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TRANSCRIPT
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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
Section 1: The Evolution of Risk Measures
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Volatility Is Not Always An Ideal Risk Measure but not as bad as you think
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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!
Section 1: The Evolution of Risk Measures
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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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 1: The Evolution of Risk Measures
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
Section 2: Postmodern Portfolio Theory
[ ] ( )
( )[ ]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
= =+ =
Section 2: Postmodern Portfolio Theory
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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
Section 2: Postmodern Portfolio Theory
Source: Scherer (2007, chapter 9 )
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Example: Replication of CPPI Strategies
Section 2: Postmodern Portfolio Theory
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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Example: Replication of CPPI Strategies
Section 2: Postmodern Portfolio Theory
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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Example: Replication of CPPI Strategies
Section 2: Postmodern Portfolio Theory
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.
Source: Scherer (2007, chapter 8 )
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Example: Replication of CPPI Strategies
Section 2: Postmodern Portfolio Theory
# 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.
Source: Scherer (2007, chapter 8 )
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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
Section 2: Postmodern Portfolio Theory
( )( ) ( )( ) ( )( )
( )
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
+ +
+
+
=
Source: Scherer (2007, chapter 8 )
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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
Section 2: Postmodern Portfolio Theory
Source: Scherer (2007, chapter 8 )
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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
Section 2: Postmodern Portfolio Theory
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
Section 3: The Robust Contra-Revolution
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
Section 3: The Robust Contra-Revolution
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
Section 3: The Robust Contra-Revolution
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
=
Section 3: The Robust Contra-Revolution
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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.
Section 3: The Robust Contra-Revolution
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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
Section 3: The Robust Contra-Revolution
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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
Out of sample utility
Some corner portfolios dontdo well out of sample
Section 3: The Robust Contra-Revolution
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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.
Out of sample utility
Section 3: The Robust Contra-Revolution
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Summary
Section 3: The Robust Contra-Revolution
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
Section 4: Fairness In Asset Management
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
Section 4: Fairness In Asset Management
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
Section 4: Fairness In Asset Management
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
Section 4: Fairness In Asset Management
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
Section 4: Fairness In Asset Management
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
Section 4: Fairness In Asset Management
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
+
=