Tài chính doanh nghiệp - Financial modeling - Topic 4: Portfolio risk - Return optimization

Financial ModelingTopic #4: Portfolio Risk-Return OptimizationL. Gattis12ReferencesFinancial Modeling 3rd Edition by Simon BenningaModeling SupportCh. 8: Portfolio ModelsCh. 9: Efficient Portfolios with Short SalesCh. 12: Efficient Portfolios without Short SalesLearning ObjectivesCompute optimal portfolio weights that combine risky portfolios and risk free assetsCompute efficient (max return/min risk) and optimal risky portfolios Compute optimal complete portfolios that combine a risk free asset or borrowing.3Data4EnvelopeAn efficient risky portfolio is one thatmaximizes return for a given volatility or minimizes volatility for a given returnA set of efficient portfolios is called the portfolio envelope or efficient frontierYou find an efficient portfolio by running Excel’s Solver add-in.Objective: Max Expected (Mean) Return (or minimize vol)Constraint: Vol = X (or return = Y)Constraint: Sum of Weights = 1 (100%)Optional Constraint: Weights non-negative (no short sales)5Envelope – Optimal Risky Portfolio(Minimize σ, given µ, S.T. Ʃwi=1, Wi=0)6Insert XY Scatter with Lines (for Envelope Vols and ReturnsAdding Asset Data Points on graph- Right click on graph, Select data / add- Add series label by editing dataPortfolio ChoiceWith No Risk-Free Asset7Select an efficient portfolio that meets your risk or return preferencesNotice how EFA is dominatedOptimal Portfolios With a Risk-Free AssetWhen you combine a risk free asset and a risky asset (or portfolio), the combined portfolio mean is the weighted average mean and the volatility is the weight in the risky asset times the volatility of the risky assetσCombined = Wrisky x σRiskyPortfolio combination of risky and risk free plot along a straight lineThere is only one portfolio that you will choose to combine with a risk free assets -- the portfolio is at a tangency between a line extending from Rf and the top surface of the envelope.8EnvelopeRfTangency – Optimal Risky PortfolioOptimal Risky PortfolioThe tangency line maximizes the slopeSlope = Sharpe Ratio = (Rp – Rf) / σpRun Solver: Max Sharpe s.t. ƩWi=1, Wi>=0The line is called the capital allocation line (CAL)9Optimal Portfolios With BorrowingBorrowing allows you to leverage the returns (and risk) of an envelope portfolioThere is only one portfolio that you will choose to combine with borrowing -- the portfolio is at a tangency between a line extending from Rb and the top surface of the envelope. 10RfTangency – Optimal Borrowing PortfolioRbCALCAL-BOptimal Borrowing Portfolio – CAL-BThe tangency line maximizes the slopeSlope = Sharpe Ratio = (Rp – Rb) / σpRun Solver: Max BSharpe s.t. ƩW=0, W>=011Finding Optimal Complete PortfolioAn optimal complete portfolio is a portfolio that combines ORP + Rf orOBP + Rb orEnvelope Portfolio12RfσOn CAL-B: Combine OBP and RbOn CAL: Combine ORP and RbOn Envelope: Run Solver to Max μ or Min σOBP=7.44ORP=6.80OBP=21.09ORP=16.23Finding Optimal Complete (Combining Rf/Rb and risky) PortfolioFind optimal complete portfolios for complete portfolio volatility of 9%, 18%, and 27%Complete (or combined) portfolios may combine risky assets with risk-free assets and borrowing18%: Envelope Portfolio: Contains SPY, EFA, EEM9:%: ORP + Rf: Contains SPY, EFA, EEM, and Rf27%: OBP + Rb: Contains SPY, EFA, EEM, and -Rb13RfσOBP=7.44ORP=6.80OBP=21.09ORP=16.23RbRisky+RfRisky+borrowingOn Envelope Solution: 18% VolRun solver: Max Mean, S.t. 18% Vol, ƩWi=1, Wi>=014On CAL Solution: 9% Complete VolSince σComplete = WRisky x σRiskyWRisky = σComplete / σRiskyWRisky = 9%/16.23% = 55.47% RRisky =55.47% x RORP + (1-55.37%) * RfWSPY, EEM, EFA= WRisky x (WSPY, EEM, EFA in ORP)15On CAL-B Solution: 27% Combined VolWrisky = σCombined /σRiskyWrisky = 27%/21.09% = 128.04% Rrisky = 128.04% x ROBP + (1- 128.04%) * RfWSPY, EEM, EFA= Wrisky x (WSPY, EEM, EFA in OBP)16Finding Optimal Complete PortfolioWith Expected Return TargetsFind optimal complete portfolios for complete portfolio expected returns 5%, 7%, 8%7%: Envelope Portfolio: Contains SPY, EFA, EEM5:%: ORP + Rf: Contains SPY, EFA, EEM, and Rf8%: OBP + Rb: Contains SPY, EFA, EEM, and -Rb17RfσOBP=7.44ORP=6.80OBP=21.09ORP=16.23RbRisky+RfRisky+borrowingFinding Optimal Complete PortfolioFind optimal complete portfolios for complete portfolio expected returns 5%, 7%, 8%7%: Envelope Portfolio: Contains SPY, EFA, EEMRun Solver: Minimize Vol, S.t. mean = 7%Which happens to be envelope portfolio #35:%: ORP + Rf: Contains SPY, EFA, EEM, and RfRcomplete=WRiskyRrisky+(1-WRisky)RfWrisky=(Rcomplete-Rf)/(Rrisky-Rf)8%: OBP + Rb: Contains SPY, EFA, EEM, and –RbRcomplete=WRiskyRrisky+(1-WRisky)RbWrisky=(Rcomplete-Rb)/(Rrisky-Rb)19SolutionNotes on Assignments and ExamsAssignment and exam problems may require much less effort to find an optimal complete portfolio.ORP and/or OBP may be givenNo Rf and/or borrowing (only envelope solutions)Short sales may be allowed. (do not restrict constraint values >= 0 (checked box in solver)20Want More? ObjectivesCompute efficient (max return/min risk) and optimal risky portfolios Compute optimal complete portfolios that combine a risk free asset or borrowing.22