Estimating Return and Risk

1
Estimating Return and Risk

• Chapter 7
• Jones, Investments Analysis and Management

1
2
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

2
3
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

3
4
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

4
5
Calculating 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

5
6
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

6
7
Portfolio 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

7
8
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

8
9
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

9
10
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

10
11
Portfolio Risk and Diversification
?p 35 20 0
Portfolio risk
Market Risk
10 20 30 40 ...... 100
Number of securities in portfolio
12
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

12
13
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

13
14
Correlation 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

14
15
Correlation Coefficient

• 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

15
16
Covariance

• 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

16
17
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

17
18
Calculating Portfolio Risk

• 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 securitys risk
  decreases

18
19
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

19
20
Efficient 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

21
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

22
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
  
  
  
  

23
CAPM

• Required rate of return
• ki Risk free rate Risk premium
• RF ßiE(Rm) - RF
• Security Market Line (SML) is the linear
  relationship between an assets risk and its
  required rate of return