Finance Models: CAPM and APT Overview
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Questions and Answers

What is the primary purpose of the Capital Asset Pricing Model (CAPM)?

To determine the expected return of an asset based on its systematic risk relative to the market.

How does Arbitrage Pricing Theory (APT) differ from the Capital Asset Pricing Model (CAPM)?

APT allows for multiple factors affecting asset returns, while CAPM focuses solely on market risk.

What assumptions underlie the capital market in the context of CAPM and APT models?

The capital market is assumed to be perfect and complete, with no frictions or imperfections.

What role do investors' risk aversion and homogeneous expectations play in the decision-making process according to the models discussed?

<p>Investors' risk aversion dictates their portfolio choices while homogeneous expectations ensure consistent predictions about asset returns.</p> Signup and view all the answers

In a one-period model, what is meant by maximizing the expected utility of end-of-period wealth?

<p>It refers to selecting portfolio weights that lead to the highest expected satisfaction from future wealth.</p> Signup and view all the answers

What determines if a firm should invest in a project according to CAPM?

<p>The firm should invest if the expected return on the project is greater than the required rate of return calculated using CAPM.</p> Signup and view all the answers

What is the relationship between E(R e project) and E(R e CAPM) when a project is funded entirely with equity?

<p>E(R e project) equals E(R e CAPM) under the condition that the project does not involve any debt.</p> Signup and view all the answers

Given the provided market conditions, should the firm invest in the project with a return of 2.25%?

<p>Yes, the firm should invest because 2.25% (expected return) is greater than 1.99% (required return).</p> Signup and view all the answers

Why is it crucial to transform theoretical CAPM to an ex ante form for empirical testing?

<p>Transforming to an ex ante form allows practitioners to apply CAPM using actual observed data, making it relevant for real-world applications.</p> Signup and view all the answers

How does the covariance Cov(R e project, R e M) factor into the decision process for investment?

<p>Covariance indicates how the project return moves in relation to the market return, influencing the expected risk and return profile of the project.</p> Signup and view all the answers

What does the equation Rjt = E(Rjt) + βj δmt + ϵjt represent in relation to asset returns?

<p>It represents the expected return on an asset j, accounting for the market return's excess and a random error term.</p> Signup and view all the answers

How does the empirical version of CAPM differ from its theoretical version?

<p>The empirical version uses actual return data for analysis, rather than relying on expected returns.</p> Signup and view all the answers

In the context of CAPM, what does the term βj indicate?

<p>βj represents the sensitivity of asset j's returns to the overall market returns.</p> Signup and view all the answers

What is the significance of the condition αj = 0 in the CAPM framework?

<p>It implies that there are no abnormal returns for asset j after adjusting for market risks.</p> Signup and view all the answers

What is represented by the term δmt in the CAPM equation?

<p>δmt indicates the excess return of the market over its expected value.</p> Signup and view all the answers

Explain the relationship captured by the equation Rjt - Rft = (Rmt - Rft)βj + ϵjt.

<p>This equation shows how the excess return of asset j over the risk-free rate is influenced by the excess return of the market and its beta.</p> Signup and view all the answers

What role does the error term ϵjt play in the CAPM equation?

<p>ϵjt captures the random variations in asset returns that are not explained by market movements.</p> Signup and view all the answers

What does the term λ represent in the context of the cross-sectional analysis of CAPM?

<p>λ represents the market risk premium, defined as the expected market return minus the risk-free rate.</p> Signup and view all the answers

What are the two main components that affect asset returns in factor models?

<p>Common factors and firm specific noise.</p> Signup and view all the answers

What does the term $b_{ik}$ represent in the context of the asset return equation?

<p>$b_{ik}$ represents the sensitivity of the ith asset's returns to the k th factor.</p> Signup and view all the answers

In the factor model for portfolios, how is the expected return of a portfolio $R_{ep}$ expressed?

<p>It's expressed as $\alpha_p = \sum_{i=1}^{N} w_i \alpha_i = E(R_{ep})$.</p> Signup and view all the answers

What does the term $\epsilon_{ep}$ signify in the portfolio return variance equation?

<p>$\epsilon_{ep}$ represents the unsystematic risk of the portfolio.</p> Signup and view all the answers

What is the formula for the variance of a portfolio return $R_{ep}$ with uncorrelated factors?

<p>The formula is $Var(R_{ep}) = \sum_{k=1}^{K} b_{pk}^2 Var(F_{ek}) + Var(\epsilon_{p})$.</p> Signup and view all the answers

What does $ ext{Cov}(\epsilon_i, \epsilon_j) = 0$ imply in the context of factor models?

<p>It implies that the noise terms for different assets are uncorrelated.</p> Signup and view all the answers

What does the notation $F_{ek}$ signify in the factor models?

<p>$F_{ek}$ signifies the k th factor that is common to the returns on all assets.</p> Signup and view all the answers

Explain the significance of the noise term $\epsilon_{ei}$ in asset return equations.

<p>$\epsilon_{ei}$ represents the firm-specific risk that is unique to asset i.</p> Signup and view all the answers

How is systematic risk quantified in the factor model for portfolios?

<p>Systematic risk is quantified by $b_{pk} F_{ek}$, where $b_{pk}$ is the portfolio's sensitivity to the k th factor.</p> Signup and view all the answers

What does the assumption of uncorrelated factors imply for factor models?

<p>It implies that the covariance between different factors is zero, simplifying the calculation of portfolio variance.</p> Signup and view all the answers

What is a pure factor portfolio regarding factor m?

<p>A pure factor portfolio for factor m has a beta of 1 with respect to m and a beta of 0 with respect to all other factors.</p> Signup and view all the answers

How is the return of a pure factor portfolio expressed mathematically?

<p>The return of a pure factor portfolio is defined as $R_{e}^{pm} = \alpha_{pm} + F_{em}$.</p> Signup and view all the answers

What does the factor risk-premium define in the context of risk-free return?

<p>The factor risk-premium is defined as $\lambda_{m} = E(R_{e^{pm}}) - R_{f} = \alpha_{pm} - R_{f}$.</p> Signup and view all the answers

What is a tracking portfolio and how is it related to pure factor portfolios?

<p>A tracking portfolio is a weighted average of pure factor portfolios and the risk-free asset, designed to have no firm-specific risk.</p> Signup and view all the answers

How is the expected return of a tracking portfolio calculated?

<p>The expected return is calculated as $E(R_{e}) = \sum_{m=1}^{K} b_{im} (\lambda_{m} + R_{f}) + (1 - \sum_{m=1}^{K} b_{im}) R_{f}$.</p> Signup and view all the answers

Under what assumptions does the Asset Pricing Theory (APT) operate?

<p>APT operates under the assumptions of a factor model defining returns, elimination of firm-specific risk through diversification, and a perfect capital market in equilibrium.</p> Signup and view all the answers

What are the key components of the Fama and French three-factor model?

<p>The key components are the market risk premium (bi), size premium (si), and value premium (hi).</p> Signup and view all the answers

In the context of APT, how is the expected return for asset i determined?

<p>The expected return for asset i is determined by $E(R_{i}) = b_{i1} \lambda_{1} + \ldots + b_{iK} \lambda_{K} + R_{f}$.</p> Signup and view all the answers

How did Fama and French empirically test their factor model?

<p>They used a cross-sectional regression approach via Fama-MacBeth method.</p> Signup and view all the answers

What is meant by having no firm-specific risk in a pure factor portfolio?

<p>No firm-specific risk implies that the returns are solely influenced by systematic risk factors and not by idiosyncratic risks associated with individual firms.</p> Signup and view all the answers

What role does the risk-free asset play in the tracking portfolio?

<p>The risk-free asset provides a baseline level of return and enables the tracking portfolio to achieve lower overall risk and different return profiles.</p> Signup and view all the answers

What does the term SMB represent in the Fama and French model?

<p>SMB stands for 'Small Minus Big,' which measures the size premium in returns between small and large firms.</p> Signup and view all the answers

How does the concept of arbitrage relate to APT?

<p>In APT, the concept of no arbitrage implies that assets are correctly priced based on their risk exposures; thus, no risk-free profit can be made.</p> Signup and view all the answers

What does HML indicate in the context of the Fama and French three-factor model?

<p>HML stands for 'High Minus Low,' reflecting the value premium by comparing returns between firms with high and low book-to-market ratios.</p> Signup and view all the answers

From what period did Fama and French analyze data to develop their three-factor model?

<p>They analyzed data from July 1963 to December 1990.</p> Signup and view all the answers

What debate exists regarding the empirical validity of factor models, particularly CAPM?

<p>The debate centers around potential errors in the execution and design of empirical tests.</p> Signup and view all the answers

What are the three main directions in the research of empirical testing of CAPM mentioned?

<p>The three directions are errors in execution, design of empirical tests, and variations among different tests.</p> Signup and view all the answers

What notable databases were used for the analysis conducted by Fama and French?

<p>The analysis utilized daily individual stock returns from NYSE, AMEX, and NASDAQ from 1963 to 1990.</p> Signup and view all the answers

Flashcards

Market Equilibrium Asset Pricing Models

A model explaining market equilibrium and how assets are priced based on their risk and return. It determines the optimal portfolio for investors by considering the trade-off between risk and return, and establishes a relationship between expected return and risk for all assets.

One-Period Model

Focuses on a single period, assuming a perfect and complete capital market with no frictions or imperfections. All assets are perfectly divisible and marketable, and investors are rational and risk-averse.

Perfect and Complete Capital Market

This refers to a market where any asset can be traded freely with no barriers to entry or exit, and where information is readily available to all investors.

Capital Asset Pricing Model (CAPM)

A model that determines how to price a single asset by considering its relationship with the market portfolio, its beta, and the risk-free rate.

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Arbitrage Pricing Theory (APT)

A model that extends the CAPM by incorporating multiple factors that can affect asset prices, beyond just beta. It allows for the existence of multiple risk factors, such as inflation, interest rates, and economic growth.

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Theoretical CAPM Equation

The Capital Asset Pricing Model (CAPM) equation in its theoretical form, assuming perfect markets and rational investors, connects the expected return of an asset to its systematic risk, represented by beta, and the risk-free rate. It also includes the market risk premium, essentially the difference between the expected return on the market portfolio and the risk-free rate.

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Transforming CAPM to Ex Ante Form

The process involves transforming the theoretical CAPM equation, which deals with expected returns, into an ex ante form that can be applied to historical data, allowing empirical analysis and testing of the CAPM model on real-world observations.

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CAPM for Cost of Equity (No Debt)

The CAPM is employed to determine the cost of equity, or the required rate of return that investors expect on a company's equity. This is particularly useful when a company has no debt, as the cost of equity directly translates to the required rate of return for a project.

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Project Investment Decision

Comparing the project's expected return to the required rate of return derived from the CAPM allows a company to make informed investment decisions. If a project's expected return exceeds the required rate of return, it is considered a worthwhile investment.

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Cost of Equity = Required Return on Project (100% Equity)

When a company finances a project purely with equity, the required rate of return on the project directly aligns with the cost of equity determined by the CAPM formula.

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Rate of return on an asset

The rate of return on an asset, represented as the sum of expected return, beta multiplied by the market's deviation from its expected return, and a random error term.

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Transformation of the rate of return equation

Expressing the rate of return on an asset as a function of the risk-free rate, the market risk premium, and the asset's beta.

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Ex post form of CAPM

An empirical version of the CAPM that utilizes observed return data instead of expected returns, enabling analysis of historical performance.

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Time-series regression

Analyzing the variation in returns over time for a specific asset, aiming to understand how its return fluctuates.

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Cross-sectional analysis

Investigating average returns across assets within a specified time period, comparing them to their respective betas.

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Lambda (λ)

The slope of the regression line in a cross-sectional analysis, representing the risk premium per unit of beta.

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Alpha (α)

The intercept of the regression line in a cross-sectional analysis, indicating the average excess return for an asset with zero beta.

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Alpha equals zero

A scenario where the average excess return for an asset with zero beta is zero, indicating that the CAPM holds true.

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CAPM (Capital Asset Pricing Model)

A model that describes the expected return of an asset based on its sensitivity to systematic risk, represented by beta, and the market risk premium.

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SMB (Size Premium)

The difference in returns between small and large firms.

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HML (Value Premium)

The difference in returns between firms with a high book-to-market value ratio and those with a low ratio.

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Fama-French Three-Factor Model

This model expands on CAPM by adding two additional factors: SMB (Size Premium) and HML (Value Premium).

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Fama-MacBeth Regression

A statistical technique that uses cross-sectional regression to estimate the relationship between a dependent variable (e.g., stock returns) and several independent variables (e.g., beta, size, value).

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Empirical Testing of Factor Models

The process of analyzing the historical performance of a model and comparing it to real-world data to see how well it predicts future returns.

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Tracking Portfolio

A portfolio designed to track a specific factor, such as size or value.

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Factor Model

A model that explains asset returns based on common factors and firm-specific noise. Asset returns are influenced by both systematic factors that affect all assets and idiosyncratic factors unique to individual assets.

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Factor Loading (bik)

The sensitivity of an asset's return to a specific factor. It measures how much an asset's return is expected to change for every unit change in the factor.

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Common Factor (Fk)

A common factor that influences the returns of all assets in a portfolio. These factors are often macroeconomic or industry-wide forces that affect the performance of multiple assets.

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Noise Term (εei)

The noise term represents the unique, firm-specific factors that affect an asset's return. It captures the random fluctuations that cannot be explained by the common factors.

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Expected Return (E(Rei))

The expected return on an asset based on its sensitivity to common factors and the expected values of those factors. It's the return you would expect based on the asset's systematic risk.

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Portfolio Factor Loading (bpk)

The weighted average of the factor loadings for all assets in a portfolio. It represents the overall sensitivity of the portfolio to the factors.

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Systematic Risk

The portion of a portfolio's return that is due to the common factors. It reflects the portfolio's exposure to systematic risk.

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Unsystematic Risk

The portion of a portfolio's return that is specific to the individual assets and not explained by the common factors. It represents the portfolio's exposure to unsystematic risk.

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Portfolio Return Variance

The variance of the portfolio's return, which measures the overall volatility or dispersion of potential returns. It takes into account the contribution of both systematic and unsystematic risk.

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Uncorrelated Factors

A simplifying assumption in factor models that the common factors are not correlated with each other. This means that the changes in one factor do not influence the changes in other factors.

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Pure Factor Portfolio

A portfolio designed to replicate the returns of a specific factor. It has a beta of 1 for that factor and 0 for all other factors, eliminating firm-specific risk.

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Return of a Pure Factor Portfolio

The return of a pure factor portfolio is solely determined by the factor's risk premium and the risk-free rate. It doesn't have any firm-specific risk, so its return is solely driven by the chosen factor.

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Factor Risk Premium (λm)

The risk premium of a factor is the expected return above the risk-free rate that investors demand for bearing the risk associated with that factor. It represents the compensation for exposure to a certain factor.

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No Firm-Specific Risk in APT

The APT model assumes no firm-specific risk because diversification eliminates it, implying that the asset's return is solely driven by its exposures to various factors.

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Perfect Capital Market in APT

The APT model assumes a perfect capital market, meaning that all assets are perfectly divisible and marketable, and information is readily available to all investors.

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Market Equilibrium in APT

The APT model requires market efficiency to hold. This means that the expected return of an asset should be determined by the factors that influence its risk, and no arbitrage opportunities should exist.

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Asset Expected Return in APT

The APT model suggests that an asset’s expected return is determined by the linear combination of its factor betas (sensitivities to various factors) and the corresponding factor risk premiums.

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Return Defined by a Factor Model in APT

The APT model assumes that the return of an asset can be represented by a factor model, capturing its relationship with various market factors and firm-specific risk.

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Study Notes

Principles of Finance Lecture 9&10

  • Lecture covers Market Equilibrium, CAPM, and APT.
  • Resources include CWS chapter 6 and Fama and French (1992).

Introduction

  • Last time: Mean-variance analysis by Markowitz (1952, 1959)
  • Today covers Market equilibrium asset pricing models
  • Topics include: Determining asset pricing, determining market price of risk, and single asset pricing.
  • Two models are discussed:
    • Capital asset pricing model (CAPM) introduced by Sharpe (1963, 1964), and Treynor (1961).
    • Arbitrage pricing theory (APT) introduced by Ross (1976).

Settings

  • One-period model
  • Perfect capital market (no frictions or imperfections) and complete.
  • N risky assets, with return R = (R1, ..., RN), where R~N(E(R), σ²).
  • A risk-free asset with return of Rf.
  • All assets are marketable and perfectly divisible.
  • Investors are greedy and risk-averse. All investors have homogeneous expectations about asset returns.
  • Investment decision involves optimizing a portfolio to maximize the expected utility of end-period wealth.
  • Optimal portfolios are described by weights w = (w1, ..., wN) and wf.

Mean-variance efficient PFs in equilibrium with a risk-free asset

  • Equilibrium prices in a market are established such that supply equals demand for any asset.
  • Total market value of assets (supply) equals total demand for assets.
  • Market portfolio = Tangency portfolio.
  • The Capital Market Line (CML) represents the relationship between expected return and standard deviation in a market in equilibrium with a risk-free asset.

Capital asset pricing model (CAPM)

  • An equilibrium model where tangency portfolio = Market portfolio.

  • Concerned with pricing all assets and portfolios

  • New risk measure accounts only for market risk.

    • Idiosyncratic (asset-specific) risk is independent of market risk.
    • Example: Changes in oil prices
    • Systematic (market) risk is affected by, for example, changes in interest rates and recessions

  • Only idiosyncratic risk is diversifiable.

  • Investors are only compensated for market risk.

Capital Asset Pricing Model (CAPM)

  • New risk measure accounts for only market risk
  • Beta (βᵢ)= Cov(Rᵢ, Rₘ)/Var(Rₘ)= σᵢₘ/σ²ₘ
  • Security Market Line (SML): E(Rᵢ) = Rf + βᵢ(E(Rₘ) – Rf)

First property of CAPM: Systematic risk and Idiosyncratic risk

  • Total risk = systematic risk + idiosyncratic risk (σᵢ = σᵢ, sys + σᵢ, unsys)
  • Using the SML to find the systematic risk, σᵢ, sys, for assets/portfolios.
  • SML pricing based on systematic risk, σp = σᵢ, sys.
  • SML prices all assets/portfolios based on its systematic risk only.
  • Systematic risk: σᵢ, sys = βᵢσₘ (Where β is from calculating the SML which was defined before.)

Second property of CAPM: Portfolio betas - measure the systematic risk of portfolios

  • Portfolio betas equal a weighted average of the individual security betas, weighted by their portfolio weights. βp = Σ wiβᵢ

CAPM - usage for investment decision for firms

  • Determine cost of equity/required rate of return on equity in the absence of debt.
  • Required rate of return on equity = required rate of return on project.
  • A firm should invest in a project if its expected rate of return is greater than its required rate of return.

Exercise

  • Assume a market in equilibrium. The market portfolio equals the tangent portfolio and the risk-free asset rate is 0.2%.
  • Assume a project funded by 100% equity with a 2.25% expected return and Cov(Rproject, RM) = 0.003.
  • Should the firm invest?

Empirical test of CAPM

  • Expectations cannot be measured directly.
  • Transforming theoretical CAPM for ex ante application to observed data.
  • Assumed rate of return on any asset is a fair gamble: Rjt = E(Rjt) + βj dmt + e jt.

Empirical test of CAPM

  • Time series regression tests return variation over time for a given asset.
  • Cross-sectional analysis to understand average returns in relation to betas for a given time period.
  • For CAPM to hold, x=0.

Frequently used technique

  • Estimate beta over a pre-period (five-year).
  • Construct portfolios from the pre-ranking of their beta values.
  • Estimate beta (Bᵢ) and calculate E(Rᵢ) for a post-period.
  • Use one cross-sectional regression for post-period, where E(R₁) - Rf = (γ) + βρ + αρ
  • Or use the Fama-MacBeth cross-sectional regression for different time periods.

Empirical findings

  • For the CAPM to hold:
    • a should not be significantly different from 0.
    • Coefficient (1) should equal E(Rₘ) – Rf.
    • Over long periods, market portfolio's excess returns more than risk-free rate.
    • Linear relationship between excess returns and beta.
    • Beta should be only factor that explains return in risky asset.

Factor models

  • Asset returns affected by common factors and firm-specific noise.
  • Asset return for asset i with K factors: Rᵢ = aᵢ + bᵢ₁ F₁ + ... + bᵢₖ Fₖ + eᵢ.
  • Expected asset returns are determined using beta values for common factors in the return of the entire market

Factor model for portfolios

  • Variance of a portfolio with N assets calculated using: Var(Rp) = Σk=1Σm=1 bpkp,m Cov(Fk,Fm) + Var(Ep).
  • For well-diversified portfolios, the variance is equal to the sum of the variances of each factor.

Tracking portfolios

  • Perfect tracking: Portfolio perfectly replicates benchmark portfolio returns in all scenarios.
  • Imperfect tracking: Portfolio exhibiting same systematic risk as benchmark portfolio.

Use of Tracking portfolios

  • Performance evaluation of various investment portfolios.
  • Corporate hedging to reduce exposure to various risks.
  • Capital allocation of varied amounts of capital to the various portfolios and assets.

Use of Tracking portfolios: Example - Corporate Hedging

  • BMW example with exchange rate risk and growth (BNP) factors as relevant.
  • Hedging (tracking) portfolio involves setting up separate portfolios so the overall risk is zero relating to various factors.

Portfolio weights for tracking portfolio

  • Calculating portfolio weights involves determining factor weights that produce the target exposure across various factors.
  • N assets needed, where N = K + 1 for a general case.

Pure factor portfolios

  • Portfolio that only reflects changes in a single factor.
  • Return of the pure factor portfolio, Rpm, defined as: Rpm = αpm + Fₘ. Factor Rpm is the risk premium for pure factors.

Asset pricing Theory (APT)

  • APT assumes returns are determined by a factor model to account for multiple common factors and the absence of firm-specific risk.
  • Asset i's expected return is determined by the tracking portfolio: E(Rᵢ) = bᵢ₁ λ₁ + ... + bᵢₖ λₖ + Rf, where Aᵢ is the factor m's risk premium

Comments about APT

  • No assumptions about how asset returns are empirically distributed.
  • Equilibrium returns allow for many different factors.
  • APT does not rely on the market portfolio being efficient, unlike CAPM.

Empirical test of Asset pricing Theory (APT)

  • The method for testing APT is similar to approach for CAPM.
  • Identify relevant factors that explain financial risk variability.

Fama and French (1992)

  • Example using the value-weighted portfolio of NYSE, AMEX, and NASDAQ stocks to find cross-sectional relationships between return and risk (β).
  • Firm size (ln(ME)), book-to-market equity (ln(BE/ME)).
  • Databases used include daily individual stock returns.

Fama and French (1992) Table Properties of Portfolios

  • Data from 1963-1990 on various stock properties, including returns, size (ln(ME)), book-to-market equity (ln(BE/ME)) and firm characteristics.

Fama and French (1992) Table Two-Pass

  • Data from 1963-1990 representing various averages from two separate portfolios' sorted groups of company sizes and beta values.
  • Low, third, average of fifth and sixth, eighth, and high decile subgroups included for each size and beta characteristic.

Fama and French (1992) Table III

  • Data from 1963–1990 averages slopes (t-statistics) for returns on different characteristic factors (β, Size, Book-to-Market Equity, etc.).
  • Uses cross-sectional regressions.

Fama and French's three factor model

  • Introduces three-factor model:
    • E(Rᵢ) - Rf = bᵢ[E(Rₘ) – Rf] + sᵢE(SMB) + hᵢE(HML)
    • SMB = size premium (difference in returns between small and big firms)
    • HML = value premium (difference in returns between high and low book-to-market value ratio firms).

Comments regarding empirical testing of Factor models

  • Debate around empirical validity of factor models continue.
  • Issues include potential errors due to inconsistencies in execution of empirical testing.
  • Lack of intuitive explanation behind factors.
  • Possibility that market risk premiums and beta values change over time.

References

  • Provides citations for sources and further learning materials.

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Principles of Finance Lecture 9&10: Market Equilibrium PDF

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Explore key concepts related to the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) in this quiz. Test your understanding of investment decision-making, risk aversion, and the assumptions behind these financial models. Perfect for finance students looking to solidify their grasp of essential theories in asset management.

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