Chapter 12.1: Modern Portfolio Theory Fan Longzhen Outline ? Mean-variance analysis; ? Mean-variance analysis and utility maximization; ? Does high moment matter? Eliciting preference ? Through experiment: ? Consider gaining 50000 vs losing 10000 ? U(50000)=1, U(-10000)=0 ? Let G1 be a 50/50 gamble: ? Finding certainty equivalent X1 of G1: X1=?, then ? U(x1)=E[U(G1)]; ? Define G2 and G3 similarly: ? This yield five points: ? U(-10000)=0 ? U(x2)=0.75 ? U(x1)=0.5 ? U(x3)=0.25 ? U(50000)=1 ? ? ? ? = 5.0000,10 5.0000,50 1 G ? ? ? = 5.0 5.0000,50 1 2 X G ? ? ? ? = 5.0000,10 5.0 1 3 X G Maximize expected utility and mean-variance analysis ? What about mean-variance preference? ? Investors like mean, dislike variance: ? Consistent with expected utility? ? Consider second order Taylor expansion 22 ),( σμσμ baV ?= ...)(''2/1)(')())1(()( );1( 2 0000 0 +++=+= += rwUrwUwUrwUwU rww )( 2 ...))var())()((('' 2 1 )()(')()(),( 22 2 000 2 σμμ σμ +?∝ ++++== b rrEwUrEwUwUwEUV 0,/10 <<> σμ μ VbifV Version 1 of the investment problem ? Two dates: 0 and 1 (today and tomorrow); ? Current wealth W0 and future wealth W1; ? Preference U(W1); ? No consumption, no income, no dynamics; ? n assets with expected returns ? ?variance {} n RRR ,..., 21 ( )',...,, 21 n RE μμμμ == r v ? ? ? ? ? ? ? ? ? ? ? ? ? ? = 2 21 2 2 221 112 2 1 ... ... ... ... )var( nnn n n R σσσ σσσ σσσ r To be continued ? Investment problem ? Subject to {} )]([ 1 max WUE i ? ∑ ∑ = = = = += n i i n i iiP P RR RWW 1 1 01 1 )1( ω ω Expected return and variance ? Portfolio return ? Expected return ?variance RRR n i iiP ' 1 ?ω == ∑ = )(' 1 REERER n i iiP ?ω == ∑ = )( ')var( ij P R σ ?? =Σ Σ= Portfolio optimization without riskless asset ? Problem A ? Use method of lagrange ? Minimize L ? First order conditions: {} 11' ' ' 2 1 min = = Σ r v ? μμω ?? ω p tosubject i )1'1()'(' 2 1 v r ?λμ?μγ?? ?+?+Σ= p L 010 =??Σ?= ? ? r r λμγ? ? L 0'0 =??= ? ? p L μμ? γ v 011'0 =??= ? ? v ? λ L Minimum-Variance portfolio 0'1 1* 11 11 =?Σ+Σ Σ+Σ= ?? ?? p μμμγμλ μγλ? rr r r r 01'111 11 =?Σ+Σ ?? μγλ rrr Solution 0 , 2 ≥?= ? = ? = BACD D BA D BC pp μ γ μ λ Define μμ vv vvvv Σ=Σ=Σ= ?? ',11,11 11 CBA Properties of MVPs ? Characterization of MVP set: ? In the mean-standard deviation space, the curve is a hyperbola p p γμλ μγ?λ? μγλ? ??σ += += Σ+ΣΣ= Σ= ?? r r r r *'1*' 1(*' *' 11 2 ） pp γμλσ += 2 D CBA pp p +? = μμ σ 2 2 2 parabola The global minimum variance portfolio ?Let ? We have ? All MVPs are portfolios of two distinct portfolios; ? D CBA pp p +? = μμ σ 2 2 2 0 2 = p p d d μ σ 0 22 = ? D BA p μ AAA B ggg 1 , 1 , 1 2 r ? Σ === ?σμ )1,0, ' ',( 1* 11 11 =+= ? = Σ == Σ = += Σ+Σ= ?? ?? BA D B B C C BB BA ddd dg γλλ μμ μ?μ μ ? ?γ?λ μγλ? rr r v r r continued ? Both and are MVP’s ? Any two MVP’s ‘span’ the MV frontier. g ? d ? Mutual fund separation theorem ? Choose any two minimum-variance portfolios--calls them X any Y, then all minimum-variance portfolios can be expressed as a linear combination of X and Y. Alternative representation of MVP ? Covariance properties of MVP’s ? For any MVP P with weight ? Consider any two MVP’s P and Q ? )1( 1 )1( 1 1 1 11 11 11 11 μ? μ? μ??μ μμ μγλ? v v v v vv vvv v v v ?? ?? ?? ?? Σ?Σ= Σ?Σ= +=Σ ? +Σ ? = Σ+Σ= BA D BC D D BA D BC b a pba pp p ? v AA RR p pggp 1 1 '),cov( 1 = ΣΣ =Σ= ? ? ?? r r rr dpgpp ?δ?δ? vvv +?= )1( dqgqq ?δ?δ? vvv +?= )1( [ ] ? ? ? ? ? ? ??+=+= ?+?++??= A B A B D A A D ABA RR qp qp gdqpqpdqpgqpqp μμ δδ σδδδδσδδσδδ )( 11 )1()1(1)(1(),cov( 2 222 ） Portfolio correlation ? For any minimum-variance portfolio P except the global-minimum- variance portfolio, there exists a unique minimum-variance portfolio- denoted by Z which has zero covariance with P; ? Consider P and Z A B AD A B satisfieswhen A B A B D A A RR p z z zpzp ? ?= ??+= μ μ μ μμ 2 / ))(( 1 ),cov( Portfolio optimization with a riskless asset ? Expand universe of assets: ? N risky assets and 1 riskless asset; ? Return of riskless asset is Rf; ? proportion invested in riskless asset; ? proportion invested in riskless asset; ? Minimize variance becomes {}'..., 21 n ????，，= v ∑ = ? n i i 1 1 ? {} pf Rst μ?μ? ?? ? =?+ Σ )1'1(' ' 2 1 min v v rr vv v solution ? Form Lagrange ? First-order condition ))1'1('(' 2 1 fp RL v vvvvv ?μ?μλ?? ???+Σ= pff RR L μμ? λ =?+?= ? ? )1('0 v vv )1(0 v vv v f R L ?=Σ?= ? ? μλ? ? )1(* 1 v vv f R?Σ= ? μλ? ARBRC R RR R ff fp ff fp 2 1 2 )1()'1( +? ? = ?Σ? ? = ? μ μμ μ λ v v v v Solution-continued ?Set of MVP’s ? Two rays in space: ? Assume ? Sharpe ratio ? Two-fund separation still holds ? ----riskless fund; ? ----tangency portfolio M ARBRCE E R ff fp p 2 2 2 2, )( +?= ? = μ σ ),( σμ E R fp p ? = μ σ A B R f < E R p fp = ? σ μ 2 2 1 )( ,, )1( f m f f m f f m ARB E ARB BRC ARB R ? = ? ? = ? ?Σ = ? σμ μ ? v v Investor’s preference and Mean-Variance analysis ? Prefer higher expected return; ? Prefer lower variance of return; ? Why mean-variance analysis ? Multivariate normality: is sufficient; ? analytically tractable and insightful; ? second-order approximation of ? ? Ignore high-order terms; ? Is this always appropriates? ),( Σμ r )( 1 WU ...)( )( 2 1 )()([)]([ 2 0 2 2 00 00 +? ? ? +? ? ? += == WW W WU WW W U WUEWUE WWWW Investor’s preference and Mean-Variance analysis-- continued ? proportional to tangency portfolio; ? remainder invested in riskless asset; ? all investors behave similarly )'),1('(),()( 2 1 ??μ?σμ vv v v r Σ?+== ffpp RRVVWEU 0,0 21 ≤≥ VV 02)1( 21 =Σ+?= ? ? ?μ ? v v v v VRV V f )1( 2 * 1 2 1 v v f R V V ?Σ ? = ? μ? :*? Investor’s preference and Mean-Variance analysis-- continued ? no risk-free asset; ? The solution ? Optimal solutions [ ])1'1),(max 2 v v ?λσμ ?+（ pp V 1'10 120 21 v v v vv v ? λ?μ ? ?= ?Σ+= ? ? = VV L gd V BV V BV VV V ?? λ μ? vv v vv ) 2 1( 2 1 22 2 1 2 1 1 2 1 2 1 ++?= Σ+Σ?= ?? Does higher moments matter? ? Apart from tractability and traditions, there are no reasons not to consider third and higher moments in portfolio problems. Example ? Moments of return distributions 5 5 5 5 5 20 15 14.5 5 5 5 6 5 -10 -5 -5.5 R0(%) R1 R2 R3 Asset 0 Asset 1 Asset 2 Asset 3 1/31/31/3probabilityAsset Mean321state Between asset 0 and 1, which one would you choose? Between asset 1 and 2…… Between asset 2 and 3…… These returns have the following moments -0.578.165R3 08.165R2 012.255R1 00.005R0 (%) SkewnessSt. D.Mean { } xofDStExxESkewness ./][( 3/1 3 ?= Outline the equilibrium by using mean-variance- skewness analysis ?Skewnes ? Here we define skewness as ? Cosskewness of three assets: ? Skewness of portfolio ? Consider an investor with derived utility function ? Optimal portfolio characterized by moments ? 3 3 )( σ ERRE skewness ? = 3 )( ERREskewness ?= [ ]))()(( kkjjiiijk ERRERRERREm ???= ∑∑∑ ∑ === = = = n i n j n k ijkkjip n i iip mm RR 111 3 1 ωωω ? ),,( 32 ppp mV σμ 3 0 2 00 ,, mσμ Outline the equilibrium by using mean-variance- skewness analysis--continued ? The portfolio combination of asset and the optimal portfolio is characterized by the moments: ? Since at the optimum, differential changes in the optimal portfolio leave utility unchanged. 33 0 2 00 23 0 3 3 22 0 2 0 22 0 )1(3)1(3)1( )1(2)1( )1( iiiip iip ip mmmmm ωωωωωω σωσωωσωσ ωμμωμ +?+?+?= +?+?= +?= i )(3)(2)( 0 3 0003 2 00201 0 3 3 2 21 mmVVV d dm V d d V d d V iii p p ?+?+?= ++= = σσμμ ωω σ ω μ ? )1()( 3 )( 2 3 000 1 3 2 0 1 2 0 mm V V V V iioi ????= σσμμ Outline the equilibrium by using mean-variance- skewness analysis--continued ? The equation also holds for the riskless asset ? Solving (2) for and substitute for (1) ? The contribution of the skewness term ? Assume higher skewness is preferred ? A decrease in skewness requires an increase in expected return. )2( 32 3 0 1 3 2 0 1 2 0 m V V V V R f ++= σμ 3 0 00 0 2 0 0 0 00 1 3 03 0 0 , )( 3 )( m m V mV RR i i i i iififi == ?+?+= γ σ σ β γβμβμ 1 3 00 3 V V m i i ?= ? ?μ 0 3 1 3 00 <?= ? ? V V m i i μ 12 /2 VV