Bogleheads:Sandbox

CAPM and single factor models
Given a simplifying set of assumptions, CAPM provides an equation that relates the excess expected return of asset i to its expected covariance with the expected excess return of the market portfolio M:


 * $$ \operatorname{E}(r_i)- r_f = \frac {\operatorname{Cov}(i,M)}{\sigma_M^2} (r_M - r_f)$$.

Excess expected return is the expected return of the asset or market portfolio minus the risk-free rate rf. The market portfolio is a theoretical portfolio consisting of all marketable, risky assets; e.g., stocks, bonds, real estate, commodities, collectibles, etc.

Isolating the expected return of asset i on the left hand side of the equation gives:


 * $$ \operatorname{E}(r_i) = r_f + \frac {\operatorname{Cov}(i,M)}{\sigma_M^2} (r_M - r_f)$$.

Defining the term $$\frac {\operatorname{Cov}(i,M)}{\sigma_M^2}$$ as beta ($$\beta_i$$), the equation can be rewritten as:


 * $$ \operatorname{E}(r_i) = r_f + \beta_i (r_M - r_f)$$.

Beta standardizes the measure of systematic risk by dividing the covariance of the asset with the market portfolio, $${\operatorname{Cov}(i,M)}$$, by the variance of the market portfolio, $${\sigma_M^2}$$. Since the covariance of an asset with itself is the variance of the asset, this results in a beta of 1 for the market portfolio. The expected return-beta equation above is the formulation of CAPM most widely used by investment professionals.

The CAPM model relates expected return to expected risk, and involves a theoretical market portfolio consisting of all risky assets. Since expected returns are not observable, and in practice the market portfolio is not observable, factor models and index models were developed to enable investment professionals to put the theory behind CAPM into practice.

In a single-factor model it is assumed that all unexpected macroeconomic events affecting the asset returns in a market can be summarized by one common factor, and that all remaining uncertainty in asset returns is firm specific. . Recognizing that each firm will have a different sensitivity to macroeconomic events, the following equation relates the return on security i to the common macroeconomic factor F:


 * $$ r_i = \operatorname{E}(r_i) + \beta_i F + e_i $$,

where $$\operatorname{E}(r_i)$$ is the expected return on security i at the beginning of the holding period, $$\beta_i$$ is the sensitivity of security i to macroeconomic factor F, and ei is component of return due to the impact of unanticipated firm-specific events.

A single-index model uses the excess return on a market index, such as the S&P 500, as a proxy for the common macroeconomic factor. Broader proxies for the market portfolio can be used; e.g., all US stocks, all global stocks, or all global stocks and bonds. A single-index model equation for the excess holding period return on security i is:


 * $$ r_i - r_f = \alpha_i + \beta_i (r_M - r_f) + e_i \,$$,

where $$\alpha_i$$ is the security's expected return if the excess market return is 0%, $$r_M$$ is the market index return, $$\beta_i$$ is the sensitivity of security i to the market index, and ei is the component of return due to unexpected firm-specific events.

Note that some authors express the single-index model in terms of total returns rather than excess returns.

Using capital R to represent excess return, the equation can be written more simply as:


 * $$ R_i = \alpha_i + \beta_i R_M + e_i \,$$.

In this equation $$\beta_i R_M$$ represents a source of systematic risk based on the security's sensitivity to the market index (a proxy for macroeconomic factors), and $$e_i$$ represents a source of unsystematic risk based on firm-specific events.

Since expected returns are not observable, beta typically is calculated using periodic historical returns over a specified time period; e.g., weekly or monthly returns over the previous three to five years. For example, Morningstar calculates beta based on monthly returns over the previous 36 months. The above equation is used as the basis of a linear regression analysis in which the dependent variable is the excess return of the individual asset, Ri, the independent variable is the excess return of the market index, RM, and beta is the regression coefficient (slope of the line).

This can be represented graphically as follows. A scatter plot of Ri vs. RM is generated in which the vertical axis measures the excess returns of the individual asset, and the horizontal axis measures the excess returns of the market index. Each point in the scatter plot represents a pair of excess returns for one period (week or month). A best-fit line is drawn through the points. The slope of the line is beta, and the vertical intercept is alpha. The vertical distance from the line of each point for period i is ei, the deviation from the regression line.

The variance of the excess return on security i can be written as:


 * $$ \operatorname{Var}(R_i) = \sigma_i^2 = \beta_i^2 \sigma_M^2 + \sigma^2(e_i) $$.

On the right hand side of the equation, the first term is the variance due to uncertainty in the common macroeconomic factors (as represented by the market index), and the second term is the variance due to firm-specific uncertainty.

The covariance between the excess rates of return on any two securities is:


 * $$\operatorname{Cov}(R_i,R_j) = \operatorname{Cov}(\beta_i R_M, \beta_j R_M) = \beta_i \beta_j \sigma_M^2

$$.

From these equations it can be determined that (3n + 1) estimates are required to evaluate the portfolio variance for n securities. Note that this is many fewer estimates than the 2n + n(n - 1)/2 estimates required using variances and covariances directly.

Multifactor models
Although CAPM and the related single-factor model have had significant impact on finance theory and practice, empirical studies have shown significant discrepancies between historical data and predictions of the CAPM-based single factor model. This may not be surprising considering the simplification of assuming that only a single factor affects the systematic variation in security prices. Theoretical work has been done attempting to use more than one factor to explain the systematic risk of securities.

Two of the discrepancies mentioned above are the long-term returns of small company stocks and stocks with high ratios of book value to market value (value stocks), which are higher than predicted by CAPM. Motivated by these anomalies, Fama and French conducted research that resulted in what is commonly referred to as the Fama-French three-factor model.

One of the three factors is the market portfolio (more specifically, an index representing the market), as in CAPM. The other two factors are SmB (Small minus Big) and HmL (High minus Low). SmB is a theoretical portfolio represented by the returns of small-cap stocks minus the returns of large-cap stocks (hence, Small minus Big or SmB). HmL is a theoretical portfolio represented by the returns of high book-to-market stocks minus the returns of low book-to-market stocks (hence, High minus Low or HmL).

Fama and French's research indicated that regressing the empirical data against the three factors explained the returns of stock portfolios much better than did the CAPM-based single-factor model. The equation used to perform the regressions is:



r_i - r_f = \alpha_i + \beta_M (r_M - r_f) + \beta_h r_{HmL} + \beta_s r_{SmB} + e_i \,$$.

Each of the betas in the equation represent the sensitivity of a stock or portfolio of stocks to the corresponding factor. The empirically derived beta values can be considered as explaining how much of a portfolio's return can be explained by exposure to the respective factors. A portfolio more heavily weighted toward small-cap and value stocks would be expected to have higher values of $$\beta_s$$ and $$\beta_h$$.

This result is based on empirical research, not a theory related to specific risk factors. Therefore, one conclusion is that SmB and HmL may be proxies for exposure to sources of systematic risk not captured by the single CAPM factor of overall market risk.

Investors who believe that SmB and HmL are valid proxies for systematic risk factors may decide to increase their allocations to small-cap stocks and value stocks. This is informally referred to as tilting toward small and value.

Assessing risk tolerance
Assessing risk tolerance is a critical step in determining the appropriate trade off between the risk and expected return of a portfolio. Selecting a mix of risky and risk-free assets (or higher-risk and lower-risk assets) is one of the most important decisions in designing a portfolio. The investor must develop a rational assessment of risk tolerance to make rational decisions about this asset allocation during portfolio design.

Criteria
Most approaches to assessing risk tolerance consider these criteria in one way or another: Attitude toward risk, Investment time horizon. Other factors that affect ability to compensate for investment losses; e.g., net worth, stability of income, future liabilities, flexibility of goals, etc.


 * The Bogleheads' Guide To Investing explores four areas: Goals, Time frame, Risk tolerance, Personal financial situation. Tools are recommended to help factor risk into the one's asset allocation. Various sample portfolios are presented based on these "stages in life".


 * Bogleheads author Larry Swedroe suggests that investors evaluate their risk tolerance by considering their ability, willingness and need to take risk.


 * Bogleheads author William Bernstein discusses determining appropriate risk exposure in the context of deciding on one's allocation between stocks and bonds based on age and risk tolerance (attitude toward risk), at least as a starting point.


 * Bogleheads Author Rick Ferri suggests using a modified version of the age in bonds rule of thumb to help decide on the split between higher-risk and lower-risk assets, and presents asset allocation decisions in terms of a life-cycle investing framework.


 * Daniel R. Solin in his book The Smartest Money Book You'll Ever Read, has a full chapter titles Assessing Your Risk Capacity where he presents five major factors for assessing risk capacity.


 * The CFA Institute discusses Investors and Objectives, Investor constraints (Liquidity, Investment Horizon, Regulations, Tax Considerations, Unique Needs.

Self-assessment questionnaires
Various advisory services offer self-assessment questionnaires to help determine your risk tolerance and define an initial asset allocation. Such tools are useful with caveats.

Managing financial risk
Investment risk, or financial-asset risk, is only one type of financial risk facing households. Other major categories of risk are:


 * Sickness, disability, and death.
 * Unemployment risk.
 * Consumer-durable asset risk: e.g., the risk of loss from fire, theft or obsolescence related to owning a car, house, or other consumer-durable asset.
 * Liability risk: e.g., the possibility of getting sued.

Managing these types of risks, as well as managing investment risk, from a broader financial planning perspective is the topic of this section.

Short term risk vs. long term risk
Mentioned in "Risk as the possibility of loss" section, perhaps here we go into more detail on terminal wealth dispersion, how small differences in annualized return can result in large differences in terminal wealth, stocks are risky in the long run, etc.. This is covered in Bodie, Kane, Marcus, 2008.

See Bernstein's "The 15 Stock Myth" for more on terminal wealth dispersion, but there he focuses on the danger of a portfolio with only a small number of stocks. He does emphasize that such portfolios, although they may have low standard deviation of monthly returns, also have a large dispersion of terminal wealth.

Reading

 * Reilly, Frank K. (1994). Investment Analysis and Portoflio Management. The Dryden Press, Harcourt Brace College Publishers, Fourth Edition. ISBN 978-0030970528.
 * Bodie, Zvi; Kane, Alex; Marcus, Alan J. (2008). Essentials of Investments. McGraw-Hill. ISBN 978-0071263245.
 * Elton, Gruber, Brown, Goetzmann (2003). Modern Portfolio Theory and Investment Analysis. Wiley. ISBN 978-0471238546.
 * Ilmanen (2011). Expected Returns. Wiley. ISBN 978-1-119-99072-7.
 * Bodie, Merton (2000). Finance. Prentice-Hall. ISBN 978-0133108972.
 * Bernstein (2010), The Investor's Manifesto. John Wiley & Sons, Inc. ISBN 978-0470505144.
 * Bodie, Zvi; Kane, Alex; Marcus, Alan J. (1996). Investments. McGraw-Hill. ISBN 978-0256146387.