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An outline of a talk on asset allocation, discussing the importance of asset allocation decisions, developments in the theory, and new approaches using ideas from derivative pricing. It also covers numerical implementation using monte carlo and ways to speed up the process. The scope of this approach to asset allocation is also addressed.
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Asset allocation recommendations (Merrill Lynch)
Investor type Cash Stock Bonds Conservative 20% 45% 35% Moderate 5% 55% 40% Aggressive 5% 75% 20%
Where do these numbers come from?
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M in{ 1 2
πT^ V π | μT^ π = Ep, lT^ π = 1} (1)
where
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E[ u(X 1 (ω)) ] = E[ u( X 0 [ (1 + rf ) + π(R(ω) − rf ) ] ) ] (2)
EP^ [ u
′ (X 1 ) (R − rf )] = 0
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EP^ [ u(X 1 (ω)) ] = EP^ [ u( X 0 [ (1 + rf ) + π(R(ω) − rf ) ] ) ]
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Recall that we assume there is no arbitrage. Let us denote the attainable terminal wealth by X given initial wealth X 0. We know from the no arbitrage result that
EQ^ [ (^) 1 +X r f
where Q is the risk neutral measure. The characterization of the set of attainable wealths is
X = {X : EQ^ [ (^) 1 +X r f
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We seek to maximize the expected utility of final wealth over the set X.
max X { EP^ [u(X)] subject to EQ^ [ X 1 + rf
We solve this problem by introducing a Lagrange multipler λ.
max X { EP^ [u(X)] − λ( EQ^ [ (^) 1 +X r f
Last expression involves expectations over P and over Q.
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For discrete state space this can be rewritten as
max X {
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P (ωj )[u(X(ωj )) − λ(L(ωj ) X1 +(ω rj^ ) f
First order conditions
P (ωj )u
′ (X(ωj )) = P (ωj )λ 1 +L(ω rj^ ) f
f or j = 1 · · · J.
become
u
′ (X(ω)) = λ 1 +L(ω r) f
f or all ω ∈ Ω.
Since u′ is a strictly decreasing continuous function it has an inverse h: so the last equation becomes ............
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X = h(λ (^) 1 +L r f
We still have the Lagrange multiplier. To get rid of it we use the relationship
h(λ (^) 1+Lrf ) 1 + rf^ ]^ =^ X^0 (3)
Call the solution λˆ. Hence we have
Xˆ = h(λˆ L 1 + rf^ )
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