SLIDE 1
An Index of (Absolute) Correlation Aversion Theory and Some - - PowerPoint PPT Presentation
An Index of (Absolute) Correlation Aversion Theory and Some - - PowerPoint PPT Presentation
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 :
SLIDE 2
SLIDE 3
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
- f the portfolio constituting wealth ?
3
SLIDE 4
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
SLIDE 5
Preliminary Results
We first look for conditions ensuring that, for E(˜ x) = 0 :
- E(˜
yu′(z + ˜ y)) = 0 ⇒ E(˜ yu′(z + ˜ 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
SLIDE 6
Preliminary Results
The concept of Downside Risk Aversion, or DRA, is related to the quantity u′′′/u′ where m = k 2σ(˜ ǫ)2u′′′(w) u′(w) solves in the small 1 2 [u(w − k) + E[u(w + ˜ ǫ)]] = 1 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
SLIDE 7
Preliminary Results
Necessary condition for the background risk result :
- E(˜
yu′(z + ˜ y)) = 0 ⇒ E(˜ yu′(z + ˜ y + ˜ x)) ≤ 0
- ⇒ DDRA
where DDRA is ∀w ∂ ∂w
- u′′′(w)
u′(w)
- ≤ 0
7
SLIDE 8
Preliminary Results
Sufficient condition for the background risk result :
- E(˜
yu′(z + ˜ y)) = 0 ⇒ E(˜ yu′(z + ˜ y + ˜ x)) ≤ 0
- ⇐ Ross-DDRA
where Ross-DDRA is ∀t ∀w ∂ ∂w
- u′′′(t + w)
u′(w)
- ≤ 0
8
SLIDE 9
Preliminary Results
Remark 1 : ∀t u′′′(t + .) u′(.) ց ⇔ ∃λ | ∀w T(w) ≥ λ ≥ A(w) where T and A are the temperance and risk aversion coefficients.
9
SLIDE 10
Preliminary Results
Remark 2 : the results are derived using the diffidence theorem, stating that ∀˜ x of bounded support E (f1(˜ x)) = 0 ⇒ E (f2(˜ x)) ≤ 0 is equivalent to ∀x ∈ [a, b] f2(x) ≤ f′
2(x0)
f′
1(x0)f1(x)
provided – ∃x0 | f1(x0) = f2(x0) = 0 – f1 and f2 are twice differentiable at x0 – f′
1(x0) = 0
10
SLIDE 11
Cross Background Risks and DRA
We first look for conditions ensuring that, for E(˜ x) = 0 : E[u1(z + (˜ y − i), h)˜ y] = 0 ⇒ E[u1(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
- n 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
SLIDE 12
Cross Background Risks and CDRA
The concept of Cross Downside Risk Aversion, or CDRA, is related to the quantity u122/u1 where
- m = l
4σ(˜ ǫ)2u122(x, y) u1(x, y) solves in the small 1 2 [u(x − l, y) + E [u(x, y + ǫ)]] = 1 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
SLIDE 13
Cross Background Risks and CDRA
Necessary condition for the background risk result : [E[u1(z + (˜ y − i), h)˜ y] = 0 ⇒ E[u1(z + (˜ y − i), h + ˜ x)˜ y] ≤ 0] ⇒ DCDRA where DCDRA is ∀(s, t) ∂ ∂s
- u122(s, t)
u1(s, t)
- ≤ 0
13
SLIDE 14
Cross Background Risks and CDRA
Sufficient condition for the background risk result : Ross-DCDRA ⇒ [E[u1(z + (˜ y − i), h)˜ y] = 0 ⇒ E[u1(z + (˜ y − i), h + ˜ x)˜ y] ≤ 0] where Ross-DCDRA is ∀(s, t, u) ∂ ∂s
- u122(s, t + u)
u1(s, t)
- ≤ 0
14
SLIDE 15
Alternative Approach
Remark : Ross-DCDRA ⇔ ∀x ∃λx ∀y | − u1122(x, y) u122(x, y) ≥ λx ≥ −u11(x, y) u1(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
SLIDE 16