Risk Preference and Delta Property

Risk Preference and Delta Property - Risk Preference and...

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Risk Preference and the Delta  Property
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U-curve on Wealth u(x) x ($) w w 0 Current Bank Account Increment
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General U-curve
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Example $1,000 $500 $0 0.3 0.4 0.3 ~ $450 $1,000 + $500 + $0 + ~ $450 + 0.3 0.4 0.3
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The Delta Property x 1 x 2 x 3 p 1 p 2 p 3 ~ ~x x + 1 x + 2 x + 3 p 1 p 2 p 3 ~ ~x +
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With the Delta Property, Buying and  Selling Prices are Equal  x 1 x 2 x 3 p 1 p 2 p 3 ~ x = s * Se lling Price for Owned Dea l x b 1 x b 2 x b 3 p 1 p 2 p 3 ~ 0 Buy ing Price for Unowned Dea l De lta Property x 1 x 2 x 3 p 1 p 2 p 3 ~ b Comp are ~
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Implications of the Delta Property A decision maker satisfying the delta property must: either be risk neutral and have a straight line u- curve: u(y) = a + by or have an exponential u-curve of the form: u(y) = a + br -y
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Indifference to a Deal Involving  Winning or Losing One Monetary Unit  Risk odds r = p/(1-p) u(y) = a+b r - y p 1 – p 1 –1 ~ 0
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Indifference to a Deal Involving  Winning or Losing m Monetary Units m q =r 1-q q 1 – q m – m ~ 0
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Alternate Form of Exponential γ is the risk aversion coefficient ρ is the risk tolerance Relation to risk odds, r : - - / ( ) y y u y a be a be γ ρ = + = + r r e r ln 1 , ln , = = =
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Assessing Risk Tolerance z -z/2 0.5 0.5 0 Adjust z until the decision maker is indifferent in this deal: Then risk tolerance ρ is approximately z (actually,1.04z)
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This note was uploaded on 06/16/2010 for the course MS&E 252 taught by Professor Howard during the Fall '08 term at Stanford.

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Risk Preference and Delta Property - Risk Preference and...

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