International Investment Professor André Farber Solvay Business School Université Libre de Bruxelles

28 June 2015 Invest |2 Academic contributions to investment process 1952 Markowitz: Portfolio Selection 2 dimensions: expected returns and risk Returns: normally distributed random variables Crucial role of covariances (correlation coefficients) 1965 Fama, Samuelson: Efficient Market Hypothesis Stock prices are unpredictable, move randomly Sharpe, Lintner, Mossin: Capital Asset Pricing Model Expected return function of systematic risk (  ) Black, Scholes, Merton: Option Pricing Model Pricing in a risk neutral world Source: Bernstein, P. Capital Ideas, Free Press 1992

28 June 2015 Invest |3 From Gown to Town 1971 Wells Fargo launches first index fund Portfolio insurance introduced by Leland and Rubinstein Big problem in Creation of Long Term Capital Management 1998: LTCM rescued after losing $4 billions

28 June 2015 Invest |4 A 1 slide review of Finance (no formula..yet) Expected return of portfolio Standard deviation of portfolio’s return. Risk-free rate (R f ) 4 M. 5.. Capital market line. X Y Which portfolio to choose?

28 June 2015 Invest |5 Standard & Poor 500

28 June 2015 Invest |6 Microsoft

28 June 2015 Invest |7 Normal distribution illustrated

28 June 2015 Invest |8 Normal distribution – technical (and useful) details The normal distribution is identified by two parameters: the expected value (mean) and the standard deviation. If R is a random return, we write: For the standard normal distribution, the expectation is zero and the standard deviation is equal to 1.0 The cumulative normal distribution, denoted gives the probability that the random variable will be less than or equal to x. In Excel, use NORMDIST(Value,Mean,StandardDeviation,TRUE) Example:

28 June 2015 Invest |9 Normal distribution – more details The probability that a normal variate will take on a value in the range [a,b] is: Confidence interval: the range within which the return will fall with probability 1-α (the confidence level – α is the probability of error) In Excel, use NORMINV(p,Mean,StandardDeviation)

28 June 2015 Invest |10 Value at Risk (VaR) Value at Risk is a measure of the maximum loss than can experienced over a period of time with a x% probability of exceeding this amount.

28 June 2015 Invest |11 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

28 June 2015 Invest |12 Historical Returns, Source: © Stocks, Bonds, Bills, and Inflation 2003 Yearbook™, Ibbotson Associates, Inc., Chicago (annually updates work by Roger G. Ibbotson and Rex A. Sinquefield). All rights reserved. – 90%+ 90%0% Average Standard Series Annual Return DeviationDistribution Large Company Stocks12.2%20.5% Small Company Stocks Long-Term Corporate Bonds Long-Term Government Bonds U.S. Treasury Bills Inflation3.14.4

28 June 2015 Invest |13 Is the U.S a special case?

28 June 2015 Invest |14 Market Risk Premium: The Very Long Run Common Stock Treasury Bills Risk premium Source: Ross, Westerfield, Jaffee (2005) Table 9A.1 The equity premium puzzle: Was the 20th century an anomaly?

28 June 2015 Invest |15 Siegel on the Equity Risk Premium

28 June 2015 Invest |16 And now the formulas: 2 assets portfolio Expected return Risk More formulas:

28 June 2015 Invest |17 Formulas using matrix algebra Expected return: Variance: Returns: normal distribution

28 June 2015 Invest |18 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

28 June 2015 Invest |19 Calculation in Excel Step 1. Compute covariance matrix Step 2. Compute covariances of individual securities with portfolio Step 3. Compute expected return Step 4. Compute variance Useful Excel trick: use SUMPRODUCT

28 June 2015 Invest |20 Using matrices

28 June 2015 Invest |21 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:

28 June 2015 Invest |22 Diversification

28 June 2015 Invest |23 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

28 June 2015 Invest |24 Mean-Variance Frontier Calculation: brute force Mean variance portfolio: s.t. Matrix notations:

28 June 2015 Invest |25 Some math… Lagrange: FOC: Define:

28 June 2015 Invest |26 Interpretation 1 g+h 0 H E The frontier can be spanned by two frontier returns Minimum variance portfolio MVP A/C

Beta Prof. André Farber SOLVAY BUSINESS SCHOOL UNIVERSITÉ LIBRE DE BRUXELLES

28 June 2015 Invest |28 Measuring the risk of an individual asset The measure of risk of an individual asset in a portfolio has to incorporate the impact of diversification. The standard deviation is not an correct measure for the risk of an individual security in a portfolio. The risk of an individual is its systematic risk or market risk, the risk that can not be eliminated through diversification. Remember: the optimal portfolio is the market portfolio. The risk of an individual asset is measured by beta. The definition of beta is:

28 June 2015 Invest |29 Beta Several interpretations of beta are possible: (1) Beta is the responsiveness coefficient of R i to the market (2) Beta is the relative contribution of stock i to the variance of the market portfolio (3) Beta indicates whether the risk of the portfolio will increase or decrease if the weight of i in the portfolio is slightly modified

28 June 2015 Invest |30 Beta as a slope

28 June 2015 Invest |31 A measure of systematic risk : beta Consider the following linear model R t Realized return on a security during period t  A constant : a return that the stock will realize in any period R Mt Realized return on the market as a whole during period t  A measure of the response of the return on the security to the return on the market u t A return specific to the security for period t (idosyncratic return or unsystematic return)- a random variable with mean 0 Partition of yearly return into: –Market related part ß R Mt –Company specific part  + u t

28 June 2015 Invest |32 Beta - illustration Suppose R t = 2% R Mt + u t If R Mt = 10% The expected return on the security given the return on the market E[R t |R Mt ] = 2% x 10% = 14% If R t = 17%, u t = 17%-14% = 3%

28 June 2015 Invest |33 Measuring Beta Data: past returns for the security and for the market Do linear regression : slope of regression = estimated beta

28 June 2015 Invest |34 Decomposing of the variance of a portfolio How much does each asset contribute to the risk of a portfolio? The variance of the portfolio with 2 risky assets can be written as The variance of the portfolio is the weighted average of the covariances of each individual asset with the portfolio.

28 June 2015 Invest |35 Example

28 June 2015 Invest |36 Beta and the decomposition of the variance The variance of the market portfolio can be expressed as: To calculate the contribution of each security to the overall risk, divide each term by the variance of the portfolio

28 June 2015 Invest |37 Marginal contribution to risk: some math Consider portfolio M. What happens if the fraction invested in stock I changes? Consider a fraction X invested in stock i Take first derivative with respect to X for X = 0 Risk of portfolio increase if and only if: The marginal contribution of stock i to the risk is

28 June 2015 Invest |38 Marginal contribution to risk: illustration

28 June 2015 Invest |39 Beta and marginal contribution to risk Increase (sightly) the weight of i: The risk of the portfolio increases if: The risk of the portfolio is unchanged if: The risk of the portfolio decreases if:

28 June 2015 Invest |40 Inside beta Remember the relationship between the correlation coefficient and the covariance: Beta can be written as: Two determinants of beta –the correlation of the security return with the market –the volatility of the security relative to the volatility of the market

28 June 2015 Invest |41 Properties of beta Two importants properties of beta to remember (1) The weighted average beta across all securities is 1 (2) The beta of a portfolio is the weighted average beta of the securities

28 June 2015 Invest |42 Modeling choices under uncertainty We need to specify how an investor will choose. –Economist use a utility function: a number associated with each possible choice (here each possible portfolio) First a few word about a very general specification. You have €100 to invest. You face 2 possible portfolios Futures values (Proba) Portfolio 1: 90 (0.50) 120 (0.50) Portfolio 2: 50 (0.50) 200 (0.50) Which portfolio would you choose?

28 June 2015 Invest |43 Expected utility General formulation: choice based on Expected utility(Future Wealth) a weighted average of utilities of future wealth E(u) = p 1 u(W 1 ) + p 2 u(W 2 ) + p 3 u(W 3 ) + … + p n u(W n ) Utility function u(W) –an increasing function of W (more wealth is preferred) –shape captures attitude toward risk constant marginal utility u’ (linear) : risk neutrality decreasing marginal utility (concave): risk aversion Wealth Utility Risk neutrality Risk aversion

28 June 2015 Invest |44 Back to our example Lisa is risk neutral: u(W) = W John is risk averse: u(W) = ln(W) What will they choose? –Lisa Expected utility portfolio 1 = 0.50   120 = 105 Expected utility portfolio 2 = 0.50   200 = 125 Lisa would choose portfolio 2 –John Expected utility portfolio 1 = 0.50  ln(90)  ln(120) = 4.64 Expected utility portfolio 2 = 0.50  ln(50)  ln(200) = 4.60 John would choose portfolio 1

28 June 2015 Invest |45 Why different choices? Lisa is risk neutral, only expected value matters John is risk averse: –He will always prefer a sure value over a risky one with the same expected value –The greater expected value of portfolio 2 is not sufficient to compensate for the additional risk

28 June 2015 Invest |46 Mean-variance utility In a more general setting, a pretty good approximation of the expected utility of a portfolio can be obtained with the following formulation It combines into one number the expected return and the risk of the portfolio The degree of risk aversion is captured by a Expected return Standard deviation U A B

28 June 2015 Invest |47 Using mean-variance utility Back to our example: Lisa : a = 0 (she is risk neutral) John : a = 4 (he is risk averse) Portfolio 1: Expected return = 5% Standard deviation = 12.75% Portfolio 2: Expected return = 25% Standard deviation = 79.06% Utilities Lisa John Portfolio  4 .1275² =.0325 Portfolio  4 .7906² = Choice 2 1

28 June 2015 Invest |48 Portfolio Selection & Risk Aversion E(r) Efficient frontier of risky assets More risk-averse investor U’’’ U’’ U’ Q P S St. Dev Less risk-averse investor

28 June 2015 Invest |49 Finding the optimal risky asset allocation Risk-free asset : R F Risky portfolio : Expected return = R Standard deviation =  Invest fraction X in the risky portfolio Choose X to maximize U Optimal allocation : (Proof on demand) The fraction invested in the risky portfolio is decreasing with risk aversion (the higher risk aversion, the lower the fraction invested in the risky portfolio)

28 June 2015 Invest |50 Asset Allocation and Risk Aversion Expected return Standard deviation Optimal asset allocation Optimal risky portfolio U RFRF Efficient frontier

28 June 2015 Invest |51 Risk aversion: a crude estimate Let ’s start from the historical for the US Arithmetic Standard Mean Deviation Large company stocks 12.5% 20.4% US Treasury Bills 3.8% 3.3% Historical risk premium 8.7% Set X=1 (average stock holding in equilibrium) Warning: the debate on the expected risk premium is still on

28 June 2015 Invest |52 Historical risk premium: long term perspective Real Historical Standard Sharpe Equity Premium Deviation ratio Fama, French « The Equity Premium » University of Chicago, WP 522, July 2000 What happened ?