Risk and Return: Portfolio Theory and Assets Pricing Models

Chapter Objectives• Discuss the concepts of portfolio risk and

return.• Determine the relationship between risk and

return of portfolios.• Highlight the difference between systematic

and unsystematic risks.• Examine the logic of portfolio theory .• Show the use of capital asset pricing model

(CAPM) in the valuation of securities.

Introduction• A portfolio is a bundle or a combination of

individual assets or securities.• The portfolio theory provides a normative

approach to investors to make decisions to invest their wealth in assets or securities under risk.– It is based on the assumption that investors are risk-averse.– An investor may want to maximize the returns from his

investments for a given level of risk.– The third assumption of the portfolio theory is that the

returns of assets are normally distributed.

Markowitz Portfolio Theory• The basic portfolio model was developed by Harry

Markowitz, who derived the expected rate of return for a portfolio of assets and an expected risk measure. The Markowitz model is based on several assumptions regarding investor behavior:

1. Investors consider each investment alternative as being represented by a probability distribution of expected returns over some holding period.

2. Investors maximize one-period expected utility, and their utility curves demonstrate diminishing marginal utility of wealth

Markowitz Portfolio TheoryInvestors estimate the risk of the portfolio on the

basis of the variability of expected returns.4. Investors base decisions solely on expected return

and risk, so their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.

5. For a given risk level, investors prefer higher returns to lower returns. Similarly, for a given level of expected return, investors prefer less risk to more risk

Markowitz Portfolio Theory• Under these assumptions, a single asset or

portfolio of assets is considered to be efficient if no other asset or portfolio of assets offers higher expected return with the same (or lower) risk, or lower risk with the same (or higher) expected return

Portfolio Return: Two-Asset Case• The return of a portfolio is equal to the

weighted average of the returns of individual assets (or securities) in the portfolio with weights being equal to the proportion of investment value in each asset.

Expected return on portfolio weight of security × expected return on security

weight of security × expected return on security

X X

Y Y

Portfolio Return• Consider the following stock returns

Stocks Weights Expected ReturnA 0.2 0.1B 0.3 0.11C 0.3 0.12D 0.2 0.13

Calculate the portfolio return

Portfolio Risk: Two-Asset Case• The portfolio variance or standard deviation

depends on the co-movement of returns on two assets. Covariance of returns on two assets measures their co-movement.

• The formula for calculating covariance of returns of the two securities X and Y is as follows:

Portfolio Risk: Two-Asset Case• The portfolio variance or standard deviation depends on the

co-movement of returns on two assets. Covariance of returns on two assets measures their co-movement.

• The formula for calculating covariance of returns of the two securities X and Y is as follows:Covariance XY = Standard deviation X ´ Standard deviation Y ´

Correlation XY• The variance of two-security portfolio is given by the following

equation:

2 2 2 2 2

2 2 2 2

2 Co var

2 Cor

p x x y y x y xy

x x y y x y x y xy

w w w w

w w w w

Portfolio Risk Depends on Correlation between Assets

• When correlation coefficient of returns on individual securities is perfectly positive (i.e., cor = 1.0), then there is no advantage of diversification.

• The weighted standard deviation of returns on individual securities is equal to the standard deviation of the portfolio.

• We may therefore conclude that diversification always reduces risk provided the correlation coefficient is less than 1.

Portfolio Risk

Month Coca-Cola PepsiJan 0.12 0.24Feb 0.23 0.08Mar 0.15 0.21Apr 0.07 0.15May 0.25 0.03Jun 0.32 -0.09Jul 0.11 -0.35Aug -0.09 0.21Sep 0.26 -0.15Oct -0.12 0.2Nov 0.29 0.06Dec 0.03 0.31

Returns

Portfolio Risk• Calculate the covariance between the two

assets• Calculate the portfolio risk and return

assuming the assets are equally weighted in the portfolio

Portfolio Risk• Covariance = -0.0123

Coca-Cola Pepsi (R-Rbar)C* (R-Rbar)PMonth Coca-Cola Pepsi (R-Rbar) (R-Rbar)Jan 0.12 0.24 -0.015 0.165 -0.002475Feb 0.23 0.08 0.095 0.005 0.000475Mar 0.15 0.21 0.015 0.135 0.002025Apr 0.07 0.15 -0.065 0.075 -0.004875May 0.25 0.03 0.115 -0.045 -0.005175Jun 0.32 -0.09 0.185 -0.165 -0.030525Jul 0.11 -0.35 -0.025 -0.425 0.010625Aug -0.09 0.21 -0.225 0.135 -0.030375Sep 0.26 -0.15 0.125 -0.225 -0.028125Oct -0.12 0.2 -0.255 0.125 -0.031875Nov 0.29 0.06 0.155 -0.015 -0.002325Dec 0.03 0.31 -0.105 0.235 -0.024675

0.135 0.075 -0.1473

Returns

Portfolio Risk• Portfolio return = (0.5*0.135) + (0.5*0.075)

=0.105

Portfolio Risk• Coca-Cola

Month Returns (R-Rbar) (R-Rbar)̂ 2Jan 0.12 -0.015 0.000225Feb 0.23 0.095 0.009025Mar 0.15 0.015 0.000225Apr 0.07 -0.065 0.004225May 0.25 0.115 0.013225Jun 0.32 0.185 0.034225Jul 0.11 -0.025 0.000625Aug -0.09 -0.225 0.050625Sep 0.26 0.125 0.015625Oct -0.12 -0.255 0.065025Nov 0.29 0.155 0.024025Dec 0.03 -0.105 0.011025

0.135 0.2281

variance= 0.0207 Std = 0.1440

Portfolio Risk• Pepsi

Month Returns (R-Rbar) (R-Rbar)̂ 2Jan 0.24 0.165 0.027225Feb 0.08 0.005 2.5E-05Mar 0.21 0.135 0.018225Apr 0.15 0.075 0.005625May 0.03 -0.045 0.002025Jun -0.09 -0.165 0.027225Jul -0.35 -0.425 0.180625Aug 0.21 0.135 0.018225Sep -0.15 -0.225 0.050625Oct 0.2 0.125 0.015625Nov 0.06 -0.015 0.000225Dec 0.31 0.235 0.055225

0.075 0.4009

variance= 0.0364 std= 0.1909

Portfolio Risk

𝝈𝒑 𝟐 = 𝝈𝒙 𝟐𝒘𝒙 𝟐 + 𝝈𝒚 𝟐𝒘𝒚 𝟐 + 𝟐𝒘𝒙𝒘𝒚 𝑪𝒐𝒗𝒂𝒓𝒙𝒚 𝝈𝒑 𝟐 = (𝟎.𝟎𝟐𝟎𝟕∗𝟎.𝟓𝟐) + (𝟎.𝟎𝟑𝟔𝟒∗𝟎.𝟓𝟐) + 𝟐(𝟎.𝟓∗𝟎.𝟓∗ −𝟎.𝟎𝟏𝟐𝟑) =𝟎.𝟎𝟎𝟖𝟐

𝝈𝒑 = ξ𝟎.𝟎𝟎𝟖𝟐 = 𝟎.𝟎𝟗𝟎𝟔

Example

• Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The expected return on your portfolio is:

Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected dollar return on your Exxon Mobil stock is 10% and on Coca Cola is 15%. The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

Risk Diversification: Systematic and Unsystematic Risk

05 10 15

Number of Securities

Po

rtfo

lio

sta

nd

ard

dev

iati

on

Market risk

Uniquerisk

Beta and Unique Risk

beta

Expected

return

Expectedmarketreturn

10%10%- +

-10%+10%

stock

-10%

1. Total risk = diversifiable risk + market risk2. Market risk is measured by beta, the sensitivity to market changes

Beta and Unique Risk

• Market Portfolio - Portfolio of all assets in the economy. In practice a broad stock market index, such as the GSE All Share Index, S&P Composite are used to represent the market.

• Beta - Sensitivity of a stock’s return to the return on the market portfolio.

Beta and Unique Risk

2m

imiB

Investment Opportunity Sets (2 Assets) given Different Correlations

0

5

10

15

20

0 5 10 15 20 25 30

Porfolio risk (Stdev, %)

Po

rtfo

lio

retu

rn,

%

Cor = - 1.0

Cor = - 0.25

Cor = + 1.0

Cor = + 0.50

Cor = - 1.0

L

R

Mean-Variance Criterion

• A risk-averse investor will prefer a portfolio with the highest expected return for a given level of risk or prefer a portfolio with the lowest level of risk for a given level of expected return. In portfolio theory, this is referred to as the principle of dominance

Investment Opportunity Set: The N-Asset Case

• An efficient portfolio is one that has the highest expected returns for a given level of risk. The efficient frontier is the frontier formed by the set of efficient portfolios. All other portfolios, which lie outside the efficient frontier, are inefficient portfolios.

Investment Opportunity Set: The N-Asset Case

Risk,

Return

A

P

QB

C

D

x

x

xx

x

x

x

R

Efficient Frontier

A

BN

Return

Risk

AB

Goal is to move up and left.

WHY?

ABN

Efficient Frontier

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

Efficient Frontier

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

A Risk-Free Asset and a Risky Asset • A risk-free asset or security has a zero

variance or standard deviation.• Return and risk when we combine a risk-free

and a risky asset:( ) ( ) (1 )p j fE R wE R w R

p jw

Security Market LineReturn

Risk

.

rf

Risk Free

Return =

Efficient Portfolio

Market Return = rm

For a given amount of systematic risk, SML shows the required rate of return.

Security Market LineReturn

.

rf

Risk Free

Return =

Efficient Portfolio

Market Return = rm

BETA1.0

A security market line (SML) is a line that visually represents the relationship between risk and the expected or the required rate of return on an asset.

Security Market LineReturn

BETA

rf

1.0

SML

SML Equation = rf + B ( rm - rf )

Capital Asset Pricing Model

R = rf + B ( rm - rf )

CAPM

Capital Asset Pricing ModelGiven beta of 2.5 and risk premium on the market as 5%, if the risk free rate is 19%, use the CAPM to find the opportunity cost.

Capital Asset Pricing Model• Ama is considering the following investments.

The current rate on Tbill is 5.5%, and the expected return for the market is 11%. Using CAPM, what rates of return should Ama require for each individual security?

Stock BetaA 0.75B 1.4C 0.95D 1.25

Capital Asset Pricing Model (CAPM)• The capital asset pricing model (CAPM) is a

model that provides a framework to determine the required rate of return on an asset and indicates the relationship between return and risk of the asset.

• Assumptions of CAPM– Market efficiency– Risk aversion and mean-variance optimisation – Homogeneous expectations – Single time period – Risk-free rate

Implications of CAPM• Investors will always combine a risk-free asset with a

market portfolio of risky assets. They will invest in risky assets in proportion to their market value.

• Investors will be compensated only for that risk which they cannot diversify. This is the market-related (systematic) risk.

• Beta, which is a ratio of the covariance between the asset returns and the market returns divided by the market variance, is the most appropriate measure of an asset’s risk.

• Investors can expect returns from their investment according to the risk. This implies a linear relationship between the asset’s expected return and its beta.

Limitations of CAPM • It is based on unrealistic assumptions.• It is difficult to test the validity of CAPM.• Betas do not remain stable over time.

The Arbitrage Pricing Theory (APT)

• In APT, the return of an asset is assumed to have two components: predictable (expected) and unpredictable (uncertain) return. Thus, return on asset j will be:

( ) + j fE R R URwhere Rf is the predictable return (risk-free return on a zero-beta asset) and UR is the unanticipated part of the return. The uncertain return may come from the firm specific information and the market related information:

1 1 2 2 3 3( ) ( )j f n n sE R R F F F F UR

Steps in Calculating Expected Return under APT

• Factors:– industrial production– changes in default premium– changes in the structure of interest rates– inflation rate– changes in the real rate of return

• Risk premium• Factor beta