# risk p nf

An Operational Measure of Riskiness Sergiu Hart June 2007 An Operational Measure of Riskiness Sergiu Hart Center for t...

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."