1. INTRODUCTION

Market is arguably the most known and widely used factor - think about values of broad-based equity indices such as the S&P 500 reported in news, index-tracking ETFs, and the whole universe of derivatives with indices' values as underlying assets. The market is also the most empirically studied factor with solid theoretical foundations - the famous Capital Asset Pricing Model (CAPM), which links returns of individual assets to the market return, is the cornerstone of modern finance and factor theory. Moreover, the model is still extensively used by practitioners, mainly in the corporate sector. For us it is important, that the logic of the CAPM can be easily generalized to a multifactor framework, so in the next section we review the CAPM implications, which are relevant to the factor theory. A rigorous technical derivation of the CAPM can be found in Cochrane (2005) [1] or in Cvianic and Zapatero (2004) [2].

2. CAPM

The Capital Asset Pricing Model of Sharpe (1964) [3] and Lintner (1965) [4] was the first factor pricing theory with only a single factor - market portfolio. The CAPM stems from the mean-variance portfolio theory developed by Markowitz (1952)  [5]. In the Markowitz's model investors, who care about mean and variance of returns only, hold the mean-variance-efficient portfolios, that either maximize expected return for a given level of variance, or minimize variance, given expected return. So, if there are two portfolios $A$ and $B$ with equal expected returns $E[R_a]=E[R_b]$, but with different variances, for instance $\sigma^2_a<\sigma^2_b$, then portfolio $A$ is mean-variance efficient, since it allows investors to earn the same return with lower risk. The CAPM in Sharpe's and Linter's version employs Markowitz's approach and further assumes that i.) there is unrestricted borrowing and lending at the risk-free rate, ii.) investors have homogeneous expectations (or the same perception of the joint distribution of returns). Under these conditions, in equilibrium investors hold different combinations of the same portfolio (that has the highest Sharpe ratio - return per unit of risk) and the risk-free asset according to their individual attitudes towards risk. Since the portfolio of risky assets is the same for each investor, in equilibrium, the weights of the individual assets must equal their market value divided by the total market value of all risky assets. This value-weighted portfolio is the market portfolio. Furthermore, the expected excess return of asset $i$ can be expressed as:

$E[R_i]-r_f = \beta_i(E[R_M] - r_f)  \text{  with }  \beta_i=\dfrac{Cov(R_i, R_M)}{\sigma^2_M}\tag{1}$

So the expected excess return depends only on its beta, which measures sensitivity to variation in the market return. In other words, the risk of an asset is measured by its factor exposure - beta; in case of the CAPM, the only factor is the market risk premium. High-beta securities depend more on market movements and offer higher expected returns in order to compensate investors for losses during bad times (The bad times are defined in terms of the factor, here the 'bad times' mean the poor performance of the market portfolio). Low-beta assets, on the other hand, have low risk premia - such assets are attractive for investors, who buy them as an 'insurance' for losses in the distressed market, pushing their prices up, or equivalently, decreasing their expected returns.

Consider the following simple linear regression implied by the CAPM:

$R_i - r_f = \alpha_i + \beta_i (R_M - r_f) + \varepsilon_i  \text{,         } \quad  \varepsilon_i \sim i.i.d \ N(0, \sigma_{\varepsilon}^2)\tag{2}$

Taking the variance of equation (2) we get:

$\sigma^2_i= \beta^2_i \sigma^2_M + \sigma^2_\varepsilon \tag{3}$

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The risk of a security has two components: the first term on the right hand side of (3), $\beta^2_i \sigma^2_M$, is the systematic risk which affects all investors (except those who hold the risk-free asset only); the second term represents idiosyncratic risk which is security-specific and not rewarded with higher return, because it can be diversified away. In the CAPM framework investors are rewarded for the systematic or factor risk only, which can not be avoided by diversification, and, therefore, compensates investors with higher return. In other words, investors are better off holding the factor rather than individual assets. The main contribution of the CAPM is that its logic can be generalized to a framework of multiple tradable factors. In a multifactor setting (See Ang (2014) [6] Ch. 6 for an excellent discussion of single factor vs. multifactor frameworks), risk of an asset is measured by its factor exposures (or factor betas with interpretation similar to the CAPM beta), furthermore, the factors diversify idiosyncratic risk away just as the market portfolio does in the CAPM.

3. MARKET RISK PREMIUM IN PRACTICE

The market risk premium is ubiquitous in practical applications: from M&A, where it is used to estimate cost of equity, to sophisticated portfolio strategies, which attempt to completely avoid market risk (market neutral, or zero beta strategies). Furthermore, in contrast to other factors, it is also employed as a benchmark in asset management. For the period 1991-2013 return on the US equities was 8.2% p.a. with the Sharpe ratio of 0.54 (see Table 1 in Israel et al. (2014) [7]), higher than for value, size and momentum -- so, beating the market is a challenging endeavor. However, the market portfolio possesses characteristics, which are undesirable for many investors: procyclicality (co-movement of asset prices with business cycle), significant drawdowns during crashes, high volatility, exposure to funding liquidity risk - all of these lead to increasing popularity of defensive or low risk (see the Low Risk article for more details) and market neutral strategies. For example, the betting-against-beta (BAB) factor of Frazzini and Pedersen (2014) [8] (which takes long position in low beta stocks and shorts high beta stocks - so the ex ante beta is zero) earns 8.4% p.a. and has the Sharpe ratio of 0.78 for US stocks for the period 1926-2012, with realized beta of -0.06. Overall, the optimal exposure to the market risk should be determined according to investors' needs. For example, very large institutional investors such as sovereign funds have long investment horizons and may simply wait until a distressed market rebounds, since in the long run asset prices will return to fundamental values. Obviously, the same strategy may be not acceptable for individual investors with short horizons, who, therefore, may wish to mitigate their market exposure.

4.MARKET RISK

The mean-variance framework assumes constant correlations between assets. However, empirically  asset prices tend to move together during downturns. In other words, diversification benefits vanish exactly when they are needed. The one way to address this issue is to introduce dynamic correlations and volatilities, for instance, in a regime-switching framework (see Ang and Bekaert (2002) [9] for an application to the international equity market) or in GARCH DCC framework (see Engle (2002) [10]). Another way is to understand fundamental reasons for correlations to increase. For example, Brunnermeier and Pedersen (2009) [11] suggest an appealing liquidity-based explanation - an initial price shock and increased volatility lead to tightening of margin requirements (or reduction in funding liquidity), as the result investors become more constrained to fund their positions and are forced to liquidate, thus moving prices even further away from fundamentals and decreasing supply of market liquidity, which in turn leads to higher margins. Such liquidity spirals result in cross-asset contagion, rising correlations, and trigger 'flight to quality'. So the market risk may be partially avoided by investing in factors with low funding liquidity risk exposure, for example the quality-minus-junk factor or a combination of value and momentum.

5. SUMMARY

The market risk was the first discovered factor and it marks the single source of risk in the (still widely used) CAPM. Despite its theoretical beauty, the CAPM is not capable of adequately describing asset returns in practice (for a discussion on empirical tests of CAPM as a single factor model we refer readers to Fama and French (2004) [12]). However, it is important that the basic intuition of the CAPM is still relevant -- factor risks drive assets' risk premia, and asset returns may be represented as a combination of factor risks. Furthermore, the exposure to a specific factor should be determined according to the needs and characteristics of a particular investor.  

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