Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 6
Content:
• Relaxing the CAPM assumptions
• Arbitrage pricing theory
• Multifactor models
Relaxing the CAPM assumptions
The standard CAPM has strict and unrealistic assumptions:
• All investors have homogeneous expectations (they
expect the same probability distribution of returns).
• All investors want to invest in an optimal portfolio based
on Markowitz’s mean-variance framework (lowest
volatility for a given expected return).
• All investors can lend and borrow any amount of money
at the risk-free rate.
• All investors have the same one-period horizon.
• All assets are infinitely divisible.
• There are no taxes and transaction costs.
Relaxing the CAPM assumptions
• There is no inflation or any change in interest rates, or
inflation is fully anticipated.
• Capital markets are efficient (they are in equilibrium).
Alternative versions of the CAPM have been developed by
relaxing some assumptions.
No alternative model is required if we disallow short
selling, as it can be shown that the model does not change.
Relaxing the CAPM assumptions
Different borrowing and lending rates
• Assuming an investor can lend at the risk-free rate is
realistic (can buy bonds that pay a saving rate 𝑅 𝑠)
• But borrowing at the risk-free rate is generally not
possible: investors typically pay a higher rate 𝑅 𝐵
• Different rates imply different tangent portfolios: 𝑇𝑆 for
the saving rate and 𝑇𝐵 for the borrowing rate
• The market portfolio is no longer the unique efficient
portfolio of risky securities
• The SML still holds for some rate 𝑅∗ between 𝑅 𝑠 and 𝑅 𝐵:
𝐸 𝑅𝑖 = 𝑅∗ + 𝛽𝑖(𝐸 𝑅 𝑚 − 𝑅∗)
Relaxing the CAPM assumptions
Different investors choose different portfolios on the line
between 𝑇𝑆 and 𝑇𝐵.
Relaxing the CAPM assumptions
Zero-Beta CAPM
• In the standard CAPM the return of zero-beta assets is the
risk-free rate, but this was rejected empirically
• Black (1972) drops the assumption of unlimited lending
and borrowing at the risk-free rate, and considers
portfolios uncorrelated with the markets (i.e., with 𝛽 = 0)
• The return of these 𝑅 𝑍 portfolios replaces 𝑅𝑓 in the SML:
𝐸 𝑅𝑖 = 𝐸 𝑅 𝑍 + 𝛽𝑖(𝐸 𝑅 𝑚 − 𝐸 𝑅 𝑍 )
• Empirical support for this model is mixed
Relaxing the CAPM assumptions
Relaxing the CAPM assumptions
CAPM with transaction costs
• Investors will not correct small mispricings when the
costs of buying and selling nullify the potential gains.
• The SML is a band rather than a line, and the greater the
transaction costs, the wider the band becomes
Relaxing the CAPM assumptions
CAPM with heterogeneous expectations, investment
horizons, and Taxes
• Different expectations about risk and return, and
different investment horizons imply different CML
and/or SML.
• The composite graph would be a band instead of a line.
• Differences in taxation also have similar effects
• Taxation likely impacts on CML and SML but the exact
effect is difficult to determine
Relaxing the CAPM assumptions
Final considerations:
• Dropping single assumptions does not significantly
change the CAPM model
• The empirical support for CAPM is however very weak
• CAPM is still studied for its theoretical importance
• For practical purposes, more elaborate asset pricing
models need to be used
Arbitrage pricing theory
To overcome the shortcomings of CAPM, Ross (1976,1977)
developed the Arbitrage Pricing Theory (APT).
APT rests on three assumptions:
• Capital markets are perfectly competitive.
• Investors always prefer more wealth for a given risk.
• The stochastic process generating asset returns can be
expressed as a linear function of a set of K risk factors,
and all unsystematic risk is diversified away.
Arbitrage pricing theory
The model relies on the law of one price: equivalent
securities must have the same price.
Arbitrage opportunities are quickly exploited, bringing the
markets into equilibrium.
As in CAPM, homogeneous expectations are assumed. But,
contrary to CAPM, it is NOT assumed that:
• Investors have quadratic (i.e., mean-variance) utility
functions
• Returns of securities are normally distributed
• There is a market portfolio that contains all the assets
and is mean-variance efficient
Arbitrage pricing theory
Mathematically, APT requires that the return of stock 𝑖 is
linearly related to a set of 𝑘 factors:
𝑅𝑖 = 𝛼𝑖 + 𝑏𝑖1 𝑓1 + 𝑏𝑖2 𝑓2 + ⋯ + 𝑏𝑖𝑘 𝑓𝑘 + 𝜀𝑖
• 𝛼𝑖 is the expected return of 𝑖 if all factors have value 0
• 𝑓1, … , 𝑓𝑘 are a set of 𝑘 factors with zero mean
• 𝑏1, … , 𝑏 𝑘 (called “loadings”) specify the sensitivity of
the stock return to changes in the corresponding factor
• 𝜀𝑖 is a random error term with mean 0 which indicates
idiosyncratic shocks affecting the return of 𝑖. This risk
can be diversified away.
Arbitrage pricing theory
The factors 𝑓1, … , 𝑓𝑘 are assumed to impact the return of
all the assets.
Factors are “variables that proxy for a common source of
risk” on the market (DeMiguel et al., 2018)
The loadings 𝑏1, … , 𝑏 𝑘, on the contrary, are specific to
each asset 𝑖: each asset has different sensitivity to each
factor, which is why they have different returns.
APT does not say what this factors are. APT is therefore
difficult to be tested and used in its general form.
Multifactor models
In practice some specific multifactor model is used.
The most important is the Fama-French three-factor
model (Fama & French, 1993):
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝐸 𝑅 𝑚 − 𝑅𝑓 + 𝑏𝑖2 𝑆𝑀𝐵 + 𝑏𝑖3 𝐻𝑀𝐿
• 𝑺𝑴𝑩 (Small Minus Big) is the return difference between
a portfolio of small cap and one of large cap stocks
• 𝑯𝑴𝑳 (High Minus Low) is the return difference
between a portfolio of high Book-to-market (known as
“growth” stocks) and one of low Book-to-market stocks
(known as “value” stocks)
Multifactor models
Notice that the first factor is also in the CAPM.
CAPM is also a factor model: one with only one factor.
The Fama-French three-factor model is however empirically
based. It does not require all the restrictive theoretical
assumptions required by CAPM.
A variation is the Carhart four-factor model (Carhart, 1997):
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝐸 𝑅 𝑚 − 𝑅𝑓 + 𝑏𝑖2 𝑆𝑀𝐵 + 𝑏𝑖3 𝐻𝑀𝐿 + 𝑏𝑖4 𝑀𝑂𝑀
• 𝑴𝑶𝑴 (Momentum) is the return difference between a
portfolio of best performing and one of worst performing
stocks.
Multifactor models
In 2015 Fama and French modified their model and proposed
the Fama-French five-factor model:
𝐸 𝑅𝑖 = 𝑅𝑓 + 𝑏𝑖1 𝐸 𝑅 𝑚 − 𝑅𝑓 + 𝑏𝑖2 𝑆𝑀𝐵 + 𝑏𝑖3 𝐻𝑀𝐿 + 𝑏𝑖4 𝑅𝑀𝑊 + 𝑏𝑖5 𝐶𝑀𝐴
• 𝑹𝑴𝑾 (Robust Minus Weak) is the return difference
between a portfolio of stocks from high operating
profitability (“robust”) companies and one of stocks from
low operating profitability (“weak”) companies.
• CMA (Conservative Minus Aggressive) is the return
difference between a portfolio of stocks from high
companies that invest conservatively and one of stocks
from companies that invest aggressively.