1 estimating return and risk chapter 7 jones, investments: analysis and management
TRANSCRIPT
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Estimating Return and Risk
Estimating Return and Risk
Chapter 7Jones, Investments:
Analysis and Management
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Investment Decisions
Investment Decisions
Involve uncertainty Focus on expected returns
– Estimates of future returns needed to consider and manage risk
Goal is to reduce risk without affecting returns– Accomplished by building a portfolio– Diversification is key
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Dealing With Uncertainty
Dealing With Uncertainty
Risk that an expected return will not be realized
Investors must think about return distributions, not just a single return
Use probability distributions– A probability should be assigned to
each possible outcome to create a distribution
– Can be discrete or continuous
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Calculating Expected Return
Calculating Expected Return
Expected value – The single most likely outcome
from a particular probability distribution
– The weighted average of all possible return outcomes
– Referred to as an ex ante or expected return
i
m
1iiprR)R(E
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Calculating RiskCalculating Risk
Variance and standard deviation used to quantify and measure risk– Measures the spread in the probability
distribution
– Variance of returns: 2 = (Ri - E(R))2pri
– Standard deviation of returns: =(2)1/2
– Ex ante rather than ex post relevant
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Portfolio Expected Return
Portfolio Expected Return
Weighted average of the individual security expected returns– Each portfolio asset has a weight,
w, which represents the percent of the total portfolio value
– The expected return on any portfolio can be calculated as:
n
1iiip )R(Ew)R(E
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Portfolio RiskPortfolio Risk
Portfolio risk not simply the sum of individual security risks
Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio
Individual stocks are risky only if they add risk to the total portfolio
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Portfolio RiskPortfolio Risk
Measured by the variance or standard deviation of the portfolio’s return– Portfolio risk is not a weighted
average of the risk of the individual securities in the portfolio 2
i
2
p
n
1i iw
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Risk Reduction in Portfolios
Risk Reduction in Portfolios
Assume all risk sources for a portfolio of securities are independent
The larger the number of securities the smaller the exposure to any particular risk– “Insurance principle”
Only issue is how many securities to hold
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Risk Reduction in Portfolios
Risk Reduction in Portfolios
Random diversification– Diversifying without looking at relevant
investment characteristics– Marginal risk reduction gets smaller and
smaller as more securities are added A large number of securities is not
required for significant risk reduction International diversification is
beneficial
Portfolio Risk and Diversification
p %
35
20
0
Number of securities in portfolio10 20 30 40 ...... 100+
Portfolio risk
Market Risk
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Markowitz Diversification
Markowitz Diversification
Non-random diversification– Active measurement and
management of portfolio risk– Investigate relationships between
portfolio securities before making a decision to invest
– Takes advantage of expected return and risk for individual securities and how security returns move together
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Measuring Comovements in Security Returns
Measuring Comovements in Security Returns
Needed to calculate risk of a portfolio:– Weighted individual security risks
»Calculated by a weighted variance using the proportion of funds in each security
»For security i: (wi i)2
– Weighted comovements between returns»Return covariances are weighted using the
proportion of funds in each security
»For securities i, j: 2wiwj ij
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Correlation CoefficientCorrelation Coefficient
Statistical measure of relative co-movements between security returns
mn = correlation coefficient between securities m and n mn =+1.0 = perfect positive correlation
mn =-1.0 = perfect negative (inverse) correlation
mn =0.0 = zero correlation
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When does diversification pay?– Combining securities with perfect positive
correlation provides no reduction in risk»Risk is simply a weighted average of the
individual risks of securities
– Combining securities with zero correlation reduces the risk of the portfolio
– Combining securities with negative correlation can eliminate risk altogether
Correlation CoefficientCorrelation Coefficient
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CovarianceCovariance Absolute measure of association
– Not limited to values between -1 and +1
– Sign interpreted the same as correlation
– The formulas for calculating covariance and the relationship between the covariance and the correlation coefficient are:
BAABAB
iBi,BA
m
1ii,AAB
pr)]R(ER)][R(ER[
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Calculating Portfolio Risk
Calculating Portfolio Risk
Encompasses three factors– Variance (risk) of each security– Covariance between each pair of
securities– Portfolio weights for each security
Goal: select weights to determine the minimum variance combination for a given level of expected return
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Generalizations – The smaller the positive
correlation between securities, the better
– As the number of securities increases:»The importance of covariance
relationships increases»The importance of each individual
security’s risk decreases
Calculating Portfolio Risk
Calculating Portfolio Risk
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Simplifying Markowitz
Calculations
Simplifying Markowitz
Calculations Markowitz full-covariance model
– Requires a covariance between the returns of all securities in order to calculate portfolio variance
– Full-covariance model becomes burdensome as the number of securites in a portfolio grows»n(n-1)/2 unique covariances for n securities
Therefore, Markowitz suggests using an index to simplify calculations
Efficient PortfoliosEfficient Portfolios
An efficient portfolio has the smallest portfolio risk for a given level of expected return
Alternatively, an efficient portfolio maximizes the expected return for a given level of portfolio risk
Porfolios located on efficient frontier dominate all other portfolios 20
Diversifiable vs. Nondiversifiable
Risk
Diversifiable vs. Nondiversifiable
Risk Diversifiable, or nonsystematic, risks
are those risks which are unique to individual companies– Adding nonperfectly correlated stocks to
a portfolio reduces its nonsystematic risk Nondiversifiable risks are called
systematic risks– systematic risks include interest rate
risk, market risk and inflation risk 21
Capital Asset Pricing Model (CAPM)
Capital Asset Pricing Model (CAPM)
Beta– Beta is a measure of the systematic risk
of a security that cannot be avoided through diversification`
– The overall market has a beta of 1»Riskier stocks, those which are more volatile
than the market have Betas greater than 1»Less risky stocks have Betas less than 1
CAPMCAPM
Required rate of return– ki = Risk free rate + Risk
premium»=RF + ßi[E(Rm) - RF]
Security Market Line (SML) is the linear relationship between an asset’s risk and its required rate of return