An Index of (Absolute) Correlation Aversion Theory and Some Implications Olivier Le Courtois (EM Lyon) A joint work with David Crainich (IESEG) and Louis Eeckhoudt (IESEG) 1
Outline of the Talk – Bibliography – Preliminary results : background risk and decreasing downside risk aversion – In this study, we look at the relation between : cross background risks and (decreasing ?) cross downside risk aversion 2
Question under Study So, we ask : For what type of agent does an additional zero-mean risk on health induce a decrease of the investment in the risky part of the portfolio constituting wealth ? 3
Bibliography – Ross (Econometrica, 1981) – Kimball (Econometrica, 1990) – Gollier, Pratt (Econometrica, 1996) – Eeckhoudt, Gollier, Schlesinger (Econometrica, 1996) – Gollier (MIT, 2001) – Courbage (Theory and Decision, 2001) – Eeckhoudt, Rey, Schlesinger (MS, 2007) – Malevergne, Rey (IME, 2009) – Crainich, Eeckhoudt, Le Courtois (JME, 2014, forthcoming) 4
Preliminary Results We first look for conditions ensuring that, for E (˜ x ) = 0 : � � yu ′ ( z + ˜ yu ′ ( z + ˜ E (˜ y )) = 0 ⇒ E (˜ y + ˜ x )) ≤ 0 In plain words, what are the conditions on the utility u such that the introduction of a so-called background risk ˜ x to a portfolio made of a risk-free asset z and a risky asset ˜ y reduces the proportion invested in the risky asset ? 5
Preliminary Results The concept of Downside Risk Aversion, or DRA, is related to the quantity u ′′′ /u ′ where ǫ ) 2 u ′′′ ( w ) m = k 2 σ (˜ u ′ ( w ) solves in the small 1 ǫ )]] = 1 2 [ u ( w − k ) + E [ u ( w + ˜ 2 [ E [ u ( w − k + ˜ ǫ )] + u ( w + m )] So, m is the quantity that compensates the pain attached to the lottery that combines bad ( − k ) with bad ( � x ), compared to the lottery that combines good with bad. 6
Preliminary Results Necessary condition for the background risk result : � � yu ′ ( z + ˜ yu ′ ( z + ˜ E (˜ y )) = 0 ⇒ E (˜ y + ˜ x )) ≤ 0 ⇒ DDRA where DDRA is � � u ′′′ ( w ) ∂ ∀ w ≤ 0 u ′ ( w ) ∂w 7
Preliminary Results Sufficient condition for the background risk result : � � yu ′ ( z + ˜ yu ′ ( z + ˜ E (˜ y )) = 0 ⇒ E (˜ y + ˜ x )) ≤ 0 ⇐ Ross-DDRA where Ross-DDRA is � � u ′′′ ( t + w ) ∂ ∀ t ∀ w ≤ 0 u ′ ( w ) ∂w 8
Preliminary Results Remark 1 : ∀ t u ′′′ ( t + . ) ց ⇔ ∃ λ | ∀ w T ( w ) ≥ λ ≥ A ( w ) u ′ ( . ) where T and A are the temperance and risk aversion coefficients. 9
Preliminary Results Remark 2 : the results are derived using the diffidence theorem, stating that ∀ ˜ x of bounded support E ( f 1 (˜ x )) = 0 ⇒ E ( f 2 (˜ x )) ≤ 0 is equivalent to f 2 ( x ) ≤ f ′ 2 ( x 0 ) ∀ x ∈ [ a, b ] 1 ( x 0 ) f 1 ( x ) f ′ provided – ∃ x 0 | f 1 ( x 0 ) = f 2 ( x 0 ) = 0 – f 1 and f 2 are twice differentiable at x 0 – f ′ 1 ( x 0 ) � = 0 10
Cross Background Risks and DRA We first look for conditions ensuring that, for E (˜ x ) = 0 : E [ u 1 ( z + (˜ y − i ) , h )˜ y ] = 0 ⇒ E [ u 1 ( z + (˜ y − i ) , h + ˜ x )˜ y ] ≤ 0 In plain words, what are the conditions on the utility u such that the introduction of a so-called background risk ˜ x on health (initial level : h) to a DM initially endowed with a portfolio made of a risk-free asset z and a risky asset ˜ y reduces the proportion invested in the risky asset ? Or, ‘do vapoteurs invest less in stocks ?’ 11
Cross Background Risks and CDRA The concept of Cross Downside Risk Aversion, or CDRA, is related to the quantity u 122 /u 1 where m = l ǫ ) 2 u 122 ( x, y ) 4 σ (˜ � u 1 ( x, y ) solves in the small 1 ǫ )]] = 1 2 [ u ( x − l, y ) + E [ u ( x, y + � 2 [ E [ u ( x − l, y + � ǫ )] + u ( x + � m, y )] So, � m compensates the pain attached to the lottery that combines bad on wealth ( − l ) with bad on health ( � ǫ ), compared to the lottery that combines good with bad. 12
Cross Background Risks and CDRA Necessary condition for the background risk result : [ E [ u 1 ( z + (˜ y − i ) , h )˜ y ] = 0 ⇒ E [ u 1 ( z + (˜ y − i ) , h + ˜ x )˜ y ] ≤ 0] ⇒ DCDRA where DCDRA is � � u 122 ( s, t ) ∂ ∀ ( s, t ) ≤ 0 ∂s u 1 ( s, t ) 13
Cross Background Risks and CDRA Sufficient condition for the background risk result : Ross-DCDRA ⇒ [ E [ u 1 ( z + (˜ y − i ) , h )˜ y ] = 0 ⇒ E [ u 1 ( z + (˜ y − i ) , h + ˜ x )˜ y ] ≤ 0] where Ross-DCDRA is � � u 122 ( s, t + u ) ∂ ∀ ( s, t, u ) ≤ 0 ∂s u 1 ( s, t ) 14
Alternative Approach Remark : Ross-DCDRA ⇔ ∀ x ∃ λ x ∀ y | − u 1122 ( x, y ) u 122 ( x, y ) ≥ λ x ≥ − u 11 ( x, y ) u 1 ( x, y ) where A and T are the risk aversion and cross-temperance coefficients. In plain words, cross-temperance should always be superior to risk aversion for the background risk result to prevail. 15
Conclusion To extend the results from Vapoteurs to Smokers, one needs to additionally assume that u 12 /u 1 is decreasing in wealth. 16
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