Capital Asset Pricing Theory: CAPM & APT Analysis (Q&A)

Capital Asset Pricing Theory: CAPM & APT Analysis

Question 1: Systematic vs. Unsystematic Risk in CAPM

Relevant Risk: Systematic Risk (Market Risk)

Why Systematic Risk is Relevant:

• Systematic risk cannot be eliminated through diversification
• It affects all securities in the market to varying degrees
• Investors must be compensated for bearing systematic risk through higher expected returns
• CAPM specifically measures systematic risk using beta (β)
• This is the risk premium component in the Security Market Line (SML)

Non-Relevant Risk: Unsystematic Risk (Specific Risk)

Why Unsystematic Risk is Not Relevant:

• Can be eliminated through proper diversification
• Investors are not compensated for bearing unsystematic risk
• Arbitrage opportunities would exist if investors were compensated for diversifiable risk
• In efficient markets, only non-diversifiable risk commands a risk premium

Key Insight: According to CAPM, rational investors hold diversified portfolios, thereby eliminating unsystematic risk. Therefore, the market only prices systematic risk.

Question 2: CML vs. SML Comparison

Aspect	Capital Market Line (CML)	Security Market Line (SML)
What it measures	Risk-return tradeoff for efficient portfolios	Risk-return tradeoff for individual securities
Risk measure	Total risk (standard deviation, σ)	Systematic risk (beta, β)
Equation	E(Rp) = RFR + [(E(RM) - RFR)/σM] × σp	E(Ri) = RFR + βi[E(RM) - RFR]
Application	Portfolio construction and allocation	Security valuation and pricing
Graph axis	Expected Return vs. Total Risk (σ)	Expected Return vs. Systematic Risk (β)
Efficient frontier	Represents efficient portfolios	All securities should lie on the line

Similarities: Both models relate expected return to risk, use the risk-free rate and market portfolio as benchmarks, and are based on Markowitz portfolio theory assumptions.

Question 3: Market Portfolio in CAPM

What's Included in the Market Portfolio:

• All risky assets in the economy
• Stocks, bonds, real estate, commodities, human capital
• All assets that are traded in the market
• Assets from all countries (in global CAPM)

Relative Weightings:

• Weighted by market value (capitalization weights)
• Each asset's weight = Market value of asset / Total market value of all assets
• The market portfolio represents the aggregate of all investors' portfolios

Typical Empirical Proxies:

• S&P 500 Index - Most common proxy in US studies
• Wilshire 5000 - Broader US market representation
• MSCI World Index - For global markets
• CRSP Value-Weighted Index - Academic research

Consequences of Poor Proxy:

• Beta estimates become unreliable - Systematic risk misestimated
• Security Market Line slope incorrect - Market risk premium misestimated
• Alpha calculations distorted - Performance measurement errors
• Cost of capital miscalculated - Investment and corporate finance decisions affected
• Roll's Critique - CAPM not testable if true market portfolio unobservable

Question 6: CAPM vs. APT - No-Risk, No-Wealth Investments

Why No-Risk, No-Wealth Investments Should Earn Zero Return:

This principle stems from the No-Arbitrage Condition:

• If a zero-investment, zero-risk portfolio could earn positive returns, arbitrageurs would exploit it infinitely
• This would create unlimited demand, driving returns to zero
• In efficient markets, risk-free arbitrage opportunities cannot persist
• Both CAPM and APT assume markets eliminate such opportunities

CAPM vs. APT Comparison:

Aspect	CAPM	APT
Risk Factors	Single factor (market risk)	Multiple factors (could be 3-5 or more)
Theoretical Basis	Mean-variance optimization	No-arbitrage principle
Assumptions	Strong assumptions about investor behavior	Fewer and less restrictive assumptions
Market Portfolio	Requires identification of market portfolio	Doesn't require market portfolio
Empirical Testing	Difficult due to market portfolio issue	Easier to test empirically

Establishing Superior Investment Performance:

Using CAPM: Identify securities with positive alpha (α)

α = Actual Return - [RFR + β(E(R_M) - RFR)]

Positive alpha indicates outperformance relative to systematic risk

Using APT: Identify securities mispriced relative to multiple risk factors

E(R) = RFR + β₁F₁ + β₂F₂ + ... + βₙFₙ

Superior performance comes from identifying securities with returns not fully explained by factor exposures

Question 7: CAPM vs. APT for Security Analysis

a. What CAPM and APT Attempt to Model:

CAPM: Models the relationship between systematic risk and expected return. Assumes all investors hold the market portfolio and that beta is the sole determinant of risk premium.

APT: Models expected returns as a linear function of multiple risk factors. Does not specify which factors, allowing flexibility in model specification.

Main Differences:

• Number of factors: CAPM - single factor; APT - multiple factors
• Theoretical foundation: CAPM - equilibrium model; APT - arbitrage pricing
• Practical implementation: CAPM simpler but restrictive; APT more flexible but requires factor identification
• Data requirements: CAPM needs market portfolio proxy; APT needs identified risk factors

b. When APT Would Be Preferred:

• When securities are influenced by multiple macroeconomic factors
• When the market portfolio proxy is questionable
• For specialized portfolios (sector funds, international funds)
• When analyzing assets not well-represented in market indices
• For hedge funds using multi-factor strategies
• When historical data shows multi-factor explanations better fit returns

Problem 1: SML with Changing Inflation

Initial Conditions:

• Expected inflation: 3%
• RFR (real + inflation): 6%
• Market Return (R_M): 12%
• Market Risk Premium: 6% (12% - 6%)

SML Graphs

Note: The following describes the graphical relationships

a. Initial SML (3% inflation):

Equation: E(R) = 6% + β(6%)

Y-intercept: 6% (RFR)

Slope: 6% (Market Risk Premium)

Point at β=1: (1, 12%)

b. Increased Inflation to 6%:

Fisher Effect: Nominal rates adjust for expected inflation

New RFR = 6% (original real) + 3% (additional inflation) = 9%

New R_M = 12% (original) + 3% (additional inflation) = 15%

Market Risk Premium remains: 15% - 9% = 6%

Equation: E(R) = 9% + β(6%)

Effect: Parallel upward shift of SML by 3 percentage points

c. RFR=9%, R_M=17%:

Market Risk Premium = 17% - 9% = 8%

Equation: E(R) = 9% + β(8%)

Comparison with part (b):

• Same intercept (9%)
• Steeper slope (8% vs 6%)
• Higher risk premium compensates for increased risk aversion or risk
• Investors require higher return per unit of systematic risk

Problem 2: Stock Valuation using SML

a. Expected Returns using CAPM:

Given: RFR = 10%, R_M = 14%, Market Risk Premium = 4%

E(R_i) = RFR + β_i(R_M - RFR) = 10% + β_i(4%)

Stock	Beta (β)	Expected Return	Calculation
U	0.85	13.40%	10% + 0.85(4%) = 13.40%
N	1.25	15.00%	10% + 1.25(4%) = 15.00%
D	-0.20	9.20%	10% + (-0.20)(4%) = 9.20%

b. Broker's Forecasts vs. CAPM:

Calculate Expected Returns from Price Forecasts:

Formula: Expected Return = (Expected Price + Dividend - Current Price) / Current Price

Stock	Current Price	Expected Price	Dividend	Total Return	Expected Return
U	22	24	0.75	24.75 - 22 = 2.75	12.50% (2.75/22)
N	48	51	2.00	53 - 48 = 5.00	10.42% (5/48)
D	37	40	1.25	41.25 - 37 = 4.25	11.49% (4.25/37)

Comparison and Investment Decisions:

Stock	CAPM Required Return	Broker's Expected Return	Alpha (α)	Action	Reason
U	13.40%	12.50%	-0.90%	SELL/SHORT	Undervalued - Below SML
N	15.00%	10.42%	-4.58%	SELL/SHORT	Significantly undervalued
D	9.20%	11.49%	+2.29%	BUY/LONG	Overvalued - Above SML

Graphical Interpretation: Stocks U and N plot below the SML (negative alpha), indicating they are overpriced relative to their risk. Stock D plots above the SML (positive alpha), indicating it is underpriced relative to its risk.

Problem 3: Portfolio Manager Evaluation

Given Information:

• Risk-free rate (RFR) = 4.50%
• Market risk premium = 5.00%
• Expected market return = RFR + Risk premium = 4.50% + 5.00% = 9.50%

a. Expected Returns using CAPM:

E(R) = RFR + β(R_M - RFR) = 4.50% + β(5.00%)

Manager	Beta (β)	CAPM Expected Return	Calculation
Y	1.20	10.50%	4.50% + 1.20(5.00%) = 10.50%
Z	0.80	8.50%	4.50% + 0.80(5.00%) = 8.50%

b. Alpha Calculations:

α = Actual Return - CAPM Expected Return

Manager	Actual Return	CAPM Expected	Alpha (α)
Y	10.20%	10.50%	-0.30%
Z	8.80%	8.50%	+0.30%

Graphical Representation on SML:

• Manager Y: Plots slightly below the SML (negative alpha of -0.30%)
• Manager Z: Plots slightly above the SML (positive alpha of +0.30%)
• The SML has intercept at 4.50% and slope of 5.00%
• Market portfolio plots at β=1.0, return=9.50%

c. Performance Conclusions:

1. Risk-adjusted outperformance:

• Manager Z outperformed Manager Y on a risk-adjusted basis
• Z has positive alpha (+0.30%) while Y has negative alpha (-0.30%)
• Z delivered higher returns relative to its lower systematic risk

2. Outperformance of market expectations:

• Manager Z outperformed market expectations (actual 8.80% vs expected 8.50%)
• Manager Y underperformed market expectations (actual 10.20% vs expected 10.50%)
• Neither manager dramatically outperformed or underperformed
• The alphas are relatively small in magnitude

Important Note: These conclusions are based solely on the CAPM framework. In practice, other factors (size, value, momentum) might explain part of the alpha, and statistical significance of the alphas should be considered.

Problem 5: Mutual Fund Valuation

Given Information:

• Risk-free rate (RFR) = 3.9%
• Market risk premium = 6.1%
• Expected market return = 3.9% + 6.1% = 10.0%

a. Expected Returns using CAPM:

E(R) = RFR + β(E(R_M) - RFR) = 3.9% + β(6.1%)

Fund	Beta (β)	CAPM Expected Return	Calculation
T	1.20	11.22%	3.9% + 1.20(6.1%) = 11.22%
U	0.80	8.78%	3.9% + 0.80(6.1%) = 8.78%

b. Position Relative to SML:

Fund	Forecasted Return	CAPM Expected	Position vs SML	Alpha (α)
T	9.0%	11.22%	Below SML	-2.22%
U	10.0%	8.78%	Above SML	+1.22%

c. Valuation Conclusions:

Fund T:

• Overvalued (trading above fair value)
• Market price implies 9.0% return, but CAPM says should offer 11.22% for its risk level
• Negative alpha of -2.22% indicates poor risk-adjusted performance expectation
• Action: Sell or avoid

Fund U:

• Undervalued (trading below fair value)
• Market price implies 10.0% return, but CAPM says should only offer 8.78% for its risk level
• Positive alpha of +1.22% indicates good risk-adjusted performance expectation
• Action: Buy or overweight

Investment Strategy: According to CAPM, an investor should short Fund T (overvalued) and go long on Fund U (undervalued) to create an arbitrage-like position that exploits the mispricing.