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

2. 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
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3. Today
What happens if markets are incomplete? ?
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4. 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
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5. Frontier portfolios in the E, σ spaceRf
SDF A1
σ(R)
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6. Complete markets – 3 states
Frontier portfolios 1 m
E = 1
* A2
* A3
State 1
State 2
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7. Orthogonal decomposition: graphic
Space of returns
p = 1 Ri εi Rf Rm 1 m
p = 0
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8. 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:
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9. Mean variance frontier
Rmv is on the mean-variance frontier if and only if:
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10. Decomposition in E,σ space
E(R)
Rf+w(Rm – Rf) Ri Rf Rm
Rm is the minimum second moment return
σ(R)
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11. 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
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12. 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!
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13. 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 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.
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14. 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
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15. 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
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16. 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
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17. Example
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18. 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 State State E Sigma
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19. 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:
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20. Example
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