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Sergio FocardiEDHEC Business School
The Intertek Group
Challenges in Quantitative Investment Management
Quantitative Investment ManagementThis presentation is based on a number of studies that The
Intertek Group and Frank Fabozzi (Yale University) performed in the last decade,
Including the monograph Challenges in Quantitative EquityManagement published by the CFA Research Institute (2008)
A sequel study commissioned by the CFA Research Institute (2009) will be published soon
In performing these studies, we interviewed well over 300 people in industry and academia
All studies revealed a growing interest for quantitative investing but also a number of challenges
Milestones in quantitative investing: the beginning
The conceptual foundation of quantitative investing was laid down by Harry Markowitz in the 1950s
Markowitz stated that investors should manage theirinvestments optimizing the trade-off between risk and return
In Markowitz’s approach, risk is measured by variance and returns by their expectation
Hence Markowitz’s approach is referred to as the mean-variance approach
The crucial innovationThe notion of a trade-off between risk and returns was not
new It already appeared in medieval writings on business
managementThe essential innovation is the quantification of risk and
expected returnsMarkowitz, who worked for the Rand Corporation and also
worked with Dantzig (the inventor of linear programming), foresaw the changes that computers would have broughtabout
The following stepsThe one-period, mean-variance approach was extended to
include…More general representations of the risk-return trade-off
through utility functions,…Multiple periods and…A stream of consumption In the 1960s Sharpe, Lintner, and Mossin placed the
prescriptive approach of Markowitz as the foundation of a descriptive approach of asset pricing:
The Capital Asset Pricing Model (CAPM) and General Equilibrium Theory were born
All dressed up and no place to go…However, for nearly four decadesAll these remarkable achievements remained confined to the
academic environmentUntil the 1990s, investors (often advised by their brokers)
continued to invest in companies they “liked”Analysts and asset managers continued to put the accent on
analyzing balance sheets and visiting companies But ignoring a fundamental component of Markowitz
approach: correlations and diversification Computing power and the associated skills were still too
expensive and too scarce
A different route with a caveat
Options started trading in 1848 in Chicago but only after 1985 didtrading volume pick up and the variety of options offered increase
Derivatives trading requires quantitative models for pricing: the Black-Scholes formula provided the first analytical tool
The 1987 crash showed the need for understanding risk in quantitative terms
Derivatives trading became the preserve of quants and producedan explosion in quant modelling
But with a caveat: derivatives modelling is driven more by the complexity of the contracts than by the modelling of the underlying economics
The last fifteen yearsThe last fifteen years have seen a rapid increase in both the
number of quants in finance and assets under quantitative management,
With a focus on understanding the economics behindinvesting
And with the need to cope with recurrent crisesWhat have we learned? What are the challenges today?
Let’s first set the stage
A few facts: size and interconnections
Financial markets have become truly enormous In developed countries, the turnover of financial markets is
now 10 to 15 times the turnover of the real economy Emerging markets have emerged and are poised to be a
major playerThe amount of money created by central banks and by the
banking systems to support this financial system isunprecedented
The complexity of interconnections has also reached an unprecedented level
Understanding what factors move prices and returns
The technology of factor modelling has reached considerablematurity
The mathematics of factor models - both static and dynamic -is well understood
And much experience has been gained in understandingfactors from the empirical point of view
Identifying those factors that might work
Correlations and comovementsOne lesson learned is the importance of correlations and
comovements and the associated challenges First the shere number of correlationsThe number of correlations grows with the square of the
number of assets For example, consider four years of weekly data,
approximately 1,000 data pointsWe have 500,000 observations to estimate the correlation
matrix of the S&P500 which includes about 125,000 entries That is four data points for every estimated entry
The curse of dimensionality
If we estimate the correlation matrix of the Russell 1000, wehave 1,000,000 observations but about 500,000 entries, thatis, 2 data points per covariance
Estimating correlations is difficult in local markets and…Given that the number of stocks globally exceeds 10,000
there is simply not enough data to allow a crude estimation of correlations on global markets…
Given that we need to estimate in excess of 50 million individual correlations (approximately 1/2x10000x10000)
Dimensionality reduction needed
We need to reduce the dimensionality the number of independent estimates
Otherwise random entries will prevail and will push the optimizer into corner solutions
Techniques: Shrinkage, Factor models, Random matrix theory
ShrinkageThe empirical covariance matrix is defined as:
Entries are noisy Shrinkage is a statistical technique that shrinks the empirical
covariance matrix towards a constant correlation matrix In practice shrinkage averages between the empirical
covariance matrix and the constant correlation matrixThus mitigating the effects of noise
{ } ( )( )jjt
T
tiitij XXXX
T−−= ∑
=1
1σ
Factor models Factor models of returns can be written as follows:
Under general conditions the covariance matrix becomes:
If the covariance matrix of residuals R is diagonal the numberof independent entries becomes proportional to the numberof stocks
ttt εβfr +=
NNNN RββΣ += '
Random matrix theory
Random matrix theory establishes the benchmark of a purelyrandom covariance matrix
That is, the covariance matrix of independent stock returns
The distribution of the eigenvalues of a random matrix canbe theoretically determined
The surprising result is that most eigenvalues of the empiricalcorrelation matrix must be considered random noise
The meaningful eigenvalues are outside the region of randomnoise
Theoretical distribution of eigenvalues and simulation on 500 generated random walks
Non linearities
Correlations are a linear measure of dependence
Correlations work well with normal distributions But returns have fat tailsCorrelations might be a poor representation of comovements
And might need to be replaced by copula functions
HeteroscedasticityVariances and covariances present another challenge:
heteroscedasticity or ARCH/GARCHThis means that the residual uncertainty after modeling is not
constant but fluctuates in time and is autocorrelated… So that (relatively) extended periods of high uncertainty are
followed by (relatively) extended periods of low uncertainty Similar phenomena exist for correlationsHeteroscedasticity of individual assets is well managed by
ARCH/GARCH models but the heteroscedasticity of correlations is subject to the curse of dimensionality
Six-sigma events do happen
Another lesson learned is that because returns are not normally distributed,
Very large movements are more likely to happen than if returns were normally distributed
This cannot be ignored
As we have seen, the six-sigma-event, virtually impossible under the assumption of normality, occurs every few years
Risk measurement and management based on the assumptionof normality results in a dangerous underestimation of risk
Fat tails and risk measures
The technology for handling fat tails and the associated riskmeasures exist and is viable
We have learned how to separate the tails from the bulk of distributions
And how to represent the tails
We also know how to replace measures of risk such as VaRthat work only in a normal environment
With measures of risk such as Conditional VaR (CoVaR) which work with fat tailed distributions
Momentum and mean reversion
Momentum is an anomaly of pricing, a “theory of the foolwho believes that a bigger fool will always stand ready to buy”
But still seems to be a critical ingredient of quantitative assetmanagement
While mean reversion has a solid conceptual basis though it isdifficult to correctly time
The augmented role of asset allocationThe recent financial crisis has underlined the importance of
asset allocation to generating returnsWhile there is agreement as to the central role of asset
allocation,There is less consensus as to the feasibility of truly dynamic
asset allocation In other words, profits and losses are driven primarily by
asset allocation But we do not yet have the science to allow us to forecast the
timing of asset class price movements
Liquidity riskThe crisis has heightened the attention to liquidity risk In liquid markets, it is easy to find counterparties willing to
trade Liquidity risk is the risk one cannot find a counterparty
willing to buy at a “fair” price Measuring liquidity risk is very challenging
The concept itself is vague insofar as it is difficult to separateliquidity risk from market risk
Ultimately, liquidity risk is the risk of selling at a very lowprice
Beyond counterparty risk
The 2008-2009 crisis has shown that any entity can becomefragile if it is interconnected with entities in difficulty
Banks and corporations that are individually sound by standard accounting measures…
Might become insolvent if one or more of theircounterparties should fail…
But determining the chain of counterparties is a difficult task
Possible tools include the theory of (random) networks
Systemic risk It has become clear that financial institutions are exposed to
systemic risk In the sense that the network of connections is of great
importance But it has also become clear that in the present situation it
might be extremely difficult to properly evaluate the system of connections
Hence there are now proposals from mainstream entities(e.g., UK’s Financial Services Authority) to adopt measuresof network complexity
Putting the pieces together
Though many components are in place, from factor models to the understanding of ARCH behavior, from correlations to fat tailsand though the statistical methodology is known,
Putting the components together in a coherent framework is stillvery challenging
There are many difficult trade-offs in creating a global coherentmodelling view
In particular the trade-off between in-sample acccuracy and out-of-sample performance, between learning and theory remainfundamental challenges
Models are still opportunistic problem-solving techniques
Nothing stands still Perhaps the biggest challenge is the fact that financial markets are
not stable systems: they keep changing The lifetime of models has become shorter and one needs a
portfolio of models to capture the changing nature of the underlying dynamics
There are multiple causes of model decay due to the evolution of the economy
Financial markets are not simply vehicles to optimally allocatecapital to productive activities but include a large speculativeportion
Sustained by the creation of large amount of money throughleveraging
And the complexity of interactions is difficult to understand
Conclusion
The role of measurement and of quants in asset management will continue to grow,
Motivated by competitive pressure, including the need for product innovation, risk management and cost control,
By the huge amount of available data and a growing universeof investable assets,
And thanks to the many modelling components now available
Though putting it all together given the instabilities of the financial system remains a challenge