a study of efficiency in cvar portfolio optimization · a study of efﬁciency in cvar portfolio...

A Study of Efficiency in CVaR PortfolioOptimization

chris bemisWhitebox Advisors

January 5, 2011

chris bemis Whitebox Advisors A Study of Efficiency in CVaR Portfolio Optimization

”The ultimate goal of a positive science is the development of a‘theory’ or ‘hypothesis’ that yields valid and meaningful (i.e., nottruistic) predictions about phenomena not yet observed.”

Milton Friedman

Many are familiar with the following optimization problem,

minimize w ′Σwsubject to µ ′w ⩾ α

1 ′w = 1w ⩾ 0,

suggested by Markowitz in 1952.

Many are familiar with the following optimization problem,

minimize w ′Σwsubject to µ ′w ⩾ α

1 ′w = 1w ⩾ 0,

suggested by Markowitz in 1952.

Financial data are (most likely) nonstationary, though:

For a single variable, the variance of the error in sample mean,µ̄ converges at a rate of 1

n .And the variance of the error in sample variance, σ̄ convergesat a rate of 1√

n .So that, disregarding correlation, we need very large samplesizes to obtain realistic estimates of first and second moments.

For a single variable, the variance of the error in sample mean,µ̄ converges at a rate of 1

n .And the variance of the error in sample variance, σ̄ convergesat a rate of 1√

n .So that, disregarding correlation, we need very large samplesizes to obtain realistic estimates of first and second moments.

Markowitz’ formulation for optimal portfolios also presupposesI Every investor has the same utility over a fixed horizonI That utility is quadratic in risk; viz., varianceI This necessitates (or is justified by) a geometric brownian

motion for the underlying assetsSerial independence is assumed for returns at all time levels inthe GBM case

What is important to note from the above is that further (e.g.,post 1970) studies into the dynamics of returns suggest amodification to the underlying assumption of a GBM dynamic.These new features are not compatible with, and cannot bedirectly or cogently incorporated into, the above optimizationproblem.Promising suggestions which maintain Markowitz’ frameworkinclude Goldfarb and Iyengar’s (2003) robust portfoliooptimization method.We will pursue another avenue...

For a vector of portfolio weights, w, and a ’scenario’, y, definethe function f ,

f(w, y) : Rn ×Rm → R

to be the loss of the portfolio allocated according to w underscenario y.We will call a positive value from f a loss.

For a vector of portfolio weights, w, and a ’scenario’, y, definethe function f ,

f(w, y) : Rn ×Rm → R

to be the loss of the portfolio allocated according to w underscenario y.We will call a positive value from f a loss.

Assuming that the scenarios have probability density functionp, the cumulative distribution function of losses, given portfolioweights w, is

Ψ(x,γ) =

∫f(x,y)<γ

p(y)dy

Notice, our framework is about as general as possible. This isintentional

Ψ(x,γ) =

∫f(x,y)<γ

p(y)dy

Ψ(x,γ) =

∫f(x,y)<γ

p(y)dy

We next define the value at risk for a given threshold, α:

VaRα(w) = min{γ ∈ R |Ψ(w,γ) ⩾ α}

We have that VaRα(w) is the smallest amount of loss that wecan expect with probability 1 − α

And while this particular risk measure has gained traction, weprefer a more robust measure - CVaR

The VaR construction ignores tail behavior. Conditional value atrisk, or CVaR, incorporates the tail past the VaR value; viz.,

CVaRα(w) =1

1 − α

∫f(w,y)⩾VaRα(w)

f(w, y)p(y)dy

We can discretize this in a natural way by sampling ourscenarios discretely according to p

CVaRα(w) =1

1 − α

f(w, y)p(y)dy

CVaRα(w) =1

1 − α

f(w, y)p(y)dy

Assuming we can do what was just suggested (we can, seeRockafeller (1999)), we may write another optimizationproblem:

minw∈W

CVaRα(w),

A linear programming problem.However, a problem that increases linearly with the number ofscenarios used.

minw∈W

CVaRα(w),

minw∈W

CVaRα(w),

minw∈W

CVaRα(w),

Based on what we observed in convergence of mean andvariance, we will need many, many scenarios to reflect even thefirst two moments.The LP problem may not be stable for large numbers ofscenarios, howeverWe therefore consider other formulations of the CVaR objectiveproblem. In particular

I A smoothed approximation as in Alexander, Coleman, andLi (2004)

I A fast gradient descent method proposed by Iyengar andMa (2009)

I A convolution smoothing model constructed in my IMAworkshop (2010)

We will be mainly interested inI Run time of the various methods as a function of assets

and as a function of scenariosI AccuracyI Out of sample performance

a study of efficiency in cvar portfolio optimization · a study of efﬁciency in cvar portfolio...

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