Signal Processing - May 2017 - 19
Evaluating an Active Portfolio Manager-
Is There Alpha?
CAPM
The price of a security i (or a portfolio) in a free market
can be determined by the CAPM, also called the Sharpe-
Lintner model, as follows:
The time-series CAPM regression for an investment portfolio is
R p, t - rf, t = a + b pM (R M, t - rf, t ) + f p, t ,
E [R i] = rf + b iM (E [R M] - rf ), 6i,
where
b iM =
Cov (R i, R M)
.
v 2 (R M)
Time-series regressions are used in practice to compute b. The following time-series regression is perhaps
the most famous linear regression in the world:
R i, t = rf, t + b iM (R M, t - rf, t ) + f i, t .
Insights
* The CAPM is a market equilibrium that must hold.
* There is no expected return reward for idiosyncratic
risk, the part of security risk that is uncorrelated with
the market portfolio M.
The CAPM is the foundation of asset pricing. It is commonly used in corporate finance, such as merger-andacquisition practice to evaluate a company and the cost
of capital.
Terms Related to the CAPM and Asset Pricing
* Risk asset: assets that have uncertain returns, e.g.,
stocks, bonds, real estate, gold, crude oil. Their prices
fluctuate over time. Almost all assets are in this category.
* Risk-free asset: assets that have fixed (certain)
returns. Treasuries (especially T-bills) are considered
to be risk free because they are backed by the U.S.
government. The return on risk-free assets is very
close to the current interest rate.
* Excess return: E [R i] - rf is the expected return in
excess to the risk-free rate to reward the security risk,
i.e., an RP.
* Market RP: E [R M] - rf is the excess expected return
to reward market risk (approximately 7% annually in
the U.S. market).
* Systematic risk: market risk that cannot be diversified
and is rewarded by the expected return. Represented
by b.
* Idiosyncratic risk: firm-specific risk that can be diversified, thus not rewarded by the expected return.
Represented by f i, t . Note E [f i, t ] = 0. It looks like
the noise term in SP, but it is not noise in economics.
If the market risk-free rate is r f , a market participant will
hold a combination of the risk-free asset r f and a risk portfolio
or security within or on the efficient frontier. Apparently, the
best risk portfolio to hold is on point A because the tangent
where a is the intercept of the regression. It should be
statistically insignificant according to the CAPM, i.e.,
a = 0.
The skill of an investment manager should not be evaluated by the expected return or the excess return over
the market portfolio but rather by a.
line provides the best portfolios in terms of the mean-variance
criterion. Indeed, the tangent line provides the best SNR from
the SP perspective. [The Sharpe ratio of a portfolio p can be
considered the SNR defined in finance: ^E [R p] - r f h ^v R ph.]
The efficient portfolio A is called the tangency portfolio.
In market equilibrium, the market portfolio M (i.e., the portfolio consisting of all risk securities) must be the tangency
portfolio A. A market index, such as the S&P 500 index, is
often used as a proxy of the market portfolio M. All securities must have prices that satisfy (8). Such a price relationship (or correlations) among all risk securities in the market is
described in the capital asset-pricing model (CAPM) [18]-[21]
(see "CAPM" for details).
As we show, the CAPM is consistent with the expected utility theory and portfolio optimization. According to the CAPM,
the stock return is an RP and is only related to its correlation
with the market portfolio (i.e., the market risk a stock carries).
The higher the b , the larger is the expected stock return. The
diversifiable (idiosyncratic) risk a stock carries (i.e., the company-specific risk) does not have RP rewards.
Operationally, a higher expected return can always be
achieved by leveraging the risk (i.e., by borrowing money to
buy market portfolio). In Figure 3, this means moving along
the tangency line to the right, thus generating higher risk and
higher return with a fixed Sharpe ratio (or SNR).
The CAPM provides a way to evaluate an investment
portfolio and attribute its performance (see "Evaluating an
Active Portfolio Manager-Is There Alpha?"). "An Example CAPM Regression" shows a typical empirical regression study.
Fama-French three-factor model
and multifactor asset pricing
It is difficult to have a significant alpha, i.e., to beat the market.
People have long strived to find excess returns over the market
premium. Many of the identified excess returns turned out to
be statistically chance results, though a few proved to be persistent. The most prominent model identifying excess returns
beyond the CAPM is the Fama-French (FF) three-factor
IEEE Signal Processing Magazine
|
May 2017
|
19
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
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Signal Processing - May 2017 - 111
Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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