Thorie Financire 20042005 Relation risque rentabilit attendue 1

1
Théorie Financière2004-2005Relation risque
rentabilité attendue (1)

• Professeur André Farber

2
Introduction to risk

• Objectives for this session
• 1. Review the problem of the opportunity cost of
  capital
• 2. Analyze return statistics
• 3. Introduce the variance or standard deviation
  as a measure of risk for a portfolio
• 4. See how to calculate the discount rate for a
  project with risk equal to that of the market
• 5. Give a preview of the implications of
  diversification

3
Setting the discount rate for a risky project

• Stockholders have a choice
• either they invest in real investment projects of
  companies
• or they invest in financial assets (securities)
  traded on the capital market
• The cost of capital is the opportunity cost of
  investing in real assets
• It is defined as the forgone expected return on
  the capital market with the same risk as the
  investment in a real asset

4
Uncertainty 1952 1973- the Golden Years

• 1952 Harry Markowitz
• Portfolio selection in a mean variance framework
• 1953 Kenneth Arrow
• Complete markets and the law of one price
• 1958 Franco Modigliani and Merton Miller
• Value of company independant of financial
  structure
• 1963 Paul Samuelson and Eugene Fama
• Efficient market hypothesis
• 1964 Bill Sharpe and John Lintner
• Capital Asset Price Model
• 1973 Myron Scholes, Fisher Black and Robert
  Merton
• Option pricing model

5
Three key ideas

• 1. Returns are normally distributed random
  variables
• Markowitz 1952 portfolio theory, diversification
• 2. Efficient market hypothesis
• Movements of stock prices are random
• Kendall 1953
• 3. Capital Asset Pricing Model
• Sharpe 1964 Lintner 1965
• Expected returns are function of systematic risk

6
Preview of what follow

• First, we will analyze past markets returns.
• We will
• compare average returns on common stocks and
  Treasury bills
• define the variance (or standard deviation) as a
  measure of the risk of a portfolio of common
  stocks
• obtain an estimate of the historical risk premium
  (the excess return earned by investing in a risky
  asset as opposed to a risk-free asset)
• The discount rate to be used for a project with
  risk equal to that of the market will then be
  calculated as the expected return on the market

Expected return on the market
Historical risk premium
Current risk-free rate


7
Implications of diversification

• The next step will be to understand the
  implications of diversification.
• We will show that
• diversification enables an investor to eliminate
  part of the risk of a stock held individually
  (the unsystematic - or idiosyncratic risk).
• only the remaining risk (the systematic risk) has
  to be compensated by a higher expected return
• the systematic risk of a security is measured by
  its beta (?), a measure of the sensitivity of the
  actual return of a stock or a portfolio to the
  unanticipated return in the market portfolio
• the expected return on a security should be
  positively related to the security's beta

8
Normal distribution
9
Returns

• The primitive objects that we will manipulate are
  percentage returns over a period of time
• The rate of return is a return per dollar (or ,
  DEM,...) invested in the asset, composed of
• a dividend yield
• a capital gain
• The period could be of any length one day, one
  month, one quarter, one year.
• In what follow, we will consider yearly returns

10
Ex post and ex ante returns

• Ex post returns are calculated using realized
  prices and dividends
• Ex ante, returns are random variables
• several values are possible
• each having a given probability of occurence
• The frequency distribution of past returns gives
  some indications on the probability distribution
  of future returns

11
Frequency distribution

• Suppose that we observe the following frequency
  distribution for past annual returns over 50
  years. Assuming a stable probability
  distribution, past relative frequencies are
  estimates of probabilities of future possible
  returns .

12
Mean/expected return

• Arithmetic Average (mean)
• The average of the holding period returns for the
  individual years
• Expected return on asset A
• A weighted average return each possible return
  is multiplied or weighted by the probability of
  its occurence. Then, these products are summed to
  get the expected return.

13
Variance -Standard deviation

• Measures of variability (dispersion)
• Variance
• Ex post average of the squared deviations from
  the mean
• Ex ante the variance is calculated by
  multiplying each squared deviation from the
  expected return by the probability of occurrence
  and summing the products
• Unit of measurement squared deviation units.
  Clumsy..
• Standard deviation The square root of the
  variance
• Unit return

14
Return Statistics - Example
15
Normal distribution

• Realized returns can take many, many different
  values (in fact, any real number gt -100)
• Specifying the probability distribution by
  listing
• all possible values
• with associated probabilities
• as we did before wouldn't be simple.
• We will, instead, rely on a theoretical
  distribution function (the Normal distribution)
  that is widely used in many applications.
• The frequency distribution for a normal
  distribution is a bellshaped curve.
• It is a symetric distribution entirely defined by
  two parameters
• the expected value (mean)
• the standard deviation

16
Belgium - Monthly returns 1951 - 1999
17
Normal distribution illustrated
18
Risk premium on a risky asset

• The excess return earned by investing in a risky
  asset as opposed to a risk-free asset
• U.S.Treasury bills, which are a short-term,
  default-free asset, will be used a the proxy for
  a risk-free asset.
• The ex post (after the fact) or realized risk
  premium is calculated by substracting the average
  risk-free return from the average risk return.
• Risk-free return return on 1-year Treasury
  bills
• Risk premium Average excess return on a risky
  asset

19
Total returns US 1926-1999
Source Ross, Westerfield, Jaffee (2002) Table 9.2
20
Market Risk Premium The Very Long Run
The equity premium puzzle
Source Ross, Westerfield, Jaffee (2002) Table
9A.1
Was the 20th century an anomaly?
21
Diversification
22
Covariance and correlation

• Statistical measures of the degree to which
  random variables move together
• Covariance
• Like variance figure, the covariance is in
  squared deviation units.
• Not too friendly ...
• Correlation
• covariance divided by product of standard
  deviations
• Covariance and correlation have the same sign
• Positive variables are positively correlated
• Zero variables are independant
• Negative variables are negatively correlated
• The correlation is always between 1 and 1

23
Risk and expected returns for porfolios

• In order to better understand the driving force
  explaining the benefits from diversification, let
  us consider a portfolio of two stocks (A,B)
• Characteristics
• Expected returns
• Standard deviations
• Covariance
• Portfolio defined by fractions invested in each
  stock XA , XB XA XB 1
• Expected return on portfolio
• Variance of the portfolio's return

24
Example

• Invest 100 m in two stocks
• A 60 m XA 0.6
• B 40 m XB 0.4
• Characteristics ( per year) A B
• Expected return 20 15
• Standard deviation 30 20
• Correlation 0.5
• Expected return 0.6 20 0.4 15 18
• Variance (0.6)²(.30)² (0.4)²(.20)²2(0.6)(0.4)
  (0.30)(0.20)(0.5)
• s²p 0.0532 ? Standard deviation 23.07
• Less than the average of individual standard
  deviations
• 0.6 x0.30 0.4 x 0.20 26

25
Diversification effect

• Let us vary the correlation coefficient
• Correlationcoefficient Expected return
  Standard deviation
• -1 18 10.00
• -0.5 18 15.62
• 0 18 19.7
• 0.5 18 23.07
• 1 18 26.00
• Conclusion
• As long as the correlation coefficient is less
  than one, the standard deviation of a portfolio
  of two securities is less than the weighted
  average of the standard deviations of the
  individual securities

26
The efficient set for two assets correlation 1
27
The efficient set for two assets correlation -1
28
The efficient set for two assets correlation 0
29
Choosing portfolios from many stocks

• Porfolio composition
• (X1, X2, ... , Xi, ... , XN)
• X1 X2 ... Xi ... XN 1
• Expected return
• Risk
• Note
• N terms for variances
• N(N-1) terms for covariances
• Covariances dominate

30
Some intuition
31
Example

• Consider the risk of an equally weighted
  portfolio of N "identical  stocks
• Equally weighted
• Variance of portfolio
• If we increase the number of securities ?
• Variance of portfolio

32
Diversification
33
Conclusion

• 1. Diversification pays - adding securities to
  the portfolio decreases risk. This is because
  securities are not perfectly positively
  correlated
• 2. There is a limit to the benefit of
  diversification the risk of the portfolio can't
  be less than the average covariance (cov) between
  the stocks
• The variance of a security's return can be broken
  down in the following way
• The proper definition of the risk of an
  individual security in a portfolio M is the
  covariance of the security with the portfolio

Portfolio risk
Total risk of individual security
Unsystematic or diversifiable risk
34
Efficient markets
35
Notions of Market Efficiency

• An Efficient market is one in which
• Arbitrage is disallowed rules out free lunches
• Purchase or sale of a security at the prevailing
  market price is never a positive NPV transaction.
• Prices reveal information
• Three forms of Market Efficiency
• (a) Weak Form Efficiency
• Prices reflect all information in the past
  record of stock prices
• (b) Semi-strong Form Efficiency
• Prices reflect all publicly available
  information
• (c) Strong-form Efficiency
• Price reflect all information

36
Efficient markets intuition
Price
Price change is unexpected
Time
37
Weak Form Efficiency

• Random-walk model
• Pt -Pt-1 Pt-1 (Expected return) Random
  error
• Expected value (Random error) 0
• Random error of period t unrelated to random
  component of any past period
• Implication
• Expected value (Pt) Pt-1 (1 Expected
  return)
• Technical analysis useless
• Empirical evidence serial correlation
• Correlation coefficient between current return
  and some past return
• Serial correlation Cor (Rt, Rt-s)

38
Random walk - illustration
39
Semi-strong Form Efficiency

• Prices reflect all publicly available
  information
• Empirical evidence Event studies
• Test whether the release of information
  influences returns and when this influence takes
  place.
• Abnormal return AR ARt Rt - Rmt
• Cumulative abnormal return
• CARt ARt0 ARt01 ARt02 ... ARt01

40
Strong-form Efficiency

• How do professional portfolio managers perform?
• Jensen 1969 Mutual funds do not generate
  abnormal returns
• Rfund - Rf ? ? (RM - Rf)
• Insider trading
• Insiders do seem to generate abnormal returns
• (should cover their information acquisition
  activities)

41
Portfolio selection

• Professeur André Farber

42
Portfolio selection

• Objectives for this session
• 1. Gain a better understanding of the rational
  for benefit of diversification
• 2. Identify measures of systematic risk
  covariance and beta
• 3. Analyse the choice of an optimal portfolio

43
Combining the Riskless Asset and a single Risky
Asset

• Consider the following portfolio P
• Fraction invested
• in the riskless asset 1-x (40)
• in the risky asset x (60)
• Expected return on portfolio P
• Standard deviation of portfolio

44
Relationship between expected return and risk

• Combining the expressions obtained for
• the expected return
• the standard deviation
• leads to

45
Risk aversion

• Risk aversion
• For a given risk, investor prefers more expected
  return
• For a given expected return, investor prefers
  less risk

Expected return
Indifference curve
Risk
46
Utility function

• Mathematical representation of preferences
• a risk aversion coefficient
• u certainty equivalent risk-free rate
• Example a 2
• A 6 0 0.06
• B 10 10 0.08 0.10 - 2(0.10)²
• C 15 20 0.07 0.15 - 2(0.20)²
• B is preferred

Utility
47
Optimal choice with a single risky asset

• Risk-free asset RF Proportion 1-x
• Risky portfolio S Proportion x
• Utility
• Optimum
• Solution
• Example a 2