The Capital Asset Pricing model (CAPM) was derived from Treynor (1961), Sharpe (1964), Lintner (1965) and Mossin (1966) twelve years after Harry Markowitz (1959) introduced his mean-variance portfolio theory (Bodie et al, 2014). Markowitz's modern portfolio theory laid the groundwork for the Capital Asset Pricing Model. Sharpe and Lintner applied this to an economy-wide setting where an assumption is that the portfolios of investors are held mean-variance efficient, and their views are homogeneous in a frictionless market (Campbell et al., 1996). The model provides a relationship between expected return and riskiness of the asset, which can serve as a benchmark for future investments or help estimate the expected return of assets that are not yet publicly traded (Bodie et al., 2014). The model is ultimately based on the fact that the market is in equilibrium, where assets are priced correctly when the assumption of market equilibrium is held. In other words, the definition of the market equilibrium is the adjustment of the prices, influenced by the beliefs and expectation of the investor, until the expected returns are in equilibrium where the demand matches the supply (He & Litterman, 1992). The model also allows us to use various risk measures for different kinds of assets, and also get the relationship of efficient and inefficient assets. Because the market is in equilibrium, the prices of assets are such that the tangency portfolio is the market portfolio, which is composed of all risky assets in proportion to their market capitalization.
The CAPM model is based on several assumptions which have to be fulfilled. The assumptions are quite similar to the once introduced in modern portfolio theory, and are given as follows (Jensen, 1967) (Bodie et al., 2014):
● The wealth of individual investors is small compared to the overall wealth in the total market
● Investors have the same holding horizon and homogeneous market views, i.e. the same underlying distribution of future expected returns
● The risk-free rate is the same for all investors, and they can lend and borrow at this rate
● The investor is risk averse, however, he still seeks to maximize his/her wealth
● The decision-making is determined from the risk-return perspective
● The market is in equilibrium
● There are no market frictions, taxes, etc.
Some of the assumptions mentioned above are a simplification of the real world which does not necessarily hold under true market conditions. Regardless, they are essential tools to be able to explain the market equilibrium. The assumption that all investors seek to hold or replicate the market portfolio, which in theory contains all publicly traded assets, is in reality hard to establish.
The basic CAPM-equation is defined as
𝐸(𝑟) ) = 𝑟;+ 𝛽)(𝐸(𝑟3) − 𝑟;) (Equation 3.3.1)
where 𝐸(𝑟)) is the individual asset return, 𝐸(𝑟3)is the return on the market portfolio, 𝑟; is the risk-free rate and 𝛽 is defined as a measure of the asset risk, given as
𝛽) =P9QR%&,%(
($ (Equation 3.3.1)
where 𝜎3(is the variance of the market portfolio, and 𝑐𝑜𝑣?&,?(is the covariance between the asset excess return and the excess return on the market portfolio. The equation shows that it is far from
returns. The uncertainty of the assumptions that is done to estimate the beta is, therefore, significant (Munk, 2019). Estimating beta through observations on historical returns of the asset and the market is a more conventional method. This makes it possible to build a regression model where the historical data helps us obtain a relationship between the market return and the asset return.
CAPM makes some assumptions of varying degrees of plausibility. For use in the reverse optimization of equilibrium excess returns performed in Section 3.3. Given a vector of specified market-clearing asset prices, agents must agree on the joint distribution of asset returns from this period to the next. This assumption entails that any market portfolio must be on the minimum-variance frontier if the market is to clear all positions (Fama and French, 2004).
Additionally, CAPM assumes that investors are only concerned with the asset returns and variances, the first two moments.
3.3.1 Capital allocation line
The Capital Allocation Line (CAL) is a graphical lie that illustrates the relationship of the risk-and-reward combinations of assets and is often associated with its application to find the optimal portfolio. The slope of CAL is the increase in the expected return of the portfolio per unit of additional risk, also referred to as the Sharpe ratio. The line is mathematically expressed as follows:
𝐸+𝑟!, = 𝑟;+ 𝑆ℎ𝑎𝑟𝑝𝑒!𝜎! (Equation 3.3.2)
FIGURE 5:GRAPHICAL ILLUSTRATION OF THE CAL
The different allocation options also mean that one optimal portfolio is present but does depend on the different level of risk aversions. The different levels of allocation depend on how much we
want to hold in risk-free assets and correspondingly, in risky assets. The optimal portfolio is found, when the CAL is tangent to the efficient frontier, illustrated by the graph. The point at 𝑟;, means 100% investment in risk-free assets, whereas the point (Optimal Portfolio) shows 100%
investment in the portfolio. Between the risk-free rate and the optimal portfolio, investors that lie there hold positions in both risk-free assets and the portfolio and represent an investor lending a part of their portfolio, as investors are not 100% invested in the portfolio. The point after the optimal portfolio, show a leveraged position, being more than 100% invested in the portfolio, and therefore borrowing capital to buy more portfolio (Bodie et al. , 2014).
3.3.2 Single index models
The market portfolio is in practice a mean-variance efficient portfolio consisting of all risky assets.
Testing the efficiency of the market portfolio requires construction of a value-weighted portfolio of significant size, which can be demanding and often not feasible. Hence, correcting for this issue requires additional assumptions (Bodie et al., 2014).
Sharpe (1963) developed the well-known Single-Index Model where, as opposed to regular factor models, the return on the market portfolio or a stock market index is used as a factor to explain the excess return of an individual asset (Munk, 2019). The model illustrates a relationship between the expected asset return and its respective beta, which is usually formulated as:
𝐸(𝑟)) − 𝑟;= 𝛼)+ ß)[𝐸(𝑟3) − 𝑟;]
where the notation can be recalled under Section 3.2 and 𝛼 is the abnormal return or the difference between the expected return and the realized return. The traditional CAPM is just one example of the Single Index model, but instead, the single index applies an economic variable to explain the excess return. The theoretical CAPM implicitly predicts that, for all assets, the alpha should yield zero. If the asset has an alpha that deviates from zero, it is not correctly priced according to the theoretical CAPM. Under CAPM, taking additional risks may be reduced through diversification, which is compensated from beta by taking additional systematic risks. This is not compensated when taking risks associated with alpha.