2. utility and risk aversion

8/17/2019 2. Utility and Risk Aversion

1/18

Utility and Risk

Aversion


2/18

Summary

Rational Decision: St. Petersbourg paradox

Utility and Expected Utility of Wealth

Risk Aversion, Risk Neutrality, Risk Loving

Risk premium, Insurance premium, Certainty

Equivalent

Arrow-Pratt (approximate) risk premium


3/18

St. Petersburg Paradox

Suppose someone tosses a coin

repeatedly. You receive $1 if head comesup on the first toss, $2 if head comes upfor the first time on the second toss, 4$if head comes up for the first time on the

third toss and so on…


4/18


2

1

2

1

heads

tail

$21

$20

$2n-1

2

1

2

1 2

1

2

1

2

1

2

1

21

$22


5/18


2

1

2

1

2

1

2

1

2

1

212

814

412

211)( 1

n

W E

n

n

n

2

1

2

1

heads

tail

$21$20

$2n-1

2

1

2

1 2

1

2

1

2

1

2

1

2

1

$22


6/18

St. Petersbourg Paradox- conclusion -

People should pay an infinite amount ofmoney for the right to play this game as

n increases without limit. However, this does not describe the

human behavior.

So, expected wealth is aninsufficient measure for the value ofa gamble.


7/18

Rational Investors- Expected Utility Representation -

Rational = investors are endowed withpreferences over random consumption plans.

A consumption plan looks like a lottery over thefuture possible states of nature:

x~

p4

p3

p2

p1

x1

x4

x2

x3 y~q3

q2

q1

y3

y2

y1


8/18


Preferences have expected utility representation ifthere exists a function U such that

yU E xU E y x ~~ ~~

prob xU xU E ~

4321 ,,,

)]([)]([)]([)]([~ 44332211 xU p xU p xU p xU p xU E x~ )( 2 x

)( 1 x

)( 4 x

)( 3 x

)( 1 p

)( 2 p

)( 3 p

)(4

p

“preferred to”


9/18


Investors are rational if they

Can compare gambles

Can make consistent comparisons:

Problems with the expected utility rule – Allais paradox

~~ y x

~~ z y ~~ z x


10/18

Risk aversion

Definition:

An individual is risk averse if he is unwilling toaccept any actuarially fair gamble (a gamble forwhich the expected value is 0).

Result:

Risk aversion = Utility function is concave

Proof of Risk Aversion: St. Petersburg paradox – diminishing marginal

utility

If risk averse (no actuarially fair gamble) then U

is concave (diminishing marginal utility)


11/18

U is concave - proof

For any w 1 and w 2 with w 1< w 2 and p aprobability, let be the gamble

Suppose the initial wealth is

Then, a risk averse investor will refusethe actuarially fair gamble, i.e.

h~

0)1( 21 h p ph

210 )1( W p pW W

201000 )1(~

hW U phW U phW U E W U


12/18

The concavity of the utilityfunction

2121 11 W U pW pU W p pW U

W U E W E U ~~

W U E W E U ~~

Certainty

Equivalent


13/18

Risk aversion

Money in your pockets

worth more thanexpected money in

your pockets


14/18

0

Expected Wealth

(average)

E(W)

Utility of Average – if we

have it certainly

U[E(W)]

Average

Utility

E[U(W)]

Utility

WealthW1 W2

U(W1)

U(W2)

Risk Aversion


15/18

A way to measure RiskAversion

Risk aversion:

Since U’>0, there exists a

such that:

Certainty equivalent wealth:

W U E W E U ~~

W U E W E U ~~

Risk premium

W E

~W U E U W E

~~ 1


16/18

The value of the risk premium

W E U W E U W E W E W E U W E U W E U ~'~~~~'~~

2~~~''

2

1~~~'

~~W E W W E U W E W W E U W E U E W U E

0

02

'

''

2

1

W U

W U


17/18

0

Expected Wealth

(average)E(W)

Utility of Average – if we

have it certainly

U[E(W)]

Average

Utility

E[U(W)]

Utility

WealthW1 W2

U(W1)

U(W2)

Risk Aversion

π


18/18

Finding an approximation forthe risk aversion

W U

W U ~

'

~''

2

1 2

Arrow-Pratt measure of Absolute

Risk Aversion (ARA)

It is merely an approximation; it works

only for “small” risk – all the moments of

wealth of order greater than 2 are negligible

with respect to the variance

2. utility and risk aversion

Documents