# Risk

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```					            Risk and Risk Measurement (L3)
• Following topics are covered:
– Risk
– Defining “More Risk Averse”
•   Utility functions
•   Absolute risk aversion
•   Relative risk aversion
•   Certainty equivalent
– Defining “More Risky”
• Increase in risk
• Aversion to downside risk
• Stochastic dominance

(Materials from Chapters 1&2 of EGS and Chapters 1&2 of HL)

L3: Risk and Risk Measurement       1
Risk
• Risk can be generally defined as “uncertainty”.
• Sempronius owns goods at home worth a total of 4000 ducats and in
addition possesses 8000 decats worth of commodities in foreign countries
from where they can only be transported by sea, with ½ chance that the
ship will perish.
• If he puts all the foreign commodity in 1 ship, this wealth, represented by a
lottery, is x ~ (4000, ½; 12000, ½)
• If he put the foreign commodity in 2 ships, assuming the ships follow
independent but equally dangerous routes. Sempronius faces a more
diversified lottery y ~ (4000, ¼; 8000, ½; 12000, ¼)
• In either case, Sempronius faces a risk on his wealth. What are the expected
values of these two lotteries?
• In reality, most people prefer the latter. Why?

L3: Risk and Risk Measurement                     2
• You pay a fixed fee to enter and then a fair
coin is tossed repeatedly until a tail appears,
ending the game. The pot starts at 1 dollar is
doubled every time a head appears. You win
whatever is in the pot after the game ends.
What is the fair price to pay for entering the
game?
• Are you willing to pay this?
• See page 13, HL.
L3: Risk and Risk Measurement     3
Risk Aversion and Utility Function
• A risk-averse agent is an agent who dislikes zero-mean
risks.
• Risk aversion: Eu(w  z )  u(w  Ez) .
– Utility function is the relationship between monetary outcome,
x, and the degree of satisfaction, u(x).
• Concavity
– a twice-differentiable function u is concave if and only if its second
derivative is negative, i.e., if the marginal utility u’(x) is decreasing
in x.
– E.g, u(x)=ln(x)
• A decision maker is risk-averse  the agent’s utility
function u is concave. – proposition 2, page 8, EGS
– Be able to prove this
• Jensen Inequality
L3: Risk and Risk Measurement                      4
Von Neumann_Morgenstern Utility Function

von Neumann-Morgenstern utility describes a utility function
(or perhaps a broader class of preference relations) that has the
expected utility property: the agent is indifferent between
receiving a given bundle or a gamble with the same expected
value.

L3: Risk and Risk Measurement              5
– Risk-averse investors may want to purchase risky assets if their
expected return exceed the risk-free rate.
– Eu(w+z)=u(w-Π), where z is a zero-mean risk, w is the initial wealth
– Π is risk premium – cost of risk
• Certainty equivalent
– The amount of payoff (e.g. money or utility) that an agent would
have to receive to be indifferent between that payoff and a given
gamble
– What is certainty equivalent in the above expression?
• Arrow-Pratt approximation of risk premium
1 2
–            A( w) where A(w)=-u’’(w)/u(w)
2

L3: Risk and Risk Measurement                     6
(Arrow_Pratt Approximation)
1 2
Eu( w  z )  E[u ( w)  zu ' ( w)      z u ' ' ( w)]
2
1
 u ( w)  u ' ( w) Ez  Ez 2 u ' ' ( w)
2
1
 u ( w)   2 u ' ' ( w)
2
where Ez=0 and variance σ2=Ez2.
Also note that Eu(w+z) = u(w-П), where u(w- П) = u(w)        -   П   u   ’   (   w   )   . Thus proved.

Note: The cost of risk, as measured by risk premium, is approximately
proportional to the variance of its payoffs. This is one reason why
researchers use a mean-variance decision criterion for modeling behavior
under risk. However П=1/2σ2*A(w) only holds for small risk (thus we can
apply for the 2nd-order approximation).

L3: Risk and Risk Measurement                                                7
Further discussions
• Arrow-Pratt expression of risk premium works
for small risk or special utility functions where
only mean and variance are applicable to the
expected utility
– See page 20, quadratic utility function

L3: Risk and Risk Measurement   8
Measuring Risk Aversion
• The degree of absolute risk aversion
 u ' ' ( w)
A( w)  
u ' ( w)
• For small risks, the risk premium increases with the size of the
risk proportionately to the square of the size
– Assuming z=k*ε, where E(ε)=0, σ(ε)=σ
• Accepting a small-mean risk has no effect on the wealth of
risk-averse agents
1 2 2
        k  A( w)
2
• ARA is a measure of the degree of concavity of a utility
function, i.e., the speed at which marginal utility decreases

L3: Risk and Risk Measurement             9
More Risk Averse Agent

• Consider two risk averse agents u and v. if v is more risk averse than u,
this is equivalent to that Av>Au
Proof:

Suppose v is a concave transformation of u. I.e., v(w) = φ(u(w))                                                                                             ,

We have v’(w) = φ                 . Hence,       ’       (   u       (   w           )   )   u   ’   (   w       )

2
v’’(w)= φ’   ’   (   u   ( +φw   )   )   (   u     , ’       (   w           )   )                       ’   (   u   (   w   )   )   u   ’   ’   (   w   )

  ' ' (u ( w))u ' ( w)
We have Av ( w)  Au ( w) 
 ' (u ( w))

•   Conditions leading to more risk Aversion – page 14-15

L3: Risk and Risk Measurement                           10
Example

• Two agents’ utility functions are u(w) and v(w).
u(w)= w , v(w)=ln(w);

(1) Which agent is more risk averse in small risks and in large risks?
(2) Suppose they have an initial wealth of 4000 ducats and face a risk
of (0, ½; 8000, ½). Find their respective risk premiums.

L3: Risk and Risk Measurement                     11
CARA and DARA
• CARA
• However, ARA typically decreases
– Assuming for a square root utility function, what would be the risk premium
of an individual having a wealth of dollar 101 versus a guy whose wealth is
dollar 100000 with a lottery to gain or lose \$100 with equal probability?
• What kind of utility function has a decreasing risk premium?
Eu(w+z)=u(w-π(w)) Eu’(w+z)=(1- π       ’   (   w   )   )   u   ’   (   w   -   π   (   w   )   )

u ' ( w   )  Eu ' ( w  z )
 ' ( w) 
u' (w   )
u(w) is an increasing and concave utility function, thus, u’(.)>0
to have a decreasing risk premium in w, we need u’(w- π                                             )   ≤   E   u   ’   (   w   +   z   )

Defining v=-u’, we have the following condition for decreasing risk premium
Ev(w+z) ≤v(w- π )

The above condition states that ????

L3: Risk and Risk Measurement                                                                                    12
Prudence
• Thus –u’ is a concave transformation of u.
• Defining risk aversion of (-u’) as –u’’’/u’’
• This is known as prudence, P(w)
• The risk premium associated to any risk z is decreasing in
wealth if and only if absolute risk aversion is decreasing or
prudence is uniformly larger than absolute risk aversion
• P(w)≥A(w)

L3: Risk and Risk Measurement             13
Relative Risk Aversion

• Definition
du' ( w) / u ' ( w)    wu ' ' ( w)
R( w)                                     wA( w)
dw / w             u ' ( w)
• Using z for proportion risk, the relation between relative risk premium,
ПR(z), and absolute risk premium, ПA(wz) is
 A ( wz ) 1 2
 ( z)  
R
  R( w)
w       2
• This can be used to establish a reasonable range of risk aversion: given that
(1) investors have a lottery of a gain or loss of 20% with equal probability
and (2) most people is willing to pay between 2% and 8% of their wealth
(page 18, EGS).

• CRRA

L3: Risk and Risk Measurement                    14
Some Classical Utility Function

A. Quadratic function: u(w)=aw-1/2w2 – increasing absolute risk aversion
exp(aw)
B. Exponential function: u ( w)              -- constant absolute risk aversion
a
w1
C. Power utility function: u ( w)        -- constant relative risk aversion
1 
D. log utility function: u(w) = ln(w) – constant relative risk aversion

For more of utility functions commonly used, see HL, pages 25-28
Also see HL, Page 11 for von Neumann-Morgenstern utility, i.e., utility
function having expected value
Appropriate expression of expected utility, see HL, page 6 and 7

L3: Risk and Risk Measurement                       15
Measuring Risks
• So far, we discuss investors’ attitude on risk when risk is given
– I.e., investors have different utility functions, how a give risk affects
investors’ wealth
• Now we move to risk itself – how does a risk change?
• Definition: A wealth distributions w1 is preferred to w2, when
Eu(w1)≥Eu(w2)
– Increasing risk in the sense of Rothschild and Stiglitz (1970)
– An increase in downside risk (Menezes, Geiss and Tressler (1980))
– First-order stochastic dominance

L3: Risk and Risk Measurement                        16

• w1~(4000, ½; 12000, ½)
noise]
• Then look at the expected utility (page 29)
• General form: w1 takes n possible value w1, w2,w3,…, wn. Let
ps denotes the probability that w1 takes the value of ws. If w2 =
w1+ ε  Eu(w2) ≤ Eu(w1).

L3: Risk and Risk Measurement            17
•    Definition:
– Assuming all possible final wealth levels are in interval
[a,b] and I is a subset of [a, b]
– Let fi(w) denote the probability mass of w2 (i=1, 2) at w.
w2 is a mean-preserving spread (MPS) of w1 if
1.   Ew2= Ew1
2.   There exists an interval I such that f2(w)≤ f1(w) for all w in I
•    Example: the figure in the left-handed panel of page 31
•    Increasing noise and mean preserving spread (MPS) are
equivalent

L3: Risk and Risk Measurement                      18
Single Crossing Property
• Mean Preserving spread implies that (integration by parts – page 31)
b
 [ F (s)  F (s)]ds  0
a
2         1

• This implies a “single-crossing” property: F2 must be larger than F1 to the
left of some threshold w and F2 must be smaller than F1 to its right. I.e.,
w
s( w)   [ F2 ( s)  F1 ( s)]ds  0
a

L3: Risk and Risk Measurement                       19
The Integral Condition
b                                                  b
Eu ( wi )   u ( w) f i ( w)dw  u ( w) Fi ( w) |    wb
w a     u ' ( w) Fi ( w)dw
a                                                 a
b
I.e., Eu ( wi )  u (b)   u ' ( w) Fi ( w)dw
a
b
It follows that: Eu ( w2 )  Eu ( w1 )   u ' ( w)[ F1 ( w)  F2 ( w)]dw -- page 32 for
a
interpretation

Integrating by parts yields
b             b
Eu ( w2 )  Eu ( w1 )  u ' ( w) S ( w) |b   u ' ' ( w)(  [ F1 ( w)  F2 ( w)]dw)dw
a
a             a
b
Thus Eu ( w2 )  Eu ( w1 )   u ' ' ( w) S ( w)dw
a
For a risk averse agent, the above expressive is uniformly negative.

L3: Risk and Risk Measurement                         20
MPS Conditions

Consider two random variable w1 and w2 with the same mean,
(1) All risk averse agents prefer w1 to w2 for all concave
function u
(2) w2 is obtained from w1 by adding zero-mean noise to the
possible outcome of w1
(3) w2 is obtained w1 by a sequence of mean-preserving spreads
(4) S(w)≥0 holds for all w.

L3: Risk and Risk Measurement          21
Preference for Diversification

• Suppose Sempreonius has an initial wealth of w (in term of
pounds of spicy). He ships 8000 pounds oversea. Also suppose
the probability of a ship being sunk is ½. x takes 0 if the ship
sinks and 1 otherwise. If putting spicy in 1 ship, his final
wealth is w2=w+8000*x
• If he puts spicy in 2 ships, his final wealth is
w1=w+8000(x1+x2)/2
• Sempreonius would prefer two ships as long as his utility
function is concave
• Diversification is a risk-reduction device in the sense of
Rothschild of Stiglitz (1970).

L3: Risk and Risk Measurement            22
Variance and Preference

• Two risky assets, w1 and w2, w1 is preferred to w2 iff П2> П1
• For a small risk, w1 is preferred to w2 iff the variance of w2
exceeds the variance of w1
• But this does not hold for a large risk. The correct statement is
that all risk-averse agents with a quadratic utility function prefer
w1 to w2 iff the variance of the second is larger than the variance
of the first
– See page 35.

L3: Risk and Risk Measurement             23
Aversion to Downside Risk

• Definition: agents dislike transferring a zero-mean risk from a
richer to a poor state.
• w2~(4000, ½; 12000+ε, ½)
• w3~(4000+ ε, ½; 12000, ½)
• Which is more risky
• Depends, generally agents dislike risks in lower states more.
But it depends on agents’ utility function

L3: Risk and Risk Measurement            24
First-Degree Stochastic Dominance
• Definition: w2 is dominated by w1 in the sense of the first-
degree stochastic dominance order if F2(w)≥F1(w) for all w.
• Consumers would dislike FSD-dominated shifts in the
distribution final wealth
b
Eu( w2 )  Eu( w1 )   u ( w)[ f 2 ( w)  f1 ( w)]dw
a
b
  u ' ( w)[F1 ( w)  F2 ( w)]dw  0
a                                     F(w)
W2
• No longer mean preserving
W1

w

L3: Risk and Risk Measurement               25
Equivalent Conditions on FSD
• All agents with a nondecreasing utility function prefer w1
to w2: Eu(w2)<=Eu(w1) for all nondecreasing functions u,
or stated as, w  w
1            2
FSD

• W2 is dominated w1 in the sense of FSD: w2 is obtained
from w1 by a transfer of probability mass from the high
wealth states to lower wealth states, or F2(w)≥F1(w) for all
w.
• W1 is obtained from w2 by adding nonnegative noise
d
terms to the possible outcome of w2:
w1 w2   where є
is no less than 0 with probability one.

L3: Risk and Risk Measurement        26
Second-degree Stochastic Dominance

• Definition
– w1 dominates w2 in the sense of the second-degree
stochastic dominance if and only if

E ( w1 )  E ( w2 )
y
s( y)   [ F1 ( w)  F2 ( w)]dz  0              y [0,1]
0

noise; see page 49, HL.

L3: Risk and Risk Measurement               27
• The Allais paradox is a choice problem designed by Maurice Allais to
show an inconsistency of actual observed choices with the predictions of
expected utility theory.
• Two pairs of games

L3: Risk and Risk Measurement                     28
• The Ellsberg paradox is a paradox in decision theory and experimental
economics in which people’s choices violate the expected utility
hypothesis. It is generally taken to be evidence for ambiguity aversion.
• The 1urn paradox: suppose you have an urn containing 30 red balls and 60
other balls that are either black or yellow. You don't know how many black
or yellow balls there are, but that the total number of black balls plus the
total number of yellow equals 60. The balls are well mixed so that each
individual ball is as likely to be drawn as any other.
• You are now given a choice:
– Gamble A: You receive \$100 if you draw a red ball
– Gamble B: You receive \$100 if you draw a black ball
• Also you are given options between Gamble C and D
– You receive \$100 if you draw a red or yellow ball
– You receive \$100 if you draw a black or yellow ball

L3: Risk and Risk Measurement               29
Technical Notes

• Taylor Series Expansion: f(x)=

which in a more compact form can be written

• Integration by parts

L3: Risk and Risk Measurement   30
Exercises

• EGS, 1.2; 1.4; 2.2

L3: Risk and Risk Measurement   31

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