1 Portfolio Analysis Global Financial Management Campbell R. Harvey Fuqua School of Business Duke University

2 Overview l Risk and risk aversion l How to measure risk and return »Risk measures for some classes of securities l Diversification »How to analyze the benefits from diversification »How to determine the trade-off between risk and return »Is there a limit to diversification l Minimum variance portfolios l Portfolio analysis and hedging

3 Toss two coins: Outcome GainProbabilityExp. gain H H +$600 H T + £ 100 T T - £ 400 Total Which distribution do you prefer, safe or risky? Risk and risk aversion

4 Risk Aversion l An individual is said to be risk averse if he prefers less risk for the same expected return. l Given a choice between $C for sure, or a risky gamble in which the expected payoff is $C, a risk averse individual will choose the sure payoff. l Individuals are generally risk averse when it comes to situations in which a large fraction of their wealth is at risk. »Insurance »Investing l What does this imply about the relationship between an individual ’ s wealth and utility?

5 Relationship Between Wealth and Utility Utility l Suppose an individual has: »current wealth of W 0 »the opportunity to undertake an investment which has a 50% chance of earning x and a 50% chance of earning -x. l Is this an investment the individual would voluntarily undertake? Wealth Utility Function

6 Risk Aversion Example

7 Implications of Risk Aversion l Individuals who are risk averse will try to avoid “ fair bets. ” »Hedging can be valuable. l Risk averse individuals require higher expected returns on riskier investments. l Whether an individual undertakes a risky investment will depend upon three things: »The individual ’ s utility function. »The individual ’ s initial wealth. »The payoffs on the risky investment relative to those on a riskfree investment. Issues: »How do you measure risk? »How do you compare risk and return?

8 US Equities: A Risky Investment S&P500,

9 Equities: Distribution of Returns Histogramm of distriubution of S&P 500 returns,

10 Wealth Indices US

11 How to put it into numbers l We measure the average return R P on a portfolio in period t as: where x j = fraction of the portfolio ’ s total value invested in stock j, j=1, …,N. »x j > 0 is a long position. »x j < 0 is a short position;  j x j = 1 l Stock market indices: »Equally weighted: x 1 =x 2 = … =x N =1/N »Value weighted: x j = Proportion of market capitalization l We measure the average return over the period as:

12 Measuring Risk l The variance over time of a portfolio can be measured as: l Most of the time we shall refer to the standard deviation:

13 Equities: Monthly Returns Mean:1.00% Standard Deviation:5.71%

14 Bonds: Monthly Returns Mean:0.44% Standard Deviation:2.21%

15 Risk and Return: Distributions S&P 500 SD=20.82 Small Companies SD=40.04 LT Government Bonds 30 Day Treasury Bills SD=3.28 SD=5.44

16 Average Returns and Variabilities Source: Ibbotson Associates/Own Calculations

17 Returns and Variability Small Co ’ s S&P 500 Govt ITTBills Govt LT Corporate Return Variability Variability is closely related to returns for portfolios

18 InvestmentRisk PremiumVariability Stock market index9 20 Typical individual share l The risk premium for individual shares is not closely related to their volatility. »Need to understand diversification Individual Shares and the Stock Market: A Paradox?

19 Diversification: The Basic Idea l Construct portfolios of securities that offer the highest expected return for a given level of risk. l The risk of a portfolio will be measured by its standard deviation (or variance). l Diversification plays an important role in designing efficient portfolios.

20 Measuring Portfolio Returns l The expected rate of return on a portfolio of stocks is: l The expected rate of return on a portfolio is a weighted average of the expected rates of return on the individual stocks. l In the two-asset case:

21 Measuring Portfolio Risk l The risk of a portfolio is measured by its standard deviation or variance. l The variance for the two stock case is: or, equivalently,

22 Fire Insurance Policies An example of a two-asset portfolio Asset 1:Your house, worth $100,000 Asset 2:Your fire insurance policy Two states of the world : State 1:Your house burns down and retains no value; the insurance policy pays out $100,000 (Prob. = 10%) State 2:Your house does not burn down and retains its full value. The insurance policy does not pay out. Question:What is the riskiness of each of these two assets individually, and together, if held as a portfolio?

23 Payoffs: State/AssetHouse Insurance Total , , , ,000 Insurance Policies: states and payoffs

24 Risk Analysis: AssetHouse Insurance Together Expected90,000 10, ,000 Payoff Risk30,000 30,000 0 Expected Values are additive, but Risk is not additive! Perfect correlation gives perfect insurance. Insurances: Risk Analysis

25 Two Asset Case

26 Two Asset Case l We want to know where the portfolios of stocks 1 and 2 plot in the risk-return diagram. »Using (as before): x j = fraction of the portfolio ’ s total value invested in stock j, j=1,2 »x j > 0 is a long position. »x j < 0 is a short position; »x 2 = 1- x 1 l We need to compute expectation and standard deviation of the portfolio: l We shall consider three special cases:  12 = -1  12 = 1  12 < 1

27 Minimum Variance Portfolio l What is the upper limit for the benefits from diversification? »Determine the portfolio that gives the smallest possible variance. – We call this the global minimum-variance portfolio. l For the two stock case, the global minimum variance portfolio has the following portfolio weights: l The variance of the global minimum-variance portfolio is: Note: we have not excluded short-selling here! xi<0 is possible!

28 Perfect Negative Correlation l With perfect negative correlation,  12 = -1, it is possible to reduce portfolio risk to zero. l The global minimum variance portfolio has a variance of zero. The portfolio weights for the global minimum variance portfolio are: Consider the following example

29 Perfect Negative Correlation

30 Perfect Positive Correlation l With perfect positive correlation,  12 = +1, it is only possible to reduce portfolio risk to zero if you can short-sell. l The portfolio weights for the global minimum variance portfolio are: »Short sell one of the assets »Long position in the other asset. »If one asset has low risk/low return, portfolio return is below return of the low return asset l If you cannot short sell, then put all your wealth into the lower risk asset.

31 Perfect Positive Correlation: Example l Reconsider the previous example, but assume perfect positive correlation,  12 = +1. »Then we have portfolio weights: »This gives an expected return of: »Variance is reduced to zero –check this!

32 Perfect Positive Correlation E[r] E[r 1 ] 2 ]  2  1  Asset 2 0 Minimum-variance portfolio (no short sales) E[r p ] Portfolio of mostly Asset 2 Asset 1 Portfolio of mostly Asset 1 Short selling Minimum-variance with short sales

33 Perfect Correlation: Examples l Derivatives have very high correlations with the underlying assets: »Futures and Forwards »Options l Use these assets (with short positions) to hedge risk Example: l You have $900 to invest into any combination of two assets: »A stock currently trading at $100 with an expected return of 0.2% and a volatility of 2.5% for a one-week return »A call option on the stock with an option delta of 0.4, currently trading at $4.00 »How can you minimize the risk of your portfolio for the coming week?

34 Perfect Correlation: Example l First, observe that if the stock moves by one standard deviation, this is $3.00. Then the standard deviation of the call is: Hence, the one standard deviation movement of the call is $1.00, hence  C =$1.00/$4.00=25%. l We can now use the formula for the minimum variance portfolio to give: Hence, you write 25 call options, and invest the proceeds of $100 plus your $900 into 10 stocks.

35 Imperfect Correlation l What happens in the general case where -1<  12 <  »With less than perfect correlation, -1<  12 <    , diversification helps reduce risk, but risk cannot be eliminated completely. –Minimum variance portfolio has positive weights in both assets »If correlation is large,     <   12 < 1, diversification kdoes not help to reduce risk. –Minimum variance portfolio has negative weight in one asset

36 Non-Perfect Correlation The Case of low correlation E[r] E[r 1 ] 2 ]  2  1  Asset 2 0 Minimum-variance portfolio E[r p ] Portfolio of mostly Asset 2 Asset 1 Portfolio of mostly Asset 1

37 Example l Assume  12 =0.25 What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio? l Portfolio Weights l Expected Return and Standard Deviation

38 Non-Perfect Correlation The Case of high correlation E[r 1 ] 2 ]  2  1  Asset 2 0 Minimum-variance portfolio E[r p ] Portfolio long in asset 2, short in asset 1 Asset 1 Portfolio of mostly Asset 1

39 Example l Assume  12 =  What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio? l Portfolio Weights l Expected Return and Standard Deviation

40 Limits to Diversification l Consider an equally-weighted portfolio. The variance of such a portfolio is: l As the number of stocks gets large, the variance of the portfolio approaches: l The variance of a well-diversified portfolio is equal to the average covariance between the stocks in the portfolio.

41 Limits to Diversification l What is the expected return and standard deviation of an equally- weighted portfolio, where all stocks have E(r j ) = 15%,  j = 30%, and  ij =.40?

42 Limits to Diversification Market Risk Total Risk Firm-Specific Risk Portfolio Risk,  Number of Stocks Average Covariance

43 Specific Risk and Market Risk l Examples of firm-specific risk »A firm ’ s CEO is killed in an auto accident. »A wildcat strike is declared at one of the firm ’ s plants. »A firm finds oil on its property. »A firm unexpectedly wins a large government contract. l Examples of market risk: »Long-term interest rates increase unexpectedly. »The Fed follows a more restrictive monetary policy. »The U.S. Congress votes a massive tax cut. »The value of the U.S. dollar unexpectedly declines relative to other currencies.

44 Efficient Portfolios with Multiple Assets E[r]  0 Asset 1 Asset 2 Portfolios of Asset 1 and Asset 2 Portfolios of other assets Efficient Frontier Minimum-Variance Portfolio Investors prefer

45 Efficient Portfolios with Multiple Assets l With multiple assets, the set of feasible portfolios is a hyperbola. l Efficient portfolios are those on the thick part of the curve in the figure. »They offer the highest expected return for a given level of risk. l Assuming investors want to maximize expected return for a given level of risk, they should hold only efficient portfolios. l Common sense procedures: »Invest in stocks in different industries. »Invest in both large and small company stocks. »Diversify across asset classes. –Stocks –Bonds –Real Estate »Diversify internationally.

46 Summary l It is not possible to characterize securities in terms of risk alone »Need to understand risk l Risky investments »More risky investments have higher returns »Risk premia are not related to the risk of individual assets l Diversification benefits »Depend on correlation of assets »Possiblity of short sales »Cannot eliminate market risk l Minimum variance portfolios »Riskless if correlation perfectly negative »Applications for hedging