Portfolio Risk and Return: Part II

Portfolio Management

Capital Market Theory: Risk-Free and Risky Assets

Learning Outcome Statement:

describe the implications of combining a risk-free asset with a portfolio of risky assets; explain the capital allocation line (CAL) and the capital market line (CML)

Summary:

This LOS explores the integration of risk-free assets with risky assets to form portfolios that can potentially offer a better risk-return trade-off. It discusses the construction of the Capital Allocation Line (CAL) and the Capital Market Line (CML), which are crucial in understanding how portfolios can be optimized based on different levels of risk aversion. The content also covers the systematic and nonsystematic risks, the Capital Asset Pricing Model (CAPM), and how these concepts aid in portfolio management and asset valuation.

Key Concepts:

Capital Allocation Line (CAL)

The CAL represents combinations of portfolios of risky and risk-free assets that provide the highest expected return for a given level of risk. It is a straight line in the risk-return space starting from the risk-free rate.

Capital Market Line (CML)

A special case of the CAL where the risky portfolio is the market portfolio. The CML represents the risk-return combinations available to all investors in the market who can lend or borrow at the risk-free rate.

Systematic and Nonsystematic Risk

Systematic risk is the inherent risk associated with the overall market movements and cannot be diversified away. Nonsystematic risk, also known as specific or idiosyncratic risk, is unique to a particular company or industry and can be reduced through diversification.

Capital Asset Pricing Model (CAPM)

A model that describes the relationship between systematic risk and expected return for assets, particularly stocks. It is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio.

Formulas:

Expected Return of Portfolio

E(R_p) = \sum_{i=1}^{N} w_i E(R_i)

This formula calculates the expected return of a portfolio as the weighted average of the expected returns of the individual assets in the portfolio.

Variables:

$E(R_p)$ :: Expected return on the portfolio
$w_i$ :: Fractional weight of asset i in the portfolio
$E(R_i)$ :: Expected return of asset i
$N$ :: Total number of assets in the portfolio

Units: percentage

Portfolio Variance

\sigma_p^2 = \left( \sum_{i=1}^{N} w_i^2 \sigma_i^2 + \sum_{i,j=1,i\neq j}^{N} w_i w_j \rho_{ij} \sigma_i \sigma_j \right)

This formula calculates the variance (risk) of a portfolio, which depends on the weights of the assets, their individual risks, and the correlations between each pair of assets.

Variables:

$\sigma_p^2$ :: Variance of the portfolio
$w_i, w_j$ :: Weights of assets i and j in the portfolio
$\sigma_i, \sigma_j$ :: Standard deviations (risks) of assets i and j
$\rho_{ij}$ :: Correlation coefficient between assets i and j
$N$ :: Total number of assets in the portfolio

Units: percentage squared

Capital Market Theory: The Capital Market Line

Learning Outcome Statement:

explain the capital allocation line (CAL) and the capital market line (CML)

Summary:

The Capital Market Line (CML) represents a special case of the Capital Allocation Line (CAL) where the risky portfolio is the market portfolio. The CML combines a risk-free asset and the market portfolio to depict the risk-return trade-off in a graphical format. It is a straight line starting from the risk-free rate on the y-axis to the expected return of the market portfolio, showing the relationship between the expected return and the total risk (standard deviation) of the portfolio.

Key Concepts:

Passive and Active Portfolios

Passive portfolios rely on market prices and typically track market indexes with low management costs, while active portfolios involve selecting securities based on individual analysis to outperform the market, often resulting in higher costs and potential for greater returns.

Market Definition

The 'market' in capital market theory typically refers to a broad collection of investable assets. Practically, it is often represented by a major stock market index like the S&P 500, which serves as a proxy for the overall market.

Capital Market Line (CML)

The CML is a line that shows all possible combinations of the market portfolio with a risk-free asset. It represents the optimal trade-off between risk and return, assuming all investors can lend and borrow at the risk-free rate.

Formulas:

Expected Portfolio Return

E(R_p) = w_1 R_f + (1 - w_1) E(R_m)

This formula calculates the expected return of a portfolio that combines a risk-free asset and a risky market portfolio based on their respective weights.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$w_1$ :: Proportion of the total portfolio invested in the risk-free asset
$R_f$ :: Risk-free rate of return
$E(R_m)$ :: Expected return of the market portfolio

Units: percentage

Portfolio Standard Deviation

\sigma_p = (1 - w_1) \sigma_m

This formula calculates the standard deviation of a portfolio that combines a risk-free asset and a risky market portfolio, assuming the risk-free asset has zero standard deviation.

Variables:

$\sigma_p$ :: Standard deviation of the portfolio
$w_1$ :: Proportion of the total portfolio invested in the risk-free asset
$\sigma_m$ :: Standard deviation of the market portfolio

Units: percentage

CML Equation

E(R_p) = R_f + \left(\frac{E(R_m) - R_f}{\sigma_m}\right) \times \sigma_p

This linear equation represents the Capital Market Line, showing how the expected return on the portfolio increases with its risk. The slope of the CML is the market price of risk.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_f$ :: Risk-free rate of return
$E(R_m)$ :: Expected return of the market portfolio
$\sigma_m$ :: Standard deviation of the market portfolio
$\sigma_p$ :: Standard deviation of the portfolio

Units: percentage

Capital Market Theory: CML - Leveraged Portfolios

Learning Outcome Statement:

explain the capital allocation line (CAL) and the capital market line (CML)

Summary:

The content discusses the concept of leveraged portfolios within the framework of the Capital Market Line (CML), focusing on scenarios where investors either lend or borrow at the risk-free rate or at different rates. It explains how leveraging affects the risk and return of a portfolio, and how the CML adjusts when borrowing rates differ from lending rates.

Key Concepts:

Capital Allocation Line (CAL) and Capital Market Line (CML)

CAL represents combinations of risk-free assets and a risky market portfolio. CML is a special case of CAL where the risky portfolio is the market portfolio. It shows the trade-off between expected return and the standard deviation of returns (risk) for efficient portfolios.

Leveraged Portfolios

These are portfolios where an investor borrows at the risk-free rate to invest more than their own capital into a market portfolio, aiming to increase potential returns at the cost of higher risk.

Different Lending and Borrowing Rates

When the borrowing rate is higher than the lending rate, the CML becomes kinked. The slope of the CML changes at the market portfolio point, reflecting the higher cost of borrowing compared to lending.

Formulas:

Expected Return of Leveraged Portfolio

E(R_p) = w_1 R_f + (1 - w_1) E(R_m)

This formula calculates the expected return of a leveraged portfolio, where part of the investment is financed through borrowing at the risk-free rate.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$w_1$ :: Proportion of total funds invested at the risk-free rate (negative if borrowing)
$R_f$ :: Risk-free rate
$E(R_m)$ :: Expected return of the market portfolio

Units: percentage

Standard Deviation of Leveraged Portfolio

\sigma_p = (1 - w_1) \sigma_m

This formula calculates the standard deviation (risk) of a leveraged portfolio, which increases as more is borrowed due to higher investment in the market portfolio.

Variables:

$\sigma_p$ :: Standard deviation of the portfolio
$w_1$ :: Proportion of total funds invested at the risk-free rate (negative if borrowing)
$\sigma_m$ :: Standard deviation of the market portfolio

Units: percentage

CML Equation with Different Rates

E(R_p) = R_b + \left(\frac{E(R_m) - R_b}{\sigma_m}\right) \times \sigma_p

This equation is used when the borrowing rate is different from the lending rate, affecting the slope of the CML for leveraged portfolios.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_b$ :: Borrowing rate
$E(R_m)$ :: Expected return of the market portfolio
$\sigma_m$ :: Standard deviation of the market portfolio
$\sigma_p$ :: Standard deviation of the portfolio

Units: percentage

Systematic and Nonsystematic Risk

Learning Outcome Statement:

explain systematic and nonsystematic risk, including why an investor should not expect to receive additional return for bearing nonsystematic risk

Summary:

Systematic risk, also known as market risk, affects the entire market and cannot be diversified away. Nonsystematic risk, or diversifiable risk, pertains to specific companies or industries and can be mitigated through diversification. The total risk of a security or portfolio is the sum of systematic and nonsystematic risk. Investors are compensated for bearing systematic risk but should not expect additional return for nonsystematic risk, which can be diversified away.

Key Concepts:

Systematic Risk

Systematic risk affects the entire market or economy and includes factors like interest rates, inflation, and political uncertainty. It is inherent and non-diversifiable.

Nonsystematic Risk

Nonsystematic risk is specific to a company or industry, such as the failure of a drug trial. It does not affect the entire market and can be diversified away by holding a portfolio of uncorrelated assets.

Diversification

Diversification involves creating a portfolio of assets that are not highly correlated with each other, which helps in reducing nonsystematic risk.

Pricing of Risk

In an efficient market, investors are compensated for taking on systematic risk but not for nonsystematic risk, which can be eliminated through diversification.

Formulas:

Total Variance

Total\ variance = Systematic\ variance + Nonsystematic\ variance

This formula shows how the total risk (variance) of a security or portfolio is composed of both systematic and nonsystematic components.

Variables:

$Total variance$ :: The total risk associated with a security or portfolio
$Systematic variance$ :: The part of the total variance that is due to market-wide risks
$Nonsystematic variance$ :: The part of the total variance that is due to specific risks within a company or industry

Units: Variance units (typically percentage squared)

Return Generating Models

Learning Outcome Statement:

explain return generating models (including the market model) and their uses, calculate and interpret beta

Summary:

Return generating models are tools used to estimate the expected returns of securities based on various risk factors. These models include single-index models, multi-factor models, and the market model. They help in understanding the relationship between systematic risk and expected returns, and are essential for constructing efficient portfolios. The models also allow for the decomposition of total risk into systematic and nonsystematic components, providing insights into how different factors contribute to the overall risk and return of securities.

Key Concepts:

Return-Generating Models

Models that estimate the expected return of a security based on certain parameters, primarily systematic risk. These models can be single-factor or multi-factor, incorporating various economic, fundamental, or statistical factors.

Single-Index Model

A simple form of a return-generating model that considers only the market factor. It is expressed as E(Ri) - Rf = βi[E(Rm) - Rf], linking the expected excess return of a security to the market's excess return via the beta coefficient.

Market Model

A common implementation of the single-index model where the market return is the sole factor. It is used to estimate beta risk and compute abnormal returns, expressed as Ri = αi + βiRm + ei.

Decomposition of Total Risk

In the context of the single-index model, total risk (variance) of a security can be decomposed into systematic variance and nonsystematic variance, helping to understand the contributions of market-related and individual asset-related risks.

Beta Calculation and Interpretation

Beta measures the sensitivity of an asset's returns relative to the market returns and is calculated using covariance and variance. It is a key parameter in return-generating models, indicating the level of systematic risk associated with a security.

Formulas:

Single-Index Model

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

This formula links the expected excess return of a security to the market's excess return, scaled by the beta coefficient.

Variables:

$E(R_i)$ :: Expected return of security i
$R_f$ :: Risk-free rate
$E(R_m)$ :: Expected return of the market
$β_i$ :: Beta of security i

Units: Returns are unitless (expressed as a proportion)

Market Model

R_i = \alpha_i + \beta_i R_m + e_i

This model is used to estimate the returns of a security based on the market return and specific security characteristics (alpha and beta).

Variables:

$R_i$ :: Return of security i
$α_i$ :: Alpha of security i
$β_i$ :: Beta of security i
$R_m$ :: Return of the market
$e_i$ :: Error term (residual return of security i)

Units: Returns are unitless (expressed as a proportion)

Beta Calculation

\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \rho_{i,m} \frac{\sigma_i}{\sigma_m}

Beta is calculated as the ratio of covariance between the security's returns and the market's returns to the variance of the market's returns. It can also be expressed through correlation and standard deviations.

Variables:

$β_i$ :: Beta of security i
$Cov(R_i, R_m)$ :: Covariance between returns of security i and the market
$σ_m^2$ :: Variance of the market returns
$ρ_{i,m}$ :: Correlation between security i and the market
$σ_i$ :: Standard deviation of returns for security i
$σ_m$ :: Standard deviation of market returns

Units: Beta is unitless

Calculation and Interpretation of Beta

Learning Outcome Statement:

calculate and interpret beta

Summary:

This LOS focuses on calculating and interpreting the beta coefficient, which measures an asset's sensitivity to market movements and its systematic risk. Beta is derived using the covariance between the asset's returns and the market's returns divided by the variance of the market's returns. The LOS also explores how beta influences expected returns through the Capital Asset Pricing Model (CAPM), which relates expected return to beta.

Key Concepts:

Beta Estimation

Beta (β) is estimated using regression analysis on historical security and market returns. It represents the slope of the security characteristic line in the market model, indicating how much the security's return changes for a change in the market return.

Systematic Risk

Systematic risk, captured by beta, is the portion of an asset's risk that is attributable to market-wide risks and cannot be diversified away. It is reflected in the asset's sensitivity to market movements.

Beta and Expected Return

Using the CAPM, the expected return of an asset is calculated as the risk-free rate plus the product of the asset's beta and the market risk premium (difference between expected market return and risk-free rate). A higher beta implies a higher expected return, reflecting higher systematic risk.

Formulas:

Beta Calculation

\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2} = \frac{\rho_{i,m} \sigma_i \sigma_m}{\sigma_m^2} = \frac{\rho_{i,m} \sigma_i}{\sigma_m}

This formula calculates the beta of an asset, which measures the asset's volatility relative to the market. A higher beta indicates greater volatility and systematic risk.

Variables:

$\beta_i$ :: Beta of asset i
$\text{Cov}(R_i, R_m)$ :: Covariance between the return of asset i and the market return
$\sigma_m^2$ :: Variance of the market return
$\rho_{i,m}$ :: Correlation between the return of asset i and the market return
$\sigma_i$ :: Standard deviation of the return of asset i
$\sigma_m$ :: Standard deviation of the market return

Units: dimensionless

Expected Return using CAPM

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

This formula from the CAPM shows how the expected return of an asset is influenced by its beta. It quantifies the compensation investors require for taking on additional systematic risk.

Variables:

$E(R_i)$ :: Expected return of asset i
$R_f$ :: Risk-free rate
$\beta_i$ :: Beta of asset i
$E(R_m)$ :: Expected return of the market
$[E(R_m) - R_f]$ :: Market risk premium

Units: percentage or decimal

Capital Asset Pricing Model: Assumptions and the Security Market Line

Learning Outcome Statement:

explain the capital asset pricing model (CAPM), including its assumptions, and the security market line (SML); calculate and interpret the expected return of an asset using the CAPM

Summary:

The Capital Asset Pricing Model (CAPM) is a foundational concept in finance that links risk and expected return of an asset, based on its systematic risk as measured by beta. The model simplifies the complex financial markets by making certain assumptions, such as investors being risk-averse, rational, and markets being frictionless. The Security Market Line (SML) graphically represents the CAPM, showing the relationship between beta and expected return for any security.

Key Concepts:

CAPM Formula

The CAPM formula calculates the expected return of an asset based on its systematic risk (beta), the risk-free rate, and the expected market return. It highlights that the expected return on an asset is linearly related to its beta.

Security Market Line (SML)

The SML is a graphical representation of the CAPM, plotting risk (beta) against expected return. It shows how expected return varies with an asset's systematic risk.

Assumptions of CAPM

CAPM assumes markets are frictionless, investors are rational and risk-averse, and all investments are infinitely divisible among other simplifications. These assumptions help in understanding the basic risk-return relationship though they may not hold perfectly in reality.

Portfolio Beta

Portfolio beta is a weighted average of the betas of the individual securities in the portfolio. It represents the portfolio's systematic risk.

Formulas:

CAPM Expected Return

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

This formula calculates the expected return on an asset, accounting for its systematic risk relative to the overall market.

Variables:

$E(R_i)$ :: Expected return on asset i
$R_f$ :: Risk-free rate
$\beta_i$ :: Beta of asset i
$E(R_m)$ :: Expected return of the market
$(E(R_m) - R_f)$ :: Market risk premium

Units: Percentage or Decimal

Portfolio Beta

\beta_p = \sum_{i=1}^n w_i \beta_i

This formula calculates the overall beta of a portfolio, which is a weighted average of the betas of the individual assets in the portfolio.

Variables:

$\beta_p$ :: Portfolio beta
$w_i$ :: Weight of asset i in the portfolio
$\beta_i$ :: Beta of asset i
$n$ :: Total number of assets in the portfolio

Units: Unitless

Capital Asset Pricing Model: Applications

Learning Outcome Statement:

calculate and interpret the expected return of an asset using the CAPM

Summary:

The Capital Asset Pricing Model (CAPM) is used to estimate the expected return of an asset based on its systematic risk, represented by beta (β). The model considers the risk-free rate, the expected market return, and the asset's beta to determine the expected return. This is crucial for various financial decisions including capital budgeting, performance appraisal, and security selection.

Key Concepts:

Portfolio Beta

Portfolio beta is a weighted average of the betas of the individual assets in the portfolio. It represents the portfolio's systematic risk relative to the market.

Expected Return using CAPM

The expected return on an asset as per CAPM is calculated by adding the risk-free rate to the product of the asset's beta and the market risk premium (difference between the expected market return and the risk-free rate).

Net Present Value (NPV) using CAPM

NPV in the context of CAPM is calculated by discounting future cash flows of a project at the required rate of return derived from CAPM, which reflects the project's systematic risk.

Formulas:

Portfolio Beta

\beta_p = w_1\beta_1 + w_2\beta_2 + w_3\beta_3

This formula calculates the overall beta of a portfolio, which is a measure of its systematic risk.

Variables:

$\beta_p$ :: Portfolio beta
$w_1, w_2, w_3$ :: Weights of the assets in the portfolio
$\beta_1, \beta_2, \beta_3$ :: Betas of the individual assets

Units: dimensionless

Expected Return using CAPM

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

This formula is used to estimate the expected return on an asset based on its beta and the expected market return.

Variables:

$E(R_i)$ :: Expected return of the asset
$R_f$ :: Risk-free rate
$\beta_i$ :: Beta of the asset
$E(R_m)$ :: Expected market return

Units: percentage or decimal

Net Present Value using CAPM

NPV = \sum_{t=0}^{T} \frac{CF_t}{(1 + r)^t}

This formula calculates the present value of future cash flows, discounted at the CAPM-derived required rate of return, to assess the viability of a project.

Variables:

$NPV$ :: Net Present Value
$CF_t$ :: Cash flow at time t
$r$ :: Discount rate derived from CAPM
$t$ :: Time period
$T$ :: Total number of periods

Units: currency (e.g., USD)

Beyond CAPM: Limitations and Extensions of CAPM

Learning Outcome Statement:

describe and demonstrate applications of the CAPM and the SML

Summary:

This section discusses the limitations and extensions of the Capital Asset Pricing Model (CAPM). It highlights the theoretical and practical limitations of CAPM, such as its single-factor and single-period nature, and the challenges in estimating beta and the market portfolio. Extensions to CAPM, like the Arbitrage Pricing Theory (APT) and the Fama-French four-factor model, are introduced as alternatives that address some of these limitations by incorporating multiple risk factors and providing better asset return predictions.

Key Concepts:

Theoretical Limitations of CAPM

CAPM is criticized for being a single-factor model that only considers systematic risk and for being a single-period model that does not account for multi-period investment strategies.

Practical Limitations of CAPM

Practical challenges include the difficulty in defining and observing the true market portfolio, variations in beta estimates based on different time periods and data frequencies, and the model's poor predictive power of actual returns.

Arbitrage Pricing Theory (APT)

APT extends CAPM by allowing for multiple risk factors. It suggests a linear relationship between expected return and multiple betas corresponding to different risk factors, making it more flexible than CAPM.

Fama-French Four-Factor Model

This model includes market risk, size, value, and momentum factors to explain asset returns. It has been shown to predict returns better than CAPM, especially for U.S. stocks.

Formulas:

CAPM Formula

E(R_i) = R_f + \beta_i (E(R_m) - R_f)

This formula calculates the expected return on an asset based on its beta and the excess return of the market over the risk-free rate.

Variables:

$E(R_i)$ :: Expected return on asset i
$R_f$ :: Risk-free rate
$\beta_i$ :: Beta of asset i
$E(R_m)$ :: Expected return of the market

Units: percentage or decimal

APT Formula

E(R_p) = RF + \lambda_1\beta_{p,1} + ... + \lambda_K\beta_{p,K}

APT formula calculates the expected return of a portfolio by considering multiple risk factors and their respective sensitivities.

Variables:

$E(R_p)$ :: Expected return of portfolio p
$RF$ :: Risk-free rate
$\lambda_j$ :: Risk premium for factor j
$\beta_{p,j}$ :: Sensitivity of the portfolio to factor j
$K$ :: Number of risk factors

Units: percentage or decimal

Fama-French Four-Factor Model

E(R_{it}) = \alpha_i + \beta_{i,MKT} MKT_t + \beta_{i,SMB} SMB_t + \beta_{i,HML} HML_t + \beta_{i,UMD} UMD_t

This model extends the CAPM by including additional factors that affect asset returns, such as size, value, and momentum, in addition to market risk.

Variables:

$E(R_{it})$ :: Expected return on asset i at time t
$\alpha_i$ :: Intercept term for asset i
$\beta_{i,MKT}$ :: Beta of asset i with respect to market risk
$MKT_t$ :: Market risk factor at time t
$\beta_{i,SMB}$ :: Beta of asset i with respect to size risk
$SMB_t$ :: Size risk factor at time t
$\beta_{i,HML}$ :: Beta of asset i with respect to value risk
$HML_t$ :: Value risk factor at time t
$\beta_{i,UMD}$ :: Beta of asset i with respect to momentum risk
$UMD_t$ :: Momentum risk factor at time t

Units: percentage or decimal

Portfolio Performance Appraisal Measures

Learning Outcome Statement:

calculate and interpret the Sharpe ratio, Treynor ratio, M2, and Jensen’s alpha

Summary:

This LOS focuses on the calculation and interpretation of four key performance appraisal measures used in portfolio management: the Sharpe Ratio, Treynor Ratio, M2 (Risk-Adjusted Performance), and Jensen's Alpha. These measures help in evaluating the risk-adjusted returns of investment portfolios, providing insights into the efficiency and effectiveness of investment decisions relative to market performance and risk.

Key Concepts:

Sharpe Ratio

The Sharpe Ratio measures the excess return per unit of risk (standard deviation) of a portfolio. It is used to understand how well the return of an asset compensates the investor for the risk taken.

Treynor Ratio

The Treynor Ratio is similar to the Sharpe Ratio but uses beta (systematic risk) instead of total risk. It measures the excess return per unit of systematic risk, providing a tool for comparing portfolios with different risk profiles.

M2 (Risk-Adjusted Performance)

M2, developed by Franco Modigliani, adjusts the portfolio return for the total risk relative to a benchmark, providing a percentage return advantage assuming the same level of total risk as the market.

Jensen's Alpha

Jensen's Alpha measures the difference between the actual portfolio returns and the expected returns based on the CAPM, indicating the portfolio's performance relative to the market after adjusting for systematic risk.

Formulas:

Sharpe Ratio

\text{SR} = \frac{E(R_p) - R_f}{\sigma_p}

Calculates the excess return per unit of total risk.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_f$ :: Risk-free rate
$\sigma_p$ :: Standard deviation of the portfolio's returns

Units: dimensionless

Treynor Ratio

\text{TR} = \frac{E(R_p) - R_f}{\beta_p}

Measures the excess return per unit of systematic risk.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_f$ :: Risk-free rate
$\beta_p$ :: Beta of the portfolio

Units: dimensionless

M2

M2 = \left(\frac{E(R_p) - R_f}{\sigma_p}\right) \sigma_m + R_f

Adjusts the Sharpe ratio to a scale that allows for easier comparison among different portfolios.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_f$ :: Risk-free rate
$\sigma_p$ :: Standard deviation of the portfolio's returns
$\sigma_m$ :: Standard deviation of the market's returns

Units: percentage

Jensen's Alpha

\alpha_p = R_p - \{R_f + \beta_p[E(R_m) - R_f]\}

Calculates the excess return of the portfolio over the expected return based on its beta.

Variables:

$R_p$ :: Actual return of the portfolio
$R_f$ :: Risk-free rate
$\beta_p$ :: Beta of the portfolio
$E(R_m)$ :: Expected return of the market

Units: percentage

Applications of the CAPM in Portfolio Construction

Learning Outcome Statement:

calculate and interpret the expected return of an asset using the CAPM

Summary:

This LOS explores the applications of the Capital Asset Pricing Model (CAPM) in portfolio construction, focusing on the security characteristic line, security selection, and implications for portfolio construction. It discusses how CAPM can be used to evaluate and select securities based on their expected returns relative to their risk (beta), and how it aids in constructing portfolios that maximize risk-adjusted returns.

Key Concepts:

Security Characteristic Line (SCL)

The SCL is a graphical representation of a security's excess return over the risk-free rate against the excess return of the market. It helps in visualizing the security's performance relative to the market, with Jensen's alpha as the intercept and beta as the slope.

Security Selection

Security selection using CAPM involves comparing the expected return of securities to their required return based on CAPM. Securities with a positive Jensen's alpha are considered undervalued and suitable for inclusion in a portfolio, while those with a negative alpha are overvalued.

Implications for Portfolio Construction

CAPM suggests that investors should hold a mix of the market portfolio and a risk-free asset. However, practical portfolio construction might involve selecting a subset of securities that diversify risk effectively, often using an index as a proxy for the market portfolio.

Formulas:

Security Characteristic Line Equation

R_i - R_f = \alpha_i + \beta_i(R_m - R_f)

This formula represents the linear relationship between the excess return of a security and the excess return of the market, indicating how much of the security's performance can be explained by the market's performance.

Variables:

$R_i$ :: return of security i
$R_f$ :: risk-free rate
$R_m$ :: return of the market
$\alpha_i$ :: Jensen's alpha of security i
$\beta_i$ :: beta of security i

Units: percentage

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