11.4 Beta, the Capital Asset Pricing Model & the Security Market Line

Diversification removes company-specific risk, so the risk that matters for pricing is the risk that cannot be diversified away. This section explains beta as the measure of systematic risk, portfolio beta, the Capital Asset Pricing Model, and the efficient frontier, Capital Market Line and Security Market Line.

1. Beta ($\beta$): The Measure of Systematic Risk

Because unsystematic risk can be freely eliminated through diversification at negligible transaction cost, rational capital markets do not compensate investors for holding it. Instead, expected return is driven solely by an asset's systematic risk.

The standardized metric used to measure systematic risk is Beta ($\beta$). Beta measures the sensitivity of a security's returns relative to the overall market portfolio (such as the S&P/TSX Composite Index):

$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = r_{im} \frac{\sigma_i}{\sigma_m}$

Where:

• $\text{Cov}(R_i, R_m)$ = Covariance between the returns of security $i$ and the market portfolio $m$
• $\sigma_m^2$ = Variance of the market portfolio returns
• $r_{im}$ = Correlation between security $i$ and the market
• $\sigma_i, \sigma_m$ = Standard deviations of the security and market, respectively

Interpreting Beta Values

By mathematical definition, the market portfolio has a Beta of exactly 1.0 ($\beta_m = 1.0$), and a completely risk-free asset (such as a 91-day Government of Canada Treasury bill) has a Beta of 0.0 ($\beta_f = 0.0$).

Portfolio Beta

The Beta of a portfolio is the simple weighted average of the Betas of its individual holdings:

$\beta_p = \sum_{i=1}^n w_i \beta_i = w_1 \beta_1 + w_2 \beta_2 + \dots + w_n \beta_n$

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2. The Capital Asset Pricing Model (CAPM)

Developed by William Sharpe, John Lintner, and Jan Mossin in the 1960s, the Capital Asset Pricing Model (CAPM) formalizes the relationship between systematic risk and expected return in an efficient market. The model posits that the expected (or required) rate of return on any risky asset equals the risk-free rate plus a risk premium proportional to its systematic risk (Beta):

$E(R_i) = R_f + \beta_i [E(R_m) - R_f]$

Where:

• $E(R_i)$ = Expected or required return on security $i$
• $R_f$ = Risk-free rate of return (yield on short-term Government of Canada Treasury bills)
• $\beta_i$ = Beta of security $i$
• $E(R_m)$ = Expected return on the overall market portfolio (e.g., S&P/TSX Composite)
• $[E(R_m) - R_f]$ = Market Risk Premium (MRP): The excess return required by investors above the risk-free rate for holding the market portfolio
• $\beta_i [E(R_m) - R_f]$ = Equity Risk Premium for security $i$

Worked Numeric Example: Applying CAPM to Canadian Equities

A Canadian wealth advisor evaluates a major pipeline utility stock. The following capital market parameters are established:

• Risk-free rate ($R_f$): 3.00% (3-month GoC T-bill)
• Expected market return ($E(R_m)$): 9.00%
• Stock Beta ($\beta$): 0.75 (defensive utility)

$\text{Market Risk Premium (MRP)} = 9.00\% - 3.00\% = 6.00\%$

$E(R_i) = 3.00\% + 0.75 \times (9.00\% - 3.00\%) = 3.00\% + 0.75 \times 6.00\% = 3.00\% + 4.50\% = 7.50\%$

The CAPM required rate of return for the utility stock is 7.50%. If the advisor's fundamental valuation predicts an actual return of 9.00%, the stock is undervalued and represents an attractive purchase (positive alpha of $+1.50\%$).

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3. The Efficient Frontier, CML, and SML

Modern Portfolio Theory and CAPM generate three graphical frameworks that frequently appear on licensing exams:

1. Markowitz Efficient Frontier: Plots Expected Return vs. Total Risk (Standard Deviation, σ).
   - Shows the set of portfolios offering maximum expected return for each level of standard deviation.
   - Portfolios below the frontier are inefficient (sub-optimal); portfolios above are unattainable.

2. Capital Market Line (CML): Tangency line from the risk-free rate (R_f) to the optimal market portfolio (M) on the Efficient Frontier.
   - Measures Expected Return vs. Total Risk (σ).
   - Valid ONLY for fully diversified (efficient) portfolios.

3. Security Market Line (SML): Graphical representation of the CAPM.
   - Plots Expected Return vs. Systematic Risk (Beta, β).
   - Valid for ALL individual securities and all portfolios, whether diversified or not.

The Security Market Line (SML) and Security Valuation

The SML provides a visual benchmark for determining whether an asset is properly valued:

• Fairly Priced: An asset plotting directly on the SML has an expected return exactly equal to its CAPM required return ($\alpha = 0$).
• Undervalued (Underpriced): An asset plotting above the SML offers an expected return higher than warranted by its systematic risk. The asset generates positive alpha ($\alpha > 0$) and should be bought.
• Overvalued (Overpriced): An asset plotting below the SML offers an expected return lower than required for its systematic risk. The asset generates negative alpha ($\alpha < 0$) and should be sold or avoided.

Direct Comparison: Capital Market Line (CML) vs. Security Market Line (SML)