Conditional Value-at-Risk for Uncommon Distributions

Conditional Value-at-Risk for Uncommon Distributions Valentyn Khokhlov, MBA, CFA Corporate Finance and Investments Consultant E-mail: [email protected]

Abstract Conditional Value-at-Risk (CVaR) represents a significant improvement over the Value-at-Risk (VaR) in the area of risk measurement, as it catches the risk beyond the VaR threshold. CVaR is also theoretically more solid, being a coherent risk measure, which enables building more robust risk assessment and management systems. This paper addresses the derivation of the closed-form CVaR formulas for several less known distributions, such as Burr type XII, Dagum, hyperbolic secant, as well as more popular generic extreme value distributions. It follows an unnoticed results of Patrizia Stucchi who derived CVaR formulas for Johnson’s SU/SB/SL distributions. While being uncommon for general public, those distributions represent a significant advancement in modeling financial assets returns. After having derived the closed-form CVaR formulas for the most popular elliptic distributions and log-distribution, this paper concludes the development of mathematical toolbox required for effective introduction of the CVaR into practical risk management.

Keywords: CVaR; Conditional Value at Risk; risk management; Burr distribution; Dagum distribution; generic extreme value distribution; hyperbolic secant distribution.

Introduction Conditional value-at-risk (CVaR) has recently become a popular measure of risk. While less known than value-at-risk (VaR), CVaR has an important advantage of being a coherent risk measure as defined by Artzner (1999). Another serious drawback of VaR is its inability to quantify the expected losses beyond the threshold amount, i.e. VaR only allows identifying the threshold itself but says nothing about the worst-case scenarios below it. However, VaR remains the most popular risk measure and is widely covered by academics and practitioners, such as Jorion (1996), Stambaugh (1996), Linsmeier (2000). CVaR, on the other hand, has been mostly overlooked by practitioners because it lacks an easy-to-use straightforward formula. Starting with Rockafellar and Uryasev (2000) there were attempts of simplifying the generic CVaR formula for some distributions and deriving it in a closed form. Rockafellar and Uryasev (2000) did this for the normal distribution using the error function, which unfortunately is not present in most non-mathematical software (e.g. Microsoft Excel™). Andreev et al. (2005) derived closed-form CVaR formulas for some of the most widely-used distributions, and Khokhlov (2016) provided the closed-form CVaR formulas derivation for elliptical distributions, and recently in Khokhlov (2018) the some of the most essential log-distributions were addressed. To conclude the development of the mathematical toolbox that effectively allows introducing the CVaR into the practical risk management, this paper provides the derivation of the CVaR for lesser known, somewhat uncommon distributions, such as Burr type XII, Dagum, and hyperbolic secant. This work was inspired by the brilliant result of Patrizia Stucchi (2011) who derived the closed-end CVaR for the Johnson’s SU distribution, as well as for the lognormal distribution. Independently from this research, the derivation of the closed-form CVaR formulas for the exponential, generalized Pareto, Weibull and the generic extreme value distributions was also performed by Matthew Norton in yet unpublished paper by Norton, Khokhlov, Uryasev.

Our methodology is based on the variance-covariance definition of CVaR, so all the formulas are derived under this approach. However, for some assets we also compare the results of our CVaR estimation with CVaR calculated under the empirical (historical) approach. The definitions of CVaR CVaR is defined as the conditional expectation of losses above VaR. Under the theoretical (variance-covariance) approach, we assume that the loss is a random variable that follows a certain distribution, and the CVaR is calculated as

c  E  x x  v   where

1   x f  x  dx , 1   v

(1a)

c is the CVaR @  , v is the VaR @  ,

x is the loss (random variable), f  x  is the probability density function (PDF) for the distribution of losses. Formula (1a) basically means CVaR is the conditional expectation in the right tail of the distribution of losses, where the VaR is the threshold. This formula is quite common in many fields of risk management, however it is not the most useful for financial applications. In finance, we are working with the distributions of returns as the key random variable, and the loss in its inverse. Therefore, we are mostly interested in negative returns in the left tail of their distributions. When the return is a random variable that follows a certain distribution, the CVaR per $1 of the investment, an assets or a portfolio is calculated as c  E  x x  v  

where

1 v  x f  x  dx ,  

(1b)

c is the CVaR @  per $1 of the selected asset, v is the VaR @  per $1 of the selected asset,

x is the return on the selected asset (random variable), f  x  is the probability density function (PDF) for the distribution of return. The second possible way of calculating CVaR under the theoretical approach would be integrating the quantile (inverse CDF) over the range of relevant probabilities instead of integrating the random variable multiplied by the PDF over the range of its relevant values. As

F  x   p  x  F1  p  , f  x  dx  dF  x   dp , and F     0, F  v   , F     1, we can rewrite (1a) or (1b) accordingly. Considering that for continuous distributions v  F1    , (1b) is equivalent to c 

1  1 1 F  p  dp   v p dp .  0 0





(1c)

Under the empirical (historical) approach, when working with samples of the asset returns, the CVaR is calculated as the conditional sample mean for the values that are less than the VaR: 1 chist  (2)  ri , n ri vhist where

chist is the historical CVaR @  per $1,

vhist is the historical VaR @  per $1, which is the  -th percentile, ri is the i-th value in the sample of historical returns (i = 1,…,N). n is the amount of returns that are less than vhist , i.e. n  N  . Finally, while the focus of this paper will be on the left tail of the distribution of returns, so we will derive the closed-form formulas from (1b) or (1c), we recognize that in many other risk management applications it is required to use the right tail of the distribution of losses, i.e. derive the closed-form formula from (1a). There is a simple analytical way to derive the conditional expectation for the right tail cr from the conditional expectation of the left tail cl :

  1   E  x   c  . 1  1 

 

E  x   xf  x  dx  v xf  x  dx   cl  1    cr , v





cr

l 

(3)

CVaR formula for the Johnson’s SU-distribution The Johnson’s SU-distribution is a transformation of the normal (Gaussian) distribution z using the following formula: x     sinh , where z ~ N  0,1 . The CDF of the Johnson’s  SU-distribution immediately follows from this transformation:

where

  x    F  x        sinh 1   ,     , ,   0,   0 are the distributional parameters.

If a random variable x adheres to the Johnson’s SU-distribution, 2  1 1 2  1   E  x      exp 2 sinh , var  x    exp 2  1 exp 2 cosh  1 .  2   2     The VaR @ α for the Johnson’s SU-distribution is its quantile:

 1      .  Patrizia Stucchi (2011) derived the following closed-end CVaR @ α formula: v  F1        sinh

c   

1  1 1   1  2   1  1  2   1 exp          exp         .  2   2    2     2    2  

CVaR formula for the Burr type XII distribution The Burr type XII distribution with 4 parameters has the following PDF and CDF:

where

c 1 

c   k 1

k

  x   c   ,  and F  x   1  1           c  0, k  0,   0,  are the distributional parameters. ck  x    f( x)      

 x 1       

If a random variable x adheres to the Burr type XII distribution, E  x    k   k  1 c ,1  1 c  ,



v  F1        1   

where

 is the beta function.

1 k



1

1c

,

(4)

Proposition E1. For the Burr type XII distribution the CVaR @  can be calculated as c   

where

2 F1





1c   1 1  1   1 k  1   1  2 F1  , k ;1  ;1  1   1 k  ,  c c  

(5a)

is the hypergeometric function.

Proof: Let’s define the standard Burr type XII random variable z so that x    z , and also



define z  Fz1     1   

1 k



1

1c

. First, let’s consider the indefinite integral in (1b): k

  x   c   dx    x    xf  x  dx   xdF  x   xF  x    F  x  dx  xF  x   x   1          k 1 1   xF  x   x   1  z c dz  xF  x   x  z 2 F1  , k ;1  ;  z c   c c 





c 1 1  x   xF  x   x   x    2 F1  , k ;1  ;    . c c       Then we can use (1b) to derive the CVaR formula from the indefinite integral: c 1 1  v     c    v        v F  v   v   v    2 F1  , k ;1  ;       c c      









1c 1c 1 k 1 k          1     1        1    1       1 c 1 1 k 1 k  1  1     1 2 F1  , k ;1  ;1  1        c c 









 1 1 k   1   1  2 F1  c , k ;1  c ;1  1      .□    Proposition E2. For the Burr type XII distribution the CVaR @  can be calculated as 11/ c  ck 1 1/ k   1 c    1   1/ k  1 2 F1 1  , k  1; 2  ;1  1    ,  c 1 c  c      1   

1 k

1c

1



2 F1

where



(5b)

is the hypergeometric function.

Proof: Let’s define the standard Burr type XII random variable z so that x    z , and also



define z  Fz1     1   

c    v xck

  0z ck 



 x     

1 k



1

1c

. The result can be obtained by expanding (1b) directly:

c 1

  x  1       

c

   

dx   0z  z    ck k 1



zc

dz   1  1  z c k  1  1  zc









k

z c 1



 1 z



c k 1

dz 

 | z  0 

11 c k 1 1 k   1 1   1 k  1 2 F1  1  , k  1; 2  ;1  1      . 11 c c  c 

Note that we can also obtain the expected value for the Burr type XII distribution this way:

E  x      xck

 x     

c 1

  x   c    1         

dx   0  z    ck k 1

z c 1



 1  zc



k 1

dz 

  0 zf z  z  dz  Fz  z  |0  k   k  1 c ,1  1 c   . □ CVaR formula for the Dagum distribution The Dagum distribution with 4 parameters has the following PDF and CDF:

where

ck 1 

c   k 1

k

  x    c   ,  and F  x   1           c  0, k  0,   0,  are the distributional parameters. ck  x    f( x)      

 x 1       

If a random variable x adheres to the Dagum distribution,    1 c    k  1 c  , E  x    c  k 





v  F1         1 k  1

1 c

,

where  is the gamma function. Proposition E3. For the Dagum distribution the CVaR @  can be calculated as  k 1/ c  ck 1 1 1   c     1/ k  1 2 F1  k  1, k  ; k  1  ;  1 k ,  ck  1 c c  1  



2 F1

where



(6)

is the hypergeometric function.

Proof: Let’s define the standard Dagum random variable z so that x    z , and also define





1 c

xf  x  dx  

 v 

z  Fz1     1 k  1 c  

 v 

. The result can be obtained by expanding (1b) directly: ck  x    x     



  0z  z    ckz ck 1 1  z c    



 k 1

ck 1 

 x 1       

c   k 1

  

dx 



dz  F  z  |0z   0z ckz ck 1  z c



 k 1

dz 

ck ck 1  1 1 c  z z 2 F1  k  1, k  ; k  1  ;  z  |0  ck  1 c c  









1  1 1   1, k  ; k  1  ;   1 k  1  . c c   Note that we can also obtain the expected value for the Dagum distribution this way:

   

ck  1 k  1 ck  1

 k 1 c

2 F1  k



 ck 1 E  x     1 zc  xf  x  dx   0  z    ckz

   E  z    



 k 1

dz  Fz  z  |0  0z zf z  z  dz 

   1 c    k  1 c  .□ c  k 

CVaR formula for the generalized hyperbolic secant (GHS) distribution The generalized hyperbolic secant (GHS) distribution has the following PDF and CDF

1 2    x     x    sech   ; F  x   arctan exp   , 2  2    2     is the location parameter (expected value), f  x 

where

 is the scale parameter (standard deviation). If a random variable x adheres to the GHS distribution, 2    v  F1       ln  tan .   2  Proposition E4. For the GHS distribution CVaR @  can be calculated as c   

where

2    2        ln  tan   2 i  Li 2  i tan   Li 2  i tan  ,   2     2  2   

(7)

,  are the location and the scale parameter of the distribution,

Li 2 is the dilogarithm function (a polylogarithm function with s = 2),

i is the imaginary unit, i  1 . Proof: Let’s define the standard GHS random variable z so that x    z , and also define z  Fz1    

2    ln  tan  . First, let’s consider the indefinite integral in (1b):   2 

2    x      x    xf  x  dx   xdF  x   xF  x    F  x  dx  xF  x    arctan exp    dx    2   







z 2  Li 2 iez 2 2  2  Li 2 ie  z    xF  x    arctan exp  2   dz  xF  x    i       Then we can use (1b) to derive the CVaR formula from the indefinite integral:

  .  

        Li 2  i tan  Li 2  i tan    2 2  2    i  c    v        v F  v            

2    2         ln  tan   2 i  Li 2  i tan   Li 2  i tan  .□   2     2  2   

CVaR formula for the generalized extreme value (GEV) distribution The generalized extreme value (GEV) distribution has the following PDF and CDF 1  1      x    x    exp   1     ; F  x   exp   1    ,           where ,   0,   0 are the distribution parameters. If a random variable x adheres to the GEV distribution,  E  x      1     1 , 

1 x   f  x   1      

11 

v  F1       where

  s  is the gamma function.

   ln    1 ,   

Proposition E5. For the GEV distribution with   0 CVaR @  can be calculated as

  1  ,  ln      ,   , ,  are the parameters of the distribution, c   

where

(8)

  s, x  is the upper incomplete gamma function. Proof: Let’s expand integral in (1c): с  0 F1  p  dp    

     0   ln p  dp , then  

use the following substitution y   ln p; p  e y ; dp  e y dy :     

   ln    y    y e dy    

   11  y    ln  y e dy     1  ,  ln      .□   

Note that the expected value immediately follows from (8) if we put   1 , as   s,0     s  . The special case of the GEV distribution with   0 has the following PDF and CDF f  x 

where

1    x    x     x    exp    exp   exp     ; F  x   exp   exp    ,             

,   0 are the distribution parameters. If a random variable x adheres to this special case of the GEV distribution, E  x     ,

v  F1        ln   ln   , where

 is the Euler–Mascheroni constant.

Proposition E6. For the GEV distribution with   0 CVaR @  can be calculated as  li      ln   ln    ,  ,  are the parameters of the distribution, c   

where

li is the logarithmic integral function, li  x   

(9)

dx x dt .  0 ln x ln t

Proof: Let’s expand integral in (1c): с  0 F1  p  dp    0 ln   ln p  dp . Let’s define 



d 1 d 1 d ,   x   ln   ln x   li  x    li  x   ln   ln x  , dx ln x dx ln x dx hence   x    ln   ln x  dx  C . Using limits, we can show that   0   0,  1   . Therefore,   x   x ln   ln x   li  x  . As

с            0       ln   ln    li    .□

Note that the expected value immediately follows from the above if we put   1 ,  1   . Monte-Carlo simulation of the derived formulas Monte-Carlo simulations were used to test the formulas (4)–(6), (8), as in the ideal case when a reasonably large sample is known to follow a particular distribution, the empirical (sample) VaR and CVaR should be located close enough to the theoretical (variance-covariance) VaR and CVaR. While we do not operate with confidence intervals for VaR and CVaR in this research, it should be really straightforward to see whether those values are “close enough” or not. So our methodology is generating a sample of 10,000 random returns that follow a particular

distribution with known parameters, estimate the sample CVaR with (2) and compare it with the theoretical CVaR calculated with (4)–(6), (8). For VaR a similar methodology is utilized by comparing the sample percentile with the quantile for the respective distribution. The samples were generated using the distributional parameters found by fitting the sample of the inflation-linked U.S. Treasuries ETF (ticker TIP) daily returns from March 2, 2009 to January 30, 2015 using EasyFit package. The results are presented in Table 1 and confirm that the derived formulas are correct. Table 1. Theoretical and sample VaR and CVaR comparison for Monte-Carlo simulated data Distribution

Johnson’s SU

Burr type XII

Parameters

γ=-0.14135 δ=1.29286 λ=0.003463 ξ=-2.81425E-4

k=1.12295 c=9.74892E+6 β=20499.4 γ=-20499.4

VaR @ 5% VaR @ 1% CVaR @ 5% CVaR @ 1%

sample theoretical sample theoretical sample theoretical sample theoretical

Dagum

k=1.74315 c=1.22261E+8 β=323185 γ=-323185

GEV

ξ=-0.30122 σ=0.003566 µ=-9.85572E-4

-0.5164% -0.6630% -0.3898% -0.5558% -0.5280% -0.6441% -0.4020% -0.5622% -0.9519% -1.0075% -0.6688% -0.7906% -0.9346% -0.9907% -0.6788% -0.7900% -0.7926% -0.8812% -0.5613% -0.7018% -0.7904% -0.8595% -0.5535% -0.7016% -1.2995% -1.2438% -0.8408% -0.9045% -1.2637% -1.2020% -0.8365% -0.8986% (Source: author’s calculations based on Monte-Carlo simulated samples)

Risk assessment for different asset classes This section presents VaR and CVaR estimation for real-life asset classes, such as equities, bonds, real estate, gold, and so on. It is based on findings of the original research conducted by the author and professor A. Kaminsky from Kyiv National University. Exchangetraded funds tracking the relevant indices are used to represent the following asset classes: 1. SPY — large-cap U.S. equities (tracking S&P 500 index) 2. MDY — mid-cap U.S. equities (tracking S&P 400 index) 3. IJR — small-cap U.S. equities (tracking S & P 600 index) 4. IEF — U.S. Treasury bonds 5. LQD — investment-grade corporate bonds 6. TIP — inflation-linked U.S. Treasury bonds 7. VNQ — real estate (tracking US REIT index) 8. GLD — gold 9. PSP — private equity investments 10. MBB — mortgage-backed bonds Table 2 presents the VaR @ 5% estimation results for the asset classes 1–8 using the daily returns from March 2003 to October 2007, a very long growth period on the stock market. The last row shows the root mean squared error (RMSE) of the estimation defined as RMSE 

where

1 n 2   vˆi  vi  , n i 1

n is the number of assets, vˆi is the theoretical VaR estimated with (4)–(8) for i-th asset, vi is the historical VaR for i-th asset.

Table 2. VaR @ 5% per $1 estimation for selected asset classes in 2003–2007 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD RMSE

historical 1.2362% 1.5054% 1.7213% 0.5999% 0.5359% 0.5017% 1.7522% 1.7833%

Johnson SU Burr type XII Dagum GEV GHS 1.2184% 1.3996% 1.2293% 1.2039% 1.1886% 1.4455% 1.4574% 1.5563% 1.4441% 1.3646% 1.6845% 1.7086% 1.7690% 1.6881% 1.6142% 0.5826% 0.5974% 0.5864% 0.5717% 0.5741% 0.5715% 0.5745% 0.5711% 0.5605% 0.5540% 0.5189% 0.5293% 0.5223% 0.5116% 0.5095% 1.8307% 2.1139% 1.9003% 1.8126% 1.7805% 1.8410% 1.8614% 1.0910% 1.7636% 1.7615% 0.0456% 0.1451% 0.2520% 0.0378% 0.0670% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Table 3 presents the VaR @ 1% estimation results for the asset classes 1–8 using the daily returns from March 2003 to October 2007. As we can see comparing these results with Table 2, moving further into the left tail deteriorates the VaR estimation accuracy for all distributions. Johnson’s SU-distribution provides the best overall accuracy, albeit GEV may be better in case of VaR @ 5%, but inaccurate for VaR @ 1%. Table 3. VaR @ 1% per $1 estimation for selected asset classes in 2003–2007 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD RMSE

historical 1.9488% 2.1468% 2.3828% 0.8893% 0.9185% 0.7237% 3.1119% 3.3492%

Johnson SU Burr type XII Dagum GEV GHS 2.0174% 2.1325% 2.0009% 1.7280% 1.9808% 2.2333% 2.2611% 2.2596% 2.0981% 2.2776% 2.5494% 2.5732% 2.6272% 2.4234% 2.6886% 0.9343% 0.9196% 0.9243% 0.8021% 0.9485% 0.9020% 0.9088% 0.9200% 0.7909% 0.9174% 0.7831% 0.7943% 0.8174% 0.7144% 0.8444% 2.9303% 3.2166% 2.4411% 2.5904% 2.9572% 3.2207% 2.8713% 1.9598% 2.5370% 2.9310% 0.1091% 0.2026% 0.5552% 0.3550% 0.2027% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Table 4 presents the CVaR @ 5% estimation results for the asset classes 1–8 using the daily returns from March 2003 to October 2007. Comparing VaR @ 5% and CVaR @ 5% RMSE for the GEV distribution shows it is very inaccurate in modeling the tail, albeit provide quite accurate VaR @ 5% estimation. Johnson’s SU and Burr type XII distributions, on the opposite, are more accurate for CVaR estimation than for the VaR. Table 4. CVaR @ 5% per $1 estimation for selected asset classes in 2003–2007 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD RMSE

historical 1.7152% 1.9204% 2.1644% 0.7965% 0.7633% 0.6652% 2.5917% 2.7026%

Johnson SU Burr type XII Dagum GEV GHS 1.7215% 1.8551% 1.7062% 1.5248% 1.6809% 1.9334% 1.9507% 1.9808% 1.8448% 1.9320% 2.2191% 2.2368% 2.2884% 2.1384% 2.2819% 0.8033% 0.7960% 0.7946% 0.7127% 0.8068% 0.7779% 0.7817% 0.7878% 0.7015% 0.7798% 0.6827% 0.6915% 0.7046% 0.6357% 0.7176% 2.5190% 2.7992% 2.2303% 2.2888% 2.5117% 2.7164% 2.4890% 1.6288% 2.2373% 2.4883% 0.0340% 0.1201% 0.4039% 0.2129% 0.0939% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Table 5 presents the CVaR @ 1% estimation results for the asset classes 1–8 using the daily returns from March 2003 to October 2007. Here we must admit, that the only distribution that was able to model the expected losses in the left tail is Johnson’s SU, as the accuracy for Burr type XII and the GHS distributions deteriorated substantially comparing to the estimation of the CVaR @ 5%.

Table 5. CVaR @ 1% per $1 estimation for selected asset classes in 2003–2007 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD RMSE

historical 2.5874% 2.5903% 2.9386% 1.1970% 1.1572% 0.9314% 3.6080% 4.3250%

Johnson SU Burr type XII Dagum GEV GHS 2.5562% 2.5796% 2.4530% 1.9827% 2.4724% 2.6850% 2.6902% 2.5415% 2.4197% 2.8442% 3.0317% 3.0059% 2.9981% 2.7810% 3.3555% 1.1640% 1.1000% 1.1153% 0.9121% 1.1809% 1.1081% 1.1070% 1.1328% 0.9014% 1.1429% 0.9348% 0.9306% 0.9878% 0.8105% 1.0522% 3.6323% 3.8893% 2.6711% 2.9676% 3.6874% 4.2158% 3.4867% 2.4456% 2.9140% 3.6569% 0.0658% 0.3179% 0.7453% 0.6105% 0.2998% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Table 6 presents the VaR @ 5% estimation results for all asset classes using the daily returns from November 2008 to February 2009, the period of the global financial crisis. We can note the substantial increase in the VaR and CVaR values comparing to the previous period. We can note that Johnson’s SU, the GEV and the GHS distributions model VaR @ 5% the best. Table 6. VaR @ 5% per $1 estimation for selected asset classes in 2007–2009 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD PSP MBB RMSE

historical 4.3265% 4.4549% 4.2273% 0.9189% 1.1136% 1.0606% 7.4898% 3.0697% 6.9305% 0.6040%

Johnson SU Burr type XII Dagum GEV GHS 3.7798% 3.6323% 3.8728% 3.7932% 4.1434% 4.4176% 4.0412% 4.3659% 4.3063% 4.5473% 4.4855% 4.7510% 4.3831% 4.3266% 4.3699% 0.9533% 0.9681% 0.9299% 0.9496% 0.9555% 1.6978% 1.2096% 2.9218% 1.4565% 1.9287% 1.1481% 1.3355% 1.2430% 1.1529% 1.1479% 7.2383% 5.8371% 7.0589% 6.8621% 7.4082% 2.9547% 3.0436% 3.1544% 3.0624% 3.1431% 6.2891% 6.1411% 6.4414% 6.3217% 6.5660% 0.6371% 0.6304% 0.6401% 0.6255% 0.6541% 0.3473% 0.6611% 0.6304% 0.3476% 0.2972% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Table 7 presents the VaR @ 1% estimation results for all asset classes using the daily returns from November 2008 to February 2009. It is interesting to note how the GEV distribution results deteriorated when moving from 5% to 1% quantile. Only Johnson’s SU and the GHS distributions were able to model the tail risk, albeit moderately underestimating it in most cases. Table 7. VaR @ 1% per $1 estimation for selected asset classes in 2007–2009 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD PSP MBB RMSE

historical 7.2827% 8.3208% 7.7014% 1.3144% 3.9800% 1.9225% 12.3331% 4.9604% 11.2783% 1.0433%

Johnson SU Burr type XII Dagum GEV GHS 6.2951% 5.5365% 5.9103% 5.1805% 6.6539% 7.3586% 6.2391% 6.7800% 5.9571% 7.3237% 7.1349% 7.2418% 6.8760% 5.9840% 7.0258% 1.3852% 1.3898% 1.1878% 1.3212% 1.5924% 3.4232% 1.9318% 3.8282% 2.0249% 3.1472% 1.8820% 2.0257% 1.5860% 1.6296% 1.8808% 11.6100% 9.3669% 10.7665% 9.2650% 11.9695% 4.8887% 4.7316% 4.8775% 4.2920% 5.1791% 9.9401% 9.3487% 9.6836% 8.6485% 10.4732% 1.0042% 0.9902% 0.9972% 0.8651% 1.0874% 0.6971% 1.5615% 1.0049% 1.8350% 0.5873% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Table 8 presents the CVaR @ 5% estimation results for all asset classes using the daily returns from November 2007 to February 2009. One of the surprising facts we see here in the increase of the CVaR estimation accuracy comparing to the VaR for Johnson’s SU, Dagum and

the GHS distributions. On the contrary, the GEV distribution results are much worse, and it confirms our previous conclusion that it is unable to model the actual distribution far in the tail. Table 8. CVaR @ 5% per $1 estimation for selected asset classes in 2007–2009 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD PSP MBB RMSE

historical 5.9699% 6.6207% 6.4184% 1.1953% 2.6583% 1.6434% 10.3353% 4.2483% 9.1081% 0.8963%

Johnson SU Burr type XII Dagum GEV GHS 5.3970% 4.8154% 5.1262% 4.6420% 5.7035% 6.2938% 5.4071% 5.8550% 5.3167% 6.2726% 6.1512% 6.2990% 5.9304% 5.3411% 6.0203% 1.2194% 1.2243% 1.0873% 1.1769% 1.3513% 2.8438% 1.6585% 3.4750% 1.8043% 2.6859% 1.6119% 1.7644% 1.4523% 1.4448% 1.6034% 10.0046% 8.0309% 9.3487% 8.3314% 10.2427% 4.1857% 4.0879% 4.2091% 3.8149% 4.4083% 8.6044% 8.1343% 8.4275% 7.7456% 8.9940% 0.8675% 0.8540% 0.8606% 0.7720% 0.9233% 0.3017% 1.0060% 0.6075% 1.0709% 0.2062% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Finally, Table 9 presents the CVaR @ 1% estimation results for all asset classes using the daily returns from November 2007 to February 2009. As in the case of 2003-2007 returns, the only distribution whose accuracy did not deteriorate substantially when assessing the expected losses that far in the left tail is Johnson’s SU-distribution. Table 9. CVaR @ 1% per $1 estimation for selected asset classes in 2007–2009 Ticker SPY MDY IJR IEF LQD TIP VNQ GLD PSP MBB RMSE

historical 8.4905% 9.8727% 9.0201% 1.5741% 6.3857% 2.3255% 15.0129% 5.9847% 12.4078% 1.2613%

Johnson SU Burr type XII Dagum GEV GHS 8.2761% 6.6914% 7.0445% 5.8391% 8.2119% 9.5578% 7.5764% 8.1646% 6.7496% 9.0467% 8.8991% 8.7611% 8.3900% 6.7803% 8.6740% 1.6133% 1.5813% 1.2972% 1.4959% 1.9877% 5.0743% 2.3708% 4.2133% 2.2963% 3.9033% 2.3921% 2.4468% 1.7316% 1.8606% 2.3356% 14.6728% 11.5112% 12.9039% 10.3851% 14.8003% 6.3052% 5.7151% 5.7821% 4.8796% 6.4427% 12.5385% 11.2988% 11.4022% 9.7607% 12.8981% 1.2428% 1.2088% 1.2023% 0.9772% 1.3563% 0.4606% 1.9565% 1.2672% 2.6167% 0.8785% (Source: author’s calculations based on historical prices from Yahoo! Finance)

Conclusion Conditional value-at-risk (CVaR) was considered a theoretically appealing coherent risk measure with limited practical implications because of the computational complexity. The closed-form CVaR formulas have already been derived for the key elliptical and the logdistributions. In this paper the closed-form CVaR formulas have been derived for several less common distributions — Burr type XII, Dagum, the generalized extreme value and the generalized hyperbolic secant distribution, in addition to the already derived formula for Johnson’s SU-distribution. Thus the development of the mathematical toolbox for the CVaR estimation has been concluded for all practically significant cases. The CVaR for the distributions considered in this paper require some advanced functions (hypergeometric, polylogarithm, logarithmic integral). An exception is the GEV distribution, which requires only the gamma function except one special case, and can be implemented in Microsoft Excel™. However, the closed-form CVaR for Johnson’s SU-distribution does not require any special functions besides the normal distribution ones, and since it also provided the best estimation accuracy, Johnson’s SU appears to be the best choice for modeling the CVaR.

References 1.

Artzner, P., Delbaen, F. Eber, J.-M., Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, Vol. 9, No. 3, 203–228. 2 2. Jorion, P. (1996). Risk : Measuring the Risk in Value at Risk. Financial Analysts Journal, Vol. 52, No. 6, 47–56. 3. Stambaugh, F. (1996). Risk and Value at Risk. European Management Journal, Vol. 14, No. 6, 612– 621. 4. Linsmeier, T., Pearson, N.D. (2000). Value at Risk. Financial Analysts Journal, Vol. 56, No. 2, 47–67. 5. Rockafellar, R.T., Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, Vol. 2, 21-41. 6. Andreev, A., Kanto, A., Malo, P. (2005). On closed-form calculation of CVaR. Helsinki School of Economics Working Paper W-389. (2016, October, 11) 7. Rockafellar, R.T., Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, Vol. 26, No. 7, 1443–1471. 8. Stucchi, P. (2011). Moment-Based CVaR Estimation: Quasi-Closed Formulas. Available at SSRN: . 9. Khokhlov, V. (2016). Conditional Value-at-Risk for Elliptical Distributions. Evropský Časopis Ekonomiky a Managementu, Vol. 2, No. 6, 70–79. 10. Khokhlov, V. (2018). Conditional Value-at-Risk for Log-Distributions. Available at SSRN: .

Appendix A — The distributional parameters used for modeling the asset classes The theoretical VaR and CVaR estimates presented in Tables 2–5 were calculated using the following distributional parameters, which were obtained using the EasyFit package. They are provided in Table A1 in the EasyFit notation. Table A1. EasyFit distributional parameters representing the best fit for the 2003-2007 sample Ticker

SPY

MDY

IJR

IEF

LQD

TIP

VNQ

GLD

Johnson SU Burr type XII Dagum GEV GHS γ=0.236839; k=1.21618; k=0.672199; ξ=-0.352267; δ=1.90806; c=3.23425E+6; c=48.9019; σ=0.007722; σ=0.007671; λ=0.012653; β=14393.9; β=0.176074; µ=6.10812E-4 µ=-0.001764 ξ=0.002417 γ=-14393.9 γ=-0.173069 γ=1.16688; k=2.85664; k=0.292135; ξ=-0.383818; δ=3.51793; c=12.9895; c=13.4517; σ=0.0089; σ=0.009181; λ=0.028413; β=0.092427; β=0.044854; µ=7.57272E-4 µ=-0.001915 ξ=0.010751 γ=-0.082447 γ=-0.036491 γ=0.891234; k=3.44472; k=0.363367; ξ=-0.354908; δ=3.89805; c=8.75655; c=13.1554; σ=0.010474; σ=0.010724; λ=0.038451; β=0.081958; β=0.056179; µ=8.08894E-4 µ=-0.002495 ξ=0.009974 γ=-0.067821 γ=-0.04771 γ=0.147974; k=1.53699; k=0.687858; ξ=-0.306161; δ=2.03901; c=20.6018; c=28.7951; σ=0.00365; σ=0.003583; λ=0.006548; β=0.049523; β=0.050102; µ=1.65923E-4 µ=-0.001045 ξ=7.02308e-4 γ=-0.047997 γ=-0.048953 γ=0.311789; k=1.33067; k=0.846174; ξ=-0.318283; δ=2.36454; c=61.809; c=269.774; σ=0.003542; σ=0.003527; λ=0.007567; β=0.134097; β=0.496718; µ=1.92085E-4 µ=-9.73069E-4 ξ=0.001286 γ=-0.133001 γ=-0.496006 γ=0.148441; k=2.20099; k=0.850919; ξ=-0.287167; δ=3.06415; c=10.2451; c=34.0593; σ=0.003264; σ=0.003234; λ=0.009463; β=0.025837; β=0.059749; µ=1.86990E-4 µ=-9.44692E-4 ξ=6.70672e-4 γ=-0.023215 γ=-0.059152 γ=0.265327; k=1.23079; k=89.0854; ξ=-0.347378; δ=2.19761; c=303271.0; c=38.8987; σ=0.01147; σ=0.011457; λ=0.022465; β=2030.74; β=0.441771; µ=7.57481E-4 µ=-0.002824 ξ=0.003773 γ=-2030.74 γ=-0.500822 γ=0.392694; k=1.09695; k=2.08764; ξ=-0.360833; δ=1.66587; c=856659.0; c=116193.0; σ=0.011401; σ=0.011191; λ=0.015193; β=5246.72; β=1089.95; µ=8.36141E-4 µ=-0.002572 ξ=0.005165 γ=-5246.72 γ=-1089.95 (Source: EasyFit distribution fitting for the returns based on historical prices from Yahoo! Finance)

The theoretical VaR and CVaR estimates presented in Tables 6–9 were calculated using the following distributional parameters, which were obtained using the EasyFit package. They are provided in Table A2 in the EasyFit notation. Table A2. EasyFit distributional parameters representing the best fit for the 2007-2009 sample Ticker

SPY

MDY

IJR

IEF

LQD

TIP

VNQ

GLD

PSP

MBB

Johnson SU Burr type XII Dagum GEV GHS γ=-0.181876; k=0.90001; k=0.659823; ξ=-0.290855; δ=1.34683; c=661.429; c=25.4757; σ=0.024472; σ=0.022013; λ=0.02412; β=7.66173; β=0.265554; µ=-0.00183 µ=-0.009481 ξ=-0.006133 γ=-7.66527 γ=-0.261027 γ=-0.003691; k=0.945495; k=0.670006; ξ=-0.320864; δ=1.50078; c=56149.8; c=35.1467; σ=0.027064; σ=0.025178; λ=0.032005; β=747.128; β=0.413126; µ=-0.001674 µ=-0.009951 ξ=-0.001773 γ=-747.13 γ=-0.407554 γ=0.267226; k=1.21224; k=0.809666; ξ=-0.322667; δ=1.97268; c=1.17866E+6; c=165.755; σ=0.025889; σ=0.02522; λ=0.044209; β=17825.5; β=2.11524; µ=-0.001802 µ=-0.010064 ξ=0.005028 γ=-17825.5 γ=-2.11269 γ=-0.547947; k=3.19014; k=87.953; ξ=-0.265001; δ=4.29507; c=5.87258; c=31.0358; σ=0.006209; σ=0.0061; λ=0.025735; β=0.034982; β=0.164657; µ=4.93716E-4 µ=-0.001729 ξ=-0.002889 γ=-0.027019 γ=-0.192798 γ=0.023018; k=0.84477; k=96.1557; ξ=-0.306911; δ=1.02191; c=4.04531E+8; c=36.7977; σ=0.011877; σ=0.00883; λ=0.006979; β=1.76588E+6; β=0.696791; µ=-6.63125E-5 µ=-0.003046 ξ=1.87453E-4 γ=-1.76588E+6 γ=-0.794563 γ=0.082656; k=1.26569; k=82.6245; ξ=-0.344665; δ=1.75317; c=8.01120E+7; c=29.7008; σ=0.007144; σ=0.007047; λ=0.010538; β=335861.0; β=0.208855; µ=8.21795E-5 µ=-0.002132 ξ=6.66980E-4 γ=-335861.0 γ=-0.24582 γ=-0.05077; k=0.817695; k=0.828749; ξ=-0.242505; δ=1.69811; c=8.18932E+6; c=33.5973; σ=0.044463; σ=0.040633; λ=0.06282; β=174626.0; β=0.726039; µ=-0.002126 µ=-0.017545 ξ=-0.00436 γ=-174626.0 γ=-0.723098 γ=-0.162889; k=1.2106; k=0.582405; ξ=-0.308303; δ=1.53474; c=39.7092; c=18.2693; σ=0.019847; σ=0.019068; λ=0.024105; β=0.449647; β=0.162314; µ=6.88198E-4 µ=-0.005729 ξ=-0.002481 γ=-0.445896 γ=-0.154067 γ=-0.141378; k=0.958644; k=0.586506; ξ=-0.309054; δ=1.62674; c=1859.97; c=19.2734; σ=0.038088; σ=0.036047; λ=0.050462; β=36.195; β=0.317746; µ=-0.00402 µ=-0.016134 ξ=-0.009324 γ=-36.2 γ=-0.308263 γ=-0.162986; k=0.873214; k=0.877181; ξ=-0.251788; δ=1.97959; c=6613.47; c=31.5088; σ=0.004224; σ=0.004002; λ=0.007293; β=14.3947; β=0.069237; µ=2.95281E-4 µ=-0.001198 ξ=-3.87670E-4 γ=-14.3949 γ=-0.068592 (Source: EasyFit distribution fitting for the returns based on historical prices from Yahoo! Finance)

Appendix B — Matlab code for calculating VaR/CVaR for covered distributions The theoretical VaR V and CVaR C can be calculated in Matlab™ software package using the following code fragments with the distributional parameters given as per EasyFit notation and alpha being the probability. Johnson’s SU-distribution

Input parameters: ksi, lambda>0, gamma, delta>0. Standard functions used: exp, sinh, norminv, normcdf. V = ksi+lambda*sinh((norminv(alpha)-gamma)/delta); C = ksi+1/alpha*lambda/2*(exp((1-2*gamma*delta)/2/delta^2)* normcdf(norminv(alpha)-1/delta)-exp((1+2*gamma*delta)/2/delta^2)* normcdf(norminv(alpha)+1/delta)); Burr type XII distribution

Input parameters: c>0, k>0, gamma, beta>0. Standard functions used: hypergeom. V = gamma+beta*((1-alpha)^(-1/k)-1)^(1/c); C = gamma+beta/alpha*((1-alpha)^(-1/k)-1)^(1/c)*(alpha-1+ hypergeom([1/c k], 1+1/c, 1-(1-alpha)^(-1/k))); Dagum distribution

Input parameters: c>0, k>0, gamma, beta>0. Standard functions used: hypergeom. V = gamma+beta*(alpha^(-1/k)-1)^(-1/c); C = gamma+beta/alpha*c*k/(c*k+1)*(alpha^(-1/k)-1)^(-k-1/c)* hypergeom([k+1 k+1/c], k+1+1/c, -1/(alpha^(-1/k)-1)); The generalized extreme value (GEV) distribution

Input parameters: ksi0, sigma>0, mu. Standard functions used: log, gamma, gammainc. V = mu + sigma/ksi*((-log(alpha))^(-ksi)-1); C = mu - sigma/ksi + sigma/ksi/alpha*builtin('gamma', 1-ksi)* gammainc(-log(alpha), 1-ksi, 'upper');

The generalized hyperbolic secant (GHS) distribution

Input parameters: sigma>0, mu. Standard variables used: 1i (the imaginary unit). Standard functions used: log, tan. Non-standard function used: polylog (see below). V = mu + 2*sigma/pi*log(tan(pi*alpha/2)); C = mu + 2*sigma/pi*log(tan(pi*alpha/2)) - 2*sigma/pi^2/alpha*1i* (polylog(2,-1i*tan(pi*alpha/2))-polylog(2,1i*tan(pi*alpha/2)));

A non-standard function polylog requires the following file polylog.m: function y = polylog(n,z,acc) %%POLYLOG - Computes the n-polylogarithm of z (Li_n) % % Usage: y = polylog(n,z) % y = polylog(n,z,acc) % % Input: |z|acc); k = k+1; kn = k^n; zk = zk.*z(j); err = zk./kn; y(j) = y(j)+err; end end