1.4.3 Capital Asset Pricing Model

Relative Theory and Capital Asset Pricing Model
Capital market theory
Each investor’s portfolio is just a combination of risky assets and risk-free asset, with the proportional allocation between them being a function of the individual investor’s risk appetite.
Capital asset pricing model
Sharp broke down the risk of an individual asset into specific risk and systematic risk. Investors should receive no compensation for taking on the specific.

1. Assumptions of CAPM

• Investors are expected to make decisions solely in terms of expected values and standard deviations of the returns on their portfolios.
• Investor plan for the same single holding period.
• Allocations can be made in an investment of any partial amount(infinitely divisible).
• Including no transaction cost and no tax, or other frictions.
• Short sale is allowed unlimitedly.
• All participants can borrow and lend at a common risk-free rate.
• Investor have homogeneous expectations or beliefs.
• Any individual investor’s allocation decision cannot change the market prices(price taker).
• All assets, including human capital, are tradable.

2. Systematic risk & Unsystematic risk

2.1 Systematic risk

The risk affects the entire market or economy, which cannot be avoided and is inherent in the overall market.

• Caused by macro factors: interest rates, GDP growth, supply shocks.
• Also named non-diversifiable risk or market risk,

Investor would be only rewarded for bearing systematic risk.

2.2 Unsystematic risk

The risk that can be reduced or eliminated by holding well-diversified portfolios.

• Also named firm-specific risk or idiosyncratic risk.

Investor would not be rewarded for bearing unsystematic risk as it could be eliminated through diversification.

As more diversification is made within the portfolio, systematic risk would not change while unsystematic risk would decrease.

2.3 Measurement of systematic risk

Systematic risk can be measured by Beta($\beta$) of the asset, which represents how sensitive an asset’s return is to the market as a whole.

From an investor’s perspective, $\beta$ represents the portion of an asset’s total risk that cannot be diversified away and for which investors will expect compensation.
$${\beta }_i=\frac{Cov(R_i,R_m)}{{\sigma }_{mkt}^2}=\frac{{\rho }_{i,m}{\sigma }_i{\sigma }_m}{{\sigma }_m^2}={\rho }_{i,m}\frac{{\sigma }_i}{{\sigma }_m}$$

$\beta$ can take on positive or negative values, depending on how an asset’s returns relate to those of the market portfolio.

• A risk-free asset will have a beta of zero, because its returns are completely uncorrelated to the returns on the market portfolio.
• The market portfolio has a beta of 1, because it is perfectly correlated with itself and has the same variance.
  $${\beta }_{mkt}=\frac{Cov(R_i,R_m)}{{\sigma }_{mkt}^2}=\frac{{\sigma }_{mkt}^2}{{\sigma }_{mkt}^2}=1$$

3. Capital asset pricing model(CAPM)

$$E(R_i)=R_f+{\beta }_i[E(R_m)-R_f)]$$

• $E(R_i)$: expected return on risky asset $i$,
• $E(R_m)-R_f$: market portfolio risk premium.
• ${\beta }_i$: systematical risk of asset $i$.
• ${\beta }_i[E(R_m)-R_f)]$: beta-adjusted risk premium on risky-asset $i$. expected return premium above the risk-free rate(as required by investors).

Rewriting the CAPM in terms of ${\sigma }_i$, ${\rho }_{i,m}$, ${\sigma }_m$ gives:
$$E(R_i)=R_f+{\sigma }_i{\rho }_{i,m}[\frac{E(R_m)-R_f}{{\sigma }_m}]$$

This equation shows that excess expected return is the product of the systematic component of risk (i.e., ${\sigma }_i$, ${\rho }_{i,m}$) and the unit price of risk (i.e., $[\frac{E(R_m)-R_f}{{\sigma }_m}]$).

4. Security Market Line(SML)

In a world where the market is in equilibrium and is expected to remain in equilibrium, no investor can achieve an abnormal return.(i.e., an expected return greater that return predicted by the CAPM risk-return relationship.)

All securities will lie on the Security Market Line.

In the real world, stocks and portfolios may yield a return in excess of ,or below, the return with fair compensation for risk exposure.

A graphical representation of the CAPM with beta on the x-axis and expected return on the y-axis. Intercept is $R_f$, slope is the market risk premium $(R_m-R_f)$.

• Any asset that are properly priced plots on SML( A ).
• Any asset that are overpriced plots below SML( B ).
• Any asset that are underpriced plots above SML( C ).
  

Security Market Line Application

• CAPM theory asserts that investors would increase their allocations to asset C, driving up its price and decreasing its expected return.
• Similarly, investors would decrease their allocaion to asset B, driving down its price and increasing its expected return.
• The reallocation by investors would continue until both assets fell on the capital market line. (meaning that the market has reached equilibrium).

CML vs. SML