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