long short portfolio optimisation under mean-variance-cvar framework gautam mitra carisma, brunel...

Long Short Portfolio Optimisation underMean-Variance-CVaR Framework

Gautam MitraCARISMA, Brunel University and OptiRisk systems, UK

Diana RomanCARISMA, Brunel University and OptiRisk systems, UK

Ritesh KumarIndian Institute of Management, Calcutta, India

http://www.brunel.ac.uk/

http://www.optirisk-systems.com/

Outline



Portfolio selection problem

Preliminaries

Initial Amount of capital to invest, n asset in which investment can be made.

xj : The proportion of wealth invested in asset j

Portfolio x=(x1,…,xn) has return Rx=x1R1+…+xnRn (Rx random variable).

Another portfolio y=(y1,…,yn), has return Ry=y1R1+…+ynRn (Ry random variable)

How do we choose between Rx and Ry?

Problem of deciding between Random variables when larger outcomes are preferred.



Portfolio selection problem

Preliminaries

Discrete case: Use of Scenario models to obtain a representation for Random variables Rx when making portfolio choice x.

Consider the case of T scenarios.

pi=probability of scenario i occurring; p1+p2+…+pT=1

rij = return of asset j in scenario i;

So asset j return Rj finitely distributed over {r1j, r2j,…,rTj}

The portfolio return Rx finitely distributed over {R1x,…,RTx}

Where Rix = x1ri1+…+xnrin is the return in scenario i.



Mean Risk Models

Preliminaries

Defining a preference relation for choosing a Random Variable

Distributions are described and compared using Mean and value of a risk measure

In the mean-risk approach, RX is preferred to r.v. RY (or RX dominates RY) if and only if it has greater expected value and less risk

Let be a risk measure: a function mapping random variables into real numbers.

RX is preferred to r.v. RY : (E(RX)E(RY) and (RX) (RY)) with at least one strict inequality.



Mean Risk Models

Preliminaries

The efficient (non-dominated) solutions are the Pareto optimal solutions of a two objective problem:

Max (E(RX), -(RX))Subject to: xP

P=the set of feasible decision vectors

Optimisation approach (One possible approach, not the only one!! )

Min (RX)Subject to: E(RX)d

xP



Mean Risk Models

Preliminaries

Pros:• Computationally convenient

• Ready Interpretation of results

Cons: • Over simplified approach – Distribution described by just two parameters!

• Different risk measures – Different solutions

• The question of what kind of risk measure to use is still open.



Risk Measures

Preliminaries

Dispersion Type of Risk Measures (Measure the deviation from a target)

Symmetric risk measures:

1952: Variance (Markowitz): QP Model

MAD: Due to computational difficulty of QP Models Not the case with the solvers today!!

Asymmetric risk measures:

Semi-Variance (Markowitz 1959)

Lower partial moments (Bawa 1975, Fishburn 1977)

( , ) {[max(0, )] } ( ) ( )x xLPM R E R r dF r



Risk Measures

Preliminaries

Several financial disasters raised the problem of hedging against worst case scenarios.

Risk measures concerned with the left tail of distributions:

• VaR (1993, G-30& JP Morgan): although it has several important pitfalls, it is a standard in banking. Difficult to optimise!!!

• CVaR, also called “Expected Shortfall”, or “Tail VaR” (2000, Rockafellar and Uryasev): Attractive theoretical and computational properties.



Risk Measures

Preliminaries

Variance: Measures the spread around the expected value.

Calculation:

If Rx=x1R1+…+xnRn, then

where ij= Cov(Ri,Rj)

Thus, quadratic function of x1,…,xn

]))([()( 2xxx RERER

n

jiijjix xxR

1,

,)(



Risk Measures

Preliminaries

CVaR: Let A% =(0,1). CVaR at confidence level of Rx is the mathematical transcription of the

concept “average of losses in the worst A% of cases”.

Formally:

CVaR is defined using -tail.

The -tail distribution of Rx: the lower part of the distribution of Rx (corresponding to extreme unfavourable outcomes) with distribution function rescaled to span [0,1].

CVaR is minus the mean of the -tail distribution



Risk Measures

Preliminaries

CVaR Example:

Rx a discrete random variable with 13 equally probable outcomes (arranged in ascending order) : -1, -0.75, -0.5, -0.25, -0.1, 0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6

At =0.1 (1/13,2/13) consider the 2 worst outcomes.

The -tail has outcomes –1 and –0.75 with probabilities 1/13:1/10=10/13 and 3/13.

At =0.25 (3/13,4/13) consider the 4 worst outcomes.

The -tail has outcomes –1, –0.75, -0.5 and –0.25 with probabilities 1/13:1/4=4/13, 4/13, 4/13 and 1/13.

94.0)13

375.0

13

101()( xRCVaR

71.0)13

125.0

13

45.0

13

475.0

13

41()( xRCVaR



Risk Measures

Preliminaries

CVaR Optimisation:

CVaR is calculated and optimised using an auxiliary function (Rockafellar and Uryasev 2000).

For discrete Random Variables, CVaR optimisation is a LP

vRvEvxF x }]{[1

),(

vrxvp ij

T

i

n

jji

][

1

1 1

Where [u]+ =u for u0 and [u]+ =0 for u<0.



Risk Measures

Preliminaries

CVaR Calculation:

CVaR Optimisation:

Moreover: x* minimises CVaR over P v*R such that (x*,v*) minimises F(x,v) over PxR:

),(min)( vxFRCVaRRv

x

),(min)(min),(

vxFRCVaRPxRvx

xPx



Motivation & Formulation

Mean-Variance-CVaR Model

• Variance still the most widely used risk measure by Fund Manager• CVaR is gaining popularity with Regulators

Risk quantification from different perspectives leads to differentsolution portfolios when minimised.

• Mean-Variance portfolio may have excessively large CVaR !!

• Mean-CVaR portfolio may have excessively large variance !!

Our approach: Trade-off between Variance-CVaR. Distributions arecompared using 3 parameters: Mean, Variance and CVaR (Roman et al 2007).



Motivation & Formulation


Preference relation:

Rx is preferred to Ry (or the portfolio x is preferred to portfolio y) if:

E(Rx)E(Ry), 2(Rx) 2(Ry), CVaR(Rx)CVaR(Ry)

With at-least one strict inequality.

The non-dominated (efficient) solutions are the Pareto optimal solutions of

a three-objective problem:

(MVC): max (E(Rx), -2(Rx), - CVaR(Rx))

Subject to: xP.



Optimisation approach


The -constraint method in multi-objective optimisation:

• Optimise one of the objective functions

• Impose limits on the other ones and transform to constraints

We choose to optimise Variance while limits are imposed on CVaR and Expected return.

Representing CVaR constraint: The same function F used for minimising CVaR

may be used for imposing a constraint on CVaR, while

maximising the mean (Krokhmal et al., 2002) => F may be generally used for

imposing a constraint on CVaR.

Single objective problem



Optimisation approach


(P): min 2(Rx) Subject to: F(x,v)z E(Rx)d xP, vR

A QP Model !!

If the covariance matrix is positive definite ( no replication of assets, no risk-free asset), and the constraints on mean and on CVaR are active, (P) is equivalent to (MVC).

x*P is a Pareto efficient solution of (MVC) v*R such that (x*,v*) optimal solution of (P) with z= F(x*,v*), d= E(Rx*).



Model details

Long Only Model

1 1

minn n

j k jkj k

x x

dx j

n

jj

1

:toSubject

zvypT

iii

1

1

1

n

i j ijj

y v x r

i{1,…,T}

0iy i{1,…,T}

(x1,…, xn) P



Obtaining Efficient Solution Sets

Long Only Model

• Varying the RHS in the constraints on mean and on CVaR such that these constraints are active produces the entire set of efficient solutions in the mean-variance-CVaR model.

• d must be greater than the expected returns of both the minimum variance portfolio (mean-variance efficient) and the minimum CVaR portfolio (mean-CVaR efficient).

• For d fixed, z must be less than the minimum CVaR of a mean-variance efficient portfolio with expected return d (for an active constraint); z must be greater than the minimum CVaR for expected return d (for feasibility).




Long Only Model

Generally, the mean-variance and mean-CVaR efficient solutions are just particular solutions of (MVC).

But: if there are several mean-variance efficient portfolios with different CVaR-s, only the portfolio with the lowest CVaR is an efficient solution of (MVC).

If there are several mean-CVaR efficient portfolios with different variances, only the portfolio with the lowest variance is an efficient solution of (MVC).

Thus, (MVC) does not exclude mean-variance and mean-CVaR models but it “embeds” them.




Long Only Model

CVaR

variance

PMin CVaR

Max Variance

PMax CVaR

Min Variance



Motivation- Short selling

Short Selling in Practice

• In the Long only model - Portfolio weights are positive

• Investors can sell short (portfolio weights can be negative) certain securities, which mean that they can sell investments that they do not currently own.

• Long only portfolio focuses on “Winning securities” ignoring a whole class of “losing securities”.

• A Long-Short portfolio expands the scope of the investor’s sphere of portfolio decisions -> Likely to improve performance than a Long only portfolio.

• Active portfolio selection - Freedom from the restrictions imposed by individual securities’ benchmark weights.



Short selling Constraints in Our Model


• Net Margin Requirements as dictated by regulatory restrictions.

Regulation T specifies that the sum of long positions plus the sum of short positions should not exceed twice the equity in the account.

Regulation T currently specifies H=2

n

Lj Sjj=1

(x x ) ; H

n

j Lj Sjj=1

(m x x ) 1; jn

Lj

Sj

j

x represents the portfolio weight in long position in j-th stock

x represents the portfolio weight in short position in j-th stock

m represents the margin requirements for long position in j-th stock

represents the margin requirements for short position in j-th stockjn





• Investor’s strategy: Investor may follow market neutral strategy or full market exposure strategy with 120:20 strategy etc.

Total value of Long – Total value of short, be close to some investor specified value, ν

Where τ represents a small non negative tolerance level.

For market neutral strategies ν = 0

For full market exposure ν = 1

Lj Sj1

(x - x ) ;n

j





• Budget Constraint

•Short Rebate

When a security is sold short, the proceeds of the sale are deposited with lender of the stock as collateral. The lender would invest that money in some bank account which earns interest. After taking a proportion of the interest, rest of the interest is passed back to the investor (called short rebate). The percentage of interest that the investor would get back depends on the negotiation.

n

Lj cj=1

1; bx x x c represents the cash balance.

represents the amount borrowed.b

x

x





Easily represented as: Modeled as:

0, j 1...n 1, j 1...n, ,Lj Sj Lj Sj Lj Sjx x binary

* * ; 1...

* * ; 1...

Lj lower Lj Lj upper

Sj lower Sj Sj upper

L x L j n

S x S j n

represents the lower limit for the individual long position.

represents the upper limit for the individual long position.

represents the lower limit for the individual short posit

lower

upper

lower

L

L

S ion.

represents the upper limit for the individual short position. upperS

, binaryLj Sj



Long-Short Model

Long-Short Model

• We represent n assets using 2n non negative variables (n variables representing long positions and n variables for short positions.)

• Integrated Optimisation of Long-Short positions (Jacobs et al 1999, 2006)

• Variance for Long-Short positions

Co-variance Matrix for the integrated optimisation

Where Q represents the co-variance matrix between the n assets for long only model

Q Q

Q Q

1 1{ ( ,..., , ,..., ) | , subject to constraints, 1... } LSL Ln S Sn Lj Sjx x x x x x j n



Long-Short Model

Long-Short Model

1 1

( -2 ) ;

represents covariance between stock j and k.

n n

jk Lj Lk jk Sj Sk jk Lj Skj k

jk

Li

x x x x x x

x

Si

represents long position in stock i.

represents short position in stock i.x

n

j=1

( ) ; j Lj j Sj cash Sj jx x r x h d d represents desired return,

represents stock's expected return,

represents return on cash,

represents short rebate ratio.cash

j

r

h



Long-Short Model

Long-Short Model

1

( )n

i ij Lj ij Sj cash Sj jj

y v r x r x r x h

zvypT

iii

1

1

0iy



Cardinality constraint

Cardinality constraint

1

1

1

For long only model:

For long-short model:

...( Cardinality on long positions only)

( ) ...(Cardinality on long-short positions)

Where

k = Cardinality value.

n

jj

n

Ljj

n

Lj Sjj

k

k

k



Long-Short Model: Formulation for Scenario model

Long-Short Model: Complete formulation

1 1

n

j=1

1

1

min ( - 2 )

subject to

( ) ; is the desired expected return level

( ) 1...

1

n n

jk Lj Lk jk Sj Sk jk Lj Skj k

j Lj j Sj cash Sj j

n

i ij Lj ij Sj cash Sj jj

T

i ii

x x x x x x

x x r x h d d

y v r x r x r x h i T

p y

n

Lj Sjj=1

n

Lj Sj1

Lj1

Sj1

; z is the desired CVaR level

(x x ) 2;

( ) (x - x ) ( ) (Investor's strategy)

1.2

(Investor's strategy as modeled in our case)

0.2

j

n

j

n

j

v z

x

x



Long-Short Model: Formulation for Scenario model

Long-Short Model: Complete formulation

1

n

Lj cj=1

c

* * ; 1...

* * ; 1...

( ) ; is the cardinality value

1;

1; 1...

0, 0 1...

0, 0

Lj lower Lj Lj upper

Sj lower Sj Sj upper

n

Lj SjJ

b

Lj Sj

Lj Sj

b

L x L j n

S x S j n

k k

x x x

j n

x x j n

x x

y

0 1...

, binaryi

Lj Sj

i T



Computational Results: Analysis and Discussion


Dataset

• 76 assets from FTSE 100 index

• 132 time periods, considered as equally probable scenarios

• Monthly returns: rij, i=1,…,T; j=1,…,n

• Confidence level =0.01

Short selling parameters

• Return on cash( ) = 4% per annum

• Short rebate ( ) = 80%

• Full market exposure (ν=1) and 120:20 strategy, Tolerance τ=0

cashr

jh





Exp return Long only Long-Short

Minimum 0.0093 0.0109

Maximum 0.0331 0.0395





In-sample analysis, CVaR-variance plot, Expected return level of 0.0188





In-sample analysis, CVaR-variance comparison, Expected return = 0.018801

Expected return

Long Long-ShortCVaR Variance Number of stocks CvaR Variance Number of

stocks

Long Short

0.018801 0.091855 0.002649 7 0.046845 0.001718 12 1

0.100458 0.002035 11

0.057542

0.001235 20 3

0.109060 0.001929 14

0.068240

0.001121 21 5

0.117663 0.001861 13

0.078937

0.001080 21 5

0.126266 0.001838 13

0.089635

0.001069 23 7





Long vs. Long-Short : Portfolio return distribution comparison for the in-sample analysis

Expected return Identifier CVaR Median Std Dev Skewness Kurtosis Min Max

Long

0.0188 Min CVaR-Long 0.0919

0.0200 0.0515 0.1051 0.8166 -0.0919 0.1883¼ CVaR-Long 0.1005 0.0219 0.0451 -0.2793 0.5886 -0.1005 0.1400½ CVaR-Long 0.1091 0.0219 0.0439 -0.3578 0.7488 -0.1109 0.1329¾ CVaR-Long 0.1177 0.0225 0.0431 -0.4437 1.0163 -0.1218 0.1274

Max CVaR-Long 0.1263

0.0223 0.0429 -0.5053 1.3092 -0.1322 0.1267Identifier Long-Short

Min CVaR-LS 0.0468 0.0204 0.0415 0.2896 -0.2423 -0.0468 0.1533¼ CVaR-LS 0.0575 0.0200 0.0351 -0.1647 -0.3352 -0.0575 0.1068½ CVaR-LS 0.0682 0.0222 0.0335 -0.3930 0.3162 -0.0682 0.1006¾ CVaR-LS 0.0789 0.0214 0.0329 -0.5119 0.6714 -0.0801 0.0963

Max CVaR-LS 0.0896 0.0202 0.0327 -0.6317 1.0559 -0.0930 0.0927





Long-short portfolio without cardinality: Out of Sample Analysis• Next 18 period Out of Sample data

Long vs. Long-Short : Out-of-sample return statistics

(For in-sample portfolios constructed at expected return level of 0.0188 and at different CVaR levels)

Expected return Identifier In-sample CVaR Mean Median Std Dev Min Max

Long

0.0188

Min CVaR-Long 0.0919 0.0113 0.0102 0.0287 -0.0425 0.0818

¼ CVaR-Long 0.1005 0.0125 0.0138 0.0319 -0.0326 0.0875

½ CVaR-Long 0.1091 0.0123 0.0131 0.0322 -0.0361 0.0827

¾ CVaR-Long 0.1177 0.0124 0.0124 0.0322 -0.0382 0.0780

Max CVaR-Long 0.1263 0.0123 0.0123 0.0326 -0.0401 0.0729

Identifier In-sample CVaR Long-Short

Min CVaR-LS 0.0468 0.0207 0.0242 0.0270 -0.0342 0.0784

¼ CVaR-LS0.0575

0.0153 0.0170 0.0266 -0.0256 0.0662

½ CVaR-LS0.0682

0.0136 0.0144 0.0264 -0.0246 0.0703

¾ CVaR-LS0.0789

0.0131 0.0134 0.0262 -0.0236 0.0743

Max CVaR-LS0.0896

0.0131 0.0137 0.0259 -0.0241 0.0749





Portfolio out-of-sample cumulative returns for 01-01-2004 to 06-01-2005 (In-sample portfolios constructed at expected return of 0.0188 and at different CVaR levels)




Portfolio selection models with threshold and cardinality constraints Lower limits on portfolio positions = 0.0001, Upper limit on portfolio positions = 1

Long only model: In-sample analysis

In-sample analysis, Long only portfolio, Portfolio return distribution’s statistics, Expected return level of 0.0188

Expected return CVaR Portfolio details Cardinality value

Median Std Dev Skewness

Kurtosis Min Max No. of Stocks

0.0188

0.1005

Long only without cardinality restriction

No cardinality restriction

0.0219 0.0451 -0.2793 0.5886 -0.1005 0.1400 11

Long only with cardinality restriction

10 0.0220 0.0451 -0.2580 0.5686 -0.1005 0.1400 10

8 0.0196 0.0453 -0.2436 0.5165 -0.1014 0.1438 8



Long only with threshold and cardinality constraints: Out-of-sample analysis

Long only portfolio, Out-of-sample portfolio return statistics for in-sample portfolios constructed at expected return of 0.0188 and at CVaR level of 0.1005

Expected return In-sample CVaR

Portfolio details Cardinality value

Mean Median Std dev Min Max

0.0188

0.1005

Long only without cardinality restriction


0.0125 0.0138 0.0319 -0.0326 0.0875

Long only with cardinality restriction

10 0.0125 0.0126 0.0320 -0.0326 0.0872

8 0.0124 0.0122 0.0325 -0.0346 0.0892



Long-short with threshold and cardinality constraints: In-sample analysis

In-sample analysis, Long-short portfolio, Portfolio return distribution’s statistics, Expected return level of 0.0188

Expected return

CVaR Portfolio details

Cardinality value

Median Std Dev Skewness

Kurtosis Min Max No. of stocks

Long Short

0.0188

0.0575

Long-short without cardinality restriction


0.0200 0.0351 -0.1647 -0.3352 -0.0575 0.1068 20 3

Long-short with cardinality restriction on long positions

10 0.0198 0.0364 -0.1092 -0.2880 -0.0575 0.1151 10 7

8 0.0204 0.0379 -0.0736 -0.3500 -0.0575 0.1206 8 5

Long-short with cardinality restriction on long-short positions

10 0.0223 0.0381 -0.2301 -0.4870 -0.0579 0.1119 8 2

8 0.0156 0.0406 0.1962 0.0575 -0.0575 0.1587 7 1



Portfolios with threshold and cardinality constraints: In-sample plots



Long-short with threshold and cardinality constraints: Out-of-sample analysis

Long-short portfolio, Out-of-sample portfolio return statistics for in-sample portfolios constructed at expected return level of 0.0188 and at CVaR level of 0.0575

Expected return In-sample CVaR

Portfolio details Cardinality value Mean Median Std dev Min Max

0.0188

0.0575

Long-short without cardinality

0.0153 0.0170 0.0266 -0.0256 0.0662 0.0153

Long-short with cardinality restriction on long positions

10 0.0150 0.0214 0.0298 -0.0323 0.0764

8 0.0146 0.0186 0.0297 -0.0307 0.0820

Long-short with cardinality restriction on long-short positions

10 0.0177 0.0237 0.0298 -0.0307 0.0730

8 0.0168 0.0198 0.0267 -0.0387 0.0692



Portfolios with threshold and cardinality constraints: Out-of-sample cumulative returns

Open issues – Our current research focus

Open issues

1. Mutual exclusivity of long and short positions – Is it really an industry requirement? Need further and deeper analysis of the properties of the efficient frontier.

2. Margin requirements - A very simplistic treatment in this model. A better treatment under mean-variance framework given in Kwan (2004) We need to include the interest rates of borrowing and lending for the amount borrowed (For long positions) and amount deposited (For short sales).

3. Investor’s strategy – What strategy do you follow? Is 120:20 strategy used in practice?

Conclusions

Conclusions

• In MVC model we try to address both classical fund manager’s and regulator’s point of view.

• Distributions compared using 3 statistics: Mean, Variance and CVaR.

• The Mean-Variance and Mean-CVaR models are “embedded” but the MVC model discriminates “positively” between Mean-Variance solutions with different CVaR levels.

• Some solutions (excluded by classical mean-risk models) may be an improvement: better left tail than mean-variance efficient distributions + smaller variance than mean-CVaR efficient distributions.

• Long-Short optimisation able to achieve superior optimal portfolios at same return level: lower CVaR and Variance.

References

1. Jacobs, B., I Levy K.N. , Starer, D. (1999). "Long-Short Portfolio Management: An Integrated Approach." The Journal of Portfolio Management.

2. Jacobs, B., I Levy K.N., Markowitz,H.M. (2006). "Trimability and Fast Optimization of Long-Short Portfolios " Financial Analyst Journal Vol:62(2).

3. Krokhamal, P., Palmquist, J. and Uryasev, S. (2002): “Portfolio Optimisation with Conditional Value-at-Risk Objective and Constraints”. Journal of Risk, 4, 43-68

4. Kwan, Clarence C.Y. (2004): “Long-short portfolio modeling: Critique and Extension”. International Journal of Theoretical and Applied Finance,Vol 7:1.

5. Markowitz., H. M. (1952). "Portfolio selection." Journal of Finance Vol:7: 77-91.6. Rockafellar, R. T., Uryasev, S. (2000). "Optimization of Conditional Value at

Risk." Journal of Risk Vol:2: 21-42.

References…

7. Roman, D., Mitra G., Darby-Dowman, K. (2007). "Mean-Risk Models Using Two Risk Measures: A Multi-Objective Approach.“ To appear in Quantitative Finance

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