aa r X i v : . [ q -f i n . P M ] S e p Efficient Portfolios ∗Keith A. LewisSeptember 22, 2020
Given two random realized returns on an investment, which is to be preferred?This is a fundamental problem in finance that has no definitive solution ex-cept in the case one investment always returns more than the other. In 1952Markowitz(Markowitz 1952) and Roy(Roy 1952) introduced the following crite-rion for risk vs. return in portfolio selection: if two portfolios have the sameexpected realized return then prefer the one with smaller variance. An efficientportfolio has the least variance among all portfolios having the same expectedrealized return.In the one-period model every efficient portfolio belongs to a two-dimensionalsubspace of the set of all possible realized returns and is uniquely determinedgiven its expected realized return. We show that if R is the (random) realizedreturn of any efficient portfolio and R and R are the realized returns of anytwo linearly independent efficient portfolios then R − R = β ( R − R )where β = Cov( R − R , R − R ) / Var( R − R ). This generalizes the classicalCapital Asset Pricing Model formula for the expected realized return of efficientportfolios. Taking expected values of both sides when Var( R ) = 0 and R isthe “market” portfolio gives E [ R ] − R = β ( E [ R ] − R )where β = Cov( R, R ) / Var( R ).The primary contribution of this short note is observation that the CAPM for-mula holds for realized returns as random variables, not just their expectations.This follows directly from writing down a mathematical model for one periodinvestments. ∗ Peter Carr and David Shimko gave insightful feedback to make the exposition more ac-cessible to finance professionals. Any remaining infelicities or omissions are my fault. ne-Period Model The one-period model is parameterized directly by instrument prices. Thesehave a clear financial interpretation and all other relevant financial quantitiescan be defined in terms of prices and portfolios.Let I be the set of market instruments and Ω be the set of possible marketoutcomes over a single period. The one-period model specifies the initial instru-ment prices x ∈ R I and the final instrument prices X : Ω → R I depending onthe outcome. We assume, as customary, that there are no cash flows associatedwith the instruments and transactions are perfectly liquid and divisible. The oneperiod model also specifies a probability measure P on the space of outcomes.It is common in the literature to write R n instead of R I where n is the cardinal-ity of the set of instruments I . If A B = { f : B → A } is the set of functions from B to A then x ∈ R I is a function x : I → R where x ( i ) = x i ∈ R is the priceof instrument i ∈ I . This avoids circumlocutions such as let I = { i , . . . , i n } be the set of instruments and x = ( x , . . . , x n ) be their corresponding prices x j = x ( i j ), j = 1 , . . . , n .A portfolio ξ ∈ R I is the number of shares initially purchased in each instrument.The value of a portfolio ξ given prices x is ξ · x = P i ∈ I ξ i x i . It is the cost ofattaining the portfolio ξ . The realized return is R ( ξ ) = ξ · X/ξ · x when ξ · x = 0.Note R ( ξ ) = R ( tξ ) for any non-zero t ∈ R so there is no loss in assuming ξ · x = 1 when considering returns. In this case R ( ξ ) = ξ · X is the realizedreturn on the portfolio. It is common in the literature to use returns instead ofrealized returns where the return r is defined by R = 1 + r ∆ t or R = exp( r ∆ t )where ∆ t is the time in years or a day count fraction of the period. Since we areconsidering a one period model there is no need to drag ∆ t into consideration.Although portfolios and prices are both vectors they are not the same. A port-folio turns prices into a value. The function ξ ξ · x is a linear functional fromprices to values. Mathematically we say ξ ∈ ( R I ) ∗ , the dual space of R I . If V is any vector space its dual space is V ∗ = L ( V, R ) where L ( V, W ) is the spaceof linear transformations from the vector space V to the vector space W . If wewrite ξ ′ to denote the linear functional corresponding to ξ then ξ ′ x = ξ · x is thelinear functional applied to x . We also write the dual pairing as h x, ξ i = ξ ′ x .Note that xξ ′ is a linear transformation from R I to R I defined by ( xξ ′ ) y = x ( ξ ′ y ) = ( ξ ′ y ) x since ξ ′ y ∈ R is a scalar. Matrix multiplication is just composi-tion of linear operators. Model Arbitrage
There is model arbitrage if there exists a portfolio ξ with ξ ′ x < ξ ′ X ( ω ) ≥ ω ∈ Ω: you make money on the initial investment and never lose money2hen unwinding at the end of the period. This definition does not require ameasure on Ω.The one-period Fundamental Theorem of Asset Pricing states there is no modelarbitrage if and only if there exists a positive measure Π on Ω with x = R Ω X ( ω ) d Π( ω ). We assume X is bounded, as it is in the real world, and Πis a finitely additive measure. The dual space of bounded functions on Ω is thespace of finitely additive measures on Ω with the dual pairing h X, Π i = R Ω X d
Π(Dunford and Schwartz 1963) Chapter III.If x = R Ω X d
Π for a positive measure Π then all portfolios have the sameexpected realized return R = 1 / k Π k where k Π k = R Ω d Π is the mass of Πand the expected value is with respect to the risk-neutral probability measure Q = Π / k Π k . This follows from E [ ξ ′ X ] = ξ ′ x/ k Π k = Rξ ′ x for any portfolio ξ .Note Q is not the probability of anything, it is simply a positive measure withmass 1. The above statements are geometrical, not probabilistic. Efficient Portfolio
A portfolio ξ is efficient if its variance Var( R ( ξ )) ≤ Var( R ( η )) for every portfolio η having the same expected realized return as ξ .If ξ ′ x = 1 then Var( R ( ξ )) = E [( ξ ′ X ) ] − ( E [ ξ ′ X ]) = E [ ξ ′ XX ′ ξ ] − E [ ξ ′ X ] E [ X ′ ξ ] = ξ ′ V ξ , where V = Var( X ) = E [ XX ′ ] − E [ X ] E [ X ′ ]. We canfind efficient portfolios using Lagrange multipliers. For a given realized return ρ we can solvemin 12 ξ ′ V ξ − λ ( ξ ′ x − − µ ( ξ ′ E [ X ] − ρ )for ξ , λ , and µ . The first order conditions for an extremum are V ξ − λx − µE [ X ] = 0, ξ ′ x = 1, and ξ ′ E [ X ] = ρ . Riskless Portfolio
A portfolio ζ is riskless if its realized return is constant. In this case 0 =Var( R ( ζ )) = ζ ′ V ζ assuming, as we may, ζ ′ x = 1. If another riskless portfolioexists with different realized return then arbitrage exists. By removing redun-dant assets we can assume there is exactly one riskless portfolio ζ with ζ ′ x = 1.Let P k = ζζ ′ /ζ ′ ζ . Note P k ζ = ζ and P k ξ = 0 if ζ ′ ξ = 0 so it is the orthogonalprojection onto the space spanned by ζ . Let P ⊥ = I − P k be the projection ontoits orthogonal complement, { ζ } ⊥ = { y ∈ R I : ζ ′ y = 0 } , so V = V P ⊥ + V P k .Below we analyze the first order conditions for an extremum on each subspace.Note P k commutes with V so these subspaces are invariant under V . Let y k = P k y be the component of y parallel to ζ and y ⊥ = P ⊥ y be the component of y orthogonal to ζ for y ∈ R I . 3he first order condition V ξ = λx + µE [ X ] implies V ξ k = λx k + µE [ X ] k . Since ξ k is a scalar multiple of ζ we have 0 = λ + µR so λ = − µR . On the orthogonalcomplement V ξ ⊥ = − µRx ⊥ + µE [ X ] ⊥ so ξ ⊥ = V ⊣ ( E [ X ] − Rx ) where V ⊣ is thegeneralized (Moore-Penrose) inverse of V . Letting α = ξ ⊥ = V ⊣ ( E [ X ] − Rx ),every efficient portfolio can be written ξ = µα + νζ . We can and do assume α ′ x = 1 so 1 = µ + ν and ξ = µα + (1 − µ ) ζ . Multiplying both sides by X wehave ξ ′ X = µα ′ X + (1 − µ ) R hence R ( ξ ) − R = µ ( R ( α ) − R ) . This implies the classical CAPM formula by taking expected valueswhere α is the “market portfolio”. It also shows the Lagrange multiplier µ = Cov( R ( ξ ) , R ( α )) / Var( R ( α )) is the classical beta. Non-singular Variance If V is invertible the Appendix shows solution is λ = ( C − ρB ) /D , µ = ( − B + ρA ) /D , and ξ = C − ρBD V − x + − B + ρAD V − E [ X ]where A = xV − x , B = x ′ V − E [ X ] = E [ X ′ ] V − x , C = E [ X ] V − E [ X ], and D = AC − B . The variance of the efficient portfolio isVar( R ( ξ )) = ( C − Bρ + Aρ ) /D. If ξ and ξ are any two independent efficient portfolios then they belong tothe subspace spanned by V − x and V − E [ X ]. Every efficient portfolio can bewritten ξ = β ξ + β ξ for some scalars β and β . Assuming ξ ′ j x = 1 for j = 0 , R ( ξ j ) = ξ ′ j X . Assuming ξ ′ x = 1 so R ( ξ ) = ξ ′ X then β + β = 1and ξ = (1 − β ) ξ + βξ where β = β . Multiplying both sides by X we have ξ ′ X = (1 − β ) ξ ′ X + βξ ′ X hence R ( ξ ) − R ( ξ ) = β ( R ( ξ ) − R ( ξ ))as functions on Ω where β = Cov( R ( ξ ) − R ( ξ ) , R ( ξ ) − R ( ξ )) / Var( R ( ξ ) − R ( ξ )). The classical CAPM formula follows from taking expected values ofboth sides when ξ is the “market portfolio” and ξ is a riskless portfolio .Note that A , B , C , and D depend only on x , E [ X ], and E [ XX ′ ]. Classical lit-erature focuses mainly on the latter three which may explain why prior authorsoverlooked our elementary but stronger result.4 ppendix Lagrange Multiplier Solution
Let’s find the minimum value of Var( R ( ξ )) given E [ R ( ξ )] = ρ . If ξ ′ x = 1 then R ( ξ ) = ξ ′ E [ X ] and Var( R ( ξ )) = ξ ′ V ξ where V = E [ XX ′ ] − E [ X ] E [ X ′ ].We use Lagrange multipliers and solvemin 12 ξ ′ V ξ − λ ( ξ ′ x − − µ ( ξ ′ E [ X ] − ρ )for ξ , λ , and µ .The first order conditions for an extremum are0 = V ξ − λx − µE [ X ]0 = ξ ′ x −
10 = ξ ′ E [ X ] − ρ Assuming V is invertible ξ = V − ( λx + µE [ X ]). Note every extremum lies inthe (at most) two dimensional subspace spanned by V − x and V − E [ X ].The constraints 1 = x ′ ξ and ρ = E [ X ′ ] ξ can be written (cid:20) ρ (cid:21) = (cid:20) λx ′ V − x + µx ′ V − E [ X ] λE [ X ′ ] V − x + µE [ X ′ ] V − E [ X ] (cid:21) = (cid:20) A BB C (cid:21) (cid:20) λµ (cid:21) with A = xV − x , B = x ′ V − E [ X ] = E [ X ′ ] V − x , and C = E [ X ] V − E [ X ].Inverting gives (cid:20) λµ (cid:21) = 1 D (cid:20) C − B − B A (cid:21) (cid:20) ρ (cid:21) = (cid:20) ( C − ρB ) /D ( − B + ρA ) /D (cid:21) where D = AC − B . The solution is λ = ( C − ρB ) /D , µ = ( − B + ρA ) /D , and ξ = C − ρBD V − x + − B + ρAD V − E [ X ] . A straightforward calculation shows the variance isVar( R ( ξ )) = ξ ′ V ξ = ( C − Bρ + Aρ ) /D. Fundamental Theorem of Asset Pricing
The one-period Fundamental Theorem of Asset Pricing states there is no modelarbitrage if and only if there exists a positive measure Π on Ω with x = R Ω X ( ω ) d Π( ω ). We assume X is bounded, as it is in the real world, and Πis finitely additive. 5f such a measure exists and ξ · X ≥ ξ · x = R Ω ξ · X d Π ≥ Lemma. If x ∈ R n and C is a closed cone in R n with x C then there exists ξ ∈ R n with ξ · x < and ξ · y ≥ for y ∈ C . Recall that a cone is a subset of a vector space closed under addition andmultiplication by a positive scalar, that is, C + C ⊆ C and tC ⊆ C for t > Proof.
Since C is closed and convex there exists x ∗ ∈ C with 0 < || x ∗ − x || ≤|| y − x || for all y ∈ C . Let ξ = x ∗ − x . For any y ∈ C and t > ty + x ∗ ∈ C so || ξ || ≤ || ty + ξ || . Simplifying gives t || y || + 2 tξ · y ≥
0. Dividingby t > t decrease to 0 shows ξ · y ≥
0. Take y = x ∗ then tx ∗ + x ∗ ∈ C for t ≥ −
1. By similar reasoning, letting t increase to 0 shows ξ · x ∗ ≤ ξ · x ∗ = 0. Now 0 < || ξ || = ξ · ( x ∗ − x ) = − ξ · x hence ξ · x < (cid:4) Since the set of non-negative finitely additive measures is a closed cone and X R Ω X d
Π is positive, linear, and continuous C = { R Ω X d
Π : Π ≥ } is alsoa closed cone. The contrapositive follows from the lemma.The proof also shows how to find an arbitrage when one exists. References
Dunford, Nelson, and Jacob Schwartz. 1963.
Linear Operators . Pure and Ap-plied Mathematics (John Wiley & Sons). New York: Interscience Publishers.Markowitz, Harry. 1952. “Portfolio Selection.”
The Journal of Finance