Systematic Risk and the Equity Risk Premium: Portfolio Theory and CAPM

Key Concepts

Portfolio Construction

Portfolio construction involves determining the proportion of total investment allocated to each asset. This is essential for calculating returns and risk.

• Portfolio weight (xi): The fraction of total portfolio value invested in asset i.

Formula:

$x_i = \frac{\text{Value of investment in asset } i}{\text{Total portfolio value}}$

• All portfolio weights must sum to 1 (or 100%).

• Example: 8,000 shares of Qantas ($5 each = $40,000) and 1,500 shares of Woolworths ($40 each = $60,000). $x_{Qantas} = 0.4$, $x_{Woolworths} = 0.6$

Portfolio Return

The expected return of a portfolio is the weighted average of the expected returns of its assets.

• Expected Portfolio Return ($E[R_p]$):

$E[R_p] = \sum_{i=1}^n x_i E[R_i]$

• Example: Stock A: 18%, Stock B: 25%, Weights 0.3 and 0.7 $E[R_p] = 0.3(18\%) + 0.7(25\%) = 23.9\%$

• Realized Portfolio Return ($R_p$):

$R_p = \sum x_i R_i$

• Example: Actual returns 10% and 20% $R_p = 0.3(10\%) + 0.7(20\%) = 17\%$

Diversification & Risk Reduction

Types of Risk

Diversification is a key strategy to reduce risk in a portfolio. There are two main types of risk:

• Unsystematic risk: Also called firm-specific risk; can be reduced through diversification.

• Systematic risk: Also called market risk; cannot be diversified away.

• Combining uncorrelated assets lowers overall portfolio volatility.

Example:

• Two airlines (North Air & West Air): correlated risks → little reduction (13.4% → 12.1%).

• Airline + oil stock (West Air & Tex Oil): opposite movements → major reduction (13.4% → 5.1%).

Covariance and Correlation

Covariance

Covariance measures how two assets move together. It is positive if they move in the same direction, negative if in opposite directions.

Formula:

$Cov(R_i, R_j) = E[(R_i - E[R_i])(R_j - E[R_j]) ]$

• Positive: assets move together.

• Negative: assets move in opposite directions.

• If zero: assets are uncorrelated.

Correlation (ρij)

Correlation standardizes covariance to a range between -1 and +1.

Formula:

$\rho_{ij} = \frac{Cov(R_i, R_j)}{SD(R_i) \times SD(R_j)}$

• Range: -1 ≤ ρ ≤ +1

• +1: perfectly positively correlated

• -1: perfectly negatively correlated

• 0: uncorrelated

Portfolio Variance and Standard Deviation (N=2 assets)

Portfolio variance measures the risk of a portfolio, considering both individual asset variances and their covariances.

Formula:

$Var(R_p) = x_1^2 \sigma_1^2 + x_2^2 \sigma_2^2 + 2x_1x_2\rho_{12}\sigma_1\sigma_2$

$SD(R_p) = \sqrt{Var(R_p)}$

• Example: 50% Woodside (SD=0.051), 50% Tex Oil (SD=0.071), ρ=0.46 $Var = 0.5^2(0.051)^2 + 0.5^2(0.071)^2 + 2(0.5)(0.5)(0.46)(0.051)(0.071)$ $SD(R_p) = 0.051 = 5.1\%$

Portfolio Risk with Many Assets

As the number of assets increases, unsystematic risk approaches zero, leaving only systematic risk.

Formula:

$Var(R_P) = \frac{1}{n}(\text{Average Var}) + \left(1 - \frac{1}{n}\right)(\text{Average Cov})$

• Total risk = Systematic + Unsystematic

• Diversification flattens after a certain point (efficient frontier).

Efficient Portfolio & Efficient Frontier

The efficient frontier represents the set of optimal portfolios offering the highest expected return for a given level of risk.

• Efficient portfolio: Cannot reduce volatility without lowering expected return.

• Inefficient portfolio: Higher risk for same or lower return.

• Minimum Variance Portfolio (MVP): Lowest risk portfolio.

• Efficient Frontier: Upward-sloping curve of optimal portfolios.

Graph interpretation:

• X-axis: Standard deviation (risk)

• Y-axis: Expected return

• Portfolios below MVP are inefficient; above are efficient.

• Investors choose along the efficient frontier based on risk tolerance.

Capital Asset Pricing Model (CAPM)

CAPM links expected return to systematic risk (beta). It is a foundational model in financial accounting and investment analysis.

Formula:

$E[R_i] = r_f + \beta_i (E[R_M] - r_f)$

• $r_f$ = risk-free rate

• $E[R_M] - r_f$ = market risk premium

• $\beta_i$ = sensitivity of stock i to market portfolio

Beta (β)

Beta measures the sensitivity of an asset's returns to market returns.

Formula:

$\beta_i = \frac{Cov(R_i, R_M)}{Var(R_M)} = Corr(R_i, R_M) \frac{SD(R_i)}{SD(R_M)}$

• Example: SD(Market)=0.44, SD(ATP)=0.68, Corr=0.91 $\beta = 0.91(0.68/0.44) = 1.41$ Expected Return: $E[R_i] = 5\% + 1.41(12\% - 5\%) = 14.87\%$

Portfolio Beta

Portfolio beta is the weighted average of the betas of the assets in the portfolio.

Formula:

$\beta_p = \sum x_i \beta_i$

• Example: 40% 3M ($\beta$=0.69), 60% HPQ ($\beta$=1.77) $\beta_p = 0.4(0.69) + 0.6(1.77) = 1.338$ $E[R_p] = 5\% + 1.338(12\% - 5\%) = 14.37\%$

Types of Risk

Type	Description	Can be Eliminated?
Systematic (Market)	Affects all firms (interest rates, inflation)	No
Unsystematic (Firm-specific)	Unique to firm or industry	Yes (through diversification)

Key Formulas Summary

Concept	Formula
Expected Portfolio Return	$E[R_p] = \sum x_i E[R_i]$
Covariance	$Cov(R_i, R_j) = E[(R_i - E[R_i])(R_j - E[R_j])]$
Correlation	$\rho_{ij} = \frac{Cov(R_i, R_j)}{SD(R_i) SD(R_j)}$
Portfolio Variance (2 assets)	$x_1^2 \sigma_1^2 + x_2^2 \sigma_2^2 + 2x_1x_2\rho_{12}\sigma_1\sigma_2$
Beta	$\beta_i = Corr(R_i, R_M) \frac{SD(R_i)}{SD(R_M)}$
CAPM	$E[R_i] = r_f + \beta_i (E[R_M] - r_f)$
Portfolio Beta	$\beta_p = \sum x_i \beta_i$

Examples Recap

• West Air & Tex Oil: efficient frontier illustrates volatility drops from 13.4% to 5.1%.

• Intel & Coca-Cola: efficient frontier graph → correlation ↑ = risk ↑.

• ATP Oil & Gas: $\beta = 1.41$ → expected return 14.87%.

• 3M + HPQ Portfolio: $\beta_p = 1.338$ → expected return 14.37%.

Core Takeaways

• Lower correlation = greater diversification benefit.

• Efficient portfolios lie on the efficient frontier.

• CAPM connects risk (β) and expected return.

• Only systematic risk matters for required return.

• Investors optimize risk-return through portfolio weights and diversification.