Upload
melvyn-gilbert
View
221
Download
0
Embed Size (px)
DESCRIPTION
Overview World of two risky assets Indifference curves 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
Citation preview
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