Risk Analytics

Conditional Value at Risk

Estimate the average portfolio loss among outcomes in the worst-case tail of the profit and loss distribution.

Overview

Conditional Value at Risk (CVaR) measures the average portfolio loss among outcomes that fall within the lower tail of the portfolio profit and loss distribution.

The lower tail is determined using the same confidence level and lower-tail P&L threshold used in the Value at Risk calculation.

While Value at Risk identifies a loss threshold, CVaR examines the outcomes beyond that threshold and estimates their average loss.

In the Quant Risk API, CVaR is calculated directly from the generated portfolio P&L outcomes.

P&L Distribution

CVaR is calculated from the distribution of portfolio profit and loss outcomes produced by the selected risk model.

Each outcome represents the portfolio profit or loss associated with one simulated or historical scenario.

\mathrm{PnL} = \left( \mathrm{PnL}^{(1)}, \mathrm{PnL}^{(2)}, \ldots, \mathrm{PnL}^{(N)} \right)

$\mathrm{PnL}^{(j)}$ represents the profit or loss associated with scenario $j$ .
$N$ represents the total number of available portfolio outcomes.

The complete P&L distribution is used to identify the lower tail associated with the requested confidence level.

Tail Probability

The API first converts the requested confidence level into the corresponding lower-tail probability.

p_{\mathrm{tail}} = 1-c

$p_{\mathrm{tail}}$ represents the lower-tail probability.
$c$ represents the requested confidence level.

For example, at a 95% confidence level, the lower-tail probability is 5%.

VaR Threshold

The API calculates the lower-tail quantile of the portfolio P&L distribution.

q_{\mathrm{PnL}} = Q_{1-c} \left( \mathrm{PnL} \right)

$q_{\mathrm{PnL}}$ represents the lower-tail P&L quantile.
$Q_{1-c}$ represents the quantile at probability $1-c$ .

This quantile is used as the threshold for selecting outcomes in the lower tail of the distribution.

Tail Outcomes

The API selects all portfolio outcomes with P&L values at or below the lower-tail P&L quantile.

\mathrm{PnL}^{(j)} \leq q_{\mathrm{PnL}}

These outcomes represent the scenarios contained in the selected lower tail of the portfolio P&L distribution.

The API includes outcomes equal to the threshold. This is equivalent to selecting P&L values using the condition $\mathrm{PnL} \leq q_{\mathrm{PnL}}$ .

Average Tail P&L

The API calculates the arithmetic mean of all P&L outcomes in the selected lower tail.

\overline{\mathrm{PnL}}_{\mathrm{tail}} = \frac{1} {N_{\mathrm{tail}}} \sum_{j \in \mathrm{tail}} \mathrm{PnL}^{(j)}

$\overline{\mathrm{PnL}}_{\mathrm{tail}}$ represents the average P&L of the selected tail outcomes.
$N_{\mathrm{tail}}$ represents the number of outcomes contained in the selected tail.
$\mathrm{PnL}^{(j)}$ represents the P&L outcome associated with scenario $j$ .

The average tail P&L is typically negative when the selected tail contains portfolio losses.

Conditional Value at Risk

The average tail P&L is converted into a non-negative loss value.

\operatorname{CVaR} = \max \left( 0, - \overline{\mathrm{PnL}}_{\mathrm{tail}} \right)

The resulting CVaR value represents the average loss across portfolio outcomes contained in the selected lower tail.

If the average tail P&L is negative, the API converts it into a positive loss magnitude.

If the average tail P&L is non-negative, the returned CVaR value is zero.

Raw Tail P&L Value

The API also returns the raw average tail P&L value before it is converted into a non-negative loss magnitude.

\operatorname{CVaR}_{\mathrm{raw}} = \overline{\mathrm{PnL}}_{\mathrm{tail}}

This value allows clients to inspect the signed average outcome in the lower tail directly.

Method Summary

Conditional Value at Risk in the Quant Risk API follows these steps:

Start with the portfolio P&L distribution generated by the selected risk model.
Calculate the lower-tail probability $1-c$ from the requested confidence level.
Calculate the lower-tail P&L quantile.
Select all outcomes with P&L values less than or equal to the lower-tail quantile.
Calculate the arithmetic mean of the selected tail outcomes.
Convert the signed average tail P&L into a non-negative loss value.

CVaR provides additional information beyond VaR by measuring the average severity of portfolio outcomes that fall within the selected lower tail of the P&L distribution.