théorie financière 2004-2005 relation risque – rentabilité attendue (1) professeur andré...
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![Page 1: Théorie Financière 2004-2005 Relation risque – rentabilité attendue (1) Professeur André Farber](https://reader035.vdocuments.mx/reader035/viewer/2022062714/56649d5d5503460f94a3c960/html5/thumbnails/1.jpg)
Théorie Financière2004-2005Relation risque – rentabilité attendue (1)
Professeur André Farber
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Tfin 2004 07 Risk and return (1) |2August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |3August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |4August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |5August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |6August 23, 2004
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
Current risk-free rate
Historical risk premium
= +
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Tfin 2004 07 Risk and return (1) |7August 23, 2004
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
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Normal distribution
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Tfin 2004 07 Risk and return (1) |9August 23, 2004
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
1
1
1
t
tt
t
tt P
PP
P
divR
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Tfin 2004 07 Risk and return (1) |10August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |11August 23, 2004
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 .
Realized Return Absolutefrequency
Relativefrequency
-20% 2 4%
-10% 5 10%
0% 8 16%
+10% 20 40%
+20% 10 20%
+30% 5 10%
50 100%
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Tfin 2004 07 Risk and return (1) |12August 23, 2004
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.
N
RRRRMean N
...21
1...
return ofy probabilit with
...)(
21
2211
n
ii
nn
ppp
Rp
RpRpRpRE
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Tfin 2004 07 Risk and return (1) |13August 23, 2004
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
VarR R R R R R
TT
2 12
22 2
1( ) ( ) ... ( )
Var R Expected RA A A( ) ) 2 2 val ue of (RA
Var R p R R p R R p R RA A A A A A N A N A( ) ( ) ( ) ... ( ), , , 21 1
22 2
2 2
SD R Var RA A A( ) ( )
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Tfin 2004 07 Risk and return (1) |14August 23, 2004
Return Statistics - Example
Return Proba Squared Dev-20% 4% 0.08526-10% 10% 0.03686
0% 16% 0.0084610% 40% 0.0000620% 20% 0.0116630% 10% 0.04326
Exp.Return 9.20%Variance 0.01514Standard deviation 12.30%
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Tfin 2004 07 Risk and return (1) |15August 23, 2004
Normal distribution
• Realized returns can take many, many different values (in fact, any real number > -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
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Tfin 2004 07 Risk and return (1) |16August 23, 2004
Belgium - Monthly returns 1951 - 1999
Bourse de Bruxelles 1951-1999
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
-20.
00
-18.
00
-16.
00
-14.
00
-12.
00
-10.
00
-8.0
0
-6.0
0
-4.0
0
-2.0
0 0.
00
2.00
4.
00
6.00
8.
00
10.0
0
12.0
0
14.0
0
16.0
0
18.0
0
20.0
0
22.0
0
24.0
0
26.0
0
28.0
0
30.0
0
Rentabilité mensuelle
Fré
qu
en
ce
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Tfin 2004 07 Risk and return (1) |17August 23, 2004
Normal distribution illustrated
Normal distribution
0.0000
0.0050
0.0100
0.0150
0.0200
0.0250
68.26%
95.44%
Standard deviation from mean
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Tfin 2004 07 Risk and return (1) |18August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |19August 23, 2004
Total returns US 1926-1999
Arithmetic Mean
Standard Deviation
Risk Premium
Common Stocks 13.3% 20.1% 9.5%
Small Company Stocks 17.6 33.6 13.8
Long-term Corporate Bonds 5.9 8.7 2.1
Long-term government bonds 5.5 9.3 1.7
Intermediate-term government bond
5.4 5.8 1.6
U.S. Treasury bills 3.8 3.2
Inflation 3.2 4.5
Source: Ross, Westerfield, Jaffee (2002) Table 9.2
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Tfin 2004 07 Risk and return (1) |20August 23, 2004
Market Risk Premium: The Very Long Run
1802-1870 1871-1925 1926-1999 1802-1999
Common Stock 6.8 8.5 13.3 9.7
Treasury Bills 5.4 4.1 3.8 4.4
Risk premium 1.4 4.4 9.5 5.3
Source: Ross, Westerfield, Jaffee (2002) Table 9A.1
The equity premium puzzle:
Was the 20th century an anomaly?
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Diversification
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Tfin 2004 07 Risk and return (1) |22August 23, 2004
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
)])([(),cov( BBAABAAB RRRRERR
BA
BABAAB
RRCovRRCorr
),(
),(
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Tfin 2004 07 Risk and return (1) |23August 23, 2004
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:
BA RR ,
BA ,
BAABAB
BBAAP RXRXR
22222 2 BBABBAAAP XXXX
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Tfin 2004 07 Risk and return (1) |24August 23, 2004
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)²+2(0.6)(0.4)(0.30)(0.20)(0.5)
²p = 0.0532 Standard deviation = 23.07 %
• Less than the average of individual standard deviations:
• 0.6 x0.30 + 0.4 x 0.20 = 26%
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Tfin 2004 07 Risk and return (1) |25August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |26August 23, 2004
The efficient set for two assets: correlation = +1
0.00
5.00
10.00
15.00
20.00
25.00
30.00
0.00 20.00 40.00 60.00
Risk (standard deviation)
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Tfin 2004 07 Risk and return (1) |27August 23, 2004
The efficient set for two assets: correlation = -1
0.00
5.00
10.00
15.00
20.00
25.00
30.00
0.00 20.00 40.00 60.00
Risk (standard deviation)
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Tfin 2004 07 Risk and return (1) |28August 23, 2004
The efficient set for two assets: correlation = 0
0.00
5.00
10.00
15.00
20.00
25.00
30.00
0.00 20.00 40.00 60.00
Risk (standard deviation)
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Tfin 2004 07 Risk and return (1) |29August 23, 2004
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
NNP RXRXRXR ...2211
i ij i j
ijjiijjijj
jP XXXXX 222
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Tfin 2004 07 Risk and return (1) |30August 23, 2004
Some intuition
Var Cov Cov Cov CovCov Var Cov Cov CovCov Cov Var Cov CovCov Cov Cov Var CovCov Cov Cov Cov Var
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Tfin 2004 07 Risk and return (1) |31August 23, 2004
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:
NX j
1
cov)1
1(1 22
NNP
NP cov2
cov),(,, jijj RRCovRR
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Tfin 2004 07 Risk and return (1) |32August 23, 2004
Diversification
Risk Reduction of Equally Weighted Portfolios
0.00%
5.00%
10.00%
15.00%
20.00%
25.00%
30.00%
35.00%
# stocks in portfolio
Po
rtfo
lio
sta
nd
ard
de
via
tio
n
Market risk
Unique risk
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Tfin 2004 07 Risk and return (1) |33August 23, 2004
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:
Total risk of individual security
Portfolio risk
Unsystematic or diversifiable risk
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Efficient markets
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Tfin 2004 07 Risk and return (1) |35August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |36August 23, 2004
Efficient markets: intuition
Expectation
Time
Price
Realization
Price change is unexpected
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Tfin 2004 07 Risk and return (1) |37August 23, 2004
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)
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Tfin 2004 07 Risk and return (1) |38August 23, 2004
Random walk - illustration
Bourse de Bruxelles 1980-1999
-30.00
-25.00
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
-30.00 -25.00 -20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00
Rentabilité mois t
Re
nta
bili
té m
ois
t+
1
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Tfin 2004 07 Risk and return (1) |39August 23, 2004
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 + ARt0+1 + ARt0+2 +... + ARt0+1
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Tfin 2004 07 Risk and return (1) |40August 23, 2004
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)
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Portfolio selection
Professeur André Farber
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Tfin 2004 07 Risk and return (1) |42August 23, 2004
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
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Tfin 2004 07 Risk and return (1) |43August 23, 2004
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 :
Riskless asset
Risky asset
Expected return
6% 12%
Standard deviation
0% 20%
SFP RxRxR )1(
%60.912.060.006.040.0 PR
SP x
%1220.060.0 P
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Tfin 2004 07 Risk and return (1) |44August 23, 2004
Relationship between expected return and risk
• Combining the expressions obtained for :
• the expected return
• the standard deviation
• leads to
SFP RxRxR )1(
SP x
PS
FSFP
RRRR
SSPR 30.006.020.0
06.012.006.0
P
PR
S
SR
FR
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Tfin 2004 07 Risk and return (1) |45August 23, 2004
Risk aversion
• Risk aversion :
• For a given risk, investor prefers more expected return
• For a given expected return, investor prefers less risk
Expected return
Risk
Indifference curve
P
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Tfin 2004 07 Risk and return (1) |46August 23, 2004
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
2),( PPPP aRRU
PR P Utility
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Tfin 2004 07 Risk and return (1) |47August 23, 2004
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
SSR ,
22 ²])1[( SSFPP axRxRxaRu
02)( 2 SFS axRRdx
du
22
1
S
FS RR
ax
375.0)20.0(
06.012.0
22
1
2
122
S
FS RR
ax