Optimization Conditional VAR

Transcript of Optimization Conditional VAR

Introduction Description of the approach An application to portfolio optimization Remarks

Optimization of conditional Value-at-Risk

Petr Zahradnk

Matematicko- fyzikaln fakultaUniverzity Karlovy v Praze

20. Rjna 2008

Petr Zahradnk Matematicko- fyzikaln fakulta Univerzity Karlovy v Praze

Optimization of cVaR

1 IntroductionMotivationVaR is simply a WRONG approach

2 Description of the approachThe general conceptAdvantages of our formula

3 An application to portfolio optimizationGeneral exampleVaR, cVaR and 2

4 Remarks



Motivation

What are we here for?

We disclose a new approach to a technique for optimizing aportfolio, which calculates VaR and CVaR simultaneously.

Our approach is suitable for use for anybody investing intorisky assets.

Combined with analytical or scenario based methods choosingthe portfolios with constraints such as a minimal gain, thecalculations often come down to basic linear programming.



Motivation

VaR

QUESTION: How can we measure risk?

The (unfortunately) most popular measure of risk is theValue-at-Risk.

A more modern (and correct) approach are coherentmeasures.

Usual definition of VaR:

Given a confidence level and a time interval [t, t + ] theVar of a portfolio with a value process Vt is given by:

VaR = inf{v R : P(Vt Vt+ > v |Ft) < }



Motivation

VaR









Motivation

VaR









Motivation

VaR









Motivation

VaR









VaR is simply a WRONG approach

Example: VAR is a bad concept

QUESTION: Why on earth should any measure ofrisk have a same value for these two cases?




Example: sub-additivity assumption broken

Let us have an almost trivial portfolio consisting of two riskyassets (Xt ,Yt) and a portfolio value proces Vt = Xt + Yt .

Conditional on Ft let (Xt ,Yt) be :(0, 0) with prob. 1 2 and (c , 0) resp. (0, c) with prob. .What is VaR(X ) resp. VaR(Y )?

Yes, it is ZERO

What is VaR(V ) ?

Yes, it is c. Diversified portfolio led to a bigger risk.

I hope we agreed on the fact, that VaR is a deeply wrongway to measure risk. However, a lot of people still use it.







Yes, it is ZERO

What is VaR(V ) ?









Yes, it is ZERO

What is VaR(V ) ?









Yes, it is ZERO

What is VaR(V ) ?






Lets begin with something more reasonable then!The coherent measures are a reasonable concept- we assume:

monotonicity,

sub-additivity,

positive homogenity

translation invariance




Lets begin with something more reasonable then!The coherent measures are a reasonable concept- we assume:

monotonicity,

sub-additivity,

positive homogenity

translation invariance



The general concept

Definitions:

Suppose a threshold , a loss function f (x , y), a randomvector y having a density p(y):

The probability of f (x , y) not exceeding a threshold given adecision vector x is given by:

(x , ) =

f (x ,y)

p(y)dy

The definition of VaR to be used from now on:

(x) = min{ R : (x , ) }

The definition of cVaR:

(x) = (1 )1f (x ,y)(x)

f (x , y)p(y)dy



The general concept

Definitions:



(x , ) =

f (x ,y)

p(y)dy


(x) = min{ R : (x , ) }


(x) = (1 )1f (x ,y)(x)

f (x , y)p(y)dy



The general concept

Definitions:



(x , ) =

f (x ,y)

p(y)dy


(x) = min{ R : (x , ) }


(x) = (1 )1f (x ,y)(x)

f (x , y)p(y)dy



The general concept

Definitions:



(x , ) =

f (x ,y)

p(y)dy


(x) = min{ R : (x , ) }


(x) = (1 )1f (x ,y)(x)

f (x , y)p(y)dy



The general concept

Remarks

VaR still means the same, we have only rewritten it and thethreshold was named .

cVaR is a conditional expectation!

The definitions ensure that VaR is never more than cVaR!Therefore, portfolios with low cVaR must have low VaR aswell.

cVaR is still not generally a coherent measure. It can be easilyimproved to meet the criteria though, for example if thecumulative distribution function (x , ) =

f (x ,y) p(y)dy is

everywhere continuous wrt .

The density p(y) need not exist for the concept to workeither. The generalization is left for further study.



The general concept

The key of the concept

We want to characterize cVaR and VaR in terms of a function Fon X x R defined as follows:

F(x , ) = + (1 )1yRm

[f (x , y) ]+p(y)dy .

The crucial features of the function F are:

convexity and continuous differentiability wrt .

(x) argminRF(x , )(x) = F(x , (x)) = minR F(x , )

The interval argminRF(x , ) perhaps can reduce to a singlepoint. Also, the minimization of F means the same as minimizing(1 )F which will probably be more numerically feasible.



The general concept

The key of the concept

We want to characterize cVaR and VaR in terms of a function Fon X x R defined as follows:

F(x , ) = + (1 )1yRm

[f (x , y) ]+p(y)dy .

The crucial features of the function F are:

convexity and continuous differentiability wrt .

(x) argminRF(x , )(x) = F(x , (x)) = minR F(x , )

The interval argminRF(x , ) perhaps can reduce to a singlepoint. Also, the minimization of F means the same as minimizing(1 )F which will probably be more numerically feasible.



The general concept

The power of the formula at the previous slide is apparent:

Continuously differentiable functions are easy to minimize.

cVaR can obviously be computed without first calculating theVaR on which its definition depends.

VaR may be obtain as a by-product (for those, who still wantto know it...)

F(x , ) = + (1 )1yRm [f (x , y) ]+p(y)dy can be

approximated, for example by sampling from the distributionof y according to its density p(y). The correspondingapproximation, from a sample y1, . . . , yq would be:

F (x , ) = +1

q(1 )1

k1,...,q

[f (x , y) ]+.

This expression is piecewise linear and convex wrt . LP



The general concept

The power of the formula at the previous slide is apparent:

Continuously differentiable functions are easy to minimize.

cVaR can obviously be computed without first calculating theVaR on which its definition depends.

VaR may be obtain as a by-product (for those, who still wantto know it...)

F(x , ) = + (1 )1yRm [f (x , y) ]+p(y)dy can be

approximated, for example by sampling from the distributionof y according to its density p(y). The correspondingapproximation, from a sample y1, . . . , yq would be:

F (x , ) = +1

q(1 )1

k1,...,q

[f (x , y) ]+.

This expression is piecewise linear and convex wrt . LPPetr Zahradnk Matematicko- fyzikaln fakulta Univerzity Karlovy v Praze


Advantages of our formula

Minimizing cVaR- the formula

minxX(x) = min(x ,)XRF(x , )

The pair (x, ) achieves the second minimum xachieves the first minimum. Therefore, the joint minimizationof F leads to a pair (x

, ), such that x minimizes thecVaR and gives the corresponding VaR.f (x , y) and constraints for X convex leads to minimizingconvex functions. CPThe approximation of the integral to be minimized can againwork.

The minimization of F falls into category of stochasticprogramming.



Advantages of our formula

Minimizing cVaR- the formula

minxX(x) = min(x ,)XRF(x , )

The pair (x, ) achieves the second minimum xachieves the first minimum. Therefore, the joint minimizationof F leads to a pair (x

, ), such that x minimizes thecVaR and gives the corresponding VaR.f (x , y) and constraints for X convex leads to minimizingconvex functions. CPThe approximation of the integral to be minimized can againwork.

The minimization of F falls into category of stochasticprogramming.



General example

Suppose xj 0 with

xj = 1. Let yj be the return on the j-thinstrument. The density of the joint distribution of y denote p(y).The loss is therefore given by

f (x , y) = xT y .

Here, our VaR and cVaR will be defined in terms of the percentagereturns. We consider the case, where we have a one-to-onecorrespondance between percentage returns and a monetary value.(Which is of course not the case of portfolios with zero netinvestment)



General example

The performance function to focus on:

F(x , ) = + (1 )1yRn

[xT y ]+p(y)dy

fact: in this setting, F(x , ) is a convex function of both xand .

sometimes fact: it is also often differentiable in these variables(details not to be dealt with)

Hence we can often use the formulas formulated earlier!



General example

The performance function to focus on:

F(x , ) = + (1 )1yRn

[xT y ]+p(y)dy

fact: in this setting, F(x , ) is a convex function of both xand .

sometimes fact: it is also often differentiable in these variables(details not to be dealt with)

Hence we can often use the formulas formulated earlier!



General example

A closer look

For a more detailed look, let (x) and (x) denote the mean andthe variance of the loss associated with the portfolio respectively;in terms of mean m and variance V of y , introducing a constrainton minimal expected gain, we have:

(x) = xTm, 2(x) = xTVxX = {x : xj = 1, (x) R}.

A sample set y1, . . . , yq yields:

F (x , ) = +1

q(1 )1

k1,...,q

[xT yk ]+.

The problem broke into a problem of linear programming.



General example

A closer look

For a more detailed look, let (x) and (x) denote the mean andthe variance of the loss associated with the portfolio respectively;in terms of mean m and variance V of y , introducing a constrainton minimal expected gain, we have:

(x) = xTm, 2(x) = xTVxX = {x : xj = 1, (x) R}.

A sample set y1, . . . , yq yields:

F (x , ) = +1

q(1 )1

k1,...,q

[xT yk ]+.

The problem broke into a problem of linear programming.



VaR, cVaR and 2

Now we want to cope with the fact, that we directed the discussiontowards minimizing cVaR and calculated VaR in the meantime.Now we would like to know, as we introduced the variance of ourloss, what is the connection of following three problems:

minimize (x) over x Xminimize (x) over x Xminimize 2(x) over x X , a popular Markowitz (1952)problem.

These problems can yield, in at least one important case, thesame optimal portfolio!!!



VaR, cVaR and 2






VaR, cVaR and 2






VaR, cVaR and 2

The most important proposition

Suppose that the loss associated with each x isnormally distributed, as holds when y is normallydistributed. If 0.5 and the constraint (x) R isactive at solutions to any two of the problems, thenthe solutions to those two problems is the same; acommon portfolio x is optimal by both criteria.



VaR, cVaR and 2

The most important proposition- continued

The proof can be done using the capabilities of mathematicalsoftware which expresses the VaR and cVaR in terms of themean and variance of the Normal distribution. The mean isthen replaced by R as the constraint is assumed to be active.The proposition gives us an opportunity to test the methodsproposed earlier by computing the optimal portfolio usingquadratic programming solutions for the Markowitz problem.



VaR, cVaR and 2

Of course, the real life shows that the assets to deal with are oftenvery far from being normally distributed. The cVaR optimizingmethod works very well even in these cases, as it is independent ofthe normality assumption, and hence can be used broadly. As[Rockafellar and Uryasev] show on a hedging example, it worksmuch better than other commonly used parametric and simulationVaR techniques. For example, it copes much better with varyingmore then one position within a specified range etc., also usingso-called non-smooth optimization techniques. Numericalcomputations have been conducted for very large problems andproved valid results.



What did we manage?

We showed how to minimize cVaR and compute VaRsimultaneously.

We stressed why VaR shouldnt be used.

We had fun and/or slept well.



What did we manage?

We showed how to minimize cVaR and compute VaRsimultaneously.

We stressed why VaR shouldnt be used.

We had fun and/or slept well.



Voila: The End.

Any questions?

Thank you for attention.

Bibliography:Optimization of conditional valute-at-risk by R.T. Rockafellarand S. Uryasev, available online for free.A paper by Karel Janecek athttp://www.rsj.cz/index.php?id document=200576104

webpage: petr.anoel.eu

mailto: [email protected]



Voila: The End.

Any questions?

Thank you for attention.

Bibliography:Optimization of conditional valute-at-risk by R.T. Rockafellarand S. Uryasev, available online for free.A paper by Karel Janecek athttp://www.rsj.cz/index.php?id document=200576104

webpage: petr.anoel.eu

mailto: [email protected]



IntroductionMotivationVaR is simply a WRONG approach

Description of the approachThe general conceptAdvantages of our formula

An application to portfolio optimizationGeneral exampleVaR, cVaR and 2

Remarks

Optimization Conditional VAR

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Transcript of Optimization Conditional VAR