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Risk Management Tools

Definitions

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR, other notations include Expected Shortfall, Expected Tail Loss, Tail VaR) are indispensable tools of portfolio risk management.

VaR and CVaR measures become increasingly useful when the distribution of portfolio logarithmic returns substantially deviates from normal. In this case, volatility as a common risk measure appears inappropriate.

Portfolio VaR is defined as a maximum portfolio loss (measured in % of initial wealth) over a given time interval at a given level of statistical confidence.
Definition. is implicitly defined by the following expression (for simplicity we assume that portfolio returns have continuous distribution):

where

is portfolio excess simple return over

period.
To put it more formally,

corresponds to the

-quantile of the distribution of portfolio excess simple return over

period.

Portfolio CVaR is conditional expectation of losses beyond VaR.
Definition.

is an expected value (with opposite sign) of portfolio excess simple return

under the condition

In other words,

is an average value of

of highest losses. Common values for

are 0.9, 0.95, 0.975 and 0.99. The time horizon

corresponds to a period required for complete liquidation of all portfolio positions.

Figure 1. Distribution of logarithmic portfolio returns over the period

Important! Although CVaR isn’t so widespread a measure of risk as VaR, in contrast to VaR it possesses important property of subadditivity. Subadditivity of CVaR means that CVaR of simultaneously holding two
portfolios is less or equal to the sum of CVaRs for individual portfolios, considered separately. That’s why CVaR is more suitable quantity to be included in portfolio optimization (as constraint or part of an
optimization criterion) than VaR is. Essential shortfalls of VaR as a measure of risk due to a lack of subadditivity are discussed in detail in [Artzner, Delbaen, Eber, Heath; 1999].