CVaR(Conditional Value-at-Risk)也被称为Expected Shortfall(ES) 或者 Expected Tail Loss(ETL)，可以解释超过给定VaR值的期望损失，在很多风险分析中，CVaR是更具有解释力，当设置了VaR阈值后，CVaR说明了在之后的hhh天交易中的期望损失风险（即资产价值低于预先设置的VaR值）. 显然VaR和资产日回报率的分布有关.

当资产回报率的尾部较厚，使用整态分布刻画是不合适的，考虑使用t分布对资产回报率分布进行描述.

令XXX表示hhh日的回报率，则
VaRh,α=−xh,αVaR_{h, \\alpha}=-x_{h, \\alpha}VaRh,α​=−xh,α​
其中，P(X xh,α)=αP(X x_{h, \\alpha})=\\alphaP(X xh,α​)=α.
CVaR使用条件期望的形式表达
CVaRh,α(X)=−E(X∣X xh,α)=−α−1∫−∞xh,α=xf(x)dxCVaR_{h, \\alpha}(X)=-\\mathbb{E}(X\\mid X x_{h, \\alpha})=-\\alpha^{-1}\\int_{-\\infty}^{x_{h, \\alpha}}=xf(x)dxCVaRh,α​(X)=−E(X∣X xh,α​)=−α−1∫−∞xh,α​​=xf(x)dx
对于任意连续的概率密度函数f(x)f(x)f(x)，需要计算x∼100(1−α)%hx\\sim 100(1-\\alpha)\\%hx∼100(1−α)%h天VaR的xf(x)xf(x)xf(x)积分.
考虑在X∼N(μh,σh2)X\\sim N(\\mu_h, \\sigma_h^2)X∼N(μh​,σh2​)的情况下的CVaR
CVaRh,α(X)=α−1φ(Φ−1(α))σh−μhCVaR_{h, \\alpha}(X)=\\alpha^{-1}\\varphi(\\Phi^{-1}(\\alpha))\\sigma_h-\\mu_hCVaRh,α​(X)=α−1φ(Φ−1(α))σh​−μh​
其中φ(z)\\varphi(z)φ(z)表示标准正态分布的pdf，Φ(α)−1\\Phi(\\alpha)^{-1}Φ(α)−1为α\\alphaα分位数.
案例：计算h=5h=5h=5的CVaR，股价年化回报率符合σ=41%,μ=0\\sigma=41\\%, \\mu=0σ=41%,μ=0的正态分布，设置一年的交易日为252天计算出
σh=σh/252=0.415/252=0.05798\\sigma_h=\\sigma\\sqrt{h/252}=0.41\\sqrt{5/252}=0.05798σh​=σh/252

​=0.415/252

​=0.05798
python计算gaussian分布VaR和CVaR代码如下

import numpy as npimport matplotlib.pyplot as pltfrom scipy.stats import normfrom math import sqrtdef demo_1(): h, D=5, 252 mu, sig = 0, 0.41 muh, sigh=mu*(h/D), sig*sqrt(h/D) alpha=0.01 lev=100*(1-alpha) CVaRh=alpha**(-1)*norm.pdf(norm.ppf(alpha))*sigh-muh VaRh=norm.ppf(1-alpha)*sigh-muh print(\'{}% {} day Normal VaR= {:.2f}%\'.format(lev, h, VaRh*100)) print(\'{}% {} day Normal CVaR= {:.2f}%\'.format(lev, h, CVaRh*100))demo_1()

考虑更适合肥尾的t分布进行建模，标准t分布的pdf函数为
fν(x)=((ν−2)π)−1/2Γ(v2)−1Γ(ν+12)⏟A(1+(ν−2)−1⏟ax2)−(ν+1)/2⏟bf_\\nu(x)=\\underbrace{((\\nu-2)\\pi)^{-1/2}\\Gamma(\\frac{v}{2})^{-1}\\Gamma(\\frac{\\nu+1}{2})}_{A}(1+\\underbrace{(\\nu-2)^{-1}}_{a}x^2)^{\\underbrace{-(\\nu+1)/2}_{b}}fν​(x)=A

((ν−2)π)−1/2Γ(2v​)−1Γ(2ν+1​)​​(1+a

(ν−2)−1​​x2)b

−(ν+1)/2​​
其中Γ\\GammaΓ表示gamma函数，ν\\nuν表示自由度，方程一般形式为
fν(x)=A(1+ax2)bf_\\nu(x)=A(1+ax^2)^bfν​(x)=A(1+ax2)b
代入到CVaR定义中得到
CVaRα,ν=−α−1∫−∞xα,νxfν(x)dx=−α−1∫−∞xα,νxA(1+ax2)bdx=−Aα∫−∞xα,νx(1+ax2⏟y)bdx(1)\\begin{aligned}CVaR_{\\alpha, \\nu} =-\\alpha^{-1}\\int_{-\\infty}^{x_{\\alpha, \\nu}}xf_\\nu(x)dx\\\\ =-\\alpha^{-1}\\int_{-\\infty}^{x_{\\alpha, \\nu}}xA(1+ax^2)^bdx\\\\ =-\\frac{A}{\\alpha}\\int_{-\\infty}^{x_{\\alpha, \\nu}}x(\\underbrace{1+ax^2}_y)^bdx\\end{aligned}\\tag{1}CVaRα,ν​​=−α−1∫−∞xα,ν​​xfν​(x)dx=−α−1∫−∞xα,ν​​xA(1+ax2)bdx=−αA​∫−∞xα,ν​​x(y

1+ax2​​)bdx​(1)
令y=1+ax2y=1+ax^2y=1+ax2进行变换，dy=2axdx，B=1+(ν−2)−1xα,ν2dy=2axdx，B=1+(\\nu-2)^{-1}x^2_{\\alpha, \\nu}dy=2axdx，B=1+(ν−2)−1xα,ν2​，方程(1)(1)(1)变换得到
(1)=−Aα∫−∞Bx(1+ax2)b2axdy=−A2aα∫∞Bybdy=−A2aαBb+1b+1\\begin{aligned}(1) =-\\frac{A}{\\alpha}\\int_{-\\infty}^B\\frac{x(1+ax^2)^b}{2ax}dy\\\\ =-\\frac{A}{2a\\alpha}\\int_{\\infty}^{B}y^bdy\\\\ =-\\frac{A}{2a\\alpha}\\frac{B^{b+1}}{b+1}\\end{aligned}(1)​=−αA​∫−∞B​2axx(1+ax2)b​dy=−2aαA​∫∞B​ybdy=−2aαA​b+1Bb+1​​
由于b+1=(1−ν)/2b+1=(1-\\nu)/2b+1=(1−ν)/2代入得到(1)(1)(1)值为
−A(ν−2)−1α2B(1−ν)/21−ν-\\frac{A}{(\\nu-2)^{-1}\\alpha}\\frac{2B^{(1-\\nu)/2}}{1-\\nu}−(ν−2)−1αA​1−ν2B(1−ν)/2​
根据fν(x)f_\\nu(x)fν​(x)的表达式可以知道
fν(x)=ABb⇒A=fν(x)B−b=fν(x)B(1+ν)/2f_\\nu(x)=AB^b\\Rightarrow A=f_\\nu(x)B^{-b}=f_\\nu(x)B^{(1+\\nu)/2}fν​(x)=ABb⇒A=fν​(x)B−b=fν​(x)B(1+ν)/2
代入方程(1)(1)(1)得到CVaR值为
(1)=−α−1fν(x)B(1+ν)/22(ν−2)−12B(1−ν)/21−ν=−α−1fν(x)(ν−2)(1−ν)−1B=−α−1(ν−2)(1−ν)−1(1+(ν−2)−1xα,ν2)fν(xα,ν)=−α−1(1−ν)−1[ν−2+xα,ν2]fν(xα,ν)\\begin{aligned}(1) =-\\alpha^{-1}\\frac{f_\\nu(x)B^{(1+\\nu)/2}}{2(\\nu-2)^{-1}}\\frac{2B^{(1-\\nu)/2}}{1-\\nu}\\\\ =-\\alpha^{-1}f_\\nu(x)(\\nu-2)(1-\\nu)^{-1}B\\\\ =-\\alpha^{-1}(\\nu-2)(1-\\nu)^{-1}(1+(\\nu-2)^{-1}x^2_{\\alpha, \\nu})f_\\nu(x_{\\alpha, \\nu})\\\\ =-\\alpha^{-1}(1-\\nu)^{-1}[\\nu-2+x^2_{\\alpha, \\nu}]f_\\nu(x_{\\alpha, \\nu})\\end{aligned}(1)​=−α−12(ν−2)−1fν​(x)B(1+ν)/2​1−ν2B(1−ν)/2​=−α−1fν​(x)(ν−2)(1−ν)−1B=−α−1(ν−2)(1−ν)−1(1+(ν−2)−1xα,ν2​)fν​(xα,ν​)=−α−1(1−ν)−1[ν−2+xα,ν2​]fν​(xα,ν​)​
所以，hhh天，t分布下CVaR值为
CVaRh,α,ν(X)=−α−1(1−ν)−1[ν−2+xα,ν2]fν(xα,ν)σh−μhCVaR_{h, \\alpha, \\nu}(X)=-\\alpha^{-1}(1-\\nu)^{-1}[\\nu-2+x_{\\alpha, \\nu}^2]f_\\nu(x_{\\alpha, \\nu})\\sigma_h-\\mu_hCVaRh,α,ν​(X)=−α−1(1−ν)−1[ν−2+xα,ν2​]fν​(xα,ν​)σh​−μh​
案例计算自由度为666的t分布的VaR和CVaR
python计算t分布下VaR和CVaR代码如下

import numpy as npimport matplotlib.pyplot as pltfrom scipy.stats import norm, tfrom math import sqrtdef demo_2(): h, D = 10, 252 mu, sig=0, 0.41 muh, sigh=mu*(h/D), sig*sqrt(h/D) alpha=0.01 lev=100*(1-alpha) nu=6 # 设置自由度 xanu=t.ppf(alpha, nu) CVaRh=-1/alpha*(1-nu)**(-1)*(nu-2+xanu**2)*t.pdf(xanu, nu)*sigh-muh VaRh=sqrt(h/D*(nu-2)/nu)*t.ppf(1-alpha, nu)*sig-mu print(\'{}% {} day t VaR= {:.2f}%\'.format(lev, h, VaRh*100)) print(\'{}% {} day t CVaR= {:.2f}%\'.format(lev, h, CVaRh*100))demo_2()

可以发现，随着自由度的上升，t分布的VaR和CVaR逐渐收敛到gaussian分布的VaR和CVaR

import numpy as npimport matplotlib.pyplot as pltfrom scipy.stats import norm, tfrom math import sqrtdef demo_3(): h, D = 10, 252 mu, sig=0, 0.41 muh, sigh=mu*(h/D), sig*sqrt(h/D) alpha=0.01 lev=100*(1-alpha) data=[] for nu in range(5, 101): xanu=t.ppf(alpha, nu) CVaRt=-1/alpha*(1-nu)**(-1)*(nu-2+xanu**2)*t.pdf(xanu, nu)*sigh-muh VaRt=sqrt(h/D*(nu-2)/nu)*t.ppf(1-alpha, nu)*sig-muh data.append([nu, VaRt, CVaRt]) CVaRn=alpha**(-1)*norm.pdf(norm.ppf(alpha))*sigh-muh VaRn=norm.ppf(1-alpha)*sigh-muh data=np.array(data) fig, ax=plt.subplots(figsize=(8, 6)) plt.plot(data[:, 0], data[:, 1]*100, \'b-\', label=\'VaRt\') plt.plot(np.arange(5, 100), VaRn*np.ones(95)*100, \'b:\', label=\'VaRn\') plt.plot(data[:, 0], data[:, 2]*100, \'r-\', label=\'CVaRt\') plt.plot(np.arange(5, 100), CVaRn*np.ones(95)*100, \'r:\', label=\'CVaRn\') plt.xlabel(\'student t. d.o.f\') plt.ylabel(\'%\') plt.legend() plt.show() demo_3()

使用IBM数据进行实证分析， 使用gaussian分布和t分布拟合IBM日回报收益率得到拟合图像如下

可以发现，t分布更适合拟合这种存在高噪声的尖峰厚尾的数据，不同分布计算的CVaR和VaR值如下

99.0% 1 day gaussian VaR= 2.81%99.0% 1 day gaussian CVaR= 3.22%99.0% 1 day student t VaR= 2.22%99.0% 1 day student t CVaR= 3.91%

案例python代码

import numpy as npimport matplotlib.pyplot as pltfrom scipy.stats import norm, t, skew, kurtosis, kurtosistestfrom math import sqrtimport pandas_datareader.data as webimport pickle# Fetching Yahoo Finance for IBM stock datadef data_fetch(): data = web.DataReader(\'IBM\', data_source=\'yahoo\', start=\'2010-12-31\', end=\'2015-12-31\')[\'Adj Close\'] adj_close=np.array(data.values) ret=adj_close[1:]/adj_close[:-1]-1 file=open(\'ibm_ret\', \'wb\') pickle.dump(ret, file)with open(\'ibm_ret\', \'rb\') as file: ret=pickle.load(file)dx=0.0001x=np.arange(-0.1, 0.1, dx)# gaussian fitmu_norm, sig_norm=norm.fit(ret)pdf_norm=norm.pdf(x, mu_norm, sig_norm)# student t fitnu, mu_t, sig_t=t.fit(ret)nu=np.round(nu)pdf_t=t.pdf(x, nu, mu_t, sig_t)# VaR and CVaRalpha=0.01lev=100*(1-alpha)xanu=t.ppf(alpha, nu)CVaRn=alpha**(-1)*norm.pdf(norm.ppf(alpha))*sig_norm-mu_normVaRn=norm.ppf(1-alpha)*sig_norm-mu_normCVaRt=-1/alpha*(1-nu)**(-1)*(nu-2+xanu**2)*t.pdf(xanu, nu)*sig_t-h*mu_tVaRt=sqrt((nu-2)/nu)*t.ppf(1-alpha, nu)*sig_t-h*mu_tprint(\'{}% {} day gaussian VaR= {:.2f}%\'.format(lev, h, VaRn*100))print(\'{}% {} day gaussian CVaR= {:.2f}%\'.format(lev, h, CVaRn*100))print(\'{}% {} day student t VaR= {:.2f}%\'.format(lev, h, VaRt*100))print(\'{}% {} day student t CVaR= {:.2f}%\'.format(lev, h, CVaRt*100))plt.figure(figsize=(12, 8))grey=0.75, 0.75, 0.75plt.hist(ret, bins=50, density=True, color=grey, edgecolor=\'none\')plt.axis(\'tight\')plt.plot(x, pdf_norm, \'b-.\', label=\'Guassian Fit\')plt.plot(x, pdf_t, \'g-.\', label=\'Student t Fit\')plt.xlim([-0.2, 0.1])plt.ylim([0, 50])plt.legend(loc=\'best\')plt.xlabel(\'IBM daily return\')plt.ylabel(\'ret. distr.\')# plt.savefig(\'ibm_fit\')# inset localsub=plt.axes([0.22, 0.35, 0.3, 0.4])plt.hist(ret, bins=50, density=True, color=grey, edgecolor=\'none\')plt.plot(x, pdf_norm, \'b\')plt.plot(x, pdf_t, \'g\')plt.plot([-CVaRt, -CVaRt], [0, 3], \'g:\')plt.plot([-CVaRn, -CVaRn], [0, 4], \'b:\')plt.text(-CVaRt-0.015, 3.1, \'Stu. t\', color=\'g\')plt.text(-CVaRn-0.015, 4.1, \'Gaussian CVaR\', color=\'b\')plt.xlim([-0.09, -0.02])plt.ylim([0, 5])# plt.savefig(\'ibmcvar\')plt.show()

从数据和图像可以看出，Gaussian拟合计算的CVaR值存在低估风险的倾向.

CVaR Quant at risk