The
efficient market hypothesis has been challenged for many decades and there has
not been a consensus among academic researchers about whether investors can
earn excess return over the market. Many studies have given evidence that some
investment strategies can actually outperform the market. These include
technical analysis, value investing, contrarian investing, or small firm
effects, etc. Such strategies have also been adopted in real-world investment
and proved to be successful.
Concerning
risk and return in investment, investors are always risk-averse while preferring
high return. The mean-variance portfolio theory, which is initiated by Markowitz (1952), proposes that risk and return have
negative relationship. Investors cannot earn higher return if they are
unwilling to take more risk, which is measured by volatility of stock prices.
Total risk consists of systematic risk and unsystematic risk. However, as
reasoned by Sharpe (1964), investors cannot be rewarded with higher return if
they bear more unsystematic risk because such risk can be eliminated for free
through diversification. Only systematic risk is relevant for investor
decision. In the Capital Asset Pricing Model (CAPM) proposed by Sharpe (1964)
and Lintner (1965), expected return of an asset is linearly related to its
beta, which is the standardized covariance between the individual asset and the
market portfolio. That relationship is represented by the security market line.
In market equilibrium, all assets should be plotted on the security market
line.
Although the CAPM together with the efficient frontier and
the capital market line are written in most finance books and widely taught in
investment courses, there have been many debates around them. The security
market line in CAPM has been noticed to be much flatter than being proposed by
the theory (Black, Jensen, & Scholes, 1972, Fama & MacBeth, 1973, or
Fama & French, 2004), that is low-beta stocks earns higher alphas than
high-beta stocks. More recently, some studies have given evidence that low-risk
equity portfolios, as measured by volatility or beta, outperform high-risk
portfolios in terms of risk-adjusted returns, and even absolute returns
(Clarke, De Silva, & Thorley, 2006, Blitz & Van Vliet, 2007, Ang et
al., 2009, or Baker, Bradley, & Wurgler, 2011).
The evidence of low-risk effect has only been documented
mainly in the U.S. market, but has not been widely acknowledged among the
investment community. If that effect really exists in real-world, investors
will have many benefits when exploiting that anomaly. Motivated by the
unanswered question, the paper aims to investigate if the low-risk effect is
present in the U.K. stock market.
II. RESEARCH QUESTION AND OBJECTIVE
Title of the dissertation:
Low-risk effect in the U.K. stock
market.
Research question: Do
low-risk portfolios, as measured by standard deviation and beta calculated from
lagged price volatility, generate higher risk-adjusted returns in the U.K.
stock market?
Research objective:
-
This paper seeks to investigate
performance of low-risk portfolios in the U.K. stock market and compare it with
the performance of high-risk portfolios and the general market.
-
Other well-known
effects such as firm size, value, or momentum will also be analyzed in order to
identify whether the low-risk effect overlaps with them.
-
This paper will give
some possible explanations about the anomaly and implications for investors.
III. LITERATURE REVIEW
1. Traditional theory about the trade-off between risk and return
In
the mean-variance portfolio theory initiated by Markowitz
(1952), one of the major assumptions is that investors are risk-averse,
which means that among a group of investment opportunities with similar
expected returns, investors will seek for the one that has the lowest risk, and
they will only accepted a riskier investment if they are compensated by greater
expected return. Risks and returns are the only two factors being considered by
investors in their investment decisions. Expected returns are subject to a
probability distribution under different scenarios over an investment time
period, and risks are measured by the variance or standard deviation of
expected returns. A combination of different levels of expected returns for different
levels of risks maps the indifference curves of investors.
Because
of investors’ nature of risk aversion, they will only invest in portfolios
which are deemed to be efficient. A portfolio is efficient if there is no other
portfolio offering a higher expected return at the same or lower level of risk
or requiring lower risk for the same or higher amount of return. Combining a
set of efficient portfolios for different levels of risk will create the
efficient frontier, which is exhibited in Figure 1. Once investors are on the
efficient frontier, it is impossible to increase expected returns without
increasing risk (Sharpe, 1970: 32). If
the capital market is efficient and investors have homogeneous expectations
about risk and return, the efficient frontier implies that risk and return will
have a positive relationship with each other. Being on the efficient frontier,
investors will only earn higher return if they are inclined to take more risk.
Delineating the efficient frontier from a set of individual securities today is
quite a simple task with the assistance of computer program. The general
approach is to estimated expected returns and covariance matrix for individual
securities. For each level of expected return, the computer program can
determine the weight of each security in the portfolio so that the portfolio’s
variance will be minimized (Clarke, De Silva, & Thorley, 2006).
Figure 1. The efficient
frontier and the capital market line
Combining
a risk-free asset with the risky portfolio on the efficient frontier will
create the capital market line, as being shown in Figure 1. Investors at the
point Rf invest 100% of their capital into the risk-free asset,
while investors at the point M invest all their capital into the risky market
portfolio. Theoretically, the market portfolio consists of all types of risky
assets, including stocks, bonds, alternative investments, so it is completely
diversified. When moving on the capital market line to the left of point M,
investors invest part of their funds into the risk-free asset and the remaining
in the market portfolio. When switching to the right of point M, investors
borrow money to buy more of market portfolio. Risks increase as investors move
along the capital market line.
The
total risk (σ) of a portfolio or a security comprises of systematic risk and
unsystematic risk. Theoretically, unsystematic risk can be diversified away by
combining various securities which have low correlations. Therefore, in market
equilibrium, investors are not compensated for bearing unsystematic risks
because it can be removed by diversification at no cost. The only relevant source
of uncertainty about the return of an asset is its systematic risk (Sharpe, 1970: 97). The expected return will
depend on the systematic risk, which is the covariance between the return of
the asset and the whole market.
With the above
argument about systematic risk, Sharpe (1964)
and Lintner (1965) develops the famed
Capital Asset Pricing Model (CAPM), which is used to determine the value of an
asset based on its systematic risk. The CAPM proposes that the expected return
of an asset has positive linear relationship with its systematic risk. That
relationship creates the security market line, which is exhibited in Figure 2.
Figure 2. The security
market line
In
Figure 2, the systematic risk of an asset is defined as βi, which is
the standardized covariance between the return on a single portfolio or asset
and the market return. In market equilibrium, all securities and portfolios,
whether diversified or not, should be plotted on the security market line. When
βi = 0, return on the asset is equal to the risk-free rate. When 0
< βi < 1, the asset has lower risk than the market portfolio,
thus generating less return than the market, but higher than the risk-free
rate. When βi = 1, the asset is perfectly correlated with the
market, and its return is expected to be the similar the market. When βi
> 1, the asset is considered to be riskier than the market and is expected
to generate higher return than the market. The slope of the security market
line is equal to the excess return on the market portfolio above the risk-free
rate, that is E(Rm) – Rf .
In
summary, it is generally believed that investors can expect higher return when
investing in assets that have higher risk, as measured by variance in return.
According to the CAPM, the only risk investors should concern about is the
systematic risk, measured by beta. Investors should not expect additional
return when bearing unsystematic risk because it can be fully eliminated by means
of diversification.
2. Empirical tests of the relationship between return and risk
Some
early empirical tests partly support the traditional view of the risk-return
relationship. One of the studies which are in favor of the CAPM is that
conducted by Sharpe and Cooper (1972). They
create decile portfolios from all New York Stock Exchange stocks ranked on
their betas. The betas are estimated from 5-year historical data. The decile
portfolios are held in one year and rebalanced annually from 1931 to 1967. They
find that the portfolios comprising of higher beta stocks have produced higher
returns. An increase of one unit in beta will generate annual excess return of
12.75%. In addition, the variance in returns of these portfolios is mainly
explained by betas, which represent systematic risk. The coefficient of
determination in regressing portfolio returns on market returns ranked from
0.87 to 0.94.
However, other tests of the CAPM do not have consistent
results with the theory. The studies of Black, Jensen, and Scholes (1972) and
Fama and MacBeth (1973) are regarded as standard in testing the theory of CAPM.
They form portfolios ranked on their betas and conduct a test of the
traditional CAPM:
Rit – RFt = αi + βi (RMt
– RFt) + eit
In the test by Black, Jensen, and Scholes (1972), low beta portfolios
have positive alphas, and high beta portfolios have negative alphas, while the
CAPM suggests that the alphas should be zero in market equilibrium. When they
divide their data from 1931 to 1965 into four sub-periods, they find that the
slopes of the security market lines were not constant. The security market line
became flatter over time. In the last period of 1957-1965, it even had the
negative slope.
Fama and MacBeth (1973) found similar results to Black,
Jensen, and Scholes (1972). They form 20 portfolios from stocks in the NYSE
ranked on their beta and observe their returns from 1926 to 1968. Although
there is a positive relationship between beta and average return, the security
market line is too “flat” compared with what is proposed by the CAPM. The
intercept of the security market line is consistently higher than the risk-free
rate, and the slope coefficient of beta is smaller than the excess return of
the market. However, they find some evidences that are consistent with what
being suggested by the CAPM: assuming the market is efficient, the unsystematic
risk has no effect on the expected return of a security, and while the slope of
the security market line is “flat”, it shows a linear relationship between
expected return and beta. Fama and French (2004) once again confirm the “flat”
security market line using data of stocks listed on NYSE (1928-2003), AMEX
(1963-2003), and Nasdaq (1972-2003).
More
recently, studies by Blitz and Van Vliet (2007)
or by Baker, Bradley, and Wurgler (2011) show that low-beta and low-volatility
stocks outperform high beta and high volatility stocks. In the mean-variance
portfolio theory, portfolios that are on the capital market line should
dominate all other portfolios. However, Blitz and Van Vliet (2007), using data
from 1986 to 2006 in the U.S., Europe and Japan markets, demonstrate that
portfolios whose total volatility is lower than the market are more efficient than
portfolios lying on the theoretical capital market line. They also argue that
the security market line has a negative slope if that line is created by
plotting actual returns of portfolios with different level of betas. When the
Sharpe ratio is taken into account, there is no reason to invest into high-risk
portfolio because they generate low risk-adjusted returns. The volatility
effect is a separate effect and does not relate to other effect such as value,
size, or momentum.
Especially, the results found by Baker, Bradley, and Wurgler
(2011) are stunning. They sort 1,000 largest stocks in the U.S. market by their
5-year total volatility and betas. From the sorting, the stocks are put into
five portfolios, ranked by their level of risks in terms of volatility and
betas. Each month, the portfolios are rebalanced so that their levels of risk
are maintained. They find that a U.S. dollar invested in the low-volatility
portfolio grows to USD 53.81 over 41 years, while a dollar only grew to USD
7.35 if invested in the high-volatility portfolio. For beta, the numbers are
USD 78.66 and USD 4.70, respectively. The return gap has widened since the
early 1980s, with the only exception in the technology stock bubble.
Another
approach to capture the low-risk anomaly is to form the minimum-variance
portfolio. Clarke, De Silva, and Thorley (2006)
select stocks from 1,000 largest stocks in the U.S. to form a portfolio that
has the minimum variance and calculate its returns from 1968 to 2005. The mean
annual return of the minimum-variance portfolio in excess of the risk-free rate
is 6.5%, while return on the market portfolio is lower at 5.6%. The standard deviation
of the minimum-variance portfolio is also lower than the market. High excess
return and low standard deviation lead to significantly better Sharpe ratio.
The concept of efficient frontier is invalid in this case because the
minimum-variance portfolio lies outside the theoretical efficient frontier.
When returns of the minimum variance portfolio are regressed on the returns of
the market portfolio, the annualized alpha is 2.8% and beta is 0.65.
The
CAPM proposes that idiosyncratic risk (or unsystematic risk) is not relevant in
determining expected return of an asset. Investors should not expect higher
return when taking additional unsystematic risk because it can be simply
eliminated by diversification. However, it is only the case when the market is efficient
and frictionless, that is no tax, no transaction cost, no restrictions in
borrowing and short-selling, and all information is readily available at no
cost. In environment of market imperfection, unsystematic risk may be related
to expected return (Ang, Hodrick, Xing, & Zhang, 2009).
The
effect of idiosyncratic risk is a real puzzle. Studies conducted by Tinic and West (1986), Lehmann (1990), Barberis and Huang (2001), or by Malkiel and Xu (2006)
find that higher returns may be expected from greater amount of idiosyncratic
volatility. However, Ang, Hodrick, Xing, and Zhang (2006, 2009) argue that low
return is associated with both high systematic risk and indiosyncratic
volatility in the U.S. market. When regressing stock returns against total
volatility and some other factors affecting returns, they find that the
regression coefficient of total volatility carries a negative sign, which means
that stocks with high systematic risk would have low average returns. In
addition to this, when quintile portfolios are formed by sorting idiosyncratic
risks of stocks, the lowest volatility portfolio generates returns that are
higher than the highest risk portfolio by an average of 1.06% per month over
the 1963-2000 period. The result is robust after being controlled for the
effects of size, book-to-market, liquidity risk, trade volume, and momentum
effects. Changing the period for calculating the volatility or the holding
period does not diminish the robustness of the result. There are similar results
about idiosyncratic risk when the study is expanded to other developed markets.
Their criticism about previous studies is that such studies do not investigate
idiosyncratic risk at the level of firms and not group firms into portfolios sorted
by their idiosyncratic volatility, thus failing to identify the negative
correlation between unsystematic risk and expected return.
Motivated
by the finding of Ang, Hodrick, Xing, and Zhang
(2006), Bali and Cakici (2008) conduct similar research about the relationship
between idiosyncratic risk and return. They acknowledge the results of Ang,
Hodrick, Xing, and Zhang (2006). However, they argue that the results may be
biased because the highest idiosyncratic quintile represent only 2% of total
market capitalization. Those stocks are very illiquid, small and always have
low price. After modifying the methodology so that it would not be biased by
the effect of small and illiquid stocks, no significant relationship between unsystematic
risk and return is found.
In summary, most empirical studies have found that the
security market line, which represents the relationship between stock returns
and betas, is much flatter than predicted by the CAPM. While early studies in
the 1970s and 1980s acknowledge the positive relationship between stock returns
and systematic risks (but flatter than proposed by the CAPM), some studies in
the 2000s noticed that the relationship is negative. Stock returns are also
negatively correlated to total volatility. The low volatility anomaly has become more obvious since the
mid-1980s, with the exception in the technology stock bubble. Regarding the
idiosyncratic risks, there is mixed evidence about its effect on returns.
The
inefficiency in the relationship between risk and return may be explained by
the unrealistic assumptions in the mean-variance portfolio theory and CAPM,
such as investors have homogeneous expectations and investment periods,
information is readily available at no cost and risk-free borrowing, lending,
and short-selling are unrestricted. The difficulties in correctly estimating
expected returns, betas and covariance among securities may also be an
obstruction against market efficiency. Besides, in the real world, it is
impossible to determine the market portfolio, which theoretically consists of
all risky assets in the market. Baker, Bradley,
and Wurgler (2011) explain the mispricing of high-risk stock from the
perspective of behavioral finance. There is an irrational preference for high
volatility, which causes high-risk stocks to be overpriced and generate low
returns. For institutional investors, they are unlikely to exploit the low-risk
anomaly because of performance benchmarks (Blitz
& Van Vliet, 2007). Portfolio managers have an incentive
to invest in high-risk stocks, as it is easy to generate above-average returns.
In addition, many investment mandates restrict the use of leverage, thus making
low-risk stock not the choice of investment managers. Without leverage, it is
hard to earn high return on low-risk stocks, which are generally believed to
have low returns. The restrictions in the use of leverage by institutional
investors make low-risk stocks underpriced.
IV. METHODOLOGY
This
paper will examine the relationship between stock returns and risks, measured
by volatility and betas, to investigate whether the volatility effect exists in
the U.K. stock market, that is, low-risk portfolios may generate high
risk-adjusted return.
Most studies about the relationship between risk and return use
portfolios ranked on their risk level instead of calculating the correlations from
individual securities. If returns are regressed against risks for each
individual security, there will be autocorrelation problem because the
residuals of the regression are not independent (Elton, Gruber, Brown,
& Goetzmann, 2007).
Besides, the estimation of beta for each security may be incorrect. Therefore, in
order to avoid these problems, the standard in conducting this type of research
is to form portfolios and rank them on their risks, rather than working with individual
security. Then the relationship between risk and return of each portfolio is
specified in each holding period. Since the portfolios represent
diversification and incorporate the effect of any cross-sectional interdependencies,
the betas estimated for portfolios are more precise and the problem of residual
correlation is resolved.
1. Data collection
Data of all stock constituents in the FTSE All-Share Index
will be collected in 31 years, from 1980 to 2010. The main source of data will
be Reuters database. The FTSE All-Share Index currently represents
approximately 98% of the UK’s market capitalization. The large market share of
the FTSE All-Share Index ascertains that the results of this study will not be
biased.
In order to calculate the volatility and beta of stocks,
stock prices are collected on a weekly basis. Weekly returns are
log-transformed in order to make them additive over time:
Standard deviation of return is calculated using weekly return
in the past three years (156 weeks):
The FTSE All-Share is considered to be the market portfolio. Beta of each stock is calculated by regressing stock weekly
return on weekly return of FTSE All-Share Index:
Apart from price, other data that should be collected
includes book value, market capitalization, industry sector of each stock and
the risk-free rate.
2. Portfolio construction and evaluation
Stocks are ranked on standard deviation and grouped into quintiles
in order to create five equally weighted portfolios with different risk level.
The first quintile consists of lowest volatility stocks, the fifth quintile
consists of the highest volatility stocks. Portfolios are rebalanced at the
beginning of each month so that the first portfolio always has the lowest level
of volatility. Transaction costs are ignored during the analysis.
After portfolios have been constructed, monthly returns of each portfolio are calculated at the
time of rebalancing. The monthly
returns are tracked in the whole period of 31 years. In the end, the following
measures will be calculated for each portfolio and the market (represented by
FTSE All-Share Index):
-
Mean monthly
returns and standard deviations.
-
Returns
of portfolios in excess of the risk-free rate, which is determined by the yield of 3-month U.K. government bonds since such
short-term bonds have the lowest risk among all other kinds of bonds.
-
The
Sharpe ratio of portfolios:
-
Alphas
(excess return over the market) and betas of portfolios, by using the CAPM
model to regress the excess return of portfolios on the market risk premium.
Ri – Rf = αi + βi (RM
– Rf) + ei (1)
Comparisons are made among the quintile portfolios and the
market in order to determine whether investing in the low-volality portfolios
will be more efficient and generate any excess return. The following hypothesis
will be tested:
-
There
is difference among mean returns of the low-volatility portfolios and the
high-volatility portfolios and the market. A t-statistic will be used to test
the difference in mean (Kaplan, 2009):
-
If the
market is efficient, the alphas in (1) should be zero. If the alphas in (1) are
significantly different from zero for a portfolio, that portfolio is believed
to generate excess return over the market. To test that hypothesis, a
t-statistic will be used:
-
There
is difference among Sharpe ratios of the low-volatility portfolios and the
high-volatility portfolios and the market. The method of Jobson and Korkie (1981)
and Memmel (2003) is used to test the hypothesis. The test statistic follows a
standard normal distribution:
Where SRi is the Sharpe ratio of portfolio i, ρi,j
is the correlation between portfolio i and j, n is the number of observations.
To investigate the effect of beta, a similar procedure is
conducted. Stocks are allocated into five portfolios following their beta
ranking. Performance of the quintile portfolios is then examined and compared.
3. Controlling for other effects
One may suspect that the low-risk effect may be related to
other effects which have been acknowledged such as size, value, or momentum. If
the low-risk effect still exists after being controlled for other effect, it
can be determined that the low-risk effect is a separate and independent one.
3.1. Construct portfolio ranked on size, value and momentum
It has been well documented in several researches that
returns on small firms tend to outperform large firms, firms having high
book-to-market value tends to outperform low-book-to-market value, and past
winning stocks are likely to continue making excess returns in the short-term.
In this part, a method which is similar to part 2 is used to
construct portfolios. Stocks are ranked on market capitalization,
book-to-market ratio, 12-month minus 1-month total return and allocated to
quintile portfolios. For each strategy, portfolio performance is evaluated in
terms of return, Sharpe ratio, alpha, and volatility. If performance
measurements of the best-performing portfolios in each strategy are not better
than the performance of low-risk portfolio, while their volatility is higher,
the low-risk anomaly can be considered to be a separate effect.
3.2. Control for size and value effect using the Fama-French regression
In this section, alphas of the quintile portfolios ranked on
volatility and beta in part 2 are determined by the model of Fama and French
(1993). The regression model is as follows:
Ri
– Rf = αi + βi (RM – Rf)
+ si SMB + hi HML + ei (2)
Where:
-
SMB is
the return of the portfolio composed of the 30% smallest firms minus the return
of the portfolio of 30% largest firms with respect to market capitalizations in
the markets.
-
HML is
the return of the portfolio consisting of the 30% highest book-to-market firms
minus the return of the portfolio consisting of 30% lowest book-to-market firms
in the markets.
The Fama-French model captures the effects of firm size and
value. If the market is efficient, the alpha in (2) should be zero. If the
alpha of the low-risk portfolio is signficantly higher zero, the low-risk
portfolio is believed to generate excess return over the market, after being
controlled for the effect of firm size and value.
3.3. Double sorting
Control for value: Every month stocks are first grouped into
five quintiles with respect to their book-to-market ratios. Then in each value
quintiles, stocks are sorted into five groups ranked by volatility. Finally, the
lowest-volatility groups of each value quintile are allocated into to the lowest-volatility
portfolio. Four other portfolios ranking on value, then by volatility are
formed in the same way. The performance of each final quintile portfolio is
evaluated in the same way as in part 2. If the low-risk portfolio after being
controlled for value still outperforms the market, an anomaly can be identified.
Control for size and momentum: The double-sorting method is
similar as above in order to investigate the low-risk anomaly.
V. EXPECTED FINDINGS
It
is expected to find that the lowest-risk portfolio in the U.K. stock market is
more efficient than the market portfolio and has superior risk-adjusted return
over the high-risk portfolio in terms of Sharpe ratio and CAPM alpha. However,
the results may not be as expected since recent discovery of empirical evidence
has mainly concentrated in the U.S. market. There has been little knowledge of
the anomaly outside the U.S. market. Besides, the low-risk effect has not been
widely acknowledged by both academics and practitioners.
If
the anomaly really exists, investors can take advantage of it by increasing
their leverage or reducing investments in fixed-income securities. Following
that investment style, they can earn higher returns while bearing the same
level of risks.
REFERENCE
Ang, A., Hodrick, R. J., Xing, Y., & Zhang, X.
(2009). High idiosyncratic volatility and low returns: International and
further U.S. evidence. Journal of Financial Economics , 91 (1),
1-23.
Ang, A., Hodrick, R. J., Xing, Y.,
& Zhang, X. (2006). The Cross-Section of Volatility and Expected Returns. Journal
of Finance , 61 (1), 259-299.
Baker, M., Bradley, B., &
Wurgler, J. (2011). Benchmarks as Limits to Arbitrage: Understanding the
Low-Volatility Anomaly. Financial Analysts Journal , 67 (1),
40-54.
Bali, T. G., & Cakici, N.
(2008). Idiosyncratic Volatility and the Cross Section of Expected Returns. Journal
of Financial & Quantitative Analysis , 43 (1), 29-58.
Barberis, N., & Huang, M.
(2001). Mental Accounting, Loss Aversion, and Individual Stock Returns. Journal
of Finance , 56 (4), 1247-1292.
Black, F., Jensen, M. C., &
Scholes, M. (1972). The Capital Asset Pricing Model: Some Empirical Tests. In
M. C. Jensen, Studies in the Theory of Capital Markets. New York:
Praeger.
Blitz, D. C., & Van Vliet, P.
(2007). The Volatility Effect. Journal of Portfolio Management , 34
(1), 102-113.
Clarke, R., De Silva, H., &
Thorley, S. (2006). Minimum-Variance Portfolios in the U.S. Equity Market. Journal
of Portfolio Management , 33 (1), 10-24.
Elton, E. J., Gruber, M. J., Brown,
S. J., & Goetzmann, W. N. (2007). Modern Portfolio Theory and Investment
Analysis. Hoboken: John Wiley & Sons.
Fama, E. F., & French, K. R.
(1993). Common risk factors in the returns on stocks and bonds. Journal of
Financial Economics , 33 (1), 3-56.
Fama, E. F., & French, K. R.
(2004). The Capital Asset Pricing Model: Theory and Evidence. Journal of
Economic Perspectives , 18 (3), 25–46.
Fama, E. F., & MacBeth, J. D.
(1973). Risk, Return, and Equilibrium: Empirical Tests. Journal of Political
Economy , 81 (3), 607-636.
Jobson, J. D., & Korkie, B. M.
(1981). Performance Hypothesis Testing with the Sharpe and Treynor Measures. Journal
of Finance , 36 (4), 889-908.
Kaplan. (2009). CFA Level 1 Book
1: Ethical and Professional Standards, and Quantitative Methods. La Crosse:
Kaplan Schweser.
Lehmann, B. N. (1990). Residual
risk revisited. Journal of Econometrics , 45 (1), 71-97.
Lintner, J. (1965). The Valuation
of Risk Assets and The Selection of Risky Investments in Stock Portfolios and
Capital Budgets. Review of Economics & Statistics , 47 (1),
13-37.
Malkiel, B. G., & Xu, Y.
(2006). Idiosyncratic Risk and Security Returns. Working Paper, The
University of Texas at Dallas.
Markowitz, H. (1952). Portfolio
Selection. Journal of Finance , 7 (1), 77-91.
Memmel, C. (2003). Performance
Hypothesis Testing with the Sharpe Ratio. Finance Letters , 1
(1), 21-23.
Miller, M. H., & Scholes, M. S.
(1972). Rates of Return in Relation to Risk: A Re-examination of Some Recent
Findings. In M. Jensen, Studies in the Theory of Capital Markets. New
York: Praeger.
Sharpe, W. F. (1964). Capital Asset
Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of
Finance , 19 (3), 425-442.
Sharpe, W. F. (1970). Portfolio
Theory and Capital Markets. New York: McGraw-Hill.
Sharpe, W. F., & Cooper, G. M.
(1972). Risk-Return Classes of New York Stock Exchange Common Stocks,
1931-1967. Financial Analysts Journal , 28 (2), 46-81.
Tinic, S. M., & West, R. R.
(1986). Risk, Return, and Equilibrium: A Revisit. Journal of Political
Economy , 94 (1), 126-147.
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