An Operational Measure of Riskiness Sergiu Hart June 2007
An Operational Measure of Riskiness Sergiu Hart Center for the Study of Rationality Dept of Economics Dept of Mathematics The Hebrew University of Jerusalem
[email protected] http://www.ma.huji.ac.il/hart
Joint work with
Dean P. Foster The Wharton School University of Pennsylvania
Joint work with
Dean P. Foster The Wharton School University of Pennsylvania
Center for Rationality DP-454 www.ma.huji.ac.il/hart/abs/risk.html
A gamble 1/2
+$120
g= 1/2
−$100
A gamble 1/2
+$120
g= 1/2
−$100
E[g] = $10
A gamble 1/2
+$120
g= 1/2
−$100
E[g] = $10
ACCEPT
g or
REJECT
g?
A gamble 1/2
+$120
g= 1/2
−$100
E[g] = $10
ACCEPT
g or
What is the
REJECT
RISK
g?
in accepting g ?
The risk of accepting a gamble 1/2
+$120 g= 1/2
−$100
The risk of accepting a gamble 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth W is:
The risk of accepting a gamble 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth W is: W = $100: very risky (BANKRUPTCY)
The risk of accepting a gamble 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth W is: W = $100: very risky (BANKRUPTCY) W = $1 000 000: not risky
The risk of accepting a gamble 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth W is: W = $100: very risky (BANKRUPTCY) W = $1 000 000: not risky The risk of accepting a gamble depends on the current wealth
The risk of accepting a gamble 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth W is: W = $100: very risky (BANKRUPTCY) W = $1 000 000: not risky The risk of accepting a gamble depends on the current wealth Where is the “cutoff point” ?
Gamble g 1/2
+$120 g= 1/2
−$100
Gamble g at wealth W = $200 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $200:
Gamble g at wealth W = $200 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $200: 1/2
$320 $200 + g = 1/2
$100
Gamble g at wealth W = $200 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $200 yields relative returns: 1/2
$320 $200 + g = 1/2
$100
+60%
Gamble g at wealth W = $200 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $200 yields relative returns: 1/2
$320
+60%
$100
−50%
$200 + g = 1/2
Gamble g at wealth W = $200 1/2
+60% 1/2
−50%
Gamble g at wealth W = $200 1/2
+60% 1/2
−50%
Assume these returns every day, independently; proceeds fully reinvested
Gamble g at wealth W = $200 1/2
+60% 1/2
−50%
Assume these returns every day, independently; proceeds fully reinvested Proposition The wealth converges to zero (a.s.)
Gamble g at wealth W = $200 1/2
+60% 1/2
−50%
Assume these returns every day, independently; proceeds fully reinvested Proposition The wealth converges to zero (a.s.) BANKRUPTCY
Gamble g at wealth W = $200 1/2
+60% 1/2
−50%
Gamble g at wealth W = $200 1/2
+60% 1/2
Proof.
−50%
Gamble g at wealth W = $200 1/2
+60% 1/2
−50%
Proof. Let Wt = wealth at time t.
Gamble g at wealth W = $200 1/2
+60% 1/2
Wt+1 = Wt × 1.6
−50%
Proof. Let Wt = wealth at time t.
Gamble g at wealth W = $200 1/2
1/2
+60%
Wt+1 = Wt × 1.6
−50%
Wt+1 = Wt × 0.5
Proof. Let Wt = wealth at time t.
Gamble g at wealth W = $200 1/2
1/2
+60%
Wt+1 = Wt × 1.6
−50%
Wt+1 = Wt × 0.5
Proof. Let Wt = wealth at time t. Law of Large Numbers ⇒ about half the days wealth is multiplied by 1.6 about half the days wealth is multiplied by 0.5
Gamble g at wealth W = $200 1/2
1/2
+60%
Wt+1 = Wt × 1.6
−50%
Wt+1 = Wt × 0.5
Proof. Let Wt = wealth at time t. Law of Large Numbers ⇒ about half the days wealth is multiplied by 1.6 about half the days wealth is multiplied by 0.5 √ ⇒ A factor of ≈ 1.6 · 0.5 per day
Gamble g at wealth W = $200 1/2
1/2
+60%
Wt+1 = Wt × 1.6
−50%
Wt+1 = Wt × 0.5
Proof. Let Wt = wealth at time t. Law of Large Numbers ⇒ about half the days wealth is multiplied by 1.6 about half the days wealth is multiplied by 0.5 √ ⇒ A factor of ≈ 1.6 · 0.5 < 1 per day
Gamble g at wealth W = $200 1/2
1/2
+60%
Wt+1 = Wt × 1.6
−50%
Wt+1 = Wt × 0.5
Proof. Let Wt = wealth at time t. Law of Large Numbers ⇒ about half the days wealth is multiplied by 1.6 about half the days wealth is multiplied by 0.5 √ ⇒ A factor of ≈ 1.6 · 0.5 < 1 per day
⇒ Wt → 0 (a.s.)
Gamble g at wealth W = $200 1/2
1/2
+60%
Wt+1 = Wt × 1.6
−50%
Wt+1 = Wt × 0.5
Proof. Let Wt = wealth at time t. Law of Large Numbers ⇒ about half the days wealth is multiplied by 1.6 about half the days wealth is multiplied by 0.5 √ ⇒ A factor of ≈ 1.6 · 0.5 < 1 per day
⇒ Wt → 0 (a.s.)
BANKRUPTCY
Gamble g at wealth W = $1000 1/2
+$120 g= 1/2
−$100
Gamble g at wealth W = $1000 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $1000:
Gamble g at wealth W = $1000 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $1000: 1/2
$1120 $1000 + g = 1/2
$900
Gamble g at wealth W = $1000 1/2
+$120 g= 1/2
−$100
Accepting the gamble g when the wealth is W = $1000 yields relative returns: 1/2
$1120
+12%
$900
−10%
$1000 + g = 1/2
Gamble g at wealth W = $1000 1/2
+12% 1/2
−10%
Gamble g at wealth W = $1000 1/2
+12% 1/2
−10%
Assume these returns every day, independently; proceeds fully reinvested
Gamble g at wealth W = $1000 1/2
+12% 1/2
−10%
Assume these returns every day, independently; proceeds fully reinvested Proposition The wealth converges to infinity (a.s.)
Gamble g at wealth W = $1000 1/2
+12% 1/2
−10%
Assume these returns every day, independently; proceeds fully reinvested Proposition The wealth converges to infinity (a.s.) NO - BANKRUPTCY
... and infinite growth ...
Gamble g at wealth W = $1000 1/2
+12% 1/2
−10%
Assume these returns every day, independently; proceeds fully reinvested Proposition The wealth converges to infinity (a.s.) NO - BANKRUPTCY
... and infinite growth ...
Gamble g at wealth W = $1000 1/2
1/2
+12%
Wt+1 = Wt × 1.12
−10%
Wt+1 = Wt × 0.90
Gamble g at wealth W = $1000 1/2
1/2
+12%
Wt+1 = Wt × 1.12
−10%
Wt+1 = Wt × 0.90
Proof. Law of Large Numbers ⇒ ≈ half the days wealth is multiplied by 1.12 ≈ half the days wealth is multiplied by 0.90
Gamble g at wealth W = $1000 1/2
1/2
+12%
Wt+1 = Wt × 1.12
−10%
Wt+1 = Wt × 0.90
Proof. Law of Large Numbers ⇒ ≈ half the days wealth is multiplied by 1.12 ≈ half the days wealth is multiplied by 0.90 √ ⇒ A factor of ≈ 1.12 · 0.90 > 1 per day
Gamble g at wealth W = $1000 1/2
1/2
+12%
Wt+1 = Wt × 1.12
−10%
Wt+1 = Wt × 0.90
Proof. Law of Large Numbers ⇒ ≈ half the days wealth is multiplied by 1.12 ≈ half the days wealth is multiplied by 0.90 √ ⇒ A factor of ≈ 1.12 · 0.90 > 1 per day
⇒ Wt → ∞ (a.s.)
Gamble g at wealth W = $1000 1/2
1/2
+12%
Wt+1 = Wt × 1.12
−10%
Wt+1 = Wt × 0.90
Proof. Law of Large Numbers ⇒ ≈ half the days wealth is multiplied by 1.12 ≈ half the days wealth is multiplied by 0.90 √ ⇒ A factor of ≈ 1.12 · 0.90 > 1 per day
⇒ Wt → ∞ (a.s.) NO - BANKRUPTCY
... and infinite growth ...
The critical wealth level = ? Accepting the gamble g when the wealth is
The critical wealth level = $600 Accepting the gamble g when the wealth is W = $600: 1/2
$720 $600 + g = 1/2
$500
The critical wealth level = $600 Accepting the gamble g when the wealth is W = $600: 1/2
$720
×
6 5
$500
×
5 6
$600 + g = 1/2
The critical wealth level = $600 Accepting the gamble g when the wealth is W = $600: 1/2
$720
×
6 5
$500
×
5 6
$600 + g = 1/2
⇒ Factor of ≈
q
6 5
·
5 6
= 1 per day
The critical wealth level = $600 Accepting the gamble g when the wealth is W = $600: 1/2
$720
×
6 5
$500
×
5 6
$600 + g = 1/2
⇒ Factor of ≈ The
q
6 5
·
5 6
= 1 per day
of the gamble g is R(g) = $600
RISKINESS
The critical wealth level = $600
The
of the gamble g is R(g) = $600
RISKINESS
The critical wealth level = $600 Accepting the gamble g when the wealth is W < $600 gives returns that lead to BANKRUPTCY (Wt → 0 a.s. for i.i.d.)
The
of the gamble g is R(g) = $600
RISKINESS
The critical wealth level = $600 Accepting the gamble g when the wealth is W < $600 gives returns that lead to BANKRUPTCY (Wt → 0 a.s. for i.i.d.) Accepting the gamble g when the wealth is W > $600 gives returns that lead to NO - BANKRUPTCY (Wt → ∞ a.s. for i.i.d.) The
of the gamble g is R(g) = $600
RISKINESS
The General Model
Gambles
Gambles A gamble is a real-valued random variable g
Gambles A gamble is a real-valued random variable g Positive expectation: E[g] > 0
Gambles A gamble is a real-valued random variable g Positive expectation: E[g] > 0 Some negative values: P[g < 0] > 0 (loss is possible)
Gambles A gamble is a real-valued random variable g Positive expectation: E[g] > 0 Some negative values: P[g < 0] > 0 (loss is possible) [technical] Finitely many values: g takes the values x1 , x2 , ..., xm with probabilities p1 , p2 , ..., pm
Gambles and Wealth
Gambles and Wealth The initial wealth is W1 > 0
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... :
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the
CURRENT WEALTH
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the a gamble gt is
CURRENT WEALTH
OFFERED
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the a gamble gt is gt may be
CURRENT WEALTH
OFFERED
ACCEPTED
or
REJECTED
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the a gamble gt is gt may be if
CURRENT WEALTH
OFFERED
ACCEPTED
ACCEPTED
or
REJECTED
then Wt+1 = Wt + gt
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the a gamble gt is gt may be
CURRENT WEALTH
OFFERED
ACCEPTED
or
REJECTED
if
ACCEPTED
then Wt+1 = Wt + gt
if
REJECTED
then Wt+1 = Wt
Gambles and Wealth The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the a gamble gt is gt may be
CURRENT WEALTH
OFFERED
ACCEPTED
or
REJECTED
if
ACCEPTED
then Wt+1 = Wt + gt
if
REJECTED
then Wt+1 = Wt
Gambles The initial wealth is W1 > 0 At every period t = 1, 2, ... : let Wt > 0 be the a gamble gt is gt may be
CURRENT WEALTH
OFFERED
ACCEPTED
or
REJECTED
if
ACCEPTED
then Wt+1 = Wt + gt
if
REJECTED
then Wt+1 = Wt
Gambles At every period t = 1, 2, ... a gamble gt is OFFERED:
Gambles At every period t = 1, 2, ... a gamble gt is OFFERED: the sequence G = (g1 , g2 , ..., gt , ...) is arbitrary
Gambles At every period t = 1, 2, ... a gamble gt is OFFERED: the sequence G = (g1 , g2 , ..., gt , ...) is arbitrary gt may depend on the past wealths, gambles, decisions
Gambles At every period t = 1, 2, ... a gamble gt is OFFERED: the sequence G = (g1 , g2 , ..., gt , ...) is arbitrary gt may depend on the past wealths, gambles, decisions (not i.i.d., arbitrary dependence
Gambles At every period t = 1, 2, ... a gamble gt is OFFERED: the sequence G = (g1 , g2 , ..., gt , ...) is arbitrary gt may depend on the past wealths, gambles, decisions (not i.i.d., arbitrary dependence; “adversary”)
Gambles At every period t = 1, 2, ... a gamble gt is OFFERED: the sequence G = (g1 , g2 , ..., gt , ...) is arbitrary gt may depend on the past wealths, gambles, decisions (not i.i.d., arbitrary dependence; “adversary”) [technical] G is finitely generated: there is a finite collection of gambles such that every gt is a multiple of one of them
Decisions
Decisions A STRATEGY prescribes when to accept and when to reject the offered gambles
Decisions A STRATEGY prescribes when to accept and when to reject the offered gambles Consider simple strategies: The decision depends only on the current wealth W and the offered gamble g (“Markov stationary”)
Decisions A STRATEGY prescribes when to accept and when to reject the offered gambles Consider simple strategies: The decision depends only on the current wealth W and the offered gamble g (“Markov stationary”) If g is accepted at W then αg is accepted at αW for every α > 0 (“homogeneous”)
Bankruptcy
Bankruptcy BANKRUPTCY :
Wt = 0
Bankruptcy BANKRUPTCY :
limt→∞ Wt = 0
No-Bankruptcy NO - BANKRUPTCY :
{ limt→∞ Wt = 0 } has probability 0
No-Bankruptcy A strategy
GUARANTEES NO - BANKRUPTCY :
{ limt→∞ Wt = 0 } has probability 0 for every G = (g1 , g2 , ..., gt , ...) and every W1 > 0
Main Result
Main Result For every gamble g there exists a unique positive number R(g) such that:
Main Result For every gamble g there exists a unique positive number R(g) such that: A strategy guarantees no-bankruptcy
Main Result For every gamble g there exists a unique positive number R(g) such that: A strategy guarantees no-bankruptcy if and only if
Main Result For every gamble g there exists a unique positive number R(g) such that: A strategy guarantees no-bankruptcy if and only if gamble gt is rejected at wealth Wt when R(gt ) > Wt
Main Result For every gamble g there exists a unique positive number R(g) such that: A strategy guarantees no-bankruptcy if and only if gamble gt is rejected at wealth Wt when R(gt ) > Wt
Main Result For every gamble g there exists a unique positive number R(g) such that: A strategy guarantees no-bankruptcy if and only if gamble gt is rejected at wealth Wt when R(gt ) > Wt
R(g) = the
RISKINESS
of g
Main Result
Main Result No-bankruptcy is guaranteed if and only if
Main Result No-bankruptcy is guaranteed if and only if One never accepts gambles whose riskiness exceeds the current wealth
Main Result No-bankruptcy is guaranteed if and only if One never accepts gambles whose riskiness exceeds the current wealth
riskiness ∼ reserve
Main Result (continued)
Main Result (continued)
Moreover, for every gamble g, its riskiness R(g) is the unique solution R > 0 of the equation
Main Result (continued)
Moreover, for every gamble g, its riskiness R(g) is the unique solution R > 0 of the equation · µ ¶¸ 1 E log 1 + g =0 R
The riskiness of some gambles 1/2
+ X g= 1/2
− $100
The riskiness of some gambles 1/2
+ X g= 1/2
X
E [g]
− $100 R(g)
$300 $100 $150 $200 $50 $200 $120 $10 $600 $105 $2.5 $2100 $102 $1 $5100
The riskiness of some gambles 1/2
+ X g= 1/2
X
E [g]
− $100 R(g)
$300 $100 $150 $200 $50 $200 $120 $10 $600 $105 $2.5 $2100 $102 $1 $5100
The riskiness of some gambles 1/2
+ X g= 1/2
X
E [g]
− $100 R(g)
$300 $100 $150 $200 $50 $200 $120 $10 $600 $105 $2.5 $2100 $102 $1 $5100
The riskiness of some gambles 1/2
+ X g= 1/2
X
E [g]
− $100 R(g)
$300 $100 $150 $200 $50 $200 $120 $10 $600 $105 $2.5 $2100 $102 $1 $5100
The riskiness of some gambles 1/2
+ X g= 1/2
X
E [g]
− $100 R(g)
$300 $100 $150 $200 $50 $200 $120 $10 $600 $105 $2.5 $2100 $102 $1 $5100
The riskiness of some gambles p
+ $105
g= 1−p
− $100
The riskiness of some gambles p
+ $105
g= 1−p p
E [g]
− $100 R(g)
0.5 $2.5 $2100 0.6 $23 $235.23 0.8 $64 $106.93 0.9 $84.5 $100.16
The Riskiness measure R
The Riskiness measure R has a clear operational interpretation
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ...
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately is normalized (unit = $)
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately is normalized (unit = $) (... more to follow ...)
Variants of the Main Result
Variants of the Main Result homogeneous strategies:
Variants of the Main Result homogeneous strategies: g is rejected at W when W < R(g)
Variants of the Main Result Non-homogeneous strategies: g is rejected at W when W < R(g) and W is small
Variants of the Main Result Shares setup:
Variants of the Main Result Shares setup: One may accept gt in any proportion (i.e., αt gt for αt > 0)
Variants of the Main Result Shares setup: One may accept gt in any proportion (i.e., αt gt for αt > 0) Replace
NO - BANKRUPTCY
with
NO - LOSS
(i.e., Wt > W1 for all large enough t)
Variants of the Main Result Shares setup: One may accept gt in any proportion (i.e., αt gt for αt > 0) Replace
NO - BANKRUPTCY
with
NO - LOSS
(i.e., Wt > W1 for all large enough t) or: ASSURED GAIN (i.e., Wt > W1 + C for all large enough t) or: ...
Variants of the Main Result Shares setup: One may accept gt in any proportion (i.e., αt gt for αt > 0) Replace
NO - BANKRUPTCY
with
NO - LOSS
(i.e., Wt > W1 for all large enough t) or: ASSURED GAIN (i.e., Wt > W1 + C for all large enough t) or: ...
⇒ same threshold: R(g)
Properties of R
Properties of R Homogeneity: R(αg) = αR(g) for α > 0
Properties of R Homogeneity: R(αg) = αR(g) for α > 0 Subadditivity: R(g + h) ≤ R(g) + R(h)
Properties of R Homogeneity: R(αg) = αR(g) for α > 0 Subadditivity: R(g + h) ≤ R(g) + R(h)
Convexity: For 0 ≤ α ≤ 1 R(αg + (1 − α)h) ≤ αR(g) + (1 − α)R(h)
Properties of R Homogeneity: R(αg) = αR(g) for α > 0 Subadditivity: R(g + h) ≤ R(g) + R(h)
Convexity: For 0 ≤ α ≤ 1 R(αg + (1 − α)h) ≤ αR(g) + (1 − α)R(h)
First order stochastic dominance: If g ≺st1 h then R(g) > R(h)
Properties of R Homogeneity: R(αg) = αR(g) for α > 0 Subadditivity: R(g + h) ≤ R(g) + R(h)
Convexity: For 0 ≤ α ≤ 1 R(αg + (1 − α)h) ≤ αR(g) + (1 − α)R(h)
First order stochastic dominance: If g ≺st1 h then R(g) > R(h)
Second order stochastic dominance: If g ≺st2 h then R(g) > R(h)
Properties of R Homogeneity: R(αg) = αR(g) for α > 0 Subadditivity: R(g + h) ≤ R(g) + R(h)
Convexity: For 0 ≤ α ≤ 1 R(αg + (1 − α)h) ≤ αR(g) + (1 − α)R(h)
First order stochastic dominance: If g ≺st1 h then R(g) > R(h)
Second order stochastic dominance: If g ≺st2 h then R(g) > R(h) ... ...
Properties of R Homogeneity: R(αg) = αR(g) for α > 0 Subadditivity: R(g + h) ≤ R(g) + R(h)
Convexity: For 0 ≤ α ≤ 1 R(αg + (1 − α)h) ≤ αR(g) + (1 − α)R(h) First order stochastic dominance: If g ≺st1 h then R(g) > R(h)
Second order stochastic dominance: If g ≺st2 h then R(g) > R(h) ... ...
Aumann & Serrano
Aumann & Serrano Aumann & Serrano (2006) have defined an Index of Riskiness RAS (g) that corresponds to the “MORE RISKY” order between gambles:
Aumann & Serrano Aumann & Serrano (2006) have defined an Index of Riskiness RAS (g) that corresponds to the “MORE RISKY” order between gambles: gamble g is
MORE RISKY THAN
if a less risk-averse agent rejects h then a more risk-averse agent rejects g (at all wealth levels)
gamble h:
Aumann & Serrano Aumann & Serrano (2006) have defined an Index of Riskiness RAS (g) that corresponds to the “MORE RISKY” order between gambles: gamble g is
MORE RISKY THAN
gamble h:
if a less risk-averse = more risk-loving agent rejects h then a more risk-averse = less risk-loving agent rejects g (at all wealth levels)
Aumann & Serrano Aumann & Serrano (2006) have defined an Index of Riskiness RAS (g) that corresponds to the “MORE RISKY” order between gambles: gamble g is
MORE RISKY THAN
gamble h:
if a less risk-averse = more risk-loving agent rejects h then a more risk-averse = less risk-loving agent rejects g (at all wealth levels)
Aumann & Serrano: Result For each gamble g, RAS (g) is the reciprocal of the absolute risk-aversion coefficient α of that CARA individual u(x) = − exp(−αx) who is indifferent between accepting and rejecting g:
Aumann & Serrano: Result For each gamble g, RAS (g) is the reciprocal of the absolute risk-aversion coefficient α of that CARA individual u(x) = − exp(−αx) who is indifferent between accepting and rejecting g: RAS (g) is the unique solution R > 0 of · µ ¶¸ 1 E 1 − exp − g =0 R
Comparing R and R
AS
RAS (g) is the unique solution R > 0 of · µ ¶¸ 1 E 1 − exp − g =0 R
Comparing R and R
AS
R(g) is the unique solution R > 0 of · µ ¶¸ 1 E log 1 + g =0 R RAS (g) is the unique solution R > 0 of · µ ¶¸ 1 E 1 − exp − g =0 R
Comparing R and R
AS
R(g) is the unique solution R > 0 of · µ ¶¸ 1 E log 1 + g =0 R RAS (g) is the unique solution R > 0 of · µ ¶¸ 1 E 1 − exp − g =0 R log(1 + x) = x − x2 /2 + x3 /3 − ...
Comparing R and R
AS
R(g) is the unique solution R > 0 of · µ ¶¸ 1 E log 1 + g =0 R RAS (g) is the unique solution R > 0 of · µ ¶¸ 1 E 1 − exp − g =0 R log(1 + x) = x − x2 /2 + x3 /3 − ...
1 − exp(−x) = x − x2 /2 + x3 /6 − ...
Comparing R and R
AS
Proposition If E[g] is small relative to g then R(g) ∼ RAS (g)
Comparing R and R
AS
Proposition If E[g] is small relative to g then R(g) ∼ RAS (g) Example
1/2
+$105 g= 1/2
−$100
Comparing R and R
AS
Proposition If E[g] is small relative to g then R(g) ∼ RAS (g) Example
1/2
+$105 g= 1/2
R(g) = $2100
−$100
Comparing R and R
AS
Proposition If E[g] is small relative to g then R(g) ∼ RAS (g) Example
1/2
+$105 g= 1/2
R(g) = $2100
−$100
RAS (g) = $2100.42...
Comparing R and R
AS
Comparing R and R R:
AS
critical wealth for any risk aversion
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R:
measure (one gamble)
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R: measure (one gamble) RAS : index (comparing gambles)
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R: measure (one gamble) RAS : index (comparing gambles) R:
bankruptcy vs no-bankruptcy
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R: measure (one gamble) RAS : index (comparing gambles) R: bankruptcy vs no-bankruptcy RAS : expected utility, risk aversion
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R: measure (one gamble) RAS : index (comparing gambles) R: bankruptcy vs no-bankruptcy RAS : expected utility, risk aversion unit and operational interpretation
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R: measure (one gamble) RAS : index (comparing gambles) R: bankruptcy vs no-bankruptcy RAS : expected utility, risk aversion unit and operational interpretation continuity
Comparing R and R
AS
R: critical wealth for any risk aversion RAS : critical risk aversion for any wealth R: measure (one gamble) RAS : index (comparing gambles) R: bankruptcy vs no-bankruptcy RAS : expected utility, risk aversion unit and operational interpretation continuity Nevertheless: similar in many respects !!
Utility Utility function u(x)
Utility Utility function u(x): Accept g at W if and only if E [u(W + g)] ≥ u(W )
Utility Utility function u(x): Accept g at W if and only if E [u(W + g)] ≥ u(W ) LOG UTILITY :
u(x) = log(x)
The Riskiness Measure R
The Riskiness Measure R · µ E log 1 +
1 R(g)
g
¶¸
=0
The Riskiness Measure R · µ E log 1 +
⇔
1 R(g)
g
¶¸
=0
· µ ¶¸ R(g) + g E log =0 R(g)
The Riskiness Measure R · µ E log 1 +
⇔ ⇔
1 R(g)
g
¶¸
=0
· µ ¶¸ R(g) + g E log =0 R(g) E [log(R(g) + g)] = log(R(g))
The Riskiness Measure R · µ E log 1 +
⇔ ⇔ ⇔
1 R(g)
g
¶¸
=0
· µ ¶¸ R(g) + g E log =0 R(g) E [log(R(g) + g)] = log(R(g))
is indifferent between accepting and rejecting g at W = R(g) LOG UTILITY
No-bankruptcy
No-bankruptcy No-bankruptcy
No-bankruptcy No-bankruptcy
⇔
Reject when W < R(g)
No-bankruptcy No-bankruptcy
⇔
⇔
Reject when
Reject when W < R(g) LOG UTILITY
rejects
No-bankruptcy and Risk Aversion No-bankruptcy
⇔
⇔
Reject when
Reject when W < R(g) LOG UTILITY
⇓
RELATIVE RISK AVERSION
rejects
≥1
No-bankruptcy and Risk Aversion No-bankruptcy
⇔
⇔
Reject when
Reject when W < R(g) LOG UTILITY
⇓
RELATIVE RISK AVERSION
(since
rejects
≥1
LOG UTILITY has constant RELATIVE RISK AVERSION = 1)
Rabin (2000): Calibration
Rabin (2000): Calibration If a risk-averse expected-utility agent rejects the gamble g = [+$105, 1/2; − $100, 1/2] at all wealth levels W < $300 000
Rabin (2000): Calibration If a risk-averse expected-utility agent rejects the gamble g = [+$105, 1/2; − $100, 1/2] at all wealth levels W < $300 000 Then he must reject the gamble h = [+$5 500 000, 1/2; − $10 000, 1/2] at wealth level W = $290 000
Rabin (2000): Calibration If a risk-averse expected-utility agent rejects the gamble g = [+$105, 1/2; − $100, 1/2] at all wealth levels W < $300 000 Then he must reject the gamble h = [+$5 500 000, 1/2; − $10 000, 1/2] at wealth level W = $290 000
Rabin (2000): Calibration If a risk-averse expected-utility agent rejects the gamble g = [+$105, 1/2; − $100, 1/2] at all wealth levels W < $300 000 Then he must reject the gamble h = [+$5 500 000, 1/2; − $10 000, 1/2] at wealth level W = $290 000 reject g at all wealth levels W < R(g) = $2100
Rabin (2000): Calibration If a risk-averse expected-utility agent rejects the gamble g = [+$105, 1/2; − $100, 1/2] at all wealth levels W < $300 000 Then he must reject the gamble h = [+$5 500 000, 1/2; − $10 000, 1/2] at wealth level W = $290 000 reject g at all wealth levels W < R(g) = $2100 no friction, no cheating
Rabin (2000): Calibration If a risk-averse expected-utility agent rejects the gamble g = [+$105, 1/2; − $100, 1/2] at all wealth levels W < $300 000 Then he must reject the gamble h = [+$5 500 000, 1/2; − $10 000, 1/2] at wealth level W = $290 000 reject g at all wealth levels W < R(g) = $2100 no friction, no cheating what is “wealth”?
What is Wealth?
What is Wealth? Rejecting g when W < W + R(g) Guarantees a minimal wealth level of W
What is Wealth? Rejecting g when W < W + R(g) Guarantees a minimal wealth level of W (replace 0 with W )
What is Wealth? Rejecting g when W < W + R(g) Guarantees a minimal wealth level of W Back to calibration:
What is Wealth? Rejecting g when W < W + R(g) Guarantees a minimal wealth level of W Back to calibration: If W = “gambling / risky investment wealth”, then $300 000 seems excessive for g (since R(g) = $2100)
What is Wealth? Rejecting g when W < W + R(g) Guarantees a minimal wealth level of W Back to calibration: If W = “gambling / risky investment wealth”, then $300 000 seems excessive for g (since R(g) = $2100) If W = total wealth, then rejecting g at all W < $300 000 is consistent with a required minimal wealth level W ≥ $297 900,
What is Wealth? Rejecting g when W < W + R(g) Guarantees a minimal wealth level of W Back to calibration: If W = “gambling / risky investment wealth”, then $300 000 seems excessive for g (since R(g) = $2100) If W = total wealth, then rejecting g at all W < $300 000 is consistent with a required minimal wealth level W ≥ $297 900, and then one rejects h at $290 000
The Riskiness measure R
The Riskiness measure R (recall) has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately is normalized (unit = $)
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately is normalized (unit = $) has good properties (e.g. monotonic with respect to first-order stochastic dominance)
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately is normalized (unit = $) has good properties (e.g. monotonic with respect to first-order stochastic dominance) may replace other measures of risk (variance-based)
The Riskiness measure R has a clear operational interpretation is independent of utilities, risk aversion, ... is defined for each gamble separately is normalized (unit = $) has good properties (e.g. monotonic with respect to first-order stochastic dominance) may replace other measures of risk (variance-based) Markowitz, CAPM, ... : E vs σ → E vs R Sharpe ratio: E/σ → E/R
The End
"We’re recommending a risky strategy for you; so we’d appreciate if you paid before you leave."