A Study in Joint Density Modeling in CVaR Optimization€¦ · A Study in Joint Density Modeling in ... cogently incorporated into, the above optimization ... Advisors A Study in

A Study in Joint Density Modeling in CVaROptimization

chris bemisWhitebox Advisors

January 7, 2010

chris bemis Whitebox Advisors A Study in Joint Density Modeling in CVaR Optimization

”The ultimate goal of a positive science is the development of a‘theory’ or ‘hypothesis’ that yields valid and meaningful (i.e.,not truistic) 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 notoriously nonstationary, though:

It is clear why this is troublesome.








Markowitz’ formulation for optimal portfolios alsopresupposesI 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














Lehmann (1990) provides evidence that returns exhibitnegative serial autocorrelation.Lo and MacKinlay (1990) provide more color and suggestpositive autocorrelation between assets is exhibited and thatthis explains Lehmann’s results.Shah (2008) shows that daily volatility, when annualized,exceeds annualized monthly volatility, and suggests negativeserial autocorrelation as an explanation.






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)<γp(y)dy

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



Ψ(x,γ) =∫

f(x,y)<γp(y)dy




Ψ(x,γ) =∫

f(x,y)<γp(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 valueat risk, 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.We see now the primacy of correct scenario generation.Our project will simulate asset returns for each scenario,necessitating a structure for the joint density.



minw∈W

CVaRα(w),




minw∈W

CVaRα(w),




minw∈W

CVaRα(w),




minw∈W

CVaRα(w),



In the tradition of Friedman’s quote above, we will examine theout of sample performance of CVaR optimized portfolios undervarious densities.We will of course examine a multivariate normal assumption.We will also look at using a multivariate Student t distributionas well as a mixed multivariate Student t.Other ideas may be entertained/entertaining.








fin.


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A Study in Joint Density Modeling in CVaR Optimization€¦ · A Study in Joint Density Modeling in ... cogently incorporated into, the above optimization ... Advisors A Study in