Ch4. Two Fund Separation and CAPM

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1 Ch4. Two Fund Separation and CAPM Two Fund Separation D portfolio frontier (i) Any portfolio on the portfolio frontier can be generated by a linear combination of two frontier portfolios or mutual funds. (ii) Two (mutual) fund separation: given any feasible portfolio, there exists a portfolio of two mutual funds such that individuals prefer at least as much as the original portfolio. â (i), (ii), u is concave 1. â SSD ªJR), ç two fund separation, two funds }Ê portfolio frontier,. 2. ÊÌ v, market should be clear,, market protfolio Ñ_Aí optimal portfolio 5 convex combination, Ä market portfolio 6}Ê portfolio frontier, v r m ªçA 1 _ fund, J riskless asset, r f ççø_ (monetary) fund. = CAPM A vector of risky asset returns r = ( r 1, r 2,..., r N ) exhibits Two Fund separation if there exists two mutual funds (or portfolio α 1, α 2 ), then for portfolio q, there exists a scalar λ, s.t E[u(λ r α1 + (1 λ) r α2 )] E[u( r q )], where u is concave SSD q λα 1 + (1 λ)α 2 E[λ r α1 + (1 λ) r α2 ] = E[ r q ] and Var(λ r α1 + (1 λ) r α2 ) Var( r q ) α 1, α 2 øìbê portfolio frontier,. ( 5-Ç, Ç2í portfolio frontier Ñ N risky assets í portfolio frontier) 1

2 α 1 = p ( mvp), α 2 = zc(p), β qp = Cov( r q, r p ) σ 2 ( r p ) VT λ, ;W (3.17) r q = (1 β qp ) r zc(p) + β qp r p + ε qp, where E[ ε qp ] = 0 = Q(β qp ) + ε qp w2 Q(β qp ) is the dominating portfolio Two Fund Separation E[u( Q(β qp ))] E[u( r q )] 2

3 b p Two fund separation E[ ε qp Q(β qp )] = 0, q (wõÿub ( ) SSD ) E[u( r q )] = E[u( Q(β qp ) + ε qp )] = E[E[u( Q(β qp ) + ε qp ) Q(β qp )]] if u is concave ;W Jensen s inequality E[u( z)] u(e[ z]) E[u(E[ Q(β qp ) + ε qp Q(β qp )])](if E[ ε qp Q(β qp ] = 0) = E[u( Q(β qp ))] ( ) J two fund separation A E[u( Q(β qp ))] E[u( r q )] max E[u(a r q + (1 a) Q(β qp ))] 5j@Ñ a = 0 a?¹ FOC = 0 when a = 0 E[u (a r q + (1 a) Q(β qp ))( r q Q(β qp ))] a=0 = E[u ( Q(β qp )) ε qp ] = 0 ; p q, E[ ε qp Q(β qp )] = 0 ( à, cq q, E[ ε qp Q(β qp )] 0, Jà } A E[u ( Q(β qp )) ε qp ] 0, ¹)}) 3

4 â E[ ε qp ] = 0 D q, s.t E[ ε qp Q(β qp )] 0 For z, E[ ε qp ] = E[E[ ε qp Q(β qp )]] = z E[ ε qp Q(β qp )] df ( Q) + z E[ ε qp Q(β qp )] df ( Q) = 0 z E[ ε qp Q(β qp )]df ( Q) = z E[ ε qp Q(β qp )]df ( Q) 0 (Ä E[ ε qp Q(β qp )] 0) assume u(y) = { K1 y if y < z K 1 z + K 2 (y z) if y z ( 5-Ç) (âç2ªõ u Å concave í K) E[u ( Q(β qp )) ε qp ] = E[E[u ( Q(β qp )) ε qp Q(β qp )]] = E[u ( Q(β qp ))E[ ε qp Q(β qp )]] = K 1 z E[ ε qp Q(β qp )]df ( Q) + K 2 z E[ ε qp Q(β qp )]df ( Q) = (K 1 K 2 ) z E[ ε qp Q(β qp )df (q)] 0 Ä E[ ε qp Q(β qp )] 0, A, ]) 4

5 One fund separation,?¹æê portfolio α U) E[u( r α )] E[u( r q )] for all q, where u is concave. For any q, r q = r α + ε q, / E[ ε q r α ] = 0 (?¹E[ r α ] = E[ r q ], Var( r α ) Var( r q )) F return øš, / α Ñ mvp (/ portfolio frontier degenerates to this point), nª? one fund separation. If r i N(( ),[ ]), and nonidentical expectations two fund separaption A r q = (1 β qp ) r zc(p) + β qp r p + ε qp, where r zc(p), r p, and ε qp are uncorrelated. OÄ multinormal 5cq A independent E[ ε qp Q(β qp )] = E[ ε qp ] = 0, ]) 5

6 If r i N(( ),[ ]) and identical expectations, one fund separation r q = r mvp + ε q where r mvp and ε q are uncorrelated. Cov( r q, r mvp ) = Cov( r mvp, r mvp ) + Cov( ε q, r mvp ) â3.12 = Var( r mvp ) Cov( ε q, r mvp ) = 0 Ä multinormal 5cq A independent E[ ε q r mvp ] = E[ ε q ] = 0 one fund separation. ( ) Two fund separation + market clearing market portfolio is on the portfolio frontier pà-: (1) i = 1, 2,..., I _ individuals (i) A W0 i wealth (â N risky assets A A5 wealth) I W0 i = W m0 = the total wealth in the economy i=1 = the value of all securities market portfolio (ii) w ij : i I w wealth Ê security j 5ª½ for any security j in the market I w ij W0 i = w mjw m0 i=1 I i=1 w ij W i 0 W m0 = w mj (where w mj : j asset 2Ò5 weight) the market portfolio weights are a convex combination of the portfolio weights of individuals. 6

7 (2) ç two fund separation A, Ä two funds.ê portfolio frontier,, _A F prefer í portfolio 6u two funds 5. (Ûb two fund separation concave 5_A, w prefer C6uzMí portfolio, ø ìê portfolio frontier,þ) (3) y, frontier portfolio 5,?Ê portfolio frontier,,?¹ two fund 5 }Ê portfolio frontier,, Ä ªJR)_A prefer 5 portfolio 6Ê portfolio frontier,. (4) 7 market portfolio u_a portfolio 5 ((1) 2F ) market portfolio Ê portfolio frontier, Ä r m Ê portfolio frontier, à r m D r zc(m) F5 r q, w2 q ÑL< portfolio. E[ r q ] = (1 β qm )E[ r zc(m) ] + β qm E[ r m ] for any security j (_$Cu ß) E[ r j ] = (1 β jm )E[ r zc(m) ] + β jm E[ r m ] (q and j.ê frontier,),hä[ý, in equilibrium, there is a linear restriction on the expected rate of return of any risky asset?ªÿa E[ r j ] = E[ r zc(m) ] + β jm (E[ r m ] E[ r zc(m) ]) (Ç) Ê E[ r j ] D β jm Þ,$A security market line (ÄÉ5? N risky assets) J E[ r m ] < E[ r zc(m) ] ( v market portfolio Ý efficient portfolio) E[ r j ] = E[ r m ] + β jzc(m) (E[ r zc(m) ] E[ r m ]) 7

(4.11) b p market portfolio is an efficient portfolio under a special case assume (i) r i N(( ),[ ]) (UA mean-variance model) (ii) u is increasing and strictly concave { individual prefers higher

8 (4.11) b p market portfolio is an efficient portfolio under a special case assume (i) r i N(( ),[ ]) (UA mean-variance model) (ii) u is increasing and strictly concave { individual prefers higher expected rate of return individual prefers a portfolio portfolio with lower s.d. IC Ê E[ r] σ( r) Þ,Ñ é0 (p Ñ p IC é0ñ )(Ç) r j N( ) r p N(E[ r p ], σ 2 ( r p )) E[u i (W i 0 (1 + r p))] = E[u i (W i 0 (1 + E[ r p] + σ( r p ) z))], where z N(0, 1) Define V i ( r p, σ( r p )) = E[u i (W i 0 (1 + E[ r p] + σ( r p ) z))] ; indifferent curve 5é0 V i E[ r p ] de[ r p] + V i σ( r p ) dσ( r p) = 0 8

9 IC 5é0 = de[ r Vi p] dσ( r p ) = σ( rp) V i E[ rp] > 0 V i σ( r p ) = E[u i ( W i ) zw 0 i] = W 0 i Cov(u i ( W i ), z) < 0 V i E[ r p ] = E[u i ( W i )W0 i] > 0 w2 W i = W0 i(1 + E[ r p] + σ( r p ) z) â IC 5 é0 FA ² efficient portfolio, 7 market portfolio u_a portfolio 5 convex market portfolio u efficient portfolio Zero-Beta Capital Asset Pricing Model (N risky assets) E[ r q ] = E[ r zc(m) ] + β qm (E[ r m ] E[ r zc(m) ]) (where (E[ r m ] E[ r zc(m) ]) > 0) 9

10 J a riskless asset (J-Ì5? Õ ) if r f < A C, ~õ portfolio e D zc(e) = r f ç α 1, α 2 ( v, e D r f ÌÊ portfolio frontier,) ( < v1 zeu market portfolio) r q = (1 β qe )r f + β qe r e + ε qe (3.19.2) (where (1 β qe )r f + β qe r e = Q(β qe )) the necessary and sufficient condition for two fund separation is: E[ ε qe Q(β qe )] Ä r f ÝÓœ = E[ ε qe r e ] = 0. (?¹E[ ε qe r e ] = 0 Ñ {( r j ) N j=1, r f}exhibit two fund separation 5 necessary and sufficient condition) If r f > A C, ~õportfolio e D zc(e ) = r f ç α 1, α 2 r q = (1 β qe )r f + β qe r e + ε qe (where (1 β qe )r f + β qe r e = Q(β qe )) necessary and sufficient condition for two fund separation E[ ε qe Q(β qe )] = E[ ε qe r e ] = 0 p5jd5 én, ~ 5 (4.4) r f +two fund separation = two fund monetary separation Two fund monetary separation + (r f < C A ) + risky assets are in strictly positive supply (risky assets 5 supply Ñ, [Jb market clearing, w demand?b Ñ, [ý ð }I risky asset,?¹.ªr I r f )+ FA5 efficient portfolio 5 weighted sum, 6ÿu market portfolio, 6}Ê N risky assets and 1 riskless asset 5 efficient portfolio, ~õ e, ßu market portfolio E[ r q ] r f = β qm (E[ r m ] r f ) (Capital Asset Pricing Model: CAPM)(independently derived by Lintner (1965), Mossin (1965), and Sharpe (1964)) zp: Ê1 riskless asset DN risky assets 58 -, FA I } r f, }e, /FA5I,, 6Ê1 riskless asset DN risky assets 5 portfolio frontier,, Ä market clear, FA5 e 5,, /Ñ market portfolio (market 10

11 portfolio uò,f securities F$A5 portfolio, FJøì6ÊÉ N risky assets F$A5 portfolio frontier,, eõ/ñ v, 5? 1 riskless asset D N risky asstes í portfolio frontier D all risky assets 5 portfolio frontier 5ñø >õ, FJeõ.Ñ market portfolio) ç r f = C A, â section 3.18ªø an individual puts all his wealth into the riskless asset and holds a zero-weighted-sum portfolio riskless asset be in strictly positive supply, risky assets be in zero net supply. OwõÊ market clearing 58 -, risky assets 5 demand D supply ÛÌÑ ([ ð } } ßI risky assets) J r f > A C, êôfi Ayu.ª?I Ñ{0ükr fí ß if E[ r] < r f E[u(W 0 (1 + r))] u(e[w 0 (1 + r)]) < u(w 0 (1 + r f )) FAI r f, U) market.? clear.uì â,hªø, ñ r f < C A, nu equilibrium, / v risk premium of the market portfolio is positive (?¹E[ r m ] > r f ) Capital Market Line 11

12 If r j N(( ),[ ]), âçøi, õ CAPM E[u i ( W i )( r j r f )] = 0, i, j w2 W i = W0 i(1 + r f + N w ij ( r j r f )) j=1 E[u i ( W i )]E[( r j r f )] = Cov(u i ( W i ), r j ) Stein s lemma if X and Ỹ are bivariate normally distributed Cov(g( X), Ỹ ) = E[g ( X)]Cov( X, Ỹ ) ('dl¾í delta method) E[u i ( W i )]E[( r j r f )] = E[u i ( W i )]Cov( W i, r j ) define i 5global ARA θ i E[u i ( W i )] E[u i( W (øo5ara ³ E[ ]) i )] 1 θ i E[ r j r f ]=Cov( W i, r j ) ( i θi 1 )E[ r j r f ] = i Cov( W i, r j )=Cov( i W i, r j )=Cov( M, r j ) (where M = i W i = W m0 (1 + r m )) E[ r j r f ] = W m0 ( i θ 1 i ) 1 Cov( r m, r j ) (where W m0 ( i θi 1 ) 1 : aggregate RRA) If r j = r m E[ r m r f ] = W m0 ( i θ 1 i ) 1 Var( r m ) W m0 ( i θ 1 i ) 1 = E[ r m r f ] Var( r m ) HŸ E[ r j r f ] = E[ r m r f ] Var( r m ) Cov( r m, r j ) = β jm E[ r m r f ] (CAPM) 12

13 Considering the quadratic utility, àçø_iõ CAPM. (4.16) u i (z) = a i z b i 2 z2 u i (z) = a i b i z HpE[u i ( W i )]E[ r j r f ] = Cov(u i ( W i ), r j ) E[a i b i Wi ]E[ r j r f ] = Cov(a i b i Wi, r j ) = b i Cov( W i, r j ) ( a i b i E[ W i ])E[ r j r f ] = Cov( W i, r j ) ( I i=1 a i b i E[ M])E[ r j r f ] = Cov( M, r j ) E[ r j r f ] = ( I i=1 E[ r m r f ] = ( I i=1 E[ r j r f ] = Cov( r m, r j ) Var( r m ) a i b i E[ M]) 1 W m0 Cov( r m, r j ) a i b i E[ M]) 1 W m0 Var( r m ) E[ r m r f ] 5? A, B s_ ß, 7/ assume they have the same expected time-1 payoffs %Èß A ß Cov(Payoff M, Payoff A ) > 0 β AM > 0 B.ß Cov(Payoff M, Payoff B ) < 0 β BM < 0 ¾ 6 B, Ä%ÈÏv, ª^ø-, Ê t = 0 v, B 5gÂòk A 5gÂ E[ r A ] > E[ r B ] r q = (1 β qe )r f + β qe r e + ε qe ( 6ª;Aø_ one factor model) Ÿ E[ ε je r e ] = 0 ª A two monetary fund separation, QO)ƒ CAPM. à ÛÊZÑ corr( ε je, ε ie ) = 0 Éb assets DÖ, s.t. ε jes can be diversified away by forming well-diversified portfolios. $A one factor model, r e is the only factor. 13

14 One (or more) factor model will hold for most of the assets approximately if there is no arbitrage opportunity (in the limit) and there are a large number of assets. APT vs. General Equilibrium Method (1).y5? state of nature 5 primitive security (2) 5?J risk factor Ñ3í primitive security (3) ÝJ General Equilibrium 5 demand supply íiv explain ß rate of return 5ZA, 7uâ %hôƒ5 primitive security 5 rate of return, Vn wf ßí rate of return. (General Equilibrium Theories are more ambitious) General Equilibrium No-Arbitrage Ê CAPM 2, â General Equilibrium R r m Ññøj b. Ê APT 2, ª pw 5 risk factor. APT: identify the sourses of systematic risk : spilt systematic risk into its fundamental components Equilibrium theories aim at providing a complete theory of value on the basis of primitives: preferences, technology, and market structure. However, arbitragebased theories can only provide a relative theory of value APT 5@à * Given the stochastic behavior of the underlying asset, APT ª option price. (à BS model) * â fundamental security market prices risk neutral measures. (Wàà complete market j binominal tree D risk neutral valuation) * â complete set of complex securities 5 prices A-D securities 5 price, 1J security or cash flow 5 value *,5, APT uâ market data Vj expected return D risk factor 5É[ 14

15 Arbitrage Pricing Theory (APT) by Ross (1976) * An arbitrage opportunity (in the limit) is a sequence of arbitrage portfolios whose expected rates of return are bounded away from zero, while their variances converge to zero. It is almost a free lunch. * ; p, if there is no arbitrage opportunity, then a linear relation among expected asset returns will hold approximately for most of the assets in a large economy. (where large means that the number of assets n is arbitrary large) r n j = an j + K βjk δ n k n + εn j, j=1, 2,..., n (the rates of return on risky assets are generated by a K-factor model) where E[ ε n j ] = 0 E[ ε n j εn l ] = 0, if l j σ 2 ( ε n j ) σ2 (cqδ k n Ñ rates of return on portfolios) assume y n j = (1 K βjk n )r f + K βjk δ n k n is a portfolio of K factors(portfolios) and the riskless asset r f which replicate the return of asset j (from general equilbrium model,?¹ñ K-factor CAPM: y n j r f = K βjk n ( δ k n r f)) (w2êr j ndyn j íβn jk uâ r j nd δ k n5ví e ),an j 6uâ r j nd δ k n5ví e )) ı p, for most of the assets in a large economy a n j + εn j = (1 K βjk n )r f Case 1: ε n j = 0 (trivial) ı p a n j = (1 K β n jk )r f if a n j < (1 K y n j, rn j β n l jk )r f (free lunch) 15

16 if a n j > (1 K y n j, rn j β n l jk )r f (free lunch) a n j + εn j = an j = (1 K β n jk )r f ). Case 2: ε n j 0 ;b pé Ö N _.Å a n j (1 K a n j (1 K β n jk )r f ε, j = 1, 2,..., N(n) N β n jk )r f, [ - Ö Å (4.21.1) ( ) J N.æÊ, [ýj {n l } = {n = K + 2, K + 3,..}, if n l, N(n l ) (N(n l ) [ý n l risky assets 2, N(n l ) _ linear valuation model.) ([ñrý M assets.å a n j (1 K we can construct N(n l )_ arbitrage portfolios (i) J a n l j (ii) Ja n l j > (1 K < (1 K β n jk )r f) β n l jk )r f, Q ò, r j n yn j 5 arbitrage portfolio β n l jk )r f, y n j rn j (i) a n l j (1 K rate of return (ii) (1 K 5 arbitrage portfolio β n l jk )r f + ε n l j β n l jk )r f a n l j εn l j ** ø N(n l )_ arbitrage portfolio J 1 N(n l ) expected rate of return ε (Ä a n j (1 K β n jk )r f ε, / E[ ε n j ] = 0) 5 weight,, expectation 16

17 variance of rate of return= 1 N(n l ) N 2 (n l ) (Ä corr( ε n j, εn l ) = 0) j=1 σ 2 ( ε n l j ) σ2 N(n l ) (ç N(n l ) v, M > 0, variance=0 øì}:â, ªP, ÝÌ 5 ÕG) çñr Öb5 asset.å a n j (1 K opportunity, almost a free lunch JbU ª arbitrage 8.æÊ,.â N, s.t. N(n) N for all n?¹ Öb5 asset Å a n j (1 K β n jk )r f ε â (4.19.1) E[ ]ª) E[ r n j K β n jk δ n k ] = an j Hpª) E[ r n j ] r f K β n jk (E[ δ n k ] r f) ε β n jk )r f, æê arbitrage When n N, a linear relation among expected asset returns holds approximately for most of the assets. This relation is called Arbitrage Princing Theory (APT) originated by Ross(1976). (4.23) (4.25) z, çn, ú Öb ß7k,APT }ú, OúÀø ß7k, ƒú.úu_½æ, Jcquúí, R Öý6u_½æ, 3ü¹bv R í upperbound. (Equilibrium rather than arbitrage arguments will be used in the derivation in sections (4.23) to (4.25). The resulting relationship among risky assets is called equilibrium APT ) (4.24), pwhen ε j 0 (1 k β jk )r f a j < 0 for j and α ij > 0 for i, j (α ij Ñ individual i I k asset j 5Àç)( M3baà 7 ε j is independent of all the other random variables5õ p.111) 17

18 (4.25), E[ r j ] r f K (E[ δ k ] r f ) S j I Ā eāsj/i Var( ε j ) (çu 0, u < 0, u 0, A i (z) u i (z)/u i (z) Ā, E[ ε j] = 0 ε j 1, ( ε 1,..., ε N, δ 1,..., δ K ) are independent) 18

ECON4510 Finance Theory Lecture 2

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