Portfolo Selecton wth Multple Rsky Securtes. Professor Lasse H. Pedersen Prof. Lasse H. Pedersen Outlne Investment opportunty set wth many rsky assets wth many rsky assets and a rsk-free securty Optmal portfolo choce and two-fund separaton Dversfable and non-dversfable rsk Prof. Lasse H. Pedersen Investment Opportunty Set wth Many Assets Expected Return Effcent Fronter Mnmum Varance Indvdual Assets Ineffcent Fronter St. Dev. of Return Prof. Lasse H. Pedersen 3
Optmal Portfolo Selecton wth Many Rsky Assets and a Rsk-Free. Create the set of possble mean-sd combnatons from dfferent portfolos of rsky assets. Fnd the tangency portfolo, that s, the portfolo wth the hghest Sharpe rato: E[R ] Rf SR 3. Choose the combnaton of the tangency portfolo and the rsk-free asset to sut your rsk-return preferences. Prof. Lasse H. Pedersen 4 Two-Fund Separaton All nvestors hold combnatons of the same two mutual funds : The rsk-free asset The tangency portfolo An nvestor s rsk averson determnes the fracton of wealth nvested n the rsk-free asset But, all nvestors should have the rest of ther wealth nvested n the tangency portfolo. Prof. Lasse H. Pedersen 5 Excell Example: Portfolo Optmzer Portfolo selecton wth 5 rsky assets and rskless asset. Why does securty have a large portfolo weght? Why do all securtes have postve portfolo weghts? Double mean return of securty : What s the effect on MVP? What s the effect on the tangency portfolo? Importance of correlaton: ρ 45 0, 0.7, 0.9. How do portfolo weghts depend on ths correlaton? Prof. Lasse H. Pedersen 6
Rsk Reducton n Dversfed Portfolos Suppose we start wth a typcal US stock. ow suppose we add stocks to the portfolo, all stock postons equally weghted. (The best mx s not, n general, equally weghted -- but ths s llustratve way of makng a general pont.) Prof. Lasse H. Pedersen 7 Rsk n Equally-Weghted Portfolos: Independent Returns Suppose we have an equally weghted portfolo (holdng weghts /) of ndependent stocks. The varance of the portfolo return s p average varance As the number of assets ncrease, the rsk s dversfed away. (The nsurance prncple.) Prof. Lasse H. Pedersen 8 Rsk n Equally-Weghted Portfolos: The General Case Suppose we have an equally weghted portfolo (holdng weghts /) of stocks. The varance of the portfolo return s: p + j> cov(r, R ) + ( -)/ average + average varance covarance j j> cov(r, R j) Prof. Lasse H. Pedersen 9
Rsk n Equally-Weghted Portfolos: The General Case What happens when goes to nfnty? Varance of portfolo return -> average covarance of returns Rsk of portfolo -> non-dversfable rsk Prof. Lasse H. Pedersen 0 Classfcatons of Rsk Part that cannot be dversfy away: `covarance rsk, systematc rsk or nondversfable rsk E.g. market rsk, macroeconomc rsk, ndustry rsk Part that can be dversfed away (n a large portfolo): dosyncratc rsk, non-systematc rsk, dversfable rsk or unque rsk E.g. Indvdual company news Prof. Lasse H. Pedersen Dversfable vs. on-dversfable Rsk When held n a portfolo some of the rsk of a stock dsappears. Or, the rsk contrbuton the stock makes to the portfolo s LESS than the rsk of the stock f held n solaton: total rsk n non - dversfable a stock rsk + dversfable rsk Investors want to be compensated for holdng whch knd of rsk? Prof. Lasse H. Pedersen
Percentage Rsk Reducton What s the percentage reducton n rsk we should expect from addng stocks to our portfolo? (In the graph 00% represents the typcal rsk of a US stock) Prof. Lasse H. Pedersen 3