Tài chính doanh nghiệp - Finance 407: Multinational financial management - Topic 19: Global portfolio optimization

Topic #19: Global Portfolio optimizationL. GattisThe Pennsylvania State UniversityFinance 407: Multinational Financial Management1Review2You plan to retire in 40 years and want to earn an annuity of $100,000 for 30 years that starts in 41 years. How much do you need to save each year (starting in on year) to have enough to buy your desired annuity? Assume a portfolio return of 6%. Hint: 1st compute how much you’ll need at retirement (PV), then compute how much you need to save annually (PMT)?$7,207$8,894$9,120$10,155$11,213Learning Objectives3Learning ObjectivesStudents understand and can recallThe process of portfolio construction and optimization Students can computeN-asset portfolio mean, volatility, and Sharpe ratiosOptimal combined portfolios that combine risky and risk-free assetsGlobal Portfolio4Suppose we would like create a portfolio that combines U.S., EAFE, and EM equity funds using the following data.Expected Return for a portfolio with N-assetsExpected rate of return of the portfolio:5i=1 to Nwiri1w1r12w2r2Sum=w1r1+w2r2Global Portfolio Return6What is the expected return of the global portfolio?Expected Return= A. 8.0%, B. 8.3%, C. 9.3% D. 9.5% Variance for a portfolio with N-assetsThe σ of a portfolio is not the weighted average σYou must account for how each asset pair is correlatedThe variance of the portfolio expected return is:where σij is the covariance between assets i and j andWhere ρij is the correlation between assets i and j78Variance of a 2-asset Portfolioi=1 to Nj =1 to Nwiwjσij11w1w1σ11= w12σ1212w1w2σ1221w2w1σ2122w2w2σ22= w22σ22Sum=w12σ12 +w22σ22 +2w1w2σ12Variance of a 3-asset Portfolio9i=1 to Nj =1 to Nwiwjσij11w1w1σ11= w12σ1212w1w2σ1213w1w3σ1321w2w1σ2122w2w2σ22= w22σ2223w2w3σ2331w3w1σ3132w3w2σ3233w3w3σ33= w32σ32Sum=w12σ12 +.+ w32σ32Covariance Matrix10The covariance matrix shows the covariances ( ij) between the returns on any pair of securities. The diagonal simply shows all the variances.Note: σ11= σ12 and σ12= σ21Compute the Missing Covariance Terms11A. .08, B. .16, C. .04, D. .016 A. .2, B. .4, C. .04, D. .02 12Three security portfolio12To find the variance of this portfolio, we need the portfolio weights and the variances and covariancesThe 3-asset portfolio variance has 9 terms: (a 4 asset return would have 16 terms)Compute the Missing Terms and Portfolio Standard Deviation13A. .02, B. .0002, C. .004, D. .0004 A. .00326, B. .002, C. .014, D. .0011 21Does This Global Portfolio Outperform US, EAFA, and EMA?14Assume a risk-free investing and borrowing rate of 2%. Is this new portfolio allocation (50/40/10) better than EAFE, EM, and US? (remember CAL)YesNoCannot DetermineEfficient Frontier15We may be able to do better than this one global portfolioRun solver in excel, to minimize portfolio standard deviation by changing the weights to apply to US, EAFE, and EMEfficient Frontier16You can see that the 50/40/10 Global is dominatedRunning solver for several more levels of return provides an “envelope” or “frontier”Extending to Include Riskless Asset17If there were no risk-free asset, you would choose one of the envelope portfolios based on your risk-return preferencesConsider what happens once we add a risk-free asset. Combining the risk-free asset with one particular portfolio of risky securities generates a new efficient frontier which is linear.This is called the Capital Allocation Line (CAL) which has the highest slope (Sharpe) attainable.To get this optimal risky portfolio, run solver to maximize the portfolio Sharpe ratioSolver Parameters: Max Sharpe Ratio18Globally Optimal Risky Portfolio19The Optimal Global Portfolio20The final step in this analysis is for an investor to select one specific portfolio of risk-free and the optimal risky portfolio which is “right” for her.This portfolio will be on the CALMaximizes Sharpe Ratio: (Rp-Rf)/σpFinding the Optimal Combined Portfolio (Rf (2%) + Optimal Risky)21Suppose an investor desires a 10% standard deviation investing in the optimal portfolio and risk free asset. What expected return should she get?A. 7.14% B 5.28% C 6.36% D. 8.65Suppose an investor desires a 5% expected return. What standard deviation will she get?A. 6.55% B 3.67% C 4.35% D. 5.84%CAL: R=Rf+SσiClicker: Class EvaluationHow would you rate today’s class? Highest Lowest22Problems231. Using the data provided for US, EAFE, and EMA, compute the expected return and standard deviation of a portfolio that invests equal amounts in each asset. What is the Sharpe ratio of this portfolio (Rf=2%)? Mean = (1/3)*(9+9.5+10)=9.5; SD = Sum(W1W2ρ12σ1σ2)^.5=((1/3)^2*(1*.16^2+.6*.16*.17+.5*.16*.2+.6*.16*.17+1*.17^2+.5*.17*.2+.5*.16*.2+.5*.17*.2+1*.2^2))^.5=.1465Sharpe =(.095-.02)/.1465=.51192. Using the data provided for US, EAFE, and EMA to compute the expected return and standard deviation of a portfolio that invests 30% in US, 50% in EAFE, and 20% in EMA. What is the Sharpe ratio of this portfolio (Rf=2%)? Mean = (.3*9+.5*9.5+.2*10)=9.45; SD =(.3^2*1*.16^2+.3*.5*.6*.16*.17+.3*.2*.5*.16*.2+.3*.5*.6*.16*.17+.5^2*1*.17^2+ .5*.2*.5*.17*.2+.3*.2*.5*.16*.2+.5*.2*.5*.17*.2+.2^2*1*.2^2))^.5=.1461Sharpe =(.0945-.02)/.1461=.50993. Using the optimal risk portfolio and CAL found on the slide 21, find the optimal combined portfolio standard deviation if you desire 3% mean? What is the weights in the global optimal portfolio and Rf? (Rf=2%)Optimal Portfolio: Mean = 9.44%, Vol = 14.48%, Sharpe =51.4%, W(US)=37.5, W(EAFA)=36.81%, W(EM)=25.69%Optimal Complete 3% Mean Portfolio vol = (.03-.02)/.514=1.95%; W(GLOBAL)=.0195/.1448=13.47%, W(Rf)=86.57%4. Using the optimal risk portfolio and CAL found on the slide 21, find the optimal combined portfolio expected return if you desire 14% standard deviation? What is the weights in Global optimal portfolio and Rf? (Rf=2%)Optimal Portfolio: Mean = 9.44%, Vol = 14.48%, Sharpe =51.4%, W(US)=37.5, W(EAFA)=36.81%, W(EM)=25.69%Optimal Complete 14% vol Portfolio Mean = .02+.14*.514=9.2%; W(GLOBAL)=.14/.1448=96.7%, W(Rf)=3.3%Exam Formulas24