1

Investment Management

Implementation of Portfolio Theory

Road Map

• Mean-variance principle (assumptions)

• Market risk premium

• The Capital Asset Pricing Model (CAPM)

• The Security Market Line (SML) and

– mispricing

– cost of capital

• Single index model

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Just a bit of theory

• The approach described in the previous lecture has been formulated by Markowitz in 1952

• It relies on a set of assumptions:

A. there are many investors whose wealth is small compared to the total market wealth (i.e. investors are price-takers)

B. all investors have got the same investment horizon (myopic)

C. all investors pay no taxes

D. all investors are rational mean-variance optimizers

E. all investors share the same view of the world (homogeneous expectations, information is costless)

F. there are no transaction costs

G. short-selling is allowed

H. unlimited borrowing and lending at the risk-free rate

Equilibrium in Markowitz’s world

• The Capital Asset Pricing Model (CAPM) is a set of predictions concerning equilibrium expected returns on risky assets:

1. all investors will choose to hold a portfolio of risky assets which replicates the representation of each traded asset in the market portfolio (including all assets)

2. the market portfolio is efficient and is also the tangency portfolio to the optimal CAL of each investor. As a results the CML is the best attainable CAL

3. the risk premium on the market portfolio is proportional to its risk and the average degree of risk aversion in the economy

4. the risk premium on individual assets is proportional to the risk premium on the market portfolio and a sensitivity between each assets returns and market returns (beta)

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All investors hold the market portfolio

• The market portfolio is the sum of all individual portfolios (lending and borrowing cancel out).

• The proportion of each (risky) asset in the market portfolio is equal to market value of each asset/ sum of market values of all assets

• Given assumptions D, E each investor must arrive to the same determination of the tangency portfolio (CML)

• The market portfolio is efficient (being tangent on the efficient frontier of all assets)

Market risk premium

( ) ( ) 22

risk premium on

1 M fM f M

MM

E r rE r r A

Aσ

σ−

= → − =

• Recall that each investor optimal weight in the (risky) tangencyportfolio is equal to

• All investors hold a market portfolio as their tangency portfolio. Differences in optimal asset allocations are due to various

individual degrees of risk aversion→ different investments in the risk free asset (borrowing and lending)

• In equilibrium borrowing and lending cancel out implying an average y = 1

( )tan2tan

fE r ry

Aσ−

=

4

Expected returns on individual assets

• Markowitz theory and the CAPM postulates that assets risk premia are determined by their contribution to the risk of the overall portfolio

• Portfolio risk is all that matters to investors

• The contribution of each asset i to the risk of the overall portfolio can be measured by its covariance with the market portfolio

• The contribution of each asset i to the return of the overall portfolio is equal to wi[ E(ri) − rf ]

( )

( )( )

( )

2

2

cov ,

cov ,

cov , cov ,

i M

M i j i ji j

i M i j i j i i Mj

r r

w w r r

w w r r w r r

σ

σ →

=

= =

∑∑

∑

• The reward-to-risk ratio for investments in asset i is therefore

• The reward-to-risk of the market portfolio (i.e. market price of risk) is

• In equilibrium the two reward-to-risk measures are equal

Expected returns on individual assets (cont’d)

( )( )

( )2cov ,

i f M f

i M M

E r r E r rr r σ

− −=

( )2

M f

M

E r rσ

−

( )( )

( )( )cov , cov ,

i i f i f

i i M i M

w E r r E r rw r r r r

⎡ ⎤− −⎣ ⎦ =

( ) ( ) ( ) ( )2

cov ,i Mi f M f i M f

M

r rE r r E r r E r rβ

σ⎡ ⎤ ⎡ ⎤− = − = −⎣ ⎦ ⎣ ⎦

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• The canonical form of the CAPM for a portfolio p is

• It indicates that expected returns on individual portfolios are:

– associated with the risk premium on the market portfolio, E(rM) – rf

– proportional to the co-movement between the portfolio returns and the market returns, beta

• Beta > 1.0 implies aggressive investments (since the portfolio returns will vary more than market returns)

• Beta < 1.0 implies conservative investments (since the portfolio returns will vary less than market returns)

The CAPM predictions

( ) ( )p f p M fE r r E r rβ ⎡ ⎤= + −⎣ ⎦

The CAPM predictions (cont’d)

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Mispricing

( ) ( )p p f p M fE r r E r rα β ⎡ ⎤= + + −⎣ ⎦

Mispricing (cont’d)

• The CAPM can be useful in identifying undervalued or overvalued assets

• This is generally done by comparing the requiredrate of return (implied by our estimates of the CAPM) on a certain asset and the estimated rate of return over a specific investment horizon (based upon other sources, i.e. fundamental analysis)

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• The implementation of the full Markowitz portfolio selection procedure requires a large number of calculations, especially for a large number of assets

• Example: a portfolio comprising only 50 assets requires the computation of 1,325 estimates (averages, variances and covariances)!

• To make our life easier we introduce an additional assumption: all relevant macroeconomic factors affecting all asset returns are summarized in one indicator only (i.e. single index)

• However, beside this common effect the asset specific uncertainty (risk) remains

A single-index model

( ) ( )i i i i i i i ir E r m e E r F eβ= + + = + +

• One reasonable proxy for the factor Fi affecting the dynamics of asset returns is rate of return on a broad index (e.g. S&P 500, HSI)

• This will allow to rewrite the model as follows

where Ri and RM are excess returns (ri – rf) and (rM – rf)

• The single-index model states that asset returns are a sum of three components

– asset expected returns if the market is neutral, αi

– component due to movements in the overall market, βiRM

– component due to firm-specific events, ei

A single-index model (cont’d)

( )i i i i i

i i i M i

r E r F eR R e

βα β

= + + →

= + +

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• Risk associated with each security comprises two components:

– market specific (systematic risk), β2iσ2

M

– firm specific (idiosyncratic risk), σ2(ei)

• Calculations are drastically reduced: for a 50 asset portfolio now only 151 estimates are required rather than 1,325!

A single-index model (cont’d)

( )( ) ( )

2 2 2 2

2cov , cov ,i i M i

i j i M j M i j M

e

R R R R

σ β σ σ

β β β β σ

= +

= =

• Risk associated with a portfolio of n securities:

Single-index model and diversification

( )

( ) ( ) ( )

( )

( )2

2 2 2 2

22 2 2 2

1

1,

1 1 1 /

p p M p p ii

n

p i ii i

e

en

e e e e nn n n

σ

σ β σ σ β β

σ σ σ σ=

= + =

⎡ ⎤⎛ ⎞= = =⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎣ ⎦

∑

∑ ∑

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Single-index model in practice

Single-index model in practice (cont’d)

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• Are the single-index model and the CAPM the same?

• The two equations convey the same information only if – α is zero (on average)

– firm-specific shocks and marketwide components are uncorrelated, i.e. cov(RM, ei) = 0

• Note that the single-index model and the CAPM betas are the same except that we replace the theoretical (unobservable) market portfolio with a realized (observable) market index

Single-index model and the CAPM

( ) ( )( )

i f i M f

i f i i M f i

E r r E r r

r r r r e

β

α β

⎡ ⎤− = −⎣ ⎦

− = + − +

• The single-index model is used in the investment industry because of its simplicity

• Investment banks publishes the Security Risk Evaluation book (Merryll Lynch, monthly) and Value Line Investment Survey(weekly) where all NYSE stocks betas and risks are reported

• They use a slightly different approach in estimating the single-index model (i.e. returns instead of excess returns)

• Merryll Lynch also reports adjusted betas calculated as follows:

Single-index model: applications

2 1 ; 13 3ii eq eqadjβ β β β= + =

::

i i i M i

i i i M i

BKM R R eML r r e

α βα β= + +

= + +

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Single-index model: applications (cont’d)

• In the CAPM and single-index model one variables proven to be especially important: the market portfolio

• The market portfolio is a reasonable concept in theory but it isa very elusive concept in practice

• How can we define a portfolio which includes all risky assets in an economy?

• Academics assume that deficiencies are minimal and do not impact on the final outcomes. However, Richard Roll pointed out that there is a benchmark problem

• If the benchmark (i.e. market portfolio) is mistakenly specifiedwe cannot measure a generic portfolio’s performance properly

An important caveat: the market portfolio

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Roll’s critique

Roll’s critique (cont’d)

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• An incorrect market proxy affects both beta risk measures and slope of the SML used to evaluate portfolio performance

• Errors tend to overestimate the performance of generic portfolios because proxy indices are not as efficient as the

true market portfolio→the slope of the SML is underestimated

• Beta is also underestimated because the variance of the proxy indices is higher of the true market portfolio’s one.

• However, this does not imply (yet) that the CAPM is not valid. It only states how difficult to test the theory and apply using real-world data!

Roll’s critique: summing up

The CAPM and its implications

• The derivation of the CAPM leads to the following equation

• The first term is the expected return on assets which are uncorrelated with the market return (rf), the second term is a risk premium

• Alternatively βi is the (standardized) covariance risk

of asset i within the market portfolio M.

( ) ( )( )

2

cov ,i f i M f

i Mi

M

E r r E r r

r r

β

βσ

⎡ ⎤= + −⎣ ⎦

=

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The CAPM and its implications

• The CAPM equation postulates that– the expected return of any risky asset is a linear function of

its tendency to co-move with the market portfolio (β). This implies that if the CAPM is an accurate description of the way assets are priced, there should be a positive linear relationship between average portfolio returns and portfolios’betas.

– betas should be all that matters in a CAPM world. If beta is included as explanatory variable, no other variables should be able to explain cross-sectional differences in average returns

• How well the CAPM explains the relationship that exists in the real world?– are beta estimates stable?

– is there any positive relationship between beta and returns on risky assets?

Is Beta Dead?

• If the CAPM holds, beta should be all that matters. No other variables should be able to explain returns on risky assets (on average)

• Some early empirical evidence provided some support for the validity of the CAPM (the relationship has been found positive, Black, Jensen and Scholes1972)

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Is Beta Dead? (cont’d)

• Subsequent evidence contradicts the predictions of the CAPM:

– Stock with high earnings/price ratios (E/P) have been found to earn significantly higher returns than stocks with low earnings/price ratios (Basu 1977, 1983)

– Firms with low market capitalisation (i.e. firm size) have higher average returns than large cap stocks (Banz, 1981)

– Stocks with high ratios of book value common equities to market value of common equities (i.e. book-to-market ratio, BtM) have significantly higher returns than stocks

with low BtM (Rosemberg et al. 1985, Chan et al. 1991)

Fama-French Criticism

• Fama and French (1992), FF from now on, brought together (among all) size, E/P, BtM and beta in a single cross-sectional study

• Their findings reject the validity of the CAPM.

– The positive linear relationship between average returns and betas is spurious

– It does not appear in the data when portfolio are ranked by both size and beta

• Given that beta does a poor job in explaining average returns, what variables do a better job?

• Their answer: size and BtM

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Fama-French Criticism (cont’d)

Empirical evidence: summing up

• Tests of the CAPM over the past thirty years found that:

– Beta coefficients for individual securities were not stable (although beta on aggregate portfolios were more stable over longer period of times)

– There was mixed support for a positive linear relationship between returns on risky assets and betas

– However additional variables (other than beta) have been found significant indicating an emerging need for different risk proxies

• The academic community has considered alternative asset pricing models

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The alternative: APT

• The search for alternatives to the CAPM that were intuitive, with limited assumptions ended up with the Arbitrage Pricing Theory (APT)

• APT assumes that:

1. Capital markets are perfectly competitive

2. There are sufficient securities to diversify away idiosyncratic risk

3. Security markets do not allow for the persistence of arbitrage opportunities

• Note that a mean-variance efficient market portfolio containing all risky assets is not required.

APT: the model

• E(ri) is the expected return on asset i if all risk factors are zero

• βj,i is the reaction of asset i to movement in the common factor Fj

• Fj is a factor with zero mean influencing returns on allassets

• ei is the firm-specific effect on asset i return (which can be fully diversified away in large portfolios)

( ) ,i i j i j ij

r E r F eβ= + +∑

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APT: the model (cont’d)

• Some clarifications:

– F are multiple risk factors affecting the whole spectrum of asset returns. What are they? In general macroeconomic factors (i.e. GDP growth, inflation, interest rate changes etc.)

– β are factor betas indicating how, over time, a specific asset reacted to a certain factor

– ei given assumption 2, the firm-specific risk can be fully diversified away in large portfolios

• One problem: No theory! No one tells us how to identify the factors (how many?) and what the factors represent.

APT and CAPM

βj,i j =1, ..,K

i = 1, .., N

βFactor risk sensitivity

Fj j=1, ..,KE(RM) – rfFactor risk premium

K(≥1)1Number of risk factors

linearlinearForm of equation

APTCAPM

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Shanken’s critique

• The APT, thanks to its set of less restrictive assumptions, delivers a quantitative implication similar to the CAPM but without incurring in the Roll’s critique

• APT test has been empirically tested providing supporting results

• However, Shanken challenged the APT legacy by proposing this similarity. It cannot be tested because:

– the CAPM suffers from the benchmark (or market portfolio problem)

– The APT suffers from the fact that there is no guidance in identifying the relevant factor structure that affect asset returns

APT in practice

• From a single index (SI) model to a multifactor (MF) model :

• As for the SI model, in the MF model we replace expected values of asset returns with realized values

• Question: Where should we look for factors?

• Risk factors can be:

– macroeconomic (economy-wide)

– microeconomic (focusing on the characteristics of the assets)

( ),

:

: i f i i M f i

i f i i j j ij

SI r r r r e

MF r r FP e

α β

α β

− = + − +

− = + +∑

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Macroeconomic-based MF models

• Chen, Roll and Ross (1986) propose the following model:

where:

– rM is the return on a market index (NYSE)

– MP is the monthly growth rate of the industrial production

– DEI is the change of CPI inflation

– UI is the difference between actual and expected inflation

– UPR is the unanticipated bond credit spread (BAA – rf)

– UTS is the unanticipated term structure shift (long-term minus short-term)

,1 ,2 ,3 ,4 ,5 ,6it i i Mt i t i t i t i t i t itr r MP DEI UI UPR UTS eα β β β β β β⎡ ⎤= + + + + + + +⎣ ⎦

Macroeconomic-based MF models (cont’d)

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Microeconomic-based MF models

• Fama and French (1993) propose a general model where the excess return on a security i can be represented as follows:

where

– rM is the excess return on a broad market portfolio

– SMB (=small minus big) is the difference between the return on a portfolio of small stocks and large stocks

– HML (=high minus low) is the difference between the return on a portfolio of stocks with high and low book to market ratio

( )it ft i i Mt ft i t i t itr r r r SMB HML e,1 ,2 ,3α β β β− = + − + + +

Microeconomic-based MF models (cont’d)

• The FF three-factor model is an equilibrium pricing model where SMB and HML are two underlying risk factors of special hedging concern to investors.

• Therefore SMB and HML captures independent sources of systematic risk, which are not embedded in betas on the (excess returns of the) market portfolio.

• SMB and HML behave like any other risk factors in the APT approach.

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Microeconomic-based MF models (cont’d)

What are the Size and Value Factors?

• SMB and HML have been found important in explaining the behaviour of stock returns. However they are mere empirical factors

– Do SMB and HML portfolios proxy macroeconomic, aggregate, nondiversifiablerisk?

– Could SMB and HML be related to a default risk premium or a distress factor?

– Do SMB and HML portfolios proxy for state variables that describe the time variation of the investment opportunity set?

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Readings

• Bodie, Kane and Marcus

– Chapters 8 (up to Section 8.3 included), 9 (up to Section 9.4 included), 10

• Other readings (optional)

– Fama, E.F., French, K.R. (1996), “MutifactorExplanations of Asset Pricing Anomalies”, Journal of Finance 51, 55-84.