Chapter 7Charles P. Jones, Investments: Analysis and Management,

Twelfth Edition, John Wiley & Sons

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Involve uncertainty Focus on expected returns

◦ Estimates of future returns needed to consider and manage risk

◦ Investors often overly optimistic about expected returns

Goal is to reduce risk without affecting returns◦ Accomplished by building a portfolio◦ Diversification is key

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Risk that an expected return will not be realized

Investors must think about return distributions

Probabilities weight outcomes◦ Assigned to each possible outcome to create a

distribution◦ History provides guide but must be modified for

expected future changes◦ Distributions can be discrete or continuous

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Expected value ◦ The weighted average of all possible return

outcomes included in the probability distribution Each outcome weighted by probability of

occurrence◦ Referred to as expected return

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i

m

1iiprR)R(E ∑

==

Variance and standard deviation used to quantify and measure risk◦ Measure the spread or dispersion around the

mean of the probability distribution◦ Variance of returns: σ² = (Ri - E(R))²pri

◦ Standard deviation of returns:σ =(σ²)1/2

◦ σ is expected (ex ante) Actual (ex post) σ useful but not perfect estimate

of future

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Modern Portfolio Theory

Provides framework for selection of portfolios based on expected return and risk

Used, to varying degrees, by financial managers

Shows benefits of diversification The risk of a portfolio does not equal the

sum of the risks of its individual securities◦ Must account for correlations among individual

risks

Weighted average of the individual security expected returns◦ Each portfolio asset has a weight, w, which

represents the percent of the total portfolio value

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∑=

=n

1iiip )R(Ew)R(E

Portfolio risk not simply the sum or the weighted average of individual security risks

Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio

Diversification almost always lowers risk Measured by the variance or standard

deviation of the portfolio’s return

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2

i

2

p

n

1i iw σσ ∑=

≠

Assume all risk sources for a portfolio of securities are independent◦ This assumption is unrealistic when investing◦ Market risk affects all firms, cannot be diversified

away If risks 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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Random (or naïve) diversification◦ Diversifying without looking at how security

returns are related to each other◦ Marginal risk reduction gets smaller and smaller

as more securities are added Beneficial but not optimal

◦ Risk reduction kicks in as soon as additional securities added

◦ Research suggests it takes a large number of securities for significant risk reduction

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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 × σi)2◦ Weighted co-movements between returns

Return covariances are weighted using the proportion of funds in each security

For securities i, j: 2wiwj × σij

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Statistical measure of relative association ij = correlation coefficient between

securities i and j◦ ij = +1.0 = perfect positive correlation

◦ ij = -1.0 = perfect negative (inverse) correlation

◦ ij = 0.0 = zero correlation

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When does diversification pay?◦ With perfectly positive correlation, risk is a

weighted average, therefore there is no risk reduction

◦ With perfectly negative correlation, diversification assures the expected return

◦ With zero correlation If many securities, provides significant risk reduction Cannot eliminate risk

Negative correlation or low positive correlation ideal but unlikely

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Absolute, not relative, measure of association ◦ Not limited to values between -1 and +1◦ Sign interpreted the same as correlation◦ Size depends on units of variables ◦ Related to correlation coefficient

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BAABAB

iBi,BA

m

1ii,AAB

pr)]R(ER)][R(ER[

σσσ

σ

=

−−=∑=

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◦ Covariance calculations grow quickly◦ As the number of securities increases:

The importance of covariance relationships increases The importance of each individual security’s risk

decreases

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Markowitz full-covariance model◦ Requires a covariance between the returns of all

securities in order to calculate portfolio variance◦ [n(n-1)]/2 set of covariances for n securities

Markowitz suggests using an index to which all securities are related to simplify

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