Microfoundations of Financial Economics 2004-2005 Professor André Farber Solvay Business School Université Libre de Bruxelles

What did we learn so far? Session 1: Complete markets Session 2: Mossin: Quadratic utility  standard CAPM EfSet Math + Efficient mkt portfolio  Zero-Beta CAPM PhD 04

Today What happens if markets are incomplete? ? PhD 04

Complete markets – 2 states proj(x|m) R1 proj(x|1) Rm Rf m A2 1 R2 p = 1 State 1 Using rescaled values p = 0 E = 0 PhD 04

Frontier portfolios in the E, σ space Rf SDF A1 σ(R) PhD 04

Complete markets – 3 states Frontier portfolios 1 m E = 1 * A2 * A3 State 1 State 2 PhD 04

Orthogonal decomposition: graphic Space of returns p = 1 Ri εi Rf Rm 1 m p = 0 PhD 04

Orthogonal decomposition: equation Every return can be expressed as: where wi is a number and εi is an excess return with E(εi) = 0 The components are orthogonal: PhD 04

Mean variance frontier Rmv is on the mean-variance frontier if and only if: PhD 04

Decomposition in E,σ space E(R) Rf+w(Rm – Rf) Ri Rf Rm Rm is the minimum second moment return σ(R) PhD 04

What happens if markets are incomplete? Incomplete markets: # states > # assets Payoff space X: a subspace of RS Does everything that we learned collapse? 2 states, 1 asset State 2 3 states, 2 assets State 3 X State 1 State 1 State 2 PhD 04

State prices are no longer unique Example: 2 states, 1 security Looking for state prices q(1), q(2) such that: p(x)=q(1) x(1) +q(2) x(1) q(1) q(2) p(x) 0.6 0.35 2.5 0.5 0.4 0.65 Looks bad! PhD 04

Riesz saves the situation Frigyes Riesz's father Ignácz Riesz was a medical man and Frigyes's his younger brother, Marcel Riesz, was himself a famous mathematician. Frigyes (or Frederic in German) Riesz studied at Budapest. He went to Göttingen and Zurich to further his studies and obtained his doctorate from Budapest in 1902. His doctoral dissertation was on geometry. He spent two years teaching in schools before being appointed to a university post. Riesz was a founder of functional analysis and his work has many important applications in physics. He built on ideas introduced by Fréchet in his dissertation, using Fréchet's ideas of distance to provide a link between Lebesgue's work on real functions and the area of integral equations developed by Hilbert and his student Schmidt. In 1907 and 1909 Riesz produced representation theorems for functional on quadratic Lebesgue integrable functions and, in the second paper, in terms of a Stieltjes integral. The following year he introduced the space of q-fold Lebesgue integrable functions and so he began the study of normed function spaces, since, for q 3 such spaces are not Hilbert spaces. Riesz introduced the idea of the 'weak convergence' of a sequence of functions ( fn(x) ). A satisfactory theory of series of orthonormal functions only became possible after the invention of the Lebesgue integral and this theory was largely the work of Riesz. Riesz's work of 1910 marks the start of operator theory. In 1918 his work came close to an axiomatic theory for Banach spaces, which were set up axiomatically two years later by Banach in his dissertation. PhD 04

Riesz Representation Theorem for Dummies If F: RS →R is a continuous linear function, then there exist a unique vector k in RS such that: Why bother? We do use continuous linear functions on RS : the pricing function p(x) for the price of x the expectation function E(x) for the expected value PhD 04

Law of One Price and Discount Factors A beautiful theorem (thanks to John Cochrane for his presentation): Let X be the payoff space A1. Portfolio formation: A2. Law of one price: Theorem: Given free portfolio formation A1, and the law of one price A2, there exist a unique payoff such that for all The theorem tells us that with incomplete markets there exist a unique portfolio whose payoff can be used to price any payoff. This portfolio is known as the pricing kernel PhD 04

How to construct the pricing kernel? Suppose there are N assets and S securities. The matrix of payoffs x is N×S The vector of prices p is N×1 Consider a portfolio defined by c, a N×1 vector PhD 04

Example PhD 04

Riez again: the expectation kernel In a similar way, one can find in X a vector e* such that: where e is the N×1 vector of expected payoffs e* is known as the expectation kernel In previous example: Price State 1 State 2 State 2 E Sigma PhD 04

Frontier returns with incomplete markets The set of frontier returns is the line passing through the return Rq of the pricing kernel and Re of the expectation kernel: PhD 04

Example PhD 04