74 lines
1.9 KiB
ReStructuredText
74 lines
1.9 KiB
ReStructuredText
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Risk Measures: VaR and CVaR
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============================
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The module :code:`risk_measures` provides Value-at-Risk and Conditional
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Value-at-Risk estimators together with a convex CVaR minimisation
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solver over the unit simplex.
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Definitions
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-----------
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For a real random variable :math:`L` (a *loss*), the Value-at-Risk at
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confidence level :math:`\alpha \in (0, 1)` is the lower :math:`\alpha`-
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quantile
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.. math::
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\mathrm{VaR}_\alpha(L)
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\;=\;
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\inf\!\big\{ \ell \in \mathbb{R} : \mathbb{P}(L \le \ell) \ge \alpha \big\}.
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The Conditional Value-at-Risk (also called Average Value-at-Risk) is
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.. math::
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\mathrm{CVaR}_\alpha(L)
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\;=\;
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\frac{1}{1-\alpha}\,
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\int_\alpha^1 \mathrm{VaR}_u(L)\,du.
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For a sample :math:`L_1, \dots, L_n` of i.i.d. losses sorted in increasing
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order, the empirical CVaR at level :math:`\alpha` is
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.. math::
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\widehat{\mathrm{CVaR}}_\alpha
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\;=\;
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\frac{1}{n - k}\, \sum_{i = k+1}^{n} L_{(i)},
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\qquad k = \lfloor \alpha\, n \rfloor.
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Convex minimisation
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-------------------
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Rockafellar--Uryasev (2000) showed that
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.. math::
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\mathrm{CVaR}_\alpha(L)
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\;=\;
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\min_{\zeta \in \mathbb{R}}\;
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\zeta + \frac{1}{1 - \alpha}\,\mathbb{E}\!\big[(L - \zeta)_+\big].
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Given samples of a vector :math:`r^{(s)} \in \mathbb{R}^d`,
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:code:`minimize_cvar` solves
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.. math::
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\min_{w \in \Delta_d,\;\zeta \in \mathbb{R}}\;
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\zeta + \frac{1}{(1 - \alpha)\, S}\, \sum_{s=1}^S
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\big(\zeta - \langle r^{(s)}, w\rangle\big)_+,
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over the unit simplex :math:`\Delta_d`, by a projected sub-gradient
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method using the Held--Wolfe--Crowder simplex projection.
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API
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---
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.. code-block:: rust
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pub fn historical_var(losses: &[f64], alpha: f64) -> Result<f64>;
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pub fn parametric_var(mu: f64, sigma: f64, alpha: f64) -> Result<f64>;
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pub fn cvar_value(losses: &[f64], alpha: f64) -> Result<f64>;
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pub fn minimize_cvar(returns: ArrayView2<f64>, cfg: &CVaRConfig)
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-> Result<CVaRResult>;
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