INVESTMENTS

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Unversité de Lausanne Master of Science in
Finance Spring 2008
INVESTMENTS Faculty Bernard DUMAS Practical
issues Multidimensional investing Sessions 3-2
and 4-2
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Overview

• Pitfalls in portfolio optimization
• Ideas that have survived
• Statistical models (much in use)
• Streamlining the investment process
• Estimating statistical factor models
• Pricing models (more questionable)
• Multi-factor pricing models

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Pitfalls in portfolio optimization

• Minor deviations in inputs (especially expected
  returns) lead to large changes in decisions.
• Generally, large, crazy positions.
• If no short sale allowed, undiversified
  portfolios.
• When estimating returns from the past, there is
  estimation risk.
• The problem is especially severe when there are
  many assets.
• In the limit, optimization becomes meaningless
  when there are more assets than observations.
• With the data we have, the asset allocation
  problem cannot be solved beyond 10-15 assets (or
  asset classes or risk dimensions).
• Beyond that number, it is not just this method
  that fails
• The problem is basically meaningless
• The information needed to address the problem is
  not available in the limited dataset.

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Ideas that have survived

• A statistical concept exposure
• beta can be seen as an exposure to market risk
• use of exposure(s) to classify assets and
  systematize portfolio construction process
• beta measures each assets contribution to total
  portfolio risk
• idea useful for risk management
• needed to develop accounting of risk (or
  breakdown of risks)
• but beta requires a generalization recognize
  more common/underlying risk factors than just
  the market
• A pricing concept systematic vs. non systematic
  risks
• In pricing, only risk factors that are common to
  many assets, matter.
• Other risks can be diversified away.

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Multifactor statistical models
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Statistical models streamlining the investment
process

• Covariance matrix is huge (many entries)
• Leads to imprecisely computed portfolios
• Let us impose structure on the matrix
• We may have 10000 assets to choose from
• In fact, there may be only 10-15 common
  underlying sources of risks (risk factors)
• A particular assets can be seen as a portfolio of
  these basic risk factors (plus idiosyncratic
  risk) and must be analyzed as such
• This is more flexible than imposing groupings of
  assets into asset classes,
• An asset can be partly exposed to one factor and
  also partly exposed to another,
• whereas asset classes force one to place one
  asset in one class or the other, with no
  intermediate classification

Assets space ? Factors space
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Example one factor model

• One-factor model ( bi called loading or
  exposure)
• where residuals ?i,t are independent across
  assets and independent from factor
• An asset is seen as a combination of two risks
  the common factor and its own specific risk
• Then

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Example construct common factor (by statistical
method (method 1 below))
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Returns Are Multi-Dimensional

• Companies possessing similar characteristics
  may, in a given month, show returns that are
  different from the other companies. The pattern
  of differing shows up as the factor relation.
• Barr Rosenberg, Extra Market Components of
  Covariance in Security Markets, Journal of
  Financial and Quantitative Analysis,1974

• A set of common factors -- not just a monolithic
  market -- influence returns

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Stocks in the same industry tend to move
togetherExample banks in Europe
Slide from BARRA
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However, there are other common factors that
simultaneously affect returns Within the banking
industry, the size factor is at work
Slide from BARRA
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Multi-factor risk models

• Multi-factor model
• where residuals ?i,t are independent across firms
  and independent of factors
• When I hold Asset i, I am truly holding b1,i of
  risk 1, b2,i of risk 2 etc..
• Then one figures out from that the risk
  statistics that are needed for portfolio
  construction

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Key insight
Slide from BARRA

• The variance of factors and the covariance
  between factors are more robust, statistically
  speaking, than the variance of individual assets
  and the covariance between assets.

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Estimating Statistical Factor Models three
approaches

• Goal capture the way in which assets returns
  move together
• Factor analysis (purely statistical)
• Factor analysis constructs a limited set of
  abstract factors that best replicate the
  estimated variances and covariances
• Throws no light on underlying economic
  determinants of the covariances
• Use of macroeconomic variables
• Use of firm specific attributes size, B/M ratio,
  consumption of oil

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Estimating Statistical Factor Models 2. Use of
macro variables

• Business cycle risk
• unanticipated growth in industrial production
• Confidence risk
• default rate-of-return spread (Baa - Aaa) which
  is a proxy for unanticipated changes in risk
  premia
• Term premium risk
• return on long bonds minus short bonds, which is
  a proxy for unanticipated shifts in slope of
  yield curve
• Get exposures (also called loadings) of each
  stock by (multiple) time series regression
  (exposures assumed constant)

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Estimating Statistical Factor Models 3. Use of
firm-specific attributes (size, B/M ratio,
consumption of oil)

• Attributes of firm
• Size
• Value vs. growth
• Etc.
• Form factor-mimicking portfolioswhich capture
  contrasting factors
• Market RM - r
• B/M High-minus-low (HML) RHML RH - RL
• Size Small-minus-big (SMB) RSMB RS - RB
• Estimate exposures by regression

This month(t)s returns Ri,t
RS,t
RB,t
This month(t)s sizes
Bottom three deciles
Top three deciles
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Example calculation of multi-factor statistical
model

• Imagine that I have observed
• that firm I is large while firm A is small (based
  on attributes)
• or that firm I is not recession prone, while firm
  A is (based on macroeconomic variables)
• or some such contrast

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Multifactor pricing models
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Multi-factor pricing models state-dependent
preferences

• Possible story investor not only cares about
  portfolio variance but also when performance
  occurs
• Example investors try to buy stocks that do
  better than others in a recession
• If stock i does ( high compared to
  other stocks)
• lower return required from it
• this drives down expected return of stock i
  (beyond the market beta effect) ?recession lt 0
  in

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Numerical example

• Take, for instance, exposures obtained by
  multiple regression of individual asset on the
  factors, as in statistical models (see above).
• Call that step the first pass regression.
• Prices of risk ? are then obtained by second-pass
  cross-sectional regression, like in CAPM.

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Example calculation multi-factor pricing model
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Example E(R) on Asset B
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Which factors?

• See BARRA list
• Get SMB factors and others from
• http/mba.tuck.dartmouth.edu/pages/faculty/ken.fre
  nch/data_library_html

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Application to fund analysis style analysis

• where
• Rp,t return on some fund
• R1,t return on strategy (or factor) 1 (e.g.,
  invest in small firms)
• R2,t return on strategy (or factor) 2 (e.g.,
  invest in value firms) etc
• the coefficients b reveal the management style
  (investment policy) of the fund.
• This is especially useful for funds that keep
  their strategy somewhat secret (for example,
  hedge funds)
• the intercept ? detects expertise, according to
  a multi-factor CAPM

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Conclusions

• Assets are analyzed as portfolios of underlying
  risk factors
• Essential method for monitoring (keeping track
  of) your risks
• Style analysis is excellent product
  definition/choice tool
• Should clearly draw the distinction between
• statistical model that uses risk factors, which
  are common to all returns, as a descriptive tool
• and a multi-factor pricing model that assigns a
  risk premium to each risk factor
• This is a less restrictive approach to asset
  pricing than the CAPM several dimensions of risk
  are priced
• general multi-factor model based, however, on
  incompletely specified investor behavior