# 1.3.1 Models of expected returns

The simplest model of expected returns is the well-known capital asset pricing model (CAPM) by Sharpe (1964) and Lintner (1965). According to this model, higher returns can only be achieved by exposing a well-diversified portfolio to higher market risk, also known as a higher market *beta*. This model serves as the basis for the discount-rate formula:

withas the market return,the risk-free rate, andthe specific risk of investment , which by definition is not correlated with.

Technically, the market *beta* is the coefficient of a single independent variable (the market return) in an ordinary least-square (OLS) regression explaining the variance of excess returns for a single asset (the dependent variable).

The CAPM implies that the 'market index' to which any asset relates is 'mean-variance efficient', that is, fully captures the asset's potential exposure to all systematic sources of risk. Empirically, this prediction is not robust and has been proven inaccurate by multiple studies for listed equities (Fama & French, 1992).

Multi-factor models have been developed to address the lack of statistical robustness of the single-factor CAPM. Important work by Fama and French (1992) established multi-factor models of asset returns as standard tools to create measures of expected returns.

Well-known industry versions of these ideas that relate expected returns in the next period () to current exposures to risk factors known today (at time ) include the BARRA® multi-factor models and can be written (omitting the time subscript for simplicity):

where is the factor exposure or *loading* of asset to factor (with being the correlation between asset and factor returns, and and being the volatility of asset and factor returns, respectively); is the return to factor during the period of time to time ; and is the asset's specific or idiosyncratic return during that period that cannot be explained by the factors.

The factors used in such models can include industrial and geographic segmentations of the data or various economic mechanisms that can be expected to have a systematic effect on average returns, such as the tendency of well-performing firms to continue to perform (the so-called momentum factor), as well as factors sometimes identified as 'anomalies' (like the outperformance of a 'low volatility' factor).

_{Fama, E.F. & French, K.R. (1992). The cross-section of expected asset returns. }_{ The Journal of Finance. 47(2),}_{ 427-465.}

_{Lintner, J. (1965). Security prices, risk and maximal gains from diversification. }_{Journal of Finance. 20(4), }_{587-615.}

_{Sharpe, W.F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. }_{The Journal of Finance. 19(3)}_{, 425-442.}