A survey on risk-return analysis
ABSTRACT This paper presents a stock-flow consistent macroeconomic model in which financial fragility in firm and household sectors evolves endogenously through the interaction between real and financial sectors. Changes in firms' and households' financial practices produce long waves. The Hopf bifurcation theorem is applied to clarify the conditions for the existence of limit cycles, and simulations illustrate stable limit cycles. The long waves are characterized by periodic economic crises following long expansions. Short cycles, generated by the interaction between effective demand and labor market dynamics, fluctuate around the long waves.
- Journal of Financial and Quantitative Analysis 06/1971; 6(03):909-912. · 1.77 Impact Factor
Article: Qualitative threshold ARCH models[Show abstract] [Hide abstract]
ABSTRACT: In this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance are endogenous stepwise functions. We first consider the probabilistic properties of these models: stationarity conditions, leptokurtic effect, linear representation, optimal prediction. In this first part most results are based on Markov chains theory. Then we derive statistical properties of this class of models; pseudo-maximum likelihood estimators, conditional homoscedasticity tests, tests of weak or strong white noise, CAPM test, factors determination, ARCH-M effects. We also discuss the introduction of exogenous variables and the case of multiple lags. Finally, an application to the Paris Stock Index is proposed.Journal of Econometrics 02/1991; · 1.71 Impact Factor
- The Journal of Finance 02/1980; 35(4):897-913. · 4.22 Impact Factor
A Review of Capital Asset Pricing Models
Department of Econometrics and Business Statistics
PO Box 197 Caulfield East
Victoria 3145 Australia
This paper provides a review of the main features of asset pricing models. The
review includes single-factor and multifactor models, extended forms of the
Capital Asset Pricing Model (CAPM) with higher order co-moments, and asset
pricing models conditional on time-varying volatility.
Key words: Asset pricing, CAPM, single-factor and multifactor models
The foundations for the development of asset pricing models were laid by Markowitz (1952)
and Tobin (1958). Early theories suggested that the risk of an individual security is the
standard deviation of its returns – a measure of return volatility. Thus, the larger the standard
deviation of security returns the greater the risk. An investor’s main concern, however, is the
risk of his/her total wealth made up of a collection of securities, the portfolio. Markowitz
observed that (i) when two risky assets are combined their standard deviations are not
additive, provided the returns from the two assets are not perfectly positively correlated and
(ii) when a portfolio of risky assets is formed, the standard deviation risk of the portfolio is
less than the sum of standard deviations of its constituents. Markowitz was the first to develop
a specific measure of portfolio risk and to derive the expected return and risk of a portfolio.
The Markowitz model generates the efficient frontier of portfolios and the investors are
* Correspondence to: Tissa.Galagedera@buseco.monash.edu.au
expected to select a portfolio, which is most appropriate for them, from the efficient set of
portfolios available to them.
The computation of risk reduction as proposed by Markowitz is tedious. Sharpe (1964)
developed a computationally efficient method, the single index model, where return on an
individual security is related to the return on a common index. The common index may be any
variable thought to be the dominant influence on stock returns and need not be a stock index
(Jones, 1991). The single index model can be extended to portfolios as well. This is possible
because the expected return on a portfolio is a weighted average of the expected returns on
When analysing the risk of an individual security, however, the individual security risk must
be considered in relation to other securities in the portfolio. In particular, the risk of an
individual security must be measured in terms of the extent to which it adds risk to the
investor’s portfolio. Thus, a security’s contribution to portfolio risk is different from the risk
of the individual security.
Investors face two kinds of risks, namely, diversifiable (unsystematic) and non-diversifiable
(systematic). Unsystematic risk is the component of the portfolio risk that can be eliminated
by increasing the portfolio size, the reason being that risks that are specific to an individual
security such as business or financial risk can be eliminated by constructing a well-diversified
portfolio. Systematic risk is associated with overall movements in the general market or
economy and therefore is often referred to as the market risk. The market risk is the
component of the total risk that cannot be eliminated through portfolio diversification.
The CAPM developed by Sharpe (1964) and Lintner (1965), discussed in the following
section, relates the expected rate of return of an individual security to a measure of its
systematic risk. Since then, a variety of models have been developed to predict asset returns.
These are discussed in Section 3. A brief summary is given in Section 4.
2 The capital asset pricing model
The CAPM conveys the notion that securities are priced so that the expected returns will
compensate investors for the expected risks. There are two fundamental relationships: the
capital market line and the security market line. These two models are the building blocks for
deriving the CAPM. Even though they are not new, it is illustrative to discuss them here
briefly. Further, since one of the aims of this thesis is to investigate various forms of CAPM,
these models deserve some attention in this paper.
2.1 Capital market line
The capital market line (CML) specifies the return an individual investor expects to receive on
a portfolio. This is a linear relationship between risk and return on efficient portfolios that can
be written as:
= portfolio return,
= risk-free asset return,
= market portfolio return,
σ = standard deviation of portfolio returns, and
σ = standard deviation of market portfolio returns.
According to (3.2.1), the expected return on a portfolio can be thought of as a sum of the
return for delaying consumption and a premium for bearing risk inherent in the portfolio. The
CML is valid only for efficient portfolios and expresses investors’ behaviour regarding the
market portfolio and their own investment portfolios.
3.2.2 Security market line
The security market line (SML) expresses the return an individual investor can expect in terms
of a risk-free rate and the relative risk of a security or portfolio. The SML with respect to
security i can be written as:
= the correlation between security return, and market portfolio return. The
i β can
be interpreted as the amount of non-diversifiable risk inherent in the security relative to the
risk of the market portfolio. Equation (2) is a version of the CAPM. The set of assumptions1
sufficient to derive the CAPM version of (2) are the following:
(i) the investor’s utility functions are either quadratic or normal,
(ii) all diversifiable risks are eliminated and
(iii) the market portfolio and the risk-free asset dominates the
opportunity set of risky assets.
The SML is applicable to portfolios as well. Therefore, SML can be used in portfolio analysis
to test whether securities are fairly priced, or not.
1 See Sinclair (1987) for a description of these assumptions.
3 Asset pricing models
3.1 Single-factor CAPM
In order to test the validity of the CAPM researchers always test the SML given in (2). The
CAPM is a single-period ex ante model. However, since the ex ante returns are unobservable,
researchers rely on realised returns. So the empirical question arises: Do the past security
returns conform to the CAPM?
The beta in such an investigation is usually obtained by estimating the security characteristic
line (SCL) that relates the excess return on security i to the excess return on some efficient
market index at time t. The ex post SCL can be written as:
i η is the constant return earned in each period and is an estimate of
i β in the
SML. The estimated
i β is then used as the explanatory variable in the following cross-
to test for a positive risk return trade-off. The coefficient
γ is the expected return of a zero
beta portfolio, expected to be the same as the risk-free rate and
1 γ is the market price of risk
(market risk premium), which is significantly different from zero and positive in order to
support the validity of the CAPM. When testing the CAPM using (4) and (5), we are actually
testing the following issues: (i)
s are true estimates of historical
i β s, (ii) the market
portfolio used in empirical studies is the appropriate proxy for the efficient market portfolio
for measuring historical risk premium and (iii) the CAPM specification is correct (Radcliffe,
Early studies (Lintner, 1965; Douglas, 1969) on CAPM were primarily based on individual
security returns. Their empirical results were discouraging. Miller and Scholes (1972)
highlighted some statistical problems encountered when using individual securities in testing
the validity of the CAPM. Most studies subsequently overcame this problem by using
portfolio returns. Black, Jensen and Scholes (1972), in their study of all the stocks of the New
York Stock Exchange over the period 1931-1965, formed portfolios and reported a linear
relationship between the average excess portfolio return and the beta, and for beta >1 (<1) the
intercept tends to be negative (positive). Therefore, they developed a zero-beta version of the
CAPM model where the intercept term is allowed to change in each period. Extending the
Black, Jensen and Scholes (1972) study, Fama and MacBeth (1973) provided evidence (i) of a
larger intercept term than the risk-free rate, (ii) that the linear relationship between the average
return and the beta holds and (iii) that the linear relationship holds well when the data covers a
long time period. Subsequent studies, however, provide weak empirical evidence on these
relationships. See, for example, Fama and French (1992), He and Ng (1994), Davis (1994)
and Miles and Timmermann (1996).
The mixed empirical findings on the return-beta relationship prompted a number of responses:
(i) The single-factor CAPM is rejected when the portfolio used as a market proxy is
inefficient. See2, for example, Roll (1977) and Ross (1977). Even very small deviations from
efficiency can produce an insignificant relationship between risk and expected returns (Roll
and Ross, 1994; Kandel and Stambaugh, 1995).
(ii) Kothari, Shanken and Sloan (1995) highlighted the survivorship bias in the data used to
test the validity of the asset pricing model specifications.
2 Also see Fama and MacBeth (1973), Black (1993) and Chan and Lakonishok (1993) and the
(iii) Beta is unstable over time. See, for example, Bos and Newbold (1984), Faff, Lee and Fry
(1992), Brooks, Faff and Lee (1994) and Faff and Brooks (1998).
(iv) There are several model specification issues: For example, (a) Kim (1995) and Amihud,
Christensen and Mendelson (1993) argued that errors in variables impact on the empirical
research, (b) Kan and Zhang (1999) focused on a time-varying risk premium, (c) Jagannathan
and Wang (1996) showed that specifying a broader market portfolio can affect the results and
(d) Clare, Priestley and Thomas (1998) argued that failing to take into account possible
correlations between idiosyncratic returns may have an impact on the results.
3.2 Multifactor models
A growing number of studies found that the cross-sectional variation in average security
returns cannot be explained by the market beta alone and showed that fundamental variables
such as size (Banz, 1981), ratio of book-to-market value (Rosenberg, Reid and Lanstein,
1985; Chan, Hamao and Lakonishok, 1991), macroeconomic variables and the price to
earnings ratio (Basu, 1983) account for a sizeable portion of the cross-sectional variation in
Fama and French (1995) observed that the two non-market risk factors SMB (the difference
between the return on a portfolio of small stocks and the return on a portfolio of large stocks)
and HML (the difference between the return on a portfolio of high-book-to-market stocks and
the return on a portfolio of low-book-to-market stocks) are useful factors when explaining a
cross-section of equity returns. Chung, Johnson and Schill (2001) observed that as higher-
order systematic co-moments are included in the cross-sectional regressions for portfolio
returns, the SMB and HML generally become insignificant. Therefore, they argued that SMB
and HML are good proxies for higher-order co-moments. Ferson and Harvey (1999) claimed
that many multifactor model specifications are rejected because they ignore conditioning
Another possibility is to construct multifactor arbitrage pricing theory (APT) models
introduced by Ross (1976). APT models allow for priced factors that are orthogonal to the
market return and do not require that all investors are mean-variance optimisers, as in the
CAPM. Groenewold and Fraser (1997) examined the validity of these models for Australian
data and compared the performance of the empirical version of APT and the CAPM. They
concluded that APT outperforms the CAPM in terms of within-sample explanatory power.
3.3 CAPM with higher-order co-moments
It is clear from well-established stylised facts that the unconditional security return
distribution is not normal (see, for example, Ané and Geman, 2000 and Chung, Johnson and
Schill, 2001) and the mean and variance of returns alone are insufficient to characterise the
return distribution completely. This has led researchers to pay attention to the third moment –
skewness3 – and the fourth moment – kurtosis.
Many researchers investigated the validity of the CAPM in the presence of higher-order co-
moments and their effects on asset prices. In particular, the effect of skewness on asset pricing
models was investigated extensively. For example, Kraus and Litzenberger (1976), Friend and
Westerfield (1980), Sears and Wei (1985) and Faff, Ho and Zhang (1998), among others,
3 Early studies examined the empirical relation of ex post returns to total skewness (see, for example,
Arditti, 1967). Subsequent studies argued that systematic skewness is more relevant to market valuation
rather than total skewness (see, for example, Kraus and Litzenberger, 1976) refuting the usefulness of
quadratic utility as a basis for positive valuation theory. The experimental evidence that most
individuals have concave utility displaying absolute risk aversion also supports inclusion of higher-
order co-moments in risk-return analysis (see, for example Gordon, Paradis and Rorke, 1972).
extended the CAPM to incorporate skewness in asset valuation models and provided mixed
Harvey and Siddique (2000) examined an extended CAPM, including systematic co-
skewness. Their model incorporates conditional skewness. The extended form of CAPM is
preferred as the conditional skewness captures asymmetry in risk, in particular downside risk4,
which has recently become considerably important in measuring value at risk. Harvey and
Siddique reported that conditional skewness explains the cross-sectional variation of expected
returns across assets and is significant even when factors based on size and book-to-market
A few studies have shown that non-diversified skewness and kurtosis play an important role in
determining security valuations. Fang and Lai (1997) derived a four-moment CAPM and it
was shown that systematic variance, systematic skewness and systematic kurtosis contribute
to the risk premium of an asset. See, also, Christie-David and Chaudhry (2001) who show
that the third and fourth moments explain the return-generating process in futures markets
Investors are generally compensated for taking high risk as measured by high systematic
variance and systematic kurtosis. Investors also forego the expected returns for taking the
benefit of a positively skewed market. It also has been documented that skewness and
kurtosis cannot be diversified away by increasing the size of portfolios (Arditti, 1971).
4 Downside risk is the risk of loss or underperformance that is considered as the appropriate measure of
risk. Variance, as a measure of risk, includes returns above and below the average return, in the same
vein. This has led to criticism of variance as a measure of risk.
3.4 Conditional asset pricing models
Testing for the instability of beta and the validity of the return-beta relationship is not new.
Following the suggestion made by Levy (1974) to compute separate betas for bull and bear
markets, Fabozzi and Francis (1977) were the first to formally estimate and test the stability of
betas over the bull and bear markets. They found no evidence supporting beta instability.
However, in an empirical analysis of the cross-sectional relationship between the expected
returns and beta, Fabozzi and Francis (1978) concluded that investors like to receive a positive
premium for accepting downside risk, while a negative premium was associated with the up
market beta, suggesting that downside risk – as measured by the beta corresponding to the
bear market – may be a more appropriate measure of portfolio risk than the conventional
Prompted by Fabozzi and Francis (1978), several studies tested for randomness of beta. Kim
and Zumwalt (1979) extended the Fabozzi-Francis design to analyse the variation of returns
on security and portfolios in up and down markets. They used three alternative measures to
determine what constituted an up and down market. Up market constituted those months in
which the market return exceeded (i) the mean market return, (ii) the mean risk-free rate or
(iii) zero. Kim and Zumwalt concluded that downside risk might be a more appropriate
measure of portfolio risk than the conventional single beta. Chen (1982) allowed beta to be
nonstationary in an examination of the risk-return relationship in the up and down markets and
concluded that (i) under the condition of either constant or changing beta, investors seek
compensation for assuming downside risk and (ii) as in the Kim and Zumwalt (1979) study,
the down market beta is a more appropriate measure of portfolio risk than the single beta.
Bhardwaj and Brooks (1993) observed that the systematic risks in bull and bear time periods
are statistically different. Their classification of bull and bear markets is based on whether the
market return exceeds the median market return or not. Studies have considered three-beta
models as well. For example, Faff and Brooks (1998), noting that there is no reason to believe
that beta is constant, especially over long estimation periods, defined three regimes relating to
two major past events.
Ferson and Harvey (1991), on the other hand, in their study of US stocks and bond returns,
revealed that the time variation in the premium for beta risk is more important than the
changes in the betas themselves. This is because equity risk premiums were found to vary
with market conditions and business cycles. Schwert (1989) attributed differential risk premia
between up and down markets to varying systematic risk over the business cycle.
Pettengill, Sundaram and Mathur (1995) highlighted that the weak and intertemporally
inconsistent results of studies testing for a systematic relation between return and beta is due
to the conditional nature of the relation between the beta and the realised return. They argued
that when realised returns are used, the relation between the beta and the expected return is
conditional on the excess market return. They postulated a positive (negative) relation
between the beta and returns during an up (down) market. See Section 6.2.2 for more details.
Their study of US stocks sampled over the period 1926-1990 reported the existence of a
systematic conditional relation between the beta and the return for the total sample period, as
well as across sub-sample periods.
Following Pettengill, Sundaram and Mathur (1995), Crombez and Vander Vennet (2000)
analysed the conditional relationship between stock returns and beta on the Brussels Stock
Exchange over the period 1990-1996. They observed that the beta factor is a strong and
consistent indicator of both upward potential in bull markets and downside risk in bear
markets. They found the results to be robust for various definitions5 of beta and different
specifications6 of up and down markets. Further, they highlighted that investors could improve
the performance of their portfolios by using up and down market betas in their asset selection
practice. A common feature in the above studies is the use of monthly data.
As far as we are aware only one study has adopted the Pettengill, Sundaram and Mathur
approach to investigate an extended CAPM with higher-order co-moments. Postulating that
the systematic risks corresponding to variance, skewness and kurtosis are different for up and
down markets, Galagedera and Silvapulle (2002) examined the relationship between the
returns and higher-order systematic co-moments in the up and down markets. They found
strong empirical evidence to suggest that in the pr4esence of skewness in the market returns
distribution, the expected excess rate of return is related not only to beta but also to systematic
3.5 CAPM conditional on time-varying volatility
5 Beta computed using different market indices.
6 Up market defined as months in which market return is non-negative and other strong criteria: (i)
market return exceeds the average value of positive market returns and (ii) market return exceeds the
average value of positive market returns plus a factor (0.5 and 0.75) of the standard deviation of
positive market returns.
Since the introduction of ARCH/GARCH7-type processes by Engle (1982) and others, testing
for, and modelling of, time-varying volatility (variance/covariance) of stock market returns
(and hence the time-varying beta) have been given considerable attention in the literature. See
Bollerslev, Engle and Wooldridge (1988) – the first study to model the beta in terms of time-
varying variance/covariance – and the survey paper by Bollerslev, Engle and Nelson (1994).
The ARCH-based empirical models appear to provide stronger evidence, though not
convincingly, of the risk-return relationship than do the unconditional models.
Using monthly data from the United Kingdom market from 1975 to 1996, Fraser, Hamelink,
Hoesli and MacGregor (2000) compared the cross-sectional risk-return relationship obtained
with an unconditional specification of the asset’s betas with betas obtained through
Quantitative Threshold ARCH (QTARCH8) and GARCH-M9 models. In all specifications,
they allowed for possible negative return-risk relationships when excess return on the market
is negative. Fraser, Hamelink, Hoesli and MacGregor observed that CAPM holds better in
downward moving markets than in upward markets and suggested that beta as a risk measure
is more appropriate in the bear markets. They observed that the QTARCH specification, in
7 The ARCH model allows the current conditional variance to be a function of the past squared error
terms. This is consistent with volatility clustering. Bollerslev (1986) later generalised the ARCH
(GARCH) model such that the current conditional variance is allowed to be a function of the past
conditional variance and past squared error terms. The return-generating process can be written as:
the information set available at time
, and the conditional variance, is defined as:
8 See Gourieroux and Monfort (1992) for details.
9 Due to Bollerslev, Engle and Wooldridge (1988).
which they allowed for asymmetries in the first and second moments of returns, yields a
significant beta without having to account for up and down markets.
Recently, several studies investigated the effect of good and bad news (leverage effects), as
measured by positive and negative returns on beta. See, for example, Braun, Nelson and
Sunier (1995) (BNS hereafter) and Cho and Engle (1999) (CE hereafter) and the references
therein. BNS investigated the variability of beta10 using bivariate Exponential GARCH
(EGARCH11) models allowing market volatility, portfolio-specific volatility and beta to
respond asymmetrically to positive and negative market and portfolio returns. CE, on the
other hand, used a two-beta model with an EGARCH variance specification and daily stock
returns of individual firms. CE concluded that news asymmetrically affects the betas while the
BNS study that used monthly data on portfolios did not uncover this relationship.
An alternative approach to capture market movements is through various market volatility
regimes. Galagedera and Faff (2003) examined the appropriateness of a conditional three-beta
model as a security return generating process. Having modelled the market return volatility as
a GARCH(1,1) process, they defined three volatility regimes based on the size of the
conditional volatilities. Even though their results overwhelmingly suggest that the betas in the
low, usual and high volatility regimes are positive and significant, most of the security/
portfolio betas were not found to be significantly different in the three regimes.
10 See also Huang (2000) for the use of a Markov regime-switching model to investigate the instability
11 Due to Nelson (1991).
For the CAPM to hold, normality of returns is a crucial assumption, and if the CAPM holds,
then only the beta should be priced. Several studies have shown that security returns are non-
normal and this is evident especially in high frequency data. When returns are normal, the
mean and the variance are sufficient to describe the return distribution. On the other hand, an
adequate description of a non-normal return distribution requires statements on higher-order
moments such as skewness and kurtosis. Prompted by the mixed results of the single-factor
CAPM studies and the non-normal nature of return distribution, the CAPM with higher-order
co-moments was proposed in the literature as an alternative to the single-factor CAPM. These
empirical studies, too, reported mixed results.
Because of the failure of market beta alone to explain cross-sectional variation in security
returns, multifactor models emerged. These models incorporate fundamental variables such as
size and the price-to-earnings ratio in addition to market beta.
Pettengill, Sundaram and Mathur (1995) argued that the studies on the beta and cross-
sectional returns relationship that used realised return as a proxy for the expected returns
might have produced biased results due to the aggregation of positive and negative market
excess returns. They postulated that when the market return in excess of the risk-free return is
negative, an inverse relationship between beta and portfolio returns is expected. Their test for
a systematic conditional relationship between the realised returns and the beta in an empirical
investigation of US data revealed a positive risk premium in the up market and a negative risk
premium in the down market. Other studies that adopted Pettengill, Sunderam and Mathur’s
conditional model to test the beta risk-return relationship on different data sets reported
stronger results than they would obtain otherwise.