Chapter 3

Managing Portfolios: Theory

Overview

• World of two risky assets– Determine the efficient frontier

• Indifference curves– Critical to determine which portfolio should be held

• World of three risky assets• World of N-risky assets• World of N-risky assets + a risk-free asset• Multifactor index models

Parameters for a Two-Security Portfolio

Where Wi = portfolio weight of asset i Wj = portfolio weight of asset j

Wi + Wj = 1

p i i j jE R W E R W E R

2 2 2 2 2p i i i j ij j jW 2W W COV W

Variances & Covariance

• = variance of the rate of return on asset i

• = variance of the rate of return on asset j

• = covariance of the rate of return on asset i with the rate of return on asset j

N

it i jt ji 1

1 [R E(R )][R E(R )]N 1

Correlation Coefficient

• Measure of co-movement tendency of two variables, such as returns on two securities

Examples of Correlation Coefficients

Three Special Cases• Correlation coefficient = +1

• Correlation coefficient = –1

• Correlation coefficient = 0

P i i j jW W

i i j jP

i i j j

W W or

W W

2 2 2 2p i i j jσ = W σ W σ

Correlation Coefficient = +1

Correlation Coefficient = -1

Correlation Coefficient = -1

Correlation Coefficient = 0

Portfolio Risk: The Two-Asset Case

Efficient Frontier

• Set of risk - expected return Tradeoffs• Each Offers Highest Expected Return for a Given

Risk and Least Risk for a Given Expected Return

Portfolio Standard Deviation: The General Case with Two Assets

2 2 2 2P i i i j ij i j j jW 2W W W

Indifference Curves• Investor indifferent between any two

portfolios on the same indifference curve• Investor prefers ANY portfolio on higher

indifference curve to one on lower one• In theory, each investor could have a unique

set of indifference curves• Cannot be scientifically measured, but

critical to all investment decision making

Indifference Curves

Indiff. Curves: Four Examples

Optimal Portfolio to HoldWhen Correlation Coefficient = –1

Why Low Correlation Coefficients Are Desirable

• NOT because they produce portfolios with least risk (or potentially no risk)

• Because they allow an investor to achieve highest possible indifference curve

Three-Asset Portfolios: Looking Only at Combinations of Two Securities

Three-Asset Portfolios: Looking Only at Pairs of Pairs

N-Asset Portfolio

E(Rp) = W1[E(R1)] + W2[E(R2)] + … + Wn[E(Rn)]

(continued)

n n n2 2 2

p i i i j iji 1 i 1 j 1

i j

W W W COV

n n n2 2

p i i i j iji=1 i = 1 j = 1

i j

σ W σ W W COV

N-Asset Portfolio (continued)

Optimal Portfolio to Hold:Risk Averse Investor

Optimal Portfolio to Hold:Aggressive Investor

Adding the Risk-free Rate

Market Portfolio

Hypothetical portfolio representing each investment asset in the world in proportion to its relative weight in the universe of investment assets

Separation Theorem• Return to any efficient portfolio and its risk can be

completely described by appropriate weighted average of two assets– the risk-free asset – the market portfolio

• Two separate decisions– What risky investments to include in the market

portfolio– How one should divide one’s money between the

market portfolio and risk-free asset

Capital Market Line:Better Efficient Frontier

M fp f p

M

E R RE(R ) R

Capital Asset Pricing Model

• Theoretical relationship that explains returns as function of risk-free rate, market risk premium, and beta

i f i M fE R R E R R

iMi 2

M

COV

Beta

• Parameter that relates stock or portfolio performance to market performance

• Example: with x percent change in market, stock or portfolio will tend to change by x percent times its beta

Implications of Beta Value

• Beta < 0 => opposite of the market• Beta = 0 => independent of the market• 0 < Beta < 1 => same as market, but less

volatile• Beta = 1 => identical to the market• Beta > 1 => same as market, but more

volatile

Portfolio Beta

p i i n nβ W x β + ... W x β

Market Model

Where Ri = return to asset i

Rm = return to the market in the same period

alpha = y-intercept value beta = slope of the line eta = random error term

i i i m iR R

Market Risk vs. Nonmarket Risk

i2 = (beta2 x M

2 ) + eta2

Total risk = market risk + nonmarket risk

Nonmarket Risk• Not related to general market movements• Diversifiable• Total risk of investment may be

decomposed into that associated with market and that which is not

• Nonsystematic risk

Coefficient of Determination• Statistic that measures how much of

variance of particular time series or sample of dependent variable is explained by movement of the independent variable(s) in a regression analysis

• Measure of diversification with respect to portfolios

Multifactor Asset Pricing Model • Model of stock pricing • Relies on arbitrage pricing multifactor model

rather than the capital asset pricing model

Arbitrage Pricing Model• Model used to explain stock pricing and

expected return • Introduces more than one factor in place of

(or in addition to) the capital asset pricing model’s market index

i i i1 1 i2 2 iM ME R F F ... F

Does MPT Matter?

• Uniform Principal and Income Act• Prudent man has evolved to prudent

investor• A model is better than no model• Departure point for how we think about

what is happening in security markets