8.0% E(R) 6.0% Lend. Borrow 4.0% 2.0% rf rf 0.0% 0.0% 1.0% 2.0% 3.0% 4.0% STD(R) E(R) Long A and Short B. Long A and Long B. Short A and Long B

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1 F8000 Valuato of Facal ssets Sprg Semester 00 Dr. Isabel Tkatch ssstat Professor of Face Ivestmet Strateges Ledg vs. orrowg rsk-free asset) Ledg: a postve proporto s vested the rsk-free asset cash outflow the preset: F 0 < 0, ad cash flow the future: F > 0) orrowg: a egatve proporto s vested the rsk-free asset cash flow the preset: F 0 > 0, ad cash outflow the future: F < 0) Ledg vs. orrowg Ivestmet Strateges Log vs. Short posto the rsky asset 8.0% 6.0% 4.0%.0% Led 0.0% 0.0%.0%.0% 3.0% 4.0% orrow STDR) Log: postve proporto s vested the rsky asset cash outflow the preset: F 0 < 0, ad cash flow the future: F > 0) Short: egatve proporto s vested the rsky asset cash flow the preset: F 0 > 0, ad cash outflow the future: F < 0) Log vs. Short Log ad Log Log ad Short Short ad Log Ivestmet Strateges Passve rsk reducto: The rsk of the portfolo s reduced f we vest a larger proporto the rsk-free asset relatve to the rsky oe The peect hedge: The rsk of asset s offset ca be reduced to zero) by formg a portfolo wth a rsky asset, such that ρ =-) Dversfcato: The rsk s reduced f we form a portfolo of at least two rsky assets ad, such that ρ <+) The rsk s reduced f we add more rsky assets to our portfolo, such that ρ j <+)

2 Oe Rsky Fud ad oe Rsk-free sset: Passve Rsk Reducto Two Rsky ssets wth ρ =-): The Peect Hedge 8.0% 6.0% 4.0%.0% Reducto portfolo rsk Icrease of portfolo Rsk Mmum Varace s zero 0.0% 0.0%.0%.0% 3.0% 4.0% STDR) P m The Peect Hedge a Example What s the mmum varace portfolo f we assume that μ =0%; μ =5%; σ =%; σ =6% ad ρ =- )? σ σ σρ w = σ + σ σ σρ 6%) %)6%) ) = = %) + 6%) %)6%) ) 3 The Peect Hedge otued What s the expected retur μ m ad the stadard devato of the retur σ m of that portfolo? μm = wμ + w) μ = 0% + 5% = 6 % σ = w σ + w ) σ + w w ) σ σ ρ m = %) + 6%) + % 6% ) = 0% Dversfcato: the orrelato oeffcet ad the Froter Dversfcato: the Number of Rsky assets ad the Froter ρ =-) -<ρ < ρ =+

3 Dversfcato: the Number of Rsky assets ad the Froter Dversfcato: the Number of Rsky assets ad the Froter Dversfcato: the Number of Rsky assets ad the Froter aptal llocato: Rsky ssets State all the possble vestmets how may possble vestmets are there? ssumg you ca use the Mea-Varace M-V) rule, whch vestmets are M-V effcet? Preset your results the μ-σ mea stadard-devato) devato) plae. The Expected Retur ad the Varace of the Retur of the Portfolo w = the proporto vested the rsky asset =, ) p = the portfolo of rsky assets w vested asset ) R p = the retur of portfolo p μ p = the expected retur of portfolo p σ p = the varace of the retur of portfolo p R = w R + w R w R = w R p = p) = μp = w μ = p) = σ p = j σj = j= ER V R w w The Set of Possble Portfolos the μ-σ Plae The Froter 3

4 The Set of Effcet Portfolos the μ-σ Plae aptal llocato: Rsky ssets The Effcet Froter The vestmet opportuty set: {all the portfolos {w, w } where Σw =} The Mea-Varace M-V or μ-σ ) effcet vestmet set: {oly portfolos o the effcet froter} The case of Rsky ssets: Fdg a Portfolo o the Froter Optmzato: Fd the mmum varace portfolo for a gve expected retur ostrats: gve expected retur; The budget costrat. The case of Rsky ssets: Fdg a Portfolo o the Froter { w,... w } M w w σ St.. = j= = = j j w μ = μ p w = aptal llocato: Rsky ssets ad a Rsk-free sset State all the possble vestmets how may possble vestmets are there? ssumg you ca use the Mea-Varace M-V) rule, whch vestmets are M-V effcet? Preset your results the μ-σ mea stadard-devato) devato) plae. The Expected Retur ad the Varace of the Retur of the Possble Portfolos w = the proporto vested the rsky asset =, ) p = the portfolo of rsky assets w vested asset ) R p = the retur of portfolo p μ p = the expected retur of portfolo p σ p = the varace of the retur of portfolo p R = w+ wr + wr wr = w+ wr p 0 0 = p) = μp = 0 + μ = p) = σ p = 0+ ww jσj = j= ER w w V R 4

5 The Set of Possble Portfolos the μ-σ Plae oly rsky assets) The Set of Possble Portfolos the μ-σ Plae rsk free asset cluded) The Froter The Set of Effcet Portfolos the μ-σ Plae The Separato Theorem μ The aptal Market Le: μ p = + [μ m -) / σ m ] σ p m The asset allocato process of the rsk-averse vestors ca be separated to two stages:.hoose the optmal portfolo of rsky assets m The allocato betwee rsky securtes s detcal for all the vestors).hoose the optmal allocato of fuds betwee the rsky portfolo m ad the rsk-free asset choose a portfolo o the ML The allocato betwee the rsky portfolo ad the rsk free asset s persoal ad depeds o the rsk prefereces of each vestor) aptal llocato: Rsky ssets ad a Rsk-free sset The vestmet opportuty set: {all the portfolos {w 0, w, w } where Σw =} The Mea-Varace M-V or μ-σ ) effcet vestmet set: {all the portfolos o the aptal Market Le - ML} Rsky ssets ad Oe Rsk-free sset: Fdg the Market Portfolo Optmzato: Fd the mmum varace portfolo for a gve expected retur ostrats: gve expected retur; The budget costrat. 5

6 Rsky ssets ad Oe Rsk-free sset: Fdg the Market Portfolo M { w,... w } = j= w w σ j j St.. wμ + -w = μ p = = Rsky ssets ad Oe Rsk-free sset: Fdg the Market Portfolo Solve the followg system of equasos ad fd the proportos { w,... w } vested the rsky assets wσ + w σ w σ = μ w σ + w σ w σ = μ... wσ + w σ w σ = μ Scale the proportos: w z = for =,... w j = ad m = { z, z,... z } s the market portfolo. j Example Fd the market portfolo f there are oly two rsky assets, ad, ad a rsk-free asset. μ =0%; μ =5%; σ =%; σ =6%; ρ =-0.5) ad =4% To fd the proportos { w, w} vested assets ad use the system of equatos for two rsky assets: w σ + w σ = μ w + w = σ σ μ Usg our data we get two equatos: w %) + w %)6%) 0.5) = 0% 4% w %)6%) 0.5) + w 6%) = 5% 4% Example otued If we solve the two equatos: w w + w = %) %)6%) 0.5) 0% 4% %)6%) 0.5) + w6%) = 5% 4% we get the proportos { w, w } = {0.0648,0.0959}. Now we have to scale the proportos w z = = = ad z = w + w ad m= { z, z } = {0.476,0.5884} s the market portfolo. Example otued The expected retur of the market portfolo s μm = zμ + zμ = % % = 7.06% The esa stadard daddeva devato of the retur eu of the market portfolo o os σ = z σ + z σ + z z σ σ ρ m = 0.476) %) ) 6%) % 6% -0.5) = 4.4% Practce Problems KM 7th Ed. h. 7: -3, 3, 7-, 5-66 KM 8th Ed. h. 7: 4-9 F: 4-6, 0- Mathematcs of Portfolo Theory: Read ad practce parts -33 6