More than you ever wanted to know about conditional value at risk optimization 287
Optimization software like NUOPT ™ for S-PLUS ™ can deal with these
problems efficiently. See Scherer and Martin (2005) for a complete set of code
for various kinds of problems including transaction costs, cardinality con-
straints, etc. A closer look at Equations (12.3) – (12.7) will reveal the mechanics.
Given that excesses are constrained to be positive via Equation (12.6), any loss
larger than value at risk will have a negative impact on the objective (12.3).
Excesses can be kept small by choosing w i in Equation (12.5) to prevent e s
from becoming large. However, if portfolio returns were all positive for a given
set of weights, how can we prevent e s from becoming negative? The optimizer
can now increase VaR in order to improve Equation (12.3) and relax Equation
(12.5). Given a linear objective function with linear inequality constraints, we
have a well-developed technology at hand to solve this type of problems. This
also applies to wider problems like the inclusion of integer variables in order to
model cardinality constraints (max number of assets) or lot sizes. Risk budgets
for CVaR are easily calculated by taking the numerical derivative:
dCVaR
dwCVaR w CVaR w
i () ()iiΔΔ
Δ−(^2)
(12.8)
for a given scenario matrix, where Δ represents an infinitesimal change in the
weight of the i th asset. For large enough scenario matrices, this function will be
sufficiently smooth. Alternatively, Tasche (1999) has shown that we can esti-
mate Equation (12.8) as expected value from a large set of scenarios (drawn
from bootstrapping or Monte Carlo simulation):
dCVaR
dw
Er r VaR r I
i
ip p S s is s
S
(| )
11
α()∑^1
(12.9)
where I s 1 for r p VaR p and zero otherwise. Again, i denotes the respec-
tive asset and VaR p stands for portfolio VaR. We simply calculate the expected
performance contribution from asset i given that portfolio return falls below
portfolio VaR.^5
The practical usefulness of CVaR for portfolio construction relies on our
ability to construct a predictive scenario matrix, i.e., a rectangular array
of dimension S n that contains returns r is for i 1, ... , n assets (columns)
across s 1, ... , S scenarios (rows). Scenarios need to be representative (reflect
expectations about future returns), approximate (approximate the distribu-
tion closely), parsimonious (offer a relatively limited set of scenarios to remain
computationally feasible), and finally arbitrage free (if the number of scenarios
is less than the number of assets involved, our hypothetical economy is not
5 Note that for VaR we would use
dVaR
dwi Er r(|ipVaRp). However, given that the equal-
ity sign will never be true for continuous distributions, we could use the kernel estimator sug-
gested by Tasche (2007). Tasche also provides R/S-Plus code on his web page, so we don’t need to
describe the procedure here.