Portfolio Risk and Return: Part I

Portfolio Management

Utility Theory and Indifference Curves

Learning Outcome Statement:

explain risk aversion and its implications for portfolio selection

Summary:

Utility theory and indifference curves are used to analyze investor behavior under risk. Risk aversion, a key concept, influences how investors choose between guaranteed outcomes and gambles. Utility functions quantify preferences, incorporating risk aversion coefficients to adjust for varying levels of risk tolerance. Indifference curves graphically represent combinations of risk and return that provide the same utility, helping investors visualize their preferences and make informed decisions about portfolio selection.

Key Concepts:

Risk Aversion

Risk aversion describes an investor's preference for a certain outcome over a risky one, even if the risky option has a higher expected return. This behavior is typical when the potential loss from the risky investment is significant relative to the investor's wealth.

Utility Function

A utility function quantifies an investor's satisfaction from different investment outcomes, factoring in both the expected return and the risk (variance) of investments. The function includes a risk aversion coefficient that scales the impact of risk on the investor's utility.

Indifference Curves

Indifference curves represent combinations of risk and return that yield the same level of utility for an investor. These curves help in understanding the trade-offs an investor is willing to make between higher risk for potentially higher returns and lower risk for lower returns.

Formulas:

Utility Function

U = E(r) - \frac{1}{2} A \sigma^2

This formula calculates the utility of an investment based on its expected return and the risk (variance) associated with it. The risk aversion coefficient (A) adjusts the impact of risk on utility, with higher values reducing utility more for a given level of risk.

Variables:

$U$ :: Utility of the investment
$E(r)$ :: Expected return of the investment
$\sigma^2$ :: Variance of the investment's returns
$A$ :: Risk aversion coefficient, higher values indicate greater risk aversion

Units: Utility is dimensionless, E(r) is typically in percentage, \sigma^2 is in percentage squared

Portfolio Expected Return

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

This formula calculates the expected return of a portfolio that includes a mix of a risk-free asset and a risky asset, based on their respective weights and returns.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$w_1$ :: Weight of the risk-free asset in the portfolio
$R_f$ :: Return of the risk-free asset
$E(R_i)$ :: Expected return of the risky asset

Units: Expected returns are typically in percentage

Portfolio Variance

\sigma_p^2 = (1 - w_1)^2 \sigma_i^2

This formula calculates the variance of a portfolio's returns when it includes a risk-free asset and a risky asset, considering only the variance contribution from the risky asset due to the zero variance of the risk-free asset.

Variables:

$\sigma_p^2$ :: Variance of the portfolio's returns
$w_1$ :: Weight of the risk-free asset in the portfolio
$\sigma_i^2$ :: Variance of the risky asset's returns

Units: Variance is in percentage squared

Capital Allocation Line

E(R_p) = R_f + \frac{(E(R_i) - R_f)}{\sigma_i} \sigma_p

This equation represents the capital allocation line, showing the relationship between the expected return of a portfolio and its risk, illustrating how additional risk must be compensated by additional return.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_f$ :: Return of the risk-free asset
$E(R_i)$ :: Expected return of the risky asset
$\sigma_i$ :: Standard deviation of the risky asset
$\sigma_p$ :: Standard deviation of the portfolio

Units: Expected returns and standard deviations are typically in percentage

Historical Return and Risk

Learning Outcome Statement:

describe characteristics of the major asset classes that investors consider in forming portfolios

Summary:

This LOS explores the historical nominal and real returns, and the associated risks of major asset classes in the US and globally. It distinguishes between historical mean returns and expected returns, and discusses the risk-return trade-off, emphasizing that higher returns generally come with higher risks. The content also covers the calculation of expected returns based on the real risk-free rate, expected inflation, and expected risk premium.

Key Concepts:

Historical vs. Expected Returns

Historical returns are actual returns earned in the past, while expected returns are what investors anticipate earning in the future based on various factors including the real risk-free rate, expected inflation, and risk premiums.

Nominal and Real Returns

Nominal returns do not account for inflation, while real returns are adjusted for inflation, providing a more accurate measure of the purchasing power gained or lost through investment.

Risk of Asset Classes

Different asset classes exhibit varying levels of risk, quantified by metrics such as standard deviation. Historically, equities have shown higher risk and higher returns compared to bonds and T-bills.

Risk-Return Trade-off

This concept describes the relationship between risk and expected return, where higher risk is associated with higher potential returns. This relationship is fundamental in financial theory and investment practice.

Formulas:

Expected Return

1 + E(R) = (1 + r_{\text{rF}}) \times [1 + E(\pi)] \times [1 + E(\text{RP})]

This formula calculates the expected return on an asset, taking into account the real risk-free rate, expected inflation, and the risk premium associated with the asset's risk.

Variables:

$E(R)$ :: Expected return
$r_{\text{rF}}$ :: Real risk-free interest rate
$E(\pi)$ :: Expected inflation rate
$E(\text{RP})$ :: Expected risk premium

Units: percentage

Application of Utility Theory to Portfolio Selection

Learning Outcome Statement:

explain risk aversion and its implications for portfolio selection; explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line

Summary:

The application of utility theory to portfolio selection involves understanding risk aversion and its implications for selecting an optimal portfolio. This includes the construction of portfolios using a combination of risk-free and risky assets, determining the portfolio's expected return and risk, and utilizing the capital allocation line to identify the optimal portfolio based on an investor's utility or risk aversion.

Key Concepts:

Risk Aversion

Risk aversion describes an investor's preference for lower risk and is a fundamental concept in utility theory. It implies that an investor requires higher returns to compensate for higher risk.

Capital Allocation Line (CAL)

The CAL represents all possible combinations of risk-free and risky assets. It is used to determine the optimal portfolio by maximizing returns for a given level of risk.

Optimal Portfolio

The optimal portfolio is the point on the CAL that maximizes an investor's utility, which is determined by their risk aversion. It is found where the investor's highest utility curve is tangent to the CAL.

Indifference Curves

These curves represent combinations of portfolio risk and return that provide the same level of utility to the investor. The optimal portfolio is found at the tangency point between an indifference curve and the CAL.

Formulas:

Expected Portfolio Return

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

This formula calculates the expected return of a portfolio consisting of a risk-free asset and a risky asset, based on their respective weights and returns.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$w_1$ :: Weight of the risk-free asset in the portfolio
$R_f$ :: Return of the risk-free asset
$E(R_i)$ :: Expected return of the risky asset

Units: percentage or decimal

Portfolio Variance

\sigma_p^2 = (1 - w_1)^2 \sigma_i^2

This formula calculates the variance of a portfolio that includes a risk-free asset and a risky asset, considering that the risk-free asset has zero variance.

Variables:

$\sigma_p^2$ :: Variance of the portfolio
$w_1$ :: Weight of the risk-free asset in the portfolio
$\sigma_i^2$ :: Variance of the risky asset

Units: squared percentage or decimal

Portfolio Standard Deviation

\sigma_p = (1 - w_1) \sigma_i

This formula calculates the standard deviation of a portfolio consisting of a risk-free and a risky asset, reflecting the portfolio's total risk.

Variables:

$\sigma_p$ :: Standard deviation of the portfolio
$w_1$ :: Weight of the risk-free asset in the portfolio
$\sigma_i$ :: Standard deviation of the risky asset

Units: percentage or decimal

Capital Allocation Line Equation

E(R_p) = R_f + \frac{(E(R_i) - R_f)}{\sigma_i} \sigma_p

This equation represents the CAL, showing how the expected return of the portfolio increases linearly with its standard deviation, based on the market price of risk.

Variables:

$E(R_p)$ :: Expected return of the portfolio
$R_f$ :: Return of the risk-free asset
$E(R_i)$ :: Expected return of the risky asset
$\sigma_i$ :: Standard deviation of the risky asset
$\sigma_p$ :: Standard deviation of the portfolio

Units: percentage or decimal

Portfolio Risk & Portfolio of Two Risky Assets

Learning Outcome Statement:

calculate and interpret the mean, variance, and covariance (or correlation) of asset returns based on historical data; calculate and interpret portfolio standard deviation; describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated

Summary:

This LOS explores the computation and implications of portfolio risk when combining two or more risky assets. It emphasizes the importance of correlation between asset returns in determining the overall risk of the portfolio. The content covers how to calculate portfolio return as a weighted average of individual asset returns, and how to compute portfolio variance and standard deviation using the covariance or correlation between assets. It also discusses the impact of diversification on portfolio risk, particularly when assets are not perfectly correlated.

Key Concepts:

Portfolio Return

Portfolio return is calculated as a weighted average of the returns of the individual assets in the portfolio. The weight of each asset reflects its proportion of the total investment.

Portfolio Variance and Standard Deviation

Portfolio variance is not simply the weighted average of individual variances due to the covariance between asset returns. It includes terms for the variance of each asset and the covariance between each pair of assets. Portfolio standard deviation is the square root of the portfolio variance.

Correlation and Diversification

Correlation measures the degree to which two assets move in relation to each other. Diversification benefits are maximized when assets are less than perfectly correlated, potentially reducing portfolio risk below that of individual assets.

Formulas:

Portfolio Return

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

This formula calculates the expected return of a portfolio by summing the products of the returns of individual assets and their respective weights in the portfolio.

Variables:

$R_p$ :: return of the portfolio
$w_i$ :: weight of asset i in the portfolio
$R_i$ :: return of asset i

Units: percentage or decimal

Portfolio Variance

\sigma_P^2 = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i w_j \text{Cov}(R_i, R_j)

This formula calculates the variance of a portfolio, which includes the weighted covariances between all pairs of assets in the portfolio.

Variables:

$\sigma_P^2$ :: variance of the portfolio
$w_i, w_j$ :: weights of assets i and j in the portfolio
$\text{Cov}(R_i, R_j)$ :: covariance between returns of assets i and j

Units: percentage squared or decimal squared

Portfolio Standard Deviation

\sigma_P = \sqrt{\sigma_P^2}

This formula calculates the standard deviation of a portfolio, which is the square root of the portfolio variance, providing a measure of total risk in the portfolio.

Variables:

$\sigma_P$ :: standard deviation of the portfolio
$\sigma_P^2$ :: variance of the portfolio

Units: percentage or decimal

Efficient Frontier: Optimal Investor Portfolio

Learning Outcome Statement:

Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line

Summary:

This LOS explores how investors select an optimal portfolio based on their risk preferences and utility functions, in relation to the capital allocation line (CAL). It discusses the impact of risk aversion on portfolio choice, the role of the capital allocation line in connecting the risk-free asset with the optimal risky portfolio, and how the utility maximization framework helps in identifying the optimal portfolio for different types of investors.

Key Concepts:

Capital Allocation Line (CAL)

The CAL represents combinations of portfolios that maximize expected return for a given level of risk. It is derived from the combination of a risk-free asset and an optimal risky portfolio.

Risk Aversion and Utility

Investors' choices are influenced by their risk aversion levels, which dictate their utility from different portfolios. Utility functions typically combine expected return and risk, with higher risk aversion leading to portfolios closer to the risk-free asset.

Optimal Risky Portfolio

This is the portfolio that offers the highest expected return per unit of risk and lies on the efficient frontier. It is combined with the risk-free asset to form the CAL.

Indifference Curves

These curves represent combinations of portfolios that provide the same level of utility to a particular investor. The optimal portfolio for an investor is where their highest possible indifference curve is tangent to the CAL.

Formulas:

Portfolio Expected Return

R_{rp} = w_A \times R_A + (1 - w_A) \times R_B

This formula calculates the expected return of a portfolio based on the returns of the assets it contains and their respective weights.

Variables:

$R_{rp}$ :: Expected return of the portfolio
$w_A$ :: Weight of asset A in the portfolio
$R_A$ :: Expected return of asset A
$R_B$ :: Expected return of asset B

Units: percentage

Portfolio Risk (Standard Deviation)

\sigma_{rp} = \sqrt{w_A^2 \sigma_A^2 + (1-w_A)^2 \sigma_B^2 + 2w_A(1-w_A)\rho_{AB}\sigma_A\sigma_B}

This formula calculates the risk (standard deviation) of a portfolio that includes two assets, considering their weights, individual risks, and correlation.

Variables:

$\sigma_{rp}$ :: Standard deviation of the portfolio
$w_A$ :: Weight of asset A in the portfolio
$\sigma_A$ :: Standard deviation of asset A
$\sigma_B$ :: Standard deviation of asset B
$\rho_{AB}$ :: Correlation coefficient between returns of asset A and B

Units: percentage

Utility of an Investor

U = E(R_p) - 0.5 \times A \times \sigma_p^2

This formula represents the utility derived by an investor from a portfolio, considering both the expected return and the risk of the portfolio adjusted by the investor's risk aversion.

Variables:

$U$ :: Utility of the investor
$E(R_p)$ :: Expected return of the portfolio
$A$ :: Risk aversion coefficient of the investor
$\sigma_p^2$ :: Variance of the portfolio

Units: utility units

Risk Aversion and Portfolio Selection

Learning Outcome Statement:

explain risk aversion and its implications for portfolio selection

Summary:

This LOS explores the concept of risk aversion and how it influences investor behavior and portfolio selection. It discusses different investor attitudes towards risk, including risk-seeking, risk-neutral, and risk-averse behaviors, and how these attitudes affect investment choices. The section also introduces utility theory and indifference curves to further explain how investors derive satisfaction from different investment outcomes based on their risk preferences.

Key Concepts:

Risk Aversion

Risk aversion describes an investor's preference for a certain outcome over a risky one, even if the risky option has a higher or equal expected return. Risk-averse investors prioritize security over potential higher returns.

Risk Seeking

Risk-seeking investors prefer taking risks for the potential of higher returns. They derive additional utility from the uncertainty and are willing to accept a lower expected return for the thrill or other benefits of risk-taking.

Risk Neutral

Risk-neutral investors are indifferent between risky and certain outcomes as long as the expected returns are the same. They focus solely on maximizing returns without regard to the level of risk involved.

Utility Theory

Utility theory in finance suggests that investors make decisions based on the utility or satisfaction derived from different investment outcomes. It helps explain why different investors may prefer different investments even if the expected returns are similar.

Indifference Curves

Indifference curves represent combinations of risk and return that provide the same level of utility to an investor. These curves help illustrate an investor's risk tolerance and preference for different investment portfolios.

Efficient Frontier: A Risk-Free Asset and Many Risky Assets

Learning Outcome Statement:

Explain the selection of an optimal portfolio, given an investor’s utility (or risk aversion) and the capital allocation line

Summary:

The learning outcome focuses on understanding how to select an optimal portfolio by incorporating a risk-free asset with many risky assets, using the capital allocation line (CAL) and considering an investor's utility or risk aversion. It involves analyzing the efficient frontier, the role of the risk-free asset in portfolio optimization, and the implications of the two-fund separation theorem.

Key Concepts:

Capital Allocation Line (CAL)

The CAL represents a straight line in a mean-variance graph that starts from the risk-free rate (Rf) and is tangent to the efficient frontier of risky assets. It shows the combination of risk-free assets and an optimal portfolio of risky assets, providing the best possible risk-return combinations available to investors.

Optimal Risky Portfolio

This is the portfolio on the efficient frontier that is tangent to the CAL. It represents the best combination of risky assets that maximizes return for a given level of risk when combined with a risk-free asset.

Two-Fund Separation Theorem

This theorem states that all investors, regardless of their risk preferences and initial wealth, will hold a combination of two funds: a risk-free asset and an optimal portfolio of risky assets. This simplifies the investment decision into selecting proportions to invest in these two funds.

Indifference Curves

These curves represent the combinations of risk and return that provide the same level of utility to an investor. The optimal portfolio for an investor is found where their highest attainable indifference curve is tangent to the CAL.

Formulas:

Utility Function

U = E(r) - 0.5 \times A \times \sigma^2

This formula calculates the utility of an investor based on the expected return of the portfolio, the investor's risk aversion, and the risk (variance) of the portfolio. It helps in determining the most suitable portfolio for an investor on the CAL.

Variables:

$U$ :: Utility of the investor
$E(r)$ :: Expected return of the portfolio
$A$ :: Risk aversion coefficient of the investor
$\sigma^2$ :: Variance of the portfolio's return

Units: Utility is dimensionless, return is in percentage, risk aversion is dimensionless, variance is in percentage squared

The Power of Diversification

Learning Outcome Statement:

describe characteristics of the major asset classes that investors consider in forming portfolios; describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated

Summary:

The content discusses the concept of diversification in investment portfolios, emphasizing the role of correlation between assets in reducing portfolio risk. It explains how diversification can mitigate risk without necessarily compromising returns, particularly when assets are not perfectly correlated. The content also covers historical risk and correlation data, different avenues for diversification, and introduces concepts like the investment opportunity set and minimum-variance portfolios.

Key Concepts:

Correlation and Risk Diversification

Correlation measures the degree to which two assets move in relation to each other. Lower correlations between assets in a portfolio can significantly reduce overall portfolio risk.

Historical Risk and Correlation

Historical data on asset returns and their correlations can provide insights but may not always predict future patterns accurately. However, historical risk levels tend to be more stable over time.

Avenues for Diversification

Diversification can be achieved through various means such as investing across different asset classes, geographic regions, or using index funds to manage costs and maintain exposure to diverse market segments.

Investment Opportunity Set

This set includes all feasible combinations of assets that an investor can hold. The set expands as more assets are added, especially those with low correlations to existing assets in the portfolio.

Minimum-Variance Portfolios

These portfolios aim to minimize risk for a given level of expected return. They form part of the efficient frontier, which represents portfolios offering the best possible expected return for a given level of risk.

Formulas:

Portfolio Risk Formula

\sigma_p = \sqrt{\frac{\sigma^2}{N} + \frac{(N-1)}{N} \rho \sigma^2}

This formula calculates the risk (standard deviation) of a portfolio based on the number of assets and the correlation between them. As N increases, the impact of individual asset variance diminishes, emphasizing the role of correlation.

Variables:

$\sigma_p$ :: portfolio standard deviation
$\sigma$ :: standard deviation of individual assets
$N$ :: number of assets in the portfolio
$\rho$ :: correlation coefficient between the assets

Units: percentage or unitless

Other Investment Characteristics

Learning Outcome Statement:

describe characteristics of the major asset classes that investors consider in forming portfolios

Summary:

This LOS explores the characteristics of major asset classes, focusing on their risk-return profiles, distributional characteristics such as skewness and kurtosis, and market characteristics including liquidity and trading costs. It emphasizes the importance of understanding these characteristics to make informed investment decisions, especially in the context of portfolio formation and risk management.

Key Concepts:

Risk–Return Trade-off

Refers to the positive relationship between the expected risk and return of an investment, indicating that higher returns are generally accompanied by higher risks.

Risk Premium

The additional return an investor expects to receive for taking on additional risk, calculated as the difference between the risky return and the risk-free rate.

Distributional Characteristics

Characteristics of return distributions such as skewness and kurtosis, which describe the asymmetry and tail behavior of the distribution, respectively.

Market Characteristics

Includes factors like liquidity, trading costs, and operational efficiency that influence investment decisions and the overall functioning of financial markets.

Formulas:

Risk Premium

RP = R_{risky} - R_{risk\_free}

Calculates the additional return expected from a risky investment over the risk-free rate.

Variables:

$RP$ :: Risk Premium
$R_{risky}$ :: Return of the risky asset
$R_{risk\_free}$ :: Risk-free rate of return

Units: percentage or decimal

Portfolio of Many Risky Assets

Learning Outcome Statement:

calculate and interpret portfolio standard deviation, describe the effect on a portfolio’s risk of investing in assets that are less than perfectly correlated

Summary:

This LOS explores the impact of correlation on the risk of a portfolio containing many risky assets. It explains how diversification can reduce portfolio risk by combining assets with less than perfect correlation. The content also introduces formulas for calculating portfolio variance and standard deviation, emphasizing the role of correlation and covariance among assets in determining portfolio risk.

Key Concepts:

Portfolio Standard Deviation

Portfolio standard deviation is a measure of the total risk of a portfolio, considering the variance of individual assets and the covariance between pairs of assets. It quantifies the volatility of the portfolio's returns.

Correlation and Portfolio Risk

Correlation measures the degree to which two assets move in relation to each other. Lower correlation between portfolio assets leads to lower overall portfolio risk due to diversification benefits.

Diversification

Diversification is a risk management strategy that mixes a wide variety of investments within a portfolio. The rationale behind this technique is that a portfolio constructed of different kinds of assets will, on average, yield higher returns and pose a lower risk than any individual investment found within the portfolio.

Formulas:

Portfolio Variance for Many Assets

\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 \text{Cov}(i, j)\right)

This formula calculates the variance of a portfolio by summing the weighted variances of individual assets and the weighted covariances between all pairs of different assets.

Variables:

$N$ :: Number of assets in the portfolio
$w_i, w_j$ :: Weights of assets i and j in the portfolio
$\sigma_i^2$ :: Variance of asset i
$\text{Cov}(i, j)$ :: Covariance between assets i and j

Units: squared units of returns

Simplified Portfolio Variance with Equal Weights

\sigma_P^2 = \frac{\overline{\sigma}^2}{N} + \frac{(N-1)}{N} \overline{\text{Cov}}

This formula simplifies the calculation of portfolio variance under the assumption of equal asset weights and average variance and covariance, showing how variance decreases and covariance dominates as the number of assets increases.

Variables:

$N$ :: Number of assets in the portfolio
$\overline{\sigma}^2$ :: Average variance of the assets
$\overline{\text{Cov}}$ :: Average covariance among the assets

Units: squared units of returns

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