Economics 487. Homework #4 Solution Key Portfolio Calculations and the Markowitz Algorithm

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1 Economics 87 Homework # Soluion Key Porfolio Calculaions and he Markowiz Algorihm A. Excel Exercises: (10 poins) 1. Download he Excel file hw.xls from he class websie. This file conains monhly closing price daa on four socks (Boeing Microsof Nordsrom and Sarbucks) over he period February 1995 January For his exercise you will use marix algebra o simplify he running of solver in Excel. a. Use he Excel funcions AVERAGE VAR STDEV COVAR and CORREL o esimae he expeced reurns variances sandard deviaions covariances and correlaions for he four socks using he hisorical reurn (Hin: you can also use he covariance feaure under ools/daa analysi) Repor you resuls in Table forma in your Word file. Commen on he esimaes paying paricular aenion o he correlaion Compue esimaed sandard errors for he means sandard deviaions and he correlaion Commen on he precision of your esimae Arrange he covariance esimaes ino a covariance marix. mu_a mu_b mu_c mu_d sigma_a sigma_b sigma_c sigma_d Esimae SE rho_ab rho_ac rho_ad rho_bc rho_bd rho_cd Esimae SE The sandard deviaions are fairly precise. Bu he expeced reurns and correlaions are much less precise. E.g. he expeced reurn for Nordsrom is quie low bu quie imprecisely esimaed. Covariance marix: Sigma b. Compue he global minimum variance porfolio allowing for shor-sale The minimizaion problem is min mʹ Σm m. mʹ 1 where m is he vecor of porfolio weighs Σ is he covariance marix and 1 is a x1 vecor of one Are here any negaive weighs in his porfolio? If so inerpre hem. Compue and repor he expeced reurn variance and sandard deviaion of his porfolio.

2 2b. m ones consrain variance E[Rp] Sd. Dev No negaive weigh c. Of he four socks deermine he sock wih he larges average hisorical reurn. Use his maximum average reurn as he arge reurn for he compuaion of an efficien porfolio allowing for shor-sale Tha is find he minimum variance porfolio ha has an expeced reurn equal o his arge reurn. The minimizaion problem is min xʹ Σx x xʹ 1 where x is he vecor of porfolio weighs is he vecor of expeced reurns and 0 is he arge expeced reurn. Are here any negaive weighs in his porfolio? Compue and repor he expeced reurn variance and sandard deviaion of his porfolio. Finally compue and repor he covariance beween he global minimum variance porfolio and he above efficien porfolio using he formula cov( R p m R p x mʹ ) = Σx. Targe porfolio opimizaion. 2c. x ones consrain consrain variance E[Rp] Sd. Dev xʹ = Cov(xm) The weighs on Boeing and Nordsom are negaive. This means an invesor should shorsell hem. 0 d. Using he fac ha all efficien porfolios can be wrien as a convex combinaion of wo efficien porfolios compue efficien porfolios as convex combinaions of he global minimum variance porfolio and he efficien porfolio compued in quesion 3. Tha is compue z = α x + ( 1 α) m for values of α beween -1 and 2 (make a grid for α = ). Compue and repor he expeced reurn variance and sandard deviaion of hese porfolios in a Table.

3 2d. alpha sigma_p mu_p E e. Plo he Markowiz bulle based on hese efficien porfolios ha you compued above. On he plo indicae he locaion of he minimum variance porfolio and he locaion of he efficien porfolio ha you found in par c. Pase his graph o your Word file.

4 f. Compue he angency porfolio assuming he risk-free rae is (i.e. r = 0.5% ) per monh. Tha is solve f r max σ σ 2. ʹ 1 where denoes he porfolio weighs in he angency porfolio. Are here any negaive weighs in he angency porfolio? If so inerpre hem. Compue and repor he expeced reurn variance and sandard deviaion of he angency porfolio. = ʹ = ʹ Σ 2f. ones consrain sd. Dev E[Rp] Sharpe Raio No negaive weigh g. On he graph wih he Markowiz bulle plo he efficien porfolios ha are combinaions of T-bills and he angency porfolio. Indicae he locaion of he angency porfolio on he graph. Pase his graph o your Word file. f

5 h. For bonus poins use he Benninga handou as a guide o compue and plo he Markowiz bulle for he daa in hw.xls when here are no shor sales allowed. On he same se of axes plo he efficien froniers wih and wihou shor sale resricion Calculae he angency porfolio imposing nonnegaiviy consrains on weighs for a variey of risk free rae The resul is ha he efficien fronier is he same as wih shor sales from he Global Min porfolio (which had only posiive weighs) up o a cerain expeced reurn. Then he efficien fronier is lower han he shor sales case beyond ha poin. See graph.

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