Risk Analytics

Value at Risk (VaR)

Estimate a portfolio loss threshold from the distribution of possible portfolio profit and loss outcomes.

Overview

Value at Risk (VaR) estimates a portfolio loss threshold over a selected time horizon and confidence level.

In the Quant Risk API, VaR is calculated from the distribution of portfolio profit and loss outcomes produced by the selected risk model.

Depending on the selected method, the portfolio P&L distribution may be generated using Monte Carlo simulation, parametric return models, or historical return scenarios.

VaR does not generate market scenarios itself. Instead, it measures the lower-tail risk of the portfolio outcomes produced by the selected risk model.

The API calculates VaR using a lower-tail quantile of the portfolio P&L distribution and returns the result as a non-negative loss value.

Profit and Loss Distribution

VaR is calculated from a collection of possible portfolio profit and loss outcomes.

For each scenario or simulated outcome, portfolio profit and loss is calculated relative to the initial portfolio value.

\operatorname{PnL}^{(j)} = V_{\text{terminal}}^{(j)} - V_0

$\operatorname{PnL}^{(j)}$ represents the portfolio profit or loss for outcome $j$ .
$V_{\text{terminal}}^{(j)}$ represents the portfolio value at the end of outcome $j$ .
$V_0$ represents the initial portfolio value.

The complete set of portfolio outcomes forms the P&L distribution used to measure portfolio risk.

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

Negative P&L values represent portfolio losses, while positive P&L values represent portfolio gains.

VaR focuses on the lower tail of this distribution, where the most unfavorable portfolio outcomes occur.

Confidence Level

VaR is calculated at a selected confidence level.

Let the confidence level be represented by $c$ .

0 < c < 1

The corresponding lower-tail probability is calculated as:

\alpha = 1-c

$c$ represents the requested confidence level.
$\alpha$ represents the lower-tail probability.

For example, for a confidence level of 95%:

c = 0.95

\alpha = 1-0.95 = 0.05

The API therefore evaluates the lower 5% region of the portfolio P&L distribution.

Lower-Tail Quantile

The API identifies the lower-tail quantile of the portfolio P&L distribution at the selected tail probability.

q_{\mathrm{PnL}} = Q_{\alpha} \left( \mathrm{PnL} \right)

Because the lower-tail probability is defined as $\alpha=1-c$ , the calculation can also be written as:

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

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

At a 95% confidence level, the API calculates the 5th percentile of the portfolio P&L distribution.

The resulting quantile may be negative, positive, or zero depending on the portfolio outcomes produced by the selected risk model.

Calculate Value at Risk

The API converts the lower-tail P&L quantile into a non-negative loss value.

\operatorname{VaR} = \max \left( 0, -q_{\mathrm{PnL}} \right)

$\operatorname{VaR}$ represents the estimated portfolio loss threshold.
$q_{\mathrm{PnL}}$ represents the lower-tail P&L quantile.

If the lower-tail P&L quantile represents a portfolio loss, its sign is inverted so that VaR is returned as a positive loss amount.

Negative P&L Quantile

For example, suppose the lower-tail P&L quantile is:

q_{\mathrm{PnL}} = -10{,}000

The resulting VaR is:

\operatorname{VaR} = \max \left( 0, -(-10{,}000) \right) = 10{,}000

Non-Negative VaR

In some cases, the lower-tail P&L quantile may be positive.

For example:

q_{\mathrm{PnL}} = 2{,}000

The API returns:

\operatorname{VaR} = \max \left( 0, -2{,}000 \right) = 0

This ensures that VaR is always represented as a non-negative loss value.

Interpreting VaR

VaR represents an estimated loss threshold associated with a selected confidence level and time horizon.

For example, suppose a portfolio has:

Portfolio value: $100,000
Time horizon: 1 day
Confidence level: 95%
Value at Risk: $8,000

The VaR result indicates that, under the assumptions and scenarios represented by the selected risk model, the estimated portfolio loss threshold at the 95% confidence level over the selected time horizon is $8,000.

The result should be interpreted in the context of the selected risk model, input data, confidence level, and time horizon.

Implementation Notes

The Quant Risk API calculates the lower-tail P&L quantile using the portfolio outcomes produced by the selected risk model.

The calculation uses the tail probability:

\alpha = 1-c

The API then evaluates the corresponding percentile of the empirical portfolio P&L distribution.

For finite datasets, the calculated quantile may depend on the percentile calculation method used by the underlying numerical implementation.

The VaR value returned by the API is always converted into a non-negative loss value.

Limitations

VaR provides information about the location of a loss threshold but does not describe the severity of losses beyond that threshold.

Two portfolios may have similar VaR values while having substantially different losses in the most extreme scenarios.

VaR results also depend on the assumptions, data, scenario generation method, confidence level, and selected time horizon.

Conditional Value at Risk (CVaR) complements VaR by measuring the average loss across outcomes in the selected lower tail of the portfolio P&L distribution.

Method Summary

Value at Risk in the Quant Risk API is calculated using the following steps:

Generate or obtain the portfolio P&L distribution using the selected risk model.
Calculate the lower-tail probability from the requested confidence level.
Calculate the corresponding lower-tail P&L quantile.
Convert the lower-tail P&L quantile into a non-negative loss value.

The resulting VaR value represents the portfolio loss threshold estimated from the lower tail of the modeled P&L distribution.