Microfoundations of Financial Economics 2004-2005

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Microfoundations of Financial Economics2004-2005

• Professor André Farber
• Solvay Business School
• Université Libre de Bruxelles

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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
Session 3
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Today

• What happens if markets are incomplete?

?
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Complete markets 2 states
A1
State 2
proj(xm)
R1
proj(x1)
Rm
Rf
A2
m
1
R2
p 1
State 1
Using rescaled values
p 0
E 0
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Frontier portfolios in the E, s space
E(R)
A2
Rf
SDF
A1
s(R)
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Complete markets 3 states
State 3
p 1
A1
Frontier portfolios
1
m
E 1
A2
State 1
A3
State 2
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Orthogonal decomposition graphic
Space of returns
p 1
Ri
ei
Rf
Rm
1
m
0
p 0
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Orthogonal decomposition equation
Every return can be expressed as
where wi is a number and ei is an excess return
with E(ei) 0
The components are orthogonal
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Mean variance frontier
Rmv is on the mean-variance frontier if and only
if
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Decomposition in E,s space
E(R)
Rfw(Rm Rf)
Ri
Rf
Rm is the minimum second moment return
Rm
s(R)
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What happens if markets are incomplete?

• Incomplete markets states gt assets
• Payoff space X a subspace of RS
• Does everything that we learned collapse?

2 states, 1 asset
3 states, 2 assets
State 2
State 3
X
State 1
State 1
State 2
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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 0.5 2.5
0.4 0.65 2.5
Looks bad!
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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.

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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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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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How to construct the pricing kernel?
Suppose there are N assets and S securities.The
matrix of payoffs x is NSThe vector of prices p
is N1Consider a portfolio defined by c, a N1
vector
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Example
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Riez again the expectation kernel
In a similar way, one can find in X a vector e
such that
where e is the N1 vector of expected payoffs e
is known as the expectation kernel
In previous example
Price State 1 State 2
State 2 E Sigma
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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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Example