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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."