I- Theoretical Framework!
Risk management and demand for risky assets!
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Literature on Portfolio choice models! •! Static model (Arrow, 1965) : optimal portfolio choice theory! •! Extensions :! –! Inter-temporal model (Mossin, 1968)! –! Inter-temporal consumption-portfolio choice model (Merton, 1969, Samuelson, 1969)! Risk management and demand for risky assets!
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Main hypothesis in Merton’s model! Complete markets! •! No transaction or holding costs (fixed or proportional), no taxes, perfect divisibility of assets, short sales allowed! •! Perfect information (no information costs…)! •! Transactions made continuously in time! Risk management and demand for risky assets!
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Main results in Merton’s model! •! If u[c(t)], the instantaneous utility function, exhibits constant relative risk aversion (CRRA) and if assets prices are log-normally distributed (yields follow a wiener process) then :!
! portfolio choice is independant of consumption! ! myopia is optimal (optimal portfolio does not depend of age).! Risk management and demand for risky assets!
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!The
Two funds
separation theorem holds :
one may consider only one risky asset (the risky mutual fund depends only on returns and covariance matrix) and the risk-free asset (portfolio are perfectly diversified)!
•! Intertemporal portfolio choice may be analysed within a standard static portfolio choice model! Risk management and demand for risky assets!
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First «"Puzzles"»!
1.! Porfolio are incomplete and very different
2.! Portfolio depend on age
Risk management and demand for risky assets!
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The standard static portfolio problem!
MaxEu(W˜ ) utc. W˜ = W [R + " ( R˜ r # R)] W : initial wealth!
! : fraction of wealth invested in risky assets ! Risk management and demand for risky assets!
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˜ r : return on risky assets R ˜ r = Rr ( Rr > R) ER
! R˜ = ! r
First order condition is written as : ˜)=0 E( R˜r ! R)u' (W Risk management and demand for risky assets!
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Demand for risky assets! •!For small risk, taking a first order Taylor expansion of FOC is (also true for CRRA utility function and log-normality of assets price) :!
Risk management and demand for risky assets!
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• Fraction of wealth invested in risky assets :
"* =
(Rr # R) 2
% (W) = Relative risk aversion
$ % (W ) • Amount invested in risky assets : " *W =
(Rr # R) 2
$ a(W )
a(W) = Absolute risk aversion
Risk management and demand for risky assets!
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Predictions! •!The proportion of risky assets in wealth is constant with wealth if u(.) exhibits CRRA! The demand for risky assets is increasing according to wealth if u(.) is DARA!
Risk management and demand for risky assets!
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“Equity Premium Puzzle”! •!In France : on the period 1896-1996 with annual data, we should obtain according to this model (same prediction in USA) :! - If "=1
!
!! = 160% (234%)"
- If "=4
!
!! = 40% (58%)!
Risk management and demand for risky assets!
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“Participation Puzzle”!
·!In 2000 in France, only 15% of households own stocks directly (only stocks) and about 23% directly or indirectly (mutual funds included)! ! How can we explain this fact in the standard portfolio choice model?! Risk management and demand for risky assets!
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How the model can explain why people holds so few stocks?!
Risk management and demand for risky assets!
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Alternative hypothesis (I)! Incomplete markets or more general utility! 1) Existence of holding and transaction costs! 2) There are other sources of risk (income, unemployment, health…)! 3) Some individuals are liquidity constrained ! 4) Some individuals have more flexible labor supply! 5) Utility function u(.) does not exhibit CRRA ! Risk management and demand for risky assets!
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Alternative hypothesis (II)! The role of other forms of wealth! 1)! Human wealth : ! (Rr # R) (Rr # R) (Rr # R) E(t) " *W (t) = [W (t) + E(t)] & "* = 2 + 2 2 $ % $ % $ % W (t)
2) Housing wealth : ratio H(T)/W(t)! Risk management and demand for risky assets!
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•! King and Leape (1998) and Arrondel and Masson (1990) introduce holding costs to explain why portfolio are incomplete! •!Gollier and Zeckhauser (1998) show that if risk tolerance (the inverse of risk aversion) is convex, then young households invest more in risky assets than old households (myopia is no longer optimal)! -if it is true, the fraction of wealth invested in wealth must increase with wealth" Risk management and demand for risky assets!
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•! Bodie, Merton and Samuelson (1992) study the impact of labor supply flexibility !
•!Flavin and Yamashita (2002) or Cocco (2002) analyze the relation between housing and risky asset demand!
Risk management and demand for risky assets!
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The optimal static portfolio composition with background risk! •!We assume now that initial wealth is random (Kimball, 1993):"
MaxE R˜ r ,y˜ u(W˜ + y˜ ) utc. W˜ = X[ R + " (R˜ r # R)] Risk management and demand for risky assets!
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y˜is
exogenous, undiversifiable and uninsurable (it is independant of portfolio risk)! •! Let us define indirect utility function:!
v(W) = Ey˜ u(W + y˜ ) = ! u(W + y)dF(y) for all W, where dF(y) is the distribution function of y.! Risk management and demand for risky assets!
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•!In consequence, portfolio choice problem can be rewritten as :!
˜) MaxER˜ v( W r
utc. W˜ = X[R + ! ( R˜ r " R)] !The introduction of an independent risk is equivalent to the transformation of the original utility function u(.) into the indirect utility function v(.) ! Risk management and demand for risky assets!
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•! Application of Arrow-Pratt theorem:! - If we can rank the degree of concavity of these two functions, we can analyse changes in attitude towards portfolio risks!
!PROPOSITION! •! Risk affects negatively the demand for stocks if v(.) is more concave than u(.)! Risk management and demand for risky assets!
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•! Technically, this means that:! "Eu' ' (W + y˜ ) "u' ' (W) E˜y = 0 ! # Eu' (W + y˜ ) u' (W)
"v' ' (W ) "u' ' (W ) $ # v' (W) u' (W) $ av # au
Risk management and demand for risky assets!
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Main results! Let us define:!
!u' '' (.) p= is absolute prudence à la Kimball (1992) u'' (.) !u' '' ' (.) t= is temperance i.e. the desire to moderate u' '' (.) total exposure to risk Risk management and demand for risky assets!
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The definitions of riskiness
!A risk y˜ is loss-aggravating when starting from initial wealth W, if and only if it satisfies
Eu'(W + y˜ ) >= u'(W)
Observe that this is equivalent to:
E y˜
