and CVaR are usually reported at a fixed probability such as 1% or 5%.

Let be the probability density of the daily portfolio pnl . If

(4.1)

is a measure of ...