Théorie Financière Relation risque – rentabilité attendue (1) Professeur André Farber

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |4 Uncertainty: 1952 – 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

August 23, 2004 Tfin Risk and return (1) |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 Capital Asset Pricing Model Sharpe 1964 Lintner 1965 Expected returns are function of systematic risk

August 23, 2004 Tfin Risk and return (1) |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 Current risk- free rate Historical risk premium =+

August 23, 2004 Tfin Risk and return (1) |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

Normal distribution

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |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.

August 23, 2004 Tfin Risk and return (1) |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.

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |14 Return Statistics - Example

August 23, 2004 Tfin Risk and return (1) |15 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

August 23, 2004 Tfin Risk and return (1) |16 Belgium - Monthly returns

August 23, 2004 Tfin Risk and return (1) |17 Normal distribution illustrated

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |19 Total returns US Arithmetic Mean Standard Deviation Risk Premium Common Stocks13.3%20.1%9.5% Small Company Stocks Long-term Corporate Bonds Long-term government bonds Intermediate-term government bond U.S. Treasury bills Inflation Source: Ross, Westerfield, Jaffee (2002) Table 9.2

August 23, 2004 Tfin Risk and return (1) |20 Market Risk Premium: The Very Long Run Common Stock Treasury Bills Risk premium Source: Ross, Westerfield, Jaffee (2002) Table 9A.1 The equity premium puzzle: Was the 20th century an anomaly?

Diversification

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (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 X A, X B X A + X B = 1 Expected return on portfolio: Variance of the portfolio's return:

August 23, 2004 Tfin Risk and return (1) |24 Example Invest $ 100 m in two stocks: A $ 60 m X A = 0.6 B $ 40 m X B = 0.4 Characteristics (% per year) A B Expected return 20% 15% Standard deviation 30% 20% Correlation 0.5 Expected return = 0.6 × 20% × 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 =  Standard deviation = % Less than the average of individual standard deviations: 0.6 x x 0.20 = 26%

August 23, 2004 Tfin Risk and return (1) |25 Diversification effect Let us vary the correlation coefficient Correlation coefficient Expected return Standard deviation 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

August 23, 2004 Tfin Risk and return (1) |26 The efficient set for two assets: correlation = +1

August 23, 2004 Tfin Risk and return (1) |27 The efficient set for two assets: correlation = -1

August 23, 2004 Tfin Risk and return (1) |28 The efficient set for two assets: correlation = 0

August 23, 2004 Tfin Risk and return (1) |29 Choosing portfolios from many stocks Porfolio composition : (X 1, X 2,..., X i,..., X N ) X 1 + X X i X N = 1 Expected return: Risk: Note: N terms for variances N(N-1) terms for covariances Covariances dominate

August 23, 2004 Tfin Risk and return (1) |30 Some intuition

August 23, 2004 Tfin Risk and return (1) |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:

August 23, 2004 Tfin Risk and return (1) |32 Diversification

August 23, 2004 Tfin Risk and return (1) |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: Total risk of individual security Portfolio risk Unsystematic or diversifiable risk

Efficient markets

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |36 Efficient markets: intuition Expectation Time Price Realization Price change is unexpected

August 23, 2004 Tfin Risk and return (1) |37 Weak Form Efficiency Random-walk model: –P t -P t-1 = P t-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 (P t ) = P t-1 * (1 + Expected return) –Technical analysis: useless Empirical evidence: serial correlation –Correlation coefficient between current return and some past return –Serial correlation = Cor (R t, R t-s )

August 23, 2004 Tfin Risk and return (1) |38 Random walk - illustration

August 23, 2004 Tfin Risk and return (1) |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: CAR t = AR t0 + AR t0+1 + AR t AR t0+1

August 23, 2004 Tfin Risk and return (1) |40 Strong-form Efficiency How do professional portfolio managers perform? Jensen 1969: Mutual funds do not generate abnormal returns R fund - R f =  +  (R M - R f ) Insider trading Insiders do seem to generate abnormal returns (should cover their information acquisition activities)

Portfolio selection Professeur André Farber

August 23, 2004 Tfin Risk and return (1) |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

August 23, 2004 Tfin Risk and return (1) |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 : Riskless asset Risky asset Expected return 6%12% Standard deviation 0%20%

August 23, 2004 Tfin Risk and return (1) |44 Relationship between expected return and risk Combining the expressions obtained for : the expected return the standard deviation leads to

August 23, 2004 Tfin Risk and return (1) |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 Risk Indifference curve

August 23, 2004 Tfin Risk and return (1) |46 Utility function Mathematical representation of preferences a: risk aversion coefficient u = certainty equivalent risk-free rate Example: a = 2 A 6% B 10% 10% 0.08 = ×(0.10)² C 15% 20% 0.07 = ×(0.20)² B is preferred Utility

August 23, 2004 Tfin Risk and return (1) |47 Optimal choice with a single risky asset Risk-free asset : R F Proportion = 1-x Risky portfolio S: Proportion = x Utility: Optimum: Solution: Example: a = 2