Attitudes Towards Risk

14.123 Microeconomic Theory III Muha met Yildiz

Model

 C = R = w eal th l e vel

 Lottery = cdf F (pdf f )

 Utility function u : R → R

 U ( F ) ≡ E F ( u ) ≡ ∫ u ( x )d F ( x )

 E F ( x ) ≡ ∫ x d F ( x )

Attitudes Towards Risk

DM is

 risk aver se if E F ( u ) ≤ u ( E F ( x )) (  F )

 strictly risk averse if E F ( u )< u ( E F ( x )) (  “risky” F )

 risk neutral if E F ( u )= u ( E F ( x )) (  F )

 risk seeking if E F ( u ) ≥ u ( E F ( x )) (  F ) DM is

 risk aver se if u is concave

 strictly risk averse if u is strictl y conca v e

 risk neutral if u is linear

 risk seeking if u is convex

Certainty Equivalence



CE ( F ) = u ⁻ ¹( U ( F ))= u ⁻ ¹( E F ( u ))

 DM is

 risk av erse if CE ( F ) ≤ E F ( x ) for all F ;

 risk neutral if CE ( F ) = E F ( x ) for all F ;

 risk seeking if CE ( F ) ≥ E F ( x ) for all F .

 Take DM1 and DM2 with u 1 and u 2 .

 DM1 is more risk averse than DM2

  u 1 is mo re concave t han u 2 ,

  u 1 = φ ◦ u 2 for some c oncave func tion φ ,

  CE 1 ( F ) ≡ u 1 ⁻ ¹( E F ( u 1 )) ≤ u 2 ⁻ ¹( E F ( u 2 )) ≡ CE 2 ( F )

Absolute Risk Aversion

 absolute risk aversion:

r A ( x ) = - u ′′ ( x )/ u ′ ( x )

 constant absolute risk aversion (CARA)

u ( x ) =- e - α x

 If x ~ N ( μ , σ ²) , CE ( F ) = μ - ασ ²/2

 Fact: More risk aversion  higher absolute risk aversion everywhere

 Fact: Decreasing absolute risk aversion (DARA)

  y >0 , u 2 with u 2 ( x ) ≡ u ( x+y ) is less risk averse

Relative risk aversion:

 relative risk aversion:

r R ( x ) = - xu ′′ ( x )/ u ′ ( x )

 constant relative risk aversion (CRRA)

u ( x )=- x 1- ρ /(1- ρ ),

 When ρ = 1 , u ( x ) = log( x ).

 Fact: Decreasing relative risk aversion (DRRA)

  t >1 , u 2 with u 2 ( x ) ≡ u ( tx ) is less risk averse

Application: Insurance

 wealth w and a loss of $1 with probability p .

 Insuran c e: pays $1 i n case of loss co sts q ;

 DM buys  units of insurance.

 Fa ct : If p = q (fair premium), then  = 1 (full insurance).

■ Ex pecte d we alt h w – p fo r a ll  .

 Fa ct : If DM1 bu ys full insuran c e, a mor e risk aver se DM2 also buys full insurance.

 CE 2 (  ) ≤ CE 1 (  ) ≤ CE 1 (  ) = CE 2 (  ).

Application: Optimal Portfolio Choice





With initial wealth w , invest α  [0, w ] in a risky a sse t that pays a return z per each $ invested; z has cdf F on [0, ∞ ) .

U ( α ) = ∫ 0 u ( w + α z - α ) d F ( z ) ; c o nc av e

It is optimal to invest α > 0 iff E[ z ] > 1 .

■ U ’( 0) = ∫ 0 u ’( w )( z -1 ) d F ( z ) = u ’( w )( E[ z ]-1) .

If agent with utility u 1 optimally inve sts α 1 , then an agent with more risk aver se u 2 (same w ) optimally inve sts α 2 ≤ α 1 .

DARA  optimal α i n crea se s in w . CARA  optimal α i s constan t in w .

CRRA (DRRA)  optimal α /w i s constant (increasing)

∞



∞









Optimal Portfolio Choice – P roof

 u 2 = g ( u 1 ) ; g is c o nc av e; g ’( u 1 ( w )) = 1 .

 U i ( α ) ≡ ∫ u i ( w + α ( z -1))( z -1) d F ( z )

 U 2 ’( α )- U 1 ’( α ) = ∫ [ u 2 ( w + α ( z -1))- u 1 ( w + α ( z -1))]( z -1)d F ( z )

≤ 0.

 g ’( u 1 ( w + α 1 z - α 1 )) < g ’( u 1 ( w )) = 1  z > 1 .

 u 2 ( w + α ( z -1) ) < u 1 ( w + α ( z -1))  z > 1 .

 α 2 ≤ α 1

Stochastic Dominance

 Goal: Compar e lotteries with mini mal assumptions on preferences

 Assume that the suppor t o f al l payof f distribution s is bounded . Support = [ a , b ].

 Two main concepts:

 First - o r de r Stoch a st ic Do min a n c e: A pay off d i stributio n is pref erred by all m onotonic Expected Utilit y pr ef er e n ce s .

 Se co nd-orde r Sto c h a s tic D o mina nce : A pa yo ff dist ribut io n is pr ef er re d by all r i sk a v erse E U pr ef er e n ce s .

FSD

Proof:

 “If:” for F ( x* ) > G ( x* ), d e fine u = 1 { x > x *} .

 “Only if”: As sume F and G ar e stric t ly inc r ea sin g a n d con t in uou s on [ a,b ] .

 Defi n e y ( x ) = F -1 ( G (x) ) ; y ( x ) ≥ x for all x

 ∫ u ( y )d F ( y ) = ∫ u ( y ( x ) )d F ( y ( x ) ) = ∫ u ( y ( x ) )d G ( x ) ≥ ∫ u ( x )d G ( x )

MPR and MLR Stochastic Orders

SSD

 Assume : F and G has the same mean

 DEF : F second-orde r stochasticall y domi n ates G  for ever y non-decreasi n g concave u , ∫ u ( x )d F ( x ) ≥ ∫ u ( x )d G ( x ) .

 DEF : G is a mean-pr eser vin g spread of F  y = x + ε for so me x ~ F , y ~ G , and ε with E [ ε |x ] = 0 .

 TH M : The following are equivalent:

 F sec o n d - o r d er st oc has tically d o m in ates G .

 G is a me a n - p r e s e rv in g s p r e ad of F .

 ∀ t ≥ 0 , ∫ 0 G ( x )d x ≥ ∫ 0 F ( x )d x .

t

t

SSD

 Example: G (dotted) is a mean-preserving spread of F (solid).

1

G

F

0

b

x