Investments: Analysis and Management, Second Canadian Edition

1
Chapter 7
Expected Return and Risk
2
Learning Objectives

• Explain how expected return and risk for
  securities are determined.
• Explain how expected return and risk for
  portfolios are determined.
• Describe the Markowitz diversification model for
  calculating portfolio risk.
• Simplify Markowitzs calculations by using the
  single-index model.

3
Investment Decisions

• Involve uncertainty
• Investors are trading a known present value for
  some expected future value that is not known with
  certainty
• Focus on expected returns
• To estimate the returns from various securities,
  investors must estimate the cash flows these
  securities are likely to provide
• Goal is to reduce risk without affecting returns
• Accomplished by building a portfolio
• Diversification is key to effective risk
  management

4
Dealing with Uncertainty

• Risk that the 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
• A probability represents the likelihood of
  various outcomes and is expressed as a decimal or
  fraction
• Sum of the probabilities of all possible outcomes
  must be 1.0
• Can be discrete or continuous (eg., normal
  distribution) (Fig.7.1 pg 188)

5
Calculating Expected Return

• Expected value (Expected rate of return)
• Calculate the expected value in order to describe
  the single most likely outcome from a particular
  probability distribution
• The weighted average of all possible return
  outcomes, where each is weighted by its
  respective probability of occurrence
• Referred to as an ex ante or expected return

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Calculating Expected Return
E(R) the expected return on a security R_i
the ith possible return pr_i the probability of
the ith return m the number of possible returns
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Calculating Risk

• Variance and standard deviation are used to
  quantify and measure risk
• Measure the spread or dispersion in the
  probability distribution
• The larger the dispersion the larger the variance
  and standard deviation
• Variance of returns ?2 ? (Ri - E(R))2pri
• Standard deviation of returns
• ? (?2)1/2

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Calculating Risk

• Calculating a standard deviation using
  probability distribution involves making
  subjective estimates of the probabilities and the
  likely returns
• We cannot avoid making estimates because future
  returns are uncertain
• The relevant ? in this situation is the ex ante
  standard deviation and not the ex post standard
  deviation based on realized returns
• In this chapter we are interested in the
  variability associated with future expected
  returns

9
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

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Example Portfolio Expected Return

• (Pg 191) Consider a three stock portfolio
  consisting of stocks G, H, and I with expected
  returns of 12, 20, and 17 respectively.
• Assume that 50 of investable funds is invested
  in security G, 30 in H, and 20 in I.
• Calculate the expected return on the portfolio

11
Portfolio Risk

• Portfolio risk is not simply the sum of
  individual security risks
• Emphasis is on the risk of the entire portfolio
  and not on the 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 Risk

• Measured by the variance or standard deviation of
  the portfolios return
• Portfolio risk is not a weighted average of the
  risk of the individual securities in the portfolio

13
Portfolio Risk

• Although the expected return of a portfolio is a
  weighted average of its expected returns,
  portfolio risk is less than the weighted average
  of the risk of the individual securities in a
  portfolio of risky securities

14
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 the insurance company
  reduces its risk by writing many policies against
  many independent sources of risk

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Risk Reduction in Portfolios

• Random (naïve) diversification
• Diversifying without looking at relevant
  investment characteristics such as expected
  return or industry classification
• An investor simply selects a relatively large
  number of securities randomly
• Marginal risk reduction gets smaller and smaller
  as more securities are added (Fig. 7.2 pg 192)
• Most finance textbooks contain similar diagrams,
  with the number of stocks required to achieve
  diversification varying depending upon the market
  and the particular empirical study referred to in
  the diagram (eg., 25 to 30 or 15 to 20)

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Risk Reduction in Portfolios

• How many securities are enough to diversify
  properly?
• A recent study by Campbell, Lettau, Malkiel, and
  Xu showed that, between 1962 and 1997 the
  markets overall volatility did not change
  whereas the volatility of individual stocks
  increased sharply. As a result, investors need
  more stocks today to adequately diversity (40
  rather than 20)
• Another recent study by Vladimir de Vassal
  examined the period from 1993 to 1999 and found
  that with a portfolio of 15 stocks, the
  probability of underperforming the market
  benchmark by 100 or more was 13.5, a
  substantial risk. With a portfolio of 40 stocks
  the probability declines to only 2.4

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Risk Reduction in Portfolios

• International diversification
• Ignoring the hazards of foreign investing, such
  as currency risk, we can conclude that if
  domestic diversification is good, international
  diversification is better (Fig 7.3 pg 195)
• Traditional thinking focused on diversifying
  across countries, but the current trend is to
  diversify across industries and across countries
  simultaneously (Fig. 7.4 pg 196)
• Since recent research suggests that industry
  factors play as big a role (if not bigger) in
  obtaining diversification benefits

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Portfolio Risk and Diversification
?p 35 20 0
Total Portfolio Risk
Market Risk
10 20 30 40 ...... 100
Number of securities in portfolio
19
International Diversification
?p 35 20 0
Domestic Stocks only
Domestic International Stocks
10 20 30 40 ...... 100
Number of securities in portfolio
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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 Co-Movements in Security Returns

• We need to consider two factors in order to
  calculate risk of a portfolio as measured by the
  variance or standard deviation
• Weighted individual security risks
• Calculated by a weighted variance using the
  proportion of funds in each security
• For security i (wi ? ?i)2
• Weighted co-movements between returns as measured
  by the covariance 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 Coefficient

• Statistical measure of relative co-movements
  between security returns (it measures how
  security returns move in relation to one another)
• Limited to values between -1 and 1
• ?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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Correlation Coefficient

• With perfect positive correlation, the returns
  have a perfect direct linear relationship.
  Knowing what the return on one security will do
  allows an investor to forecast perfectly what the
  other will do (Fig. 7.5 pg198)
• With perfect negative correlation, the
  securities returns have an inverse linear
  relationship to each other. Therefore, knowing
  the return on one security provides full
  knowledge about the return on the other (Fig. 7.6
  pg 198)
• With zero correlation, there is no relationship
  between the returns on the two securities.
  Knowledge of the return on one security is of no
  value in predicting the return of the second
  security

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Correlation Coefficient

• When does diversification pay?
• Combining securities with perfect positive
  correlation provides no reduction in risk
• Risk of the resulting portfolio is simply a
  weighted average of the individual risks of
  securities (Fig. 7.5 pg 198)
• Combining securities with zero correlation
  reduces the risk of the portfolio
• If more securities with uncorrelated returns are
  added to the portfolio, significant risk
  reduction can be achieved
• The portfolio risk cannot be eliminated in this
  case

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Correlation Coefficient

• Combining securities with negative correlation
  can eliminate risk altogether
• If the correct portfolio weights are chosen
• In the real world, securities typically have some
  positive correlation with each other since all
  security prices tend to move with changes in the
  overall economy (Fig. 7.7 pg 199, ? 0.55)
• As a result, risk can be reduced it cannot be
  eliminated
• Any reduction in risk that does not adversely
  affect return has to considered beneficial

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Example Correlation Coefficient

• (Pg 200) Over the 1998-2003 period the average
  monthly return on Abitibi Consolidated (A) common
  stock was 0.05, and the standard deviation of
  monthly returns was 10.69
• During the same period, the average monthly
  return for Air Canadas (AC) common stock was
  -0.29, and the standard deviation of monthly
  returns was 22.42
• The correlation between the returns on A and AC
  was 0.12 over this period.
• Calculate the return and standard deviation for
  an equally weighted portfolio of these two
  securities.

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Covariance

• Absolute measure of (the degree of association)
  the extent to which two random variables, such as
  the return on two securities, tend to covary, or
  move together over time
• Not limited to values between -1 and 1
• Sign interpreted the same as correlation (, -,
  0)
• The formulas for calculating covariance and the
  relationship between the covariance and the
  correlation coefficient are

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Covariance
The covariance allows us to measure the amount of
co-movement and incorporate it into any measure
of portfolio risk (covariance formula is similar
to the variance formula) s_AB the covariance
between securities A and B R_A,i one estimated
possible return on security A E(R_A) expected
return for security A m the number of likely
outcomes for a security for the period pr_i the
probability of attaining a given return R_A,i
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Calculating Portfolio Risk

• Encompasses three factors
• Variance (risk) of each security
• Covariance between each pair of securities (s_AB
  ?_AB s_A s_B)
• 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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Calculating Portfolio Risk

• Generalizations
• The smaller the positive correlation between
  securities, the better
• The only case where there are no risk reduction
  benefits obtained from two-security
  diversification occurs when the correlation
  coefficient is 1
• As the number of securities increases
• The importance of covariance relationships
  increases
• The importance of each individual securitys risk
  (variance) decreases

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Calculating Portfolio Risk

• The number of relevant covariances for an
  n-security portfolio equals n(n-1)
• For example, the number of relevant covariances
  in a 100-security portfolio would equal
  100(100-1) 9,900. On the other hand, the number
  of relevant variances will be 100

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Calculating Portfolio Risk

• Two-Security Case

• N-Security Case

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Example Portfolio Risk

• Prove that in the two-security case, the
  portfolio standard deviation will be the weighted
  average of the standard deviations of the
  individual securities when the correlation
  coefficient is equal to 1

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Example Portfolio Risk

• (Pg 202) We have an equally weighted portfolio
  that is compromised of stock A (TR 26.3, s
  37.3) and stock B (TR 11.6, s 23.3)
• Calculate the standard deviation of the portfolio
  if the correlation between A and B is 1, 0.5,
  0.15, 0, -0.5, and -1)

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Simplifying Markowitz Calculations

• Markowitz full-covariance model
• Allows us to determine the portfolio expected
  return and risk
• Can be used to determine the optimal portfolio
  combinations (Ch. 8)
• Main problem is its complexity
• Requires a covariance between the returns of all
  securities in order to calculate portfolio
  variance
• Full-covariance model becomes burdensome as
  number of securities in a portfolio grows
• n(n-1)/2 unique covariances for n securities
• Therefore, Markowitz suggests using an index to
  simplify calculations

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The Single-Index Model

• Developed by William Sharpe
• Relates returns on each security to the returns
  on a common stock index, such as the SP/TSX
  Composite Index
• Expressed by the following equation

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The Single-Index Model
R_i the total return on security i a_i the
part of security is return independent of the
market performance (intercept coefficient) ß_i
a coefficient that measures the expected change
in the dependent variable R_i given a change in
the independent variable R_M R_M the total
return on the market index e_i the random
residual error
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The Single-Index Model

• Divides return into two components
• a unique part, a_i
• Is a micro-event, affecting an individual
  company, but not all companies in general (e.g.,
  strike or resignation of CEO)
• a market-related part, ß_iR_M
• Is a macro-event, affecting all (or most) firms
  (e.g., inflation or oil prices)

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Example The Single-Index Model

• (Pg 206) Assume that the return for the market
  index for period t is 12, with a_i 3, and ß_i
  1.5
• Use the single index model to calculate the
  return for stock i for time t
• Assuming that the actual return on stock i for
  period t in the previous example is 19,
  calculate the error term

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The Single-Index Model

• ? (slope coefficient) measures the sensitivity
  of a stock to the market movements
• The single-index model assumes that
• Residuals for different securities are
  uncorrelated
• Securities are only related in their common
  response to the market
• Securities covary together only because of their
  common relationship to the market index
• Security covariances depend only on market risk
  and can be written as