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

Systematic Risk and the Equity Risk Premium

1. Key Concepts

This section introduces foundational concepts in portfolio theory, risk, and the Capital Asset Pricing Model (CAPM), essential for understanding financial accounting and investment analysis.

1.1 Portfolio Construction

• Portfolio weight (xi): The proportion 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$

1.2 Portfolio Return

• Expected Portfolio Return (E[Rp]): Weighted average of expected returns of assets.

Formula:

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

• 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 (Rp): Actual weighted return. $R_p = \sum x_i R_i$ Example: Actual returns 10% and 20% $R_p = 0.3(10\%) + 0.7(20\%) = 17\%$

1.3 Diversification & Risk Reduction

Diversification is a key strategy to reduce risk in a portfolio by combining assets with low or negative correlation.

• Unsystematic risk: Firm-specific risk, can be diversified away.

• Systematic risk: Market-related risk, cannot be diversified away.

• Combining uncorrelated assets lowers overall portfolio volatility.

• Example: Two airlines (correlated risks) reduce volatility slightly; airline and oil stock (opposite movements) reduce volatility significantly.

1.4 Covariance and Correlation

Covariance and correlation measure how asset returns move together, which is crucial for portfolio risk analysis.

• Covariance (Cov(Ri, Rj)): Measures joint variability of two assets.

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): Standardized measure of covariance.

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

1.5 Portfolio Variance and Standard Deviation (N=2 assets)

Portfolio variance and standard deviation quantify the risk (volatility) of a portfolio.

• Portfolio Variance:

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

• Portfolio Standard Deviation:

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

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

1.6 Portfolio Variance (n assets)

For portfolios with more than two assets, variance is calculated using average variance and covariance.

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

• As n increases, unsystematic risk approaches zero.

• Remaining risk is systematic (market risk).

1.7 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.

1.8 Capital Asset Pricing Model (CAPM)

CAPM links expected return to systematic risk (beta), providing a framework for asset pricing.

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

1.9 Beta (β)

Beta measures an asset's sensitivity to market movements.

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 $\beta = 0.91(0.68/0.44) = 1.41$ $E[R_i] = 5\% + 1.41(12\% - 5\%) = 14.87\%$

1.10 Portfolio Beta

Portfolio beta is the weighted average of individual asset betas.

Formula:

$\beta_P = \sum x_i \beta_i$

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

1.11 CML & SML

• CML (Capital Market Line): Relationship between expected return and total risk (σ) for efficient portfolios.

• SML (Security Market Line): Relationship between expected return and β for all securities.

• Under CAPM, only systematic risk (β) is priced.

1.12 Types of Risk

Risks in portfolio management are classified as systematic or unsystematic.

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)

1.13 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	$\beta_P = \sum x_i \beta_i$

1.14 Example 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%.

1.15 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.