2PortfoliosDefinition: A portfolio is a collection of assets (which could be a single asset if you so choose)This is the ONE CHOICE an investor hasYou don’t choose which stocks are available to invest in (you do not choose whether Microsoft stock exists)You don’t choose the how risky a stock isYou don’t choose the rate of return the stock offers
3Portfolio ChoiceHow you choose your portfolio, however, does have consequencesIf you put all your money into Treasury Bills, you will have no risk, and a low expected returnIf you put all of your money into MCI you will have a high risk and high expected return
4Portfolio WeightYour portfolio choice is measured by its portfolio weight. This is the $ proportion that you invest in each assetLet wi = portfolio weight
5Portfolio Weight Example You invest $100 in stocks, $80 in bonds, and $20 in Tbills. What are your portfolio weights? (Note the value of your portfolio is $200)Wstocks = 100/200 = 50%Wbonds = 80/200 = 40%WTBills = 20/200 = 10%Note: w1 + w2 + w wN = 1i.e. Portfolio weights must sum to 1.
6Expected Return on a Portfolio The expected return on a portfolio is weighted return of the individual securities.E(Rp) = w1E(R1)+w2E(R2)+w3E(R3) wNE(RN)For the previous example, suppose the expected return on stock is 10%, the expected return on bonds is 5%, and the expected return on Tbills is3%, the the expected portfolio return is:E(Rp) = 0.5*10% + 0.4*5% + 0.1*3%= 5% + 2% + 0.3%= 7.3%
7But this is WRONG!! Risk for a Portfolio Why? “Risk Canceling” Given that the expected return for a portfolio isE(Rp) = w1E(R1)+w2E(R2)+w3E(R3) wNE(RN)By analogy one may think the risk, as measured by variance could be computed as:Var(Rp) = w1Var(R1)+w2Var (R2)+w3Var (R3) wNVar (RN)But this is WRONG!!Why? “Risk Canceling”
8A Contrived ExampleAsset YESPUR will pay $100 if the Spurs take the NBA championship next year, and zero otherwiseAsset NOSPUR pays $100 if the Spurs do not win the NBA championship and pays zero otherwise.Both of the above assets are risky, as we do not know who will win the NBA this season
9Example, ContinuedSuppose you buy one share of YESPUR for $10, and one share of NOSPUR for $85. What will the return on your portfolio be?Whether the Spurs win or lose, this investment will have a payoff of $100. The cost to buy this portfolio is $95.
10Example - SummaryThis example shows a case where one can combine two risky assets to create a risk free asset. In other words, it shows that judicious choice of risky assets can reduce overall portfolio riskIn most cases, one cannot eliminate all risk in a portfolio, but one can reduce risk through judicious choices of which securities to include in your portfolio
11DiversificationMost individual stock prices show higher volatility than the price volatility of portfolio of all common stocks.How can the standard deviation for individual stocks be higher than the standard deviation of the portfolio?Diversification: investing in many different assets reduces the volatility of the portfolio.The ups and downs of individual stocks partially cancel each other out.
12The standard deviation of the portfolio is lower than the standard deviation of either Coke or Wendy’s
13The Impact of Additional Assets on the Risk of a Portfolio Number of StocksSystematic RiskPortfolio of 11 stocksAMDUnsystematic RiskAMD + American AirlinesAMD + American Airlines + Wal-MartPortfolio Standard Deviation
14Systematic and Unsystematic Risk Diversification reduces portfolio volatility, but only up to a point. Portfolio of all stocks still has a volatility of 21%.Systematic risk: the volatility of the portfolio that cannot be eliminated through diversification.Unsystematic risk: the proportion of risk of individual assets that can be eliminated through diversification, for example, by buying mutual funds. Because this risk can be eliminated, there is no reward for holding unsystematic riskWhat really matters is systematic risk….how a group of assets move together.
15Systematic and Unsystematic Risk Anheuser Busch stock had higher average returns than Archer-Daniels-Midland stock, with smaller volatility.American Airlines had much smaller average returns than Wal-Mart, with similar volatility.The tradeoff between standard deviation and average returns that holds for asset classes does not hold for individual stocks.Because investors can eliminate unsystematic risk through diversification, market rewards only systematic risk.Standard deviation contains both systematic and unsystematic risk.
16Risk Reduction with Diversification St. DeviationUnique RiskMarket RiskNumber of Securities
17Return on a PortfolioRule 3: The expected rate of return on a portfolio is a weighted average of the expected rates of return of each asset comprising the portfolio, with the portfolio proportions as weights.Example: A two asset portfolio of Debt and EquityE(rp ) = wE(rD) + (1-w)E(rE)w = Proportion of funds in Debt(1-w) = Proportion of funds in EquityE(rD) = Expected return on DebtE(rE) = Expected return on EquityThe return depends on the portfolio weight chosen
18Risk for a 2-asset portfolio The risk depends on the portfolio weight chosen
19Data (like page 226 of Textbook DebtEquityExpected Return8%13%Standard Deviation12%20%Consider correlations of -1, 0, 1
22Portfolio Risk Observations If the assets are perfectly correlated, a straight line describes the changing riskIf the assets are not perfectly correlated, risk is reduced for some combinations of the assetsFor perfect negative correlation, there exists a portfolio with zero riskCan use calculus to find the portfolio with the least risk.
23Use calculus to find min risk Take derivative w.r.t. w (weight in debt), then set = to zero, and solve for the w that minimizes risk. The solutions is:Note: This is not the StDev, its that ptf weight you must choose to get the minimum StDev. You then solve for the Stdev
24Can write the results in terms of Corr RecallCovariance FormCorrelation Form
26Correlation EffectsThe relationship depends on correlation coefficient.-1.0 < < +1.0The smaller the correlation, the greater the risk reduction potential.If r = +1.0, no risk reduction is possible.
31Risk and Return: Minimum Variance rp = .6087(.10) (.14) = .1157s= [(.6087)2(.15)2 + (.3913)2(.2)2 +p1/22(.6087)(.3913)(.2)(.15)(-.3)]s1/2= [.0102]= .1009p
32Adding a Risk Free Asset With the addition of the risk free asset, to create a complete portfolio, we can create an infinite number of CAL’s by drawing a straight line from the Rf to the portfolio.To prevent mean variance dominance, however, only the CAL with the maximum slope will be non dominatedFinding the efficient CAL is a calculus problem
35Calculus ProblemMaximize the slope of the CAL subject to choosing a risk portfolioDifferentiate w.r.t. w, set equal to zero, and solve to the w that maximizes slope. The results is Equation 8.7 on page 237
36Solve for optimal risky portfolio Using Equation 8.7 you can solve for the optimal risky portfolio, in terms of E(r) and Standard Deviation. You can then use this information to compute the slope of the CAL.Complete portfolios will be combination of this optimal portfolio and the risk free asset
37How should you choose y?Recall that y is the measure of the proportion of risky assets in a complete portfolioAt this points we want to choose y to maximize our utility, which we showed last chapter to be:
38Extending Concepts to All Securities First plot all risky assetsThen draw the two asset portfolios for each of the combination (depend on Corr)The outer locus of points will have the highest return for any level of risk, and are thus mean variance dominateThese optimal portfolios are called the Markowitz efficient frontier.These portfolios are mean variance dominant.
40Portfolio Selection & Risk Aversion U’’U’U’’’E(r)Efficientfrontier ofrisky assetsSPQLessrisk-averseinvestorMorerisk-averseinvestorSt. Dev
41Extending to Include Riskless Asset The set up is the same as when there was simply 2 risky assets. In other words there is a point that will maximize the slope of a CAL, and this represents the optimal mix of risky securities to hold along with the risk free asset
42Alternative CALs E(r) CAL (P) CAL (A) M M P P CAL (Global minimum variance)AAGFPP&FMA&F
43The separation property Because one CAL dominates the frontier of risky asset portfolios, all investors will choose to invest in the same risk portfolio and then will choose varying amounts of the risk free asset to maximize utility.Portfolio analysis can be separeted into two tasksDetermine the optimal risk portfolio (This mutual fund can serve all investors)Choose the asset allocation of the risk free versus the risky portfolio
44Efficient Frontier with Lending & Borrowing CALE(r)BQPArfFSt. Dev