These study notes are based on Chapter 9 the Exam 9 syllabus reading Investments by Bodie, Kane, and Marcus. This chapter introduces the Capital Asset Pricing Model, which provides a way of calculating an benchmark return for a security by considering its equiliblium expected value. This chapter corresponds to learning objectives A6 and A7 on the syllabus.
easypackages::packages("dplyr", "ggplot2")
options(scipen = 999)The Captial Asset Pricing Model is based on the following assumptions:
The core idea behind the CAPM is that there is a market price of risk given by \[ \frac{E(R_M)}{\sigma_M^2}, \] where \(R_M\) is the market risk premium and \(\sigma_M\) its standard deviation. (Note that this is distinct from the Sharpe ratio, in which the denominator is the standard deviation.) The rationale for this formula is that the optimum allocation to the risky portfolio is given by \[ y = \frac{E(R_M)}{A \sigma_M^2} \] The avearge investor holds the market portfolio, so \(y=1\) for the investor with an average risk tolerance \(\overline{A}\). Thus, \[ E[R_M] = \overline{A} \sigma_M^2 \] In other words, the expected return of the market should be proportional to its variance.
To determine a benchmark price \(R_0\) for a single security, the CAPM considers its contribution to the aggregate market variance. The implicit assumption here is that the firm-specific risk is not relevant because it can be diversified away, and we are only concerned with the security’s contribution to systematic risk. \[ \sum_{0\leq i \leq n} w_0w_i \mathrm{Cov}(R_0, R_i) = w_0 \mathrm{Cov}\left(R_0, \sum_{0\leq i\leq n} w_i R_i\right) = w_0 \mathrm{Cov}(R_0, R_M) \] Therefore, the risk-to-reward tradeoff for this security is \[ \frac{w_0 E(R_0)}{w_0 \mathrm{Cov}(R_0, R_M)} \] Setting this equal to the market price of risk, we obtain the CAPM formula: \[ E(R_0) = \frac{\mathrm{Cov}(R_0, R_M)}{\sigma_M^2} E(R_M) = \beta_0 E(R_M) \] This is more commonly expresed in terms of absolute terms: \[ E[r_0] = r_f + \beta_0(r_M - r_f) \] This formula is referred to as the mean-beta relationship.
The \(\beta\) value for the market is \[ \beta_M = \frac{\mathrm{Cov}(R_M, R_M)}{\sigma_M^2} = \frac{\sigma_M^2}{\sigma_M^2} = 1 \] which provides a benchmark for \(\beta\):
Securities with \(\beta > 1\) are considered aggressive and correspond to investments that are overly-sensitive to market swings.
Securities with \(\beta < 1\) are considered defensive, and are less sensitive to market swings.
The security market line (SML) is a graph of expected returns, with \(\beta\) on the \(x\)-axis.
It is a line whose slope is the market risk premium and whose \(y\)-intercept is the risk-free rate.
An individual security can be plotted as a point; securites that lie on the SML are fairly priced. All securities must lie on the line at equilibrium.
If a security is above the SML it has a positive \(\alpha\), which is equal to the vertical distance from the security to the SML. (Conversely, if it is below the SML it has a negative \(\alpha\).)
In conrast, the Capital Market Line (CML) plots expected returns of portfolios, comprised of the market portfolio and risk-free asset, as a function of standard deviation. The SML pertains to individual asset risk premiums.
The CAPM can also be derived from the Single Index model. Recall that in the Single Index Model, \[ R_i = \alpha_i + \beta_i R_M + \epsilon_i \] with \[ E[R_i] = \alpha_i + \beta_i E[R_M] \] and \[ \sigma_i^2 = \beta_i^2 \sigma_M^2 + \sigma^2(\epsilon_i) \] The investor can improve the risk premium by taking long positions in positive-\(\alpha\) stocks and short positions in negative-\(\alpha\) ones, shifting the price up or down. Recall that in the Single Index Model, the initial weight for each security in the active portfolio is proportional to \[ \frac{\alpha_i}{\sigma^2(\epsilon_i)} \] so as \(\alpha_i \rightarrow 0\), the weight of this stock in the active portfolio tends to zero. In particular, the weight assigned to the active portfolio will be zero, consistent with the idea that all investors would hold the market portfolio.
The Zero-Beta CAPM is based on the idea that every portfolio on the efficient frontier has a companion portfolio on the inefficient part of the frontier with which it is uncorrelated, called the zero-beta portfolio. Let \(Z\) be the zero-beta portfolio for the market portfolio \(M\). The Zero-Beta CAPM replaces the risk-free rate with the expected return on this portfolio: \[ E[r_i] = E[r_Z] + \beta_i(E[r_M] - E[r_Z]) \]
The main assumption relaxed by this model is that borrowing and lending occur at the same risk-free rate, or is otherwise restricted. The rationale is:
Investors who would normally select leveraged portfolios will instead have to select securities with high \(\beta\).
This will raise the price of these stocks, lowering their risk premium.
The slope of the SML will be flatter, consistent with the fact that \(E[r_z]\) will be greater than \(r_f\).
The CAPM can be adjusted to account for the influence of private businesses and labour income:
Of these, labour income has a greater impact on the model. The impact of private businesses can be offset by reducing the demand of similar traded assets.
The CAPM is affected by the impact of uncertainty regarding labour income on personal portfolio choice.
The idea behind the modification is that stocks of labour-intensive firms provide a hedge against labour income risk, so these stocks should have higher prices and lower expected returns.
Define the following notation:
\(P_H\) is the value of aggregate human capital
\(P_M\) is the market value of traded assets
\(R_H\) is the excess rate of return on human capital
The numerator and denominator of \(\beta\) are then adjusted based on correlations with the return on human capital, for the individual security and the market: \[ E[R_i] = E[R_M] \frac{\mathrm{Cov}(R_i, R_M) + \frac{P_H}{P_M} \mathrm{Cov}(R_i, R_H)}{\sigma_M^2 + \frac{P_H}{P_M} \mathrm{Cov}(R_M, R_H)} \]
An Interterporal Capital Asset Pricing Model (ICAPM) relaxes the single-period assumption and considers investors who optimize portfolios over a lifetime of consumption and investment, adapting as conditions change.
When the only source of risk is portfolio returns, this leads to the same result as CAPM
Other sources of risk include interest rate risk (e.g. changes in the risk-free rate over time) or inflation risk (e.g. changes in the price of consumption goods).
The approach is to identify, for each extra-market source of risk, a portfolio \(R_k\) that hedges against that risk. For example, stocks in energy companies could be a hedge against energy prices.
The general for of an ICAPM is then a multi-index model: \[ E[R_i] = \beta_{i, M} E[R_M] + \sum_{1 \leq j \leq K} \beta_{i, j} E[R_j] \]
The CAPM does not consider transaction costs, in particular, those associated with the illiquidity of assets. Sources of costs include:
The Bid-Ask spread. Electronic markets only list the inside spread, but a large purchase will require purchases deeper within the limit-order book, with progressively worse prices.
Information asymmetry: traders who post offers at a given price need to be worried about giving up value to better-informed traders.
The impact on the price of executing a large trade
Immediacy: can an asset be sold quickly without depressing the price? There is an implied discount from fair market value that a seller must accept in order to sell the asset quickly.
Noise trades are trades that occur for non-informational reasons, such as liquidating assets in order to make a large purchase.
Liquidity introduces a clientele effect, in which longer-term investors will be less concerned with liquidity risk than short-term investors.
The impact of the above on the CAPM is that, due to the illiquidity discount, the expected rate of return needs to be higher. This can be incorporated into the CAPM by introducing a market liquidity index, and calculating a liquidity \(\beta\) for this index, leading to a multi-factor model.