Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

Modeling, estimation, and optimization of equity portfolios with heavy-tailed distributions 133

where the ETL is the expected tail loss, also known as conditional value at risk
(CVaR), is defined as:

ETL Xα VaR X dqq

α

α

() ()

1

∫ 0

,

and

VaR XqX()F^1 ()q inf{ (x P Xx) q}

is the value at risk (VaR) of the random return X. If we assume a continuous dis-
tribution for the probability law of X , then ETL α ( X )   E (X | X   VaR α ( X ))
and, therefore, ETL can be interpreted as the average loss beyond VaR. Figure
5.1 shows the values of this performance ratio when α  0.01  β and the
components of three assets vary on the simplex:

SIMP(, , )x x x 123 R^3 / i 1 xi ;xi

3

{ ∈ ∑^10 }.

As we can see from Figure 5.1 , this performance ratio admits more local
maxima. In our comparison, we consider the parameters α  0.35 and
β  0.1.
The Rachev higher moments ratio (RHMR)^21 is given by:

RHMR x r

vxr r

xr r

TTb

TTb

()

()

()

,

,











11 1

ρ11 1+

where

vxrr Exrrxrr F p

aE

xr r

bbbxrr

i

i

b

x

(^11) b
2
4
()( /    ^1 ())





∑
σ  
 
rr
i
b xr r i
b
xr r F bp
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′
(^1) ();;ρ
1
1
2
4
1
()
(/ ())
xr r
Exr r xr r F q
bE
xr r
b
bbxrr
i
i
b

  





∑
bb
xr r
i
bxrri
b
σ xr r F bq

 

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⎟
,
σxr r b is the standard deviation of x r  r b , a i , ... , b i  R and p i , q i  (0,1).
21 See Ortobelli et al. (2009).