cvar¶

Defined in fynance.metrics

cvar(X, alpha=0.05, method='historical')[source]

Conditional Value-at-Risk (Expected Shortfall) of a price/equity curve.

The expected loss beyond the VaR threshold — the mean of the returns in the alpha-tail — a coherent risk measure (unlike VaR, which is not subadditive). Reported as a positive number, same convention as var.

Parameters:

Xarray_like: Time-series of price, performance or index (a single curve).
alphafloat, optional: Tail probability, in (0, 1). Default is 0.05.
method{‘historical’, ‘gaussian’, ‘cornish_fisher’}, optional: Estimation method. Default is ‘historical’.

Returns:

float: Conditional Value-at-Risk, positive for a typical loss-bearing distribution. Always \(\geq\) var at the same alpha.

See also

var, cdar, roll_cvar, tail_dependence

Notes

Let \(R\) be the per-period returns of X (see _returns_from_prices), \(n\) their count and \(\mu,\sigma\) their sample mean/std.

method='historical': mean of the k = max(1, floor(alpha n)) smallest observed returns (_tail_k) — the tail mean underlying var’s historical quantile.
method='gaussian': closed form [2] \(CVaR = -\mu + \sigma \frac{\varphi(z_\alpha)}{\alpha}\), with \(\varphi\) the standard normal pdf and \(z_\alpha = \Phi^{-1}(\alpha)\).
method='cornish_fisher': the tail mean of the Cornish-Fisher [1] -adjusted quantile function over \(p \in (0, \alpha]\),

\[CVaR_{CF} = -\frac{1}{\alpha}\int_0^{\alpha} \left(\mu + \sigma \, z_{CF}(p)\right) dp\]

approximated on a fixed probability grid (see _cf_tail_constants); collapses to the same value as 'gaussian' when sample skewness and excess kurtosis are both zero.

References

[1]

Favre, L., and Galeano, J.-A., 2002, Mean-Modified Value-at-Risk Optimization with Hedge Funds, Journal of Alternative Investments.

[2]

Rockafellar, R.T., and Uryasev, S., 2000, Optimization of Conditional Value-at-Risk, Journal of Risk, 2, 21-42.

Examples

>>> import numpy as np
>>> X = np.array([100., 99., 103., 95., 101., 98., 104., 90., 108., 97.,
...                102.])
>>> round(cvar(X, alpha=0.2, method='historical'), 4)
0.1182
>>> cvar(X, alpha=0.2, method='historical') >= var(X, alpha=0.2, method='historical')
True