View
220
Download
0
Embed Size (px)
Citation preview
L1: Risk and Risk Measurement 1
Lecture 1: Risk and Risk Measurement
• We cover the following topics in this part– Risk – Risk Aversion
• Absolute risk aversion• Relative risk aversion
– Risk premium– Certainty equivalent – Increase in risk– Aversion to downside risk– First-degree stochastic dominance
L1: Risk and Risk Measurement 2
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?
L1: Risk and Risk Measurement 3
Risk Averse Agent and Utility Function
• There is no linear relationship between wealth and the utility of consuming this wealth
• Utility function: the relationship between monetary outcome, x, and the degree of satisfaction, u(x).
• When an agent is risk averse, the relationship is concave.
L1: Risk and Risk Measurement 4
Risk Aversion
• Definition
• A decision maker with utility function u is risk-averse if and only if u is concave. – What is risk aversion? What is concavity? -- page 8
• Risk Premium
• Arrow-Pratt approximation
• Holds for small risks
)()( EzwuzwEu , where z has a non-zero payoff.
)()( wuzwEu
)(2
1 2 wA
L1: Risk and Risk Measurement 5
Deriving the Risk Premium Formula
)(''2
1)(
)(''2
1)(')(
)](''2
1)(')([)(
2
2
2
wuwu
wuEzEzwuwu
wuzwzuwuEzwEu
where Ez=0 and variance σ2=Ez2. Also note that Eu(w+z) = u(w-П), where u(w- П) = u(w)- Пu’(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).
L1: Risk and Risk Measurement 6
Measuring Risk Aversion
• The degree of absolute risk aversion
• 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
• ARA is a measure of the degree of concavity of a utility function, i.e., the speed at which marginal utility decreases
)('
)('')(
wu
wuwA
)(2
1 22 wAk
L1: Risk and Risk Measurement 7
A 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’(u(w))u’(w). Hence, v’’(w)= φ’’(u(w))(u’(w))2 +φ’(u(w))u’’(w),
We have ))(('
)('))(('')()(
wu
wuwuwAwA uv
• Conditions leading to more risk Aversion – page 14-15
L1: Risk and Risk Measurement 8
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.
L1: Risk and Risk Measurement 9
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))
)('
)(')(')('
wu
zwEuwuw
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)≤ Eu’(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 ????
L1: Risk and Risk Measurement 10
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)
L1: Risk and Risk Measurement 11
Relative Risk Aversion
• Definition
• Using z for proportion risk, the relation between relative risk premium, ПR(z), and absolute risk premium, ПA(wz) is
• 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
)()('
)(''
/
)('/)(')( wwA
wu
wwu
wdw
wuwduwR
)(2
1)()( 2 wR
w
wzz
AR
L1: Risk and Risk Measurement 12
Some Classical Utility Function
A. Quadratic function: u(w)=aw-1/2w2 – increasing absolute risk aversion
B. Exponential function: a
awwu
)exp()(
-- constant absolute risk aversion
C. Power utility function:
1)(
1wwu -- constant relative risk aversion
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
L1: Risk and Risk Measurement 13
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
L1: Risk and Risk Measurement 14
Adding Noise
• w1~(4000, ½; 12000, ½)
• w2~(4000, ½; 12000+ε, ½) [adding price risk; or an additional 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).
L1: Risk and Risk Measurement 15
Mean Preserving Spread Transformation
• 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
L1: Risk and Risk Measurement 16
Single Crossing Property• Mean Preserving spread implies that (integration by parts – page 31)
• 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.,
b
adssFsF 0)]()([ 12
w
adssFsFws 0)]()([)( 12
L1: Risk and Risk Measurement 17
The Integral Condition
b
a
b
a ibwawiii dwwFwuwFwudwwfwuwEu )()('|)()()()()(
I.e., dwwFwubuwEub
a ii )()(')()(
It follows that: dwwFwFwuwEuwEub
a)]()()[(')()( 2112 -- page 32 for
interpretation Integrating by parts yields
dwdwwFwFwuwSwuwEuwEub
a
b
a
ba ))]()([)((''|)()(')()( 2112
Thus dwwSwuwEuwEub
a )()('')()( 12
For a risk averse agent, the above expressive is uniformly negative.
L1: Risk and Risk Measurement 18
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.
L1: Risk and Risk Measurement 19
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).
L1: Risk and Risk Measurement 20
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.
L1: Risk and Risk Measurement 21
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
L1: Risk and Risk Measurement 22
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.
• No longer mean preserving
• Three equivalent conditions regarding first-degree stochastic dominance outlined in Proposition 2.5
• See page 45, HL
dwwFwFwuwEuwEub
a)]()()[(')()( 2112 <0
L1: Risk and Risk Measurement 23
Second-degree Stochastic Dominance
• Definition– See page 45, HL (it is exactly same as a MPS transformation)
• Combining FSD with increase in risk or a mean preserving spread
• Third-degree stochastic dominance
L1: Risk and Risk Measurement 24
Technical Notes
• Taylor Series Expansion: f(x)=
• Integration by parts
which in a more compact form can be written