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Morgan Stanley Investment Management

Portfolio Management Today A Tour Around Recent Advances in PortfolioConstruction

Dr. B. Scherer, September 2008


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Table of Contents

Section 1 The Evolution of Risk Measures

Section 2 Postmodern Portfolio Theory (PMPT)

Section 3 Estimation Error: Bayesian Estimates versus Robust Estimation

Section 4 Fairness in Asset Management

Section 5 Optimal Leverage: About Feathers and Stones

Overview


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Axioms on Random Variables versus Behaviour

Section 1: The Evolution of Risk Measures

Statistics versus Financial Economics

Why do we need a goodrisk measure?

- Pricing in incompletemarkets

- Portfolio selection

Can we invent riskmeasures arbitrarily?

Actuarial research / Statistics (Axioms on random variables)

define a set of reasonable axioms

deduce a risk measure or criticize risk measures

dependent on the appropriateness of axioms

Financial economics (Axioms on behaviour)

Decision making under uncertainty(NEUMAN/MORGENSTERN)

deduce a risk measure or criticize risk measures

dependent on the appropriateness of axioms

Without a good risk (better risk/reward) measure there is nomeaningful portfolio selection


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Decision theoretic Foundation

Above holds as long as returns are not too non-normal(MARKOWITZ/LEVY, 1979); many studies confirm this

Caution: An often made mistake (maximizing net presentvalue and utility optimization have nothing in common)

Modern Finance is about the separation of preferences andvaluation, see your favourite corporate finance textbook(only actuaries ignore this)

( ) ( )( ) 21/

arg max arg max 1w

w w w wnP P T T

i iiw

Neumann Morgenstern MARKOWITZ

E U Wealth E U w r =

= = +

( )

,

2 1 1! max

= + shareholder i

i

CFT T

i COEw w w

Why did we ever look at volatility?It seemed an approximation to maximizing expected utility



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Volatility Is Not Always An Ideal Risk Measure but not as bad as you think


Disadvantages

Simply not true for many return series and seriously wrong for optionedportfolios

Advantages (Why do people still use it?)

Ok for large diversified long/short portfolios and/or long time horizons Degree of non-normality is not too large for monthly (typical revision

horizon) return series

Symmetry (uses all data): small estimation error

Easy portfolio aggregation & time aggregation

Consistent with earlier asset pricing models Computational solutions readily available

Ok for elliptical distributions


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Disadvantages Quintile measure; ignores risks within the tail

Might induce taking of extreme risks (BASAK/SHAPIRO, 2001)

Diversification can lead to higher Risk (not sub-additive)

Not convex, i.e. impossible to use in optimization problems

Advantages

More perceived than real / would Value at Risk have beenpromoted if deficiencies would have been known earlier?

Value at Risk definitely worse than you think


Problem: despite all itsdeficits, no practitioner will bewilling to give it up as it isuniversal and can beexpressed in $


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Scenario Manager 1 Manager 2 Manager 1+2 Probability

1 -20% 2% -9% 3%

2 -3% 2% -0,5% 2%

3 2% -20% -9% 3%

4 2% -3% -0,5% 2%

5 2% 2% 2% 90%

Table 1. Data Manager Combination: Active Return

Risk Measure Manager 1 Manager 2 Manager 1+2

Volatility 3,80% 3,80% 2,63%

Value at Risk(95%)-3% -3% -9%

LPM0 5% 5% 10%

CVaR -13,20% -13,20% -5,60%

Table 2. Risk Measures in Multiple Manager Example

The reason for these

paradoxical results

directly lies in the

concept of value at risk.

It ignores the large -20%

losses that are waiting

undetected in the tail of

the distribution. However,when we average across

portfolios these returns

will be diversified into the

portfolio risk measure

and increase risk as they

have been ignoredbefore. )

Value at Risk gives wrong diversification advice!



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Coherent, but also economically

sensible? Are you risk neutral in the tail?

Neglected approximation issue

Nonparametric & unconditional

Large estimation error

Driven by rare events

Only uses few data points

Implicit momentum

Up markets (momentum)show little risks: Japan fallacy

Explains superiority in back-tests

Risk measure depends onreturn estimate!!

Conditional Value at RiskOnly marginally better


MSCI Japan, annual data 1975 - 1992

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

1975

1976

1977

1978

1979

1980

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

1991

1992A

nnualreturnin%


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Lower partial moments

Semi-variance

Mean absolute deviation

Omega ratio

Minimum Regret

Some Other Risk MeasuresAll of them equally problematic


Utility theory is used as ex-

post sanctification, i.e.academic fig-leave

Necessary utility functionsimplausible with observedbehaviour / preferences


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Spectral Risk Measures: Theory


Value at Risk places all weight on a single quintile (implies investordoes not care about tail risk)

Conditional Value at Risk places an equal weight to all tail quintilesand zero else (implies investor is risk neutral in the tail)

ACERBI (2004) allows us to include risk aversion in the risk measureby allowing a (subjective) weighting on quintiles

If (p) is a non-decreasing function, spectral risk measures arecoherent

Weighting quintiles

( ) ( )

( )( )1

1

0X

lossweighting

quantilefunction

M X p F p p dp =

Giving larger weights to

more extreme quintiles issuper close to

maximizing expected

utility for reasonable

utility functions

Weights all cases fromworst to best

Advantage: weighting

function allows to merge

the risk measure with risk

preferences


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Spectral Risk Measures: Example


Weighting quintiles

I assume an exponential weighting scheme derived from the

exponential utility function DOWD/COTTER (2007)

Larger risk aversion leads to a larger risk measure

This effect tails out for the normal distribution but is unchanged for

the t-distribution

( )( )( )

( )

exp 11 exp

a a p

ap

=

a Normal-Dist T-Dist1 0,27 3,8

5 1,08 18,26

10 1,50 34,69

25 1,95 79,69

100 2,51 274,79


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Portfolio Construction under Non-Normality


Use of downside risk measures

If you feel you must, but I would advise against it

Approximate Utility Optimization

Quadrature methods

Higher order expansions of utility functions: number of higher co-moments grows extremely large for large number of assets

In the co-skewness and co-kurtosis matrix we are in need to calculaten(n+1)(n+2)/6and n(n+1)(n+2)(n+3)/24 different entries

Full Scale Direct Optimization Theoretically and practically most convincing

See KRITZMAN (2007) for convincing out of sample results

What are our options?


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In Summary: A Vicious Research Cycle


After 60 years we are back to our roots

Utility OptimizationNEUMANN/MORGENSTERN, 1944

Spectral Risk MeasuresACERBI, 2004

VaR and CvaR optimization(ROCKEFELLER/URYASEV, 2000)

Mean Variance Optimizationas an approximation to Utility Optimization

(MARKOWITZ, 1955)

Lower partialMoments

(FISHBURN/SORTINO 1990)

Expected utility is a riskadjusted performancemeasure

Expected utility can not begamed

Expected utility explainswhy some people buy andothers sell options


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Case Study: Risk Measures and the Credit Crunch


Investment universe: UK equities (FTSE), Emerging market bonds

(EMBI), Commodities (GSCI), Emerging markets equities (MSCI)

Take five years of monthly data up to June 2007 to get the followingweights

Leads to the following characteristics

In Sample Optimization

Portfolioweights

EmergingMarket Bonds

UK EquitiesGSCI

CommoditiesEM Equities

Mean Variance 24,63% 34,12% 1,82% 39,43%Mean Absolute Deviation 31,83% 26,19% 0,00% 41,99%

Semi-Variance 21,35% 39,67% 1,21% 37,78%

Minimizing Regret 40,92% 13,09% 0,00% 46,00%

Conditional Value at Risk 19,31% 44,22% 0,00% 36,47%

Volatility Value at RiskConditional

Value at RiskSemivariance

CumulativeDrawdown

Worst month Return

Mean Variance 2,29% -2,73% -3,59% 1,75% -5,10% -4,89% 1,67%

Mean Absolute Deviation 2,30% -2,65% -3,77% 1,81% -5,85% -4,81% 1,67%

Semi-Variance 2,35% -2,71% -4,06% 1,88% -6,77% -4,71% 1,67%

Minimizing Regret 2,29% -2,50% -4,06% 1,72% -4,93% -4,93% 1,67%

Conditional Value at Risk 2,30% -2,34% -3,36% 1,73% -4,95% -4,95% 1,67%

See SCHERER/MARTIN(2005) for complete codeincluding URYASEV VARapproximation


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Case Study: Risk Measures and the Credit Crunch


No clear picture

Minimizing regret (close to robust and very pessimistic) does best

CVAR does worst; out of sample UK and Emerging market equitiesdid worst while Emerging bonds and Commodities did best

(sampling error seriously affects CVAR)

Out of Sample Performance

Volat ility Value at RiskConditional

Value at RiskSemivariance

CumulativeDrawdown

Worst month Return

Mean Variance 3,81% -6,83% -7,98% 2,79% -14,32% -7,98% -0,33%

Mean Absolute Deviation 3,71% -6,57% -7,88% 2,79% -14,34% -7,88% -0,30%

Semi-Variance 3,61% -6,03% -7,34% 2,42% -13,76% -7,34% -0,18%

Minimizing Regret 3,86% -7,08% -8,29% 2,88% -14,98% -8,29% -0,39%

Conditional Value at Risk 3,90% -7,30% -8,64% 2,99% -15,81% -8,64% -0,47%


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What is Postmodern Portfolio Theory?

Section 2: Postmodern Portfolio Theory

Proponents are SHARPE (2007) and COCHRANE (2007)

Asset pricing takes the distribution of wealth as given and derives

asset prices using arbitrage principles and return characteristics

Portfolio theory takes asset prices as given and derives the utilityoptimal allocation of wealth.

Postmodern portfolio theory (PMPT) unites asset pricing and portfoliotheory by aligning risk neutral and real world distribution with the use ofstate price deflators.

The new book by SHARPE (2007) is an impressive collection of realworld applications that allows for a much richer framework than the one

currently used by practitioners.


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PMPT: The Theoretical Framework

Optimize utility or other objective function under the Pmeasure while

constraints are governed by the Qmeasure

Far superior to what the industry has done so far. Some examplesinclude

Eliminates arbitrage from MV optimization (allows us toinclude options into portfolio optimization)

Implicitly solves dynamic optimization problems (we canwrite down any payoff for final wealth as long as thebudget constraint is satisfied)

Tracking dynamic trading strategies

Combines P and Q measure


[ ] ( )

( )[ ]1

arg maxQ

o

P

E W W c

E U W= +

=w


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How can we rescue implied returns?

Step 1. Build pricing kernel from utility function

Step 2. Expected returns for all assets are the same under theabove pricing measure

Solve fordi for alternative utility functions and risk aversions

What happen if we leave MV efficiency?

This approach (GRINOLD,

1999 and SHARPE, 2007)works under arbitraryutility functions and showsthe power of neo portfoliotheory by merging theliterature on asset pricingand asset allocation.

( )

( )

,*

,1

1

1

s s bs S

s s bs

U R

U R

=

+=

+

( )* *, ,1 1S S

s s i i s s bs sR d R

= =+ =



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Building More Meaningful Implied Returns

More risk averse investors

place larger weight on tailsof the distribution andhence require much largerexcess returns to bewilling to hold a negativelyskewed asset


Source: Scherer (2007, chapter 9 )


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Example: Replication of CPPI Strategies


We suggest solving the following problem. Minimize the tracking error between the

continuous CPPI trading strategy and our static buy and hold tracking portfolio

( )1

2*

, , 1,min rT

i T i c S T i c iiW w e w S w C

subject to a budget constraint

( )10 , 1,

6000 6000rT rT

i c S T i c iiW e w e w S w C = + + =

and a non-negativity constraint on individual asset holdings

10, 0, 0, , 0

mc S c cw w w w


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We also need a cardinality constraint for the number of instruments (assetsn ) involved:

{ }

{ }

{ }

{ }

1 1 1

large number , 0,1

large number , 0,1

large number , 0,1

large number , 0,1m m m

c c c

S S S

c c c

c c c

w

w

w

w

Note that each equation provides a logical switch using a binary Variable that takes

on a value of either 0 or 1. Let us review the case for cash to clarify the calculations.

As soon as 0cw > by even the smallest amount, cash enters the optimal solution in

which case 1c = to satisfy the inequality. If 0cw = instead, it must follow that 0c =

Computationally the large number should not be chosen too large, i.e. it depends

how large cw can become . Finally we need to add the dummy variables to count

the number of assets that enter the optimal solution.

1 mc S c c assetsn + + + +


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Stock market

OBPIminusCPPI

4000 6000 8000 10000 12000 14000

-1500

-100

0

-500

0

Figure 2. Hedging error of a static option hedge. We used 6assetsn = to track a given CPPI

strategy. Note that hedging errors (difference between CPPI and OBPI payoff) are small around thecurrent stock price of 6000 and much larger where tracking is less relevant, i.e. where the real worldprobability is low.



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# of admissable instruments

Tra

ckingerror

4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1

.4

Figure 3. Reduction of hedging error and number of admissible instruments. As the number ofinstruments rises, the tracking error decreases. Note that tracking error is expressed as annualpercentage volatility, meaning 0.2 equals 20%. The tracking advantage tails off quickly with more than8 admissible instruments.



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Strike

Impliedvola

tility

5000 5500 6000 6500 7000

0.1

0

0.1

5

0.2

0

0.2

5

Ordinary least squaresRobust regression

Figure 4. Fitted implied volatilities for linear and robust regression. The crosssectional regression is run using both OLS as well as robust regression. Fitted linesare either dashed or dotted, while the raw data are provided in Table 4.

Example: Utility Optimization


( )( ) ( )( ) ( )( )

( )

1 1 1 1

2

1

, , 2 , , , ,i impl i i impl i i impl i

i i

C S S C S S C S SrT

i S Se

+ +

+

+

=



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Stock price

Riskneutra

ldensity

5000 5500 6000 6500 7000

0.0

0.005

0.0

10

0.015 Lognormal PDF

Option implied PDF

Figure 5. Lognormal volatility versus option implied volatility. Implied risk neutral densities areprovided for both a lognormal model (with 11% historical volatility) as well as for the coefficientestimates of the robust regression.

Implied Density




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111

, 1u W for

=

( )1

1,1

maxi T ii

W

( )0 ,6000 6000rT rT

i c S T iiW e w e w S= + =

Risk aversion Implied volatility Lognormal volatility

2 100.00% 100.00%

5 74.54% 78.91%10 37.51% 39.45%

50 7.54% 7.89%

Optimal equity allocations

Example: Utility Optimization


Forward looking riskmeasure that uses themarkets risk perceptionrather than unconditionaldistribution


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Some Frustration with Bayesian MethodsWhere we have been stuck

The main reason for thesuccess of theBLACK/LITTERMANmodel was its anchoringin the market portfolio.

Section 3: The Robust Contra-Revolution

The BLACK/LITTERMAN model looks (very) tired Many problems yet unsolved

BAYES and Non-normality

Informative priors and missing data

Partial solutions: various ways to deal with estimation error. Whats the

best combination?

Interesting new development: Economic priors by McGULLOCH

Required subjectivity (information) is deemed too high by many users; (too)

great level of arbitrariness, too much subjectivity, too much responsibility

Practitioners want the solution to be built into the optimisation process;


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Reaction 1: Create Robust SolutionsSolutions that dont change too much

Section 3 The Robust Contra-Revolution

Upper and lower bounds: JAGANNATHAN/MA (2003), can be viewedas leveraging up the respective entries in the covariance matrix

1/n rule: DEMIGUEL/GARLAPPI/UPPAL (2007), equal weighting ishard to beat

Vector norms: DEMIGUEL/GARLAPPI/UPPAL/NOGALES (2007)

Concentration measures: KING (2008)

Most of the above can be shown to be equivalent

to Bayesian shrinkage!

Percentage contribution to risks

Weight baseddiversification constraints

Risk based diversificationconstraints


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Reaction 2: Create Robust InputsCombat estimation error impact by making forecasts that dont differ too much


Create ordinal scores (+1 for outperform and -1 for underperform). For diagonalcovariance matrix this leads to

Percentile ranks as in SATCHELL/WRIGHT (2003)

Ordering information as in CHRIS/ALMGREN (2006)

Robust statistics as reviewed in SCHERER/MARTIN, 2005

In my view this iswidespread in active quant

strategies

( )11 1

i ii iw

=


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Reaction 3: Robust OptimizationBuild solution into optimization process


An early attempt: Ttnc/Knig (2004) Try to find good solutions for all possible parameter realizations

How do we define the set of all possible parameters? Imposing shortsale constraints this simplifies to

matricescovarianceallofSet:

rsmean vectoallofSet:

minmax,

S

S

TT

SS

wwww

confidencegivenaforofelementmaximal:

confidencegivenaforofelementminmal:

max0

S

S

h

l

h

T

l

T

wwww


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The Perils of Being Too Afraid


Mean vector represents the lower 5% quintile entries, whilecovariance represents the upper 5% entries. Risk Aversion of 2.

Data From Michaud (1998)

Eq.Can Eq. Fra Eq.Ger Eq.Jap Eq.UK Eq.US Fi.US Fi.EU

0.0

0.2

0.4

0.6

0.8

1.0

Asset Class

WeightAllocation

Ad hoc way to define mean vectorand covariance matrix

Leads to very cornered portfolios

Poor rooting in classic decisiontheory


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The Structure of Estimation Error

For the purpose of this presentation we assume

Could also allow for different error structures, that allowuncorrelated estimation errors and make diversification ofestimation errors optimal

Note that as the number of assets rise the matrix becomesincreasingly difficult to estimate

/n=

21

2

2

0

0

n

n

=



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Robust Portfolio Optimization: The Contender

CERIA/STUBBS (2003) derive the following objective function

and after some tedious algebra (see SCHERER, 2006) we

can get

( ) ( )12 2

, 2, 1

T T

m p pL n

= + w w w 1

12

,

1* 2

,

12

,

1* 2

,

1 1* 11

1 1

*

min

1

1

m

p m

m

p m

Tn

rob T Tn

n

specn

+

+

= +

= +

1 1w 1

1 1 1 1

w w

12

,

1* 2

,

0 1 1m

p m

n

n

+

The careful reader will realize thatthe previous result essentiallyviews robust optimization asshrinkage estimator

As long as estimation error

aversion is positive this term willalways be smaller than one.



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Fi.EUFi.USEq.USEq.UK

Eq.JapEq.GerEq. FraEq.Can

0.0

0.2

0.4

0.6

0.8

1.0

Traditional Optimization

Fi.EUFi.USEq.USEq.UKEq.JapEq.GerEq. FraEq.Can

0.0

0.2

0.4

0.6

0.8

1.0

Robust Optimization

Figure 1. Robust versus traditional portfolio construction ( 0.01, 60, 99.99%, 1000n S = = = = ). Robust

portfolios react less sensitive to changes in expected returns. Given the high required confidence of 99.9%= ,robust portfolios invest heavily in assets with little estimation error. This is entirely different with our intuition

that error in return estimates become less and less important as we move towards the minimum risk portfolio.

The data are taken from Michaud (1998). The abbreviations used are FI.EU (fixed income Europe), FI.US (fixedincome US), Eq.US (equity US), EQ.UK (equity UK), EQ.Jap (equity Japan), EQ.Ger (equity Germany), EQ.Fra

(equity France) and EQ.Can (equity Canada).

How do solutions differ?

The least risky asset isover-weighted in the

solution



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0

10

20

30

40

50

Robust optimization

0.3 0.4 0.5 0.6 0.7

0

10

20

30

40

50

Traditional optimization

Out of sample utility

PercentofTotal


Some corner portfolios dontdo well out of sample



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Small sample size (n = 60)

= 95% = 97.5% = 99.99%

= 0.05 9.7 bps(20.16)

68.5%

8.7bps

(17.76)

63.4%

7.79bps

(16.09)

61.0%

= 0.025 -3.13bps(-18.23)

39.14%

-5.54bps

(-14.84)

32.6%

-6.96bps

(-11.85)

28.8%

= 0.01 -19.8bps(-49,6)

13.0%

-24.09bps

(-63.33)

8.1%

-26.14bps

(-70.07)

6.8%

Out of sample performance for full investment universe (m=8). The table shows the relative performance of

robust portfolio optimization relative to traditional portfolio optimization. The 1st

number is the difference inexpected utility, which we can interpret in terms of a security equivalent (i.e. basis points of monthly

performance). The 2nd

number (in round brackets) represents the t-value of the difference in expected utility (avalue of about 2 would be significant at the 5% level, for a two sided hypothesis), while the third number

represents the percentage of runs, where robust optimization generated a higher out of sample utility than

traditional optimization.




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Summary


Robust methods based on estimation error are equivalent toshrinkage estimators and leave the efficient set unchanged

Robust methods come at the expense of computational difficulties(2nd order cone programming), but at the advantage of automatedsolutions

Yet difficult to calibrate uncertainty and risk aversion


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Fairness in Asset Management

Section 4: Fairness In Asset Management

Implementation of views

Clients differ in benchmarks, constraints, investment universe,risk budgets

How do we make sure that all clients receive the sameinformation, i.e. that all client portfolios are consistent?

Trading and transaction costs

Asset management firms manage multiple accounts

Asset manager is seen as mediator and guarantor of fairness intrading decisions

Treat clients fairly

FSA Principle #6: A

firm must pay dueregard to its customersand treat them fairly


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Portfolio Factory: Consistent View Implementation


Step 1: Start with unconstrained model portfolio (otherwise views are

contaminated by constraints) Step 2: Back out implied returns

Step 3: Run optimisation with client specific constraints whereby thesame implied returns are used for all clients

Client portfolios will still differ in total and active weight (dispersion),but not in input information!

(Step 4: Overcome organisational resistance: Portfolio factory

weakens the role of the portfolio manager, portfolio constructionbecomes commodity, optimizer as the poor mans portfolio manager,portfolio manager loose out to analysts, i.e. information providers)

How can we ensure all portfolios receive the same information/attention?


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Optimization Across Multiple Accounts


Why is this a problem?

,1iw

,2iw

Trade in asset i

for account 1,2

( )w

=

( )1

ww

=

( )1

ww

=

TC allocated bytrading desk

TC in independentoptimizations

Marginal TC differ

Total, marginal,

averageTC

Nonlinear transactioncosts create an externality

from one account oneanother

The literature provides twosolutions

- NASH solution (CERIA,2007)

- Collusive, Pareto optimalsolution (OCINNEIDE/SCHERER/XU, 2006)


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Optimization Across Multiple Accounts


Two accounts of different size, s, that trade one asset n.

Quadratic transaction cost function

Standard preferences for each account (note that the cost term

reflects cost sharing, i.e. both accounts trade simultaneously

Model Set Up (joint work with Steve Satchell)

( )2

1 1 2 22n s n s = +

( ) ( )( )2 22 2i i i i i i i j j i i VA n s n s n s n s n s = +


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Stand Alone Solution


Optimal trading

Optimize accounts separately without taking interactions into account (batch job)

( ) ( )( )2

2 222 2arg max

ii

SAi i i i i i i s

nn n s n s n s

+ = =

0 0.1 0.2 0.3 0.4

n2

0

0.1

0.2

0.3

0.4

n1

( )

2 2

22

SA SAi jVA VA

+ =

+


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COURNOT/NASH-Solution


First order condition (solving leads to reaction functions below)

Interaction is accounted for but treated as given

2 2 2 12 0

i

i

dVAi i i i i j i j dn

s n s n s n s s = =

0.1 0.2 0.3 0.4

n2

0.1

0.2

0.3

0.4

n1

( ) ( )

( )( )

2 2

2 2

2

3 2

3 20

j j

j

CN SA

j j s s

s

n n

+ +

+ +

=

=


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Collusive Solution


Combined objective function (monopoly)

Leads to less trading and higher value added (risk adjusted client

performance)

Full interaction is accounted for

( )( ) ( )( )2 2 2 22 2 2 2i j i i i i j j i i j j i i j j j j VA VA VA n n n s n s n s n n n s n s n s = + = + + +

( )( )

2 2

22 2

12 2 3 2

0C CNi iVA VA

+ + = >

2

4 2

1 1/C CNi in n

+

=

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