C HOICE T HEORY, U TILITY F UNCTIONS AND R ISK
AVERSION
Szabolcs Sebestyén
[email protected]
Master in Finance
I NVESTMENTS
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Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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An Introduction to Choice Theory
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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An Introduction to Choice Theory
Dominance
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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An Introduction to Choice Theory
Dominance
Preliminaries
Risk means uncertainty in the future cash flow stream
The cash flow of an asset in any future period is typically
modelled as a random variable
Example
Consider the following asset pay-offs ($) with π1 = π2 = 1/2 for the
two states θ:
t=0
Investment 1
Investment 2
Investment 3
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−1, 000
−1, 000
−1, 000
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t=1
θ=1
θ=2
1, 050
500
1, 050
1, 200
1, 600
1, 600
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An Introduction to Choice Theory
Dominance
Dominance
State-by-state dominance: the strongest possible form of
dominance
We assume that the typical individual is non-satiated in
consumption: she prefers more rather than less of goods the
pay-offs allow her to buy
Most of the cases are not so trivial and the concept of risk enters
necessarily
Risk is not the only consideration, the ranking between
investment opportunities is typically preference dependent
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An Introduction to Choice Theory
Dominance
Mean-Variance Dominance
In our example, the mean returns and their standard deviations
are
E (r1 ) = 12.5%
σ1 = 7.5%
E (r2 ) = 5%
σ2 = 55%
E (r3 ) = 32.5%
σ3 = 27.5%
Mean-variance dominance: higher mean and lower variance
It is neither as strong nor as general as state-by-state dominance,
and it is not fully reliable
A criterion for selecting investments of equal magnitude, which
plays a prominent role in modern portfolio theory, is
For investments of the same expected return, choose the one with
the lowest variance
For investments of the same variance, choose the one with the
greatest expected return
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An Introduction to Choice Theory
Dominance
Sharpe ratio criterion
Example
Consider the state-contingent rates of return with π1 = π2 = 1/2:
Investment 4
Investment 5
θ=1
θ=2
E (r)
σ
3%
2%
5%
8%
4%
5%
1%
3%
No dominance in either state-by-state or mean-variance terms
In the mean-variance framework, one would require specifying
the terms at which the investor is willing to substitute expected
return for a given risk reduction
Sharpe ratio criterion: (E/σ)4 = 4 > (E/σ)5 = 5/3
However, decisions are again preference dependent
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An Introduction to Choice Theory
Choice Theory Under Certainty
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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An Introduction to Choice Theory
Choice Theory Under Certainty
Economic Rationality (1)
Economic rationality: individuals’ behaviour is predictable in
that it is systematic, i.e. they attempt to achieve a set objective
Definition (Preference relation)
There exists a preference relation, denoted by , that describes
investors’ ability to compare various bundles of goods, services and
money. For two bundles a and b, the expression a b means the
following: the investor either strictly prefers bundle a to bundle b, or
she is indifferent between them. Pure indifference is denoted by a ∼ b,
whereas strict preference by a b.
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An Introduction to Choice Theory
Choice Theory Under Certainty
Economic Rationality (2)
Assumptions
Economic rationality can be summarised by the following assumptions:
A.1 Every investor possesses a preference relation and it is
complete. Formally, for any two bundles a and b, either a b,
or b a, or both.
A.2 The preference relation satisfies the fundamental property of
transitivity: for any bundles a, b and c, if a b and b c,
then a c.
A.3 The preference relation is continuous in the following sense: let
{xn } and {yn } be two sequences of consumption bundles such
that xn 7−→ x and yn 7−→ y. If xn yn for all n, then x y.
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An Introduction to Choice Theory
Choice Theory Under Certainty
Existence of Utility Function
Theorem
Assumptions A.1 through A.3 are sufficient to guarantee the existence of a
continuous, time-invariant, real-valued utility function u : RN −→ R+ ,
such that for any two bundles a and b,
ab
⇐⇒
u (a) ≥ u (b) .
Remarks:
Under certainty, utility functions are unique only up to monotone
transformations
The notion of a bundle is very general, different elements of a
bundle may represent the consumption of the same good or
service in different time periods
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Introduction
Under uncertainty, the objects of choice are typically vectors of
state-contingent money pay-offs
Those vectors form an asset or an investment
Assume that individuals have no intrinsic taste for the assets
themselves (not really true in practice!), but rather they are
interested in what pay-offs these assets will yield and with what
likelihood
Also assume that investors prefer a higher pay-off to a lower one
Typically, no one investment prospect will strictly dominate the
others
There are two ingredients in the choice between two alternatives:
The probability of the states
The ex-post level of utility provided by the investment
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Example
Example
The current price of all assets is $100. The forecasted prices per share
in one period are the following:
Asset
State 1
State 2
A
B
C
$100
$90
$90
$150
$160
$150
Which stock would you invest in?
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Lotteries
Definition (Lottery)
A lottery is an object of choice, denoted by (x, y, π ), that offers pay-off
x with probability π and y with probability 1 − π.
Lottery (x, y, π ):
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π
x
1−π
y
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Example of a Compounded Lottery
Example
Lottery (x, y, π ) = (x1 , x2 , τ1 ) , (y1 , y2 , τ2 ) , π :
π
1− π
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τ1
x1
1−τ1
x2
τ2
y1
1−τ2
y2
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Axioms and Conventions
C.1 (a) (x, y, 1) = x
(b) (x, y, π ) = (y, x, 1 − π )
(c) (x, z, π ) = x, y, π + (1 − π ) τ if z = (x, y, τ ).
C.2 There exists a preference relation , which is complete
and transitive
C.3 The preference relation is continuous
C.4 Independence of irrelevant alternatives. Let (x, y, π ) and
(x, z, π ) be any two lotteries; then, y z if and only if
(x, y, π ) (x, z, π )
C.5 There exists a best (most preferred) lottery, b, as well as a
worst lottery, w
C.6 Let x, k, z be pay-offs for which x > k > z. Then there
exists a π such that (x, z, π ) ∼ k
C.7 Let x y. Then (x, y, π1 ) (x, y, π2 ) if and only if
π1 > π2
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
The Expected Utility Theorem
Theorem (The Expected Utility Theorem)
If axioms C.1 through C.7 are satisfied, then there exists a utility function U
defined on the lottery space such that
U (x, y, π ) = πu (x) + (1 − π ) u (y)
where u (x) = U (x, y, 1) . U is called the von Neumann–Morgenstern
(NM) utility function.
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An Introduction to Choice Theory
Choice Theory Under Uncertainty
Remarks on the NM Utility Function
Given the objective specification of probabilities, the NM utility
function uniquely characterises an investor
Different additional assumptions on u will identify the investor’s
tolerance for risk
We require that u be increasing for all candidate utility functions
The theorem confirms that investors are concerned only with an
asset’s final pay-off and the cumulative probabilities of achieving
them, while the structure of uncertainty resolution is irrelevant
Although the utility function is not defined over a rate of return,
but on pay-off distribution, it can be generalised
The NM representation is preserved under a certain class of
monotone affine transformations: if U (·) is a NM utility function,
then V (·) ≡ aU (·) + b with a > 0 is also such a function
A non-linear transformation does not always respect the preference
ordering
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An Introduction to Choice Theory
How Restrictive Is Expected Utility Theory? The Allais Paradox
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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An Introduction to Choice Theory
How Restrictive Is Expected Utility Theory? The Allais Paradox
The Allais Paradox (1)
Example
Consider the following four lotteries:
L1 = (10, 000; 0; 0.1)
L3 = (10, 000; 0; 1)
versus L2 = (15, 000; 0; 0.09)
versus L4 = (15, 000; 0; 0.9) .
Rank the pay-offs.
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An Introduction to Choice Theory
How Restrictive Is Expected Utility Theory? The Allais Paradox
The Allais Paradox (2)
Example (cont’d)
The following ranking is frequently observed in practice:
L2 L1
and
L3 L4 .
However, observe that
L1 = L3 , L0 , 0.1
L2 = L4 , L0 , 0.1 ,
where L0 = (0, 0, 1).
By the independence axiom, the ranking between L1 and L2 , and
between L3 and L4 , should be identical. This is the Allais paradox
(Allais, 1953).
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An Introduction to Choice Theory
How Restrictive Is Expected Utility Theory? The Allais Paradox
How to Deal With the Allais Paradox?
Possible reactions to the Allais paradox:
1
2
Yes, my choices were inconsistent; let me think again and revise
them
No, I’ll stick to my choices. The following things are missing from
the theory:
The pleasure of gambling; and/or
The notion of regret
The Allais paradox is but the first of many phenomena that
appear to be inconsistent with standard preference theory
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An Introduction to Choice Theory
How Restrictive Is Expected Utility Theory? The Allais Paradox
Example: Framing and Loss Aversion
Example (Kahneman and Tversky, 1979)
1
In addition to whatever you own, you have been given $1, 000.
You are now asked to choose between
LA = (1, 000; 0; 0.5)
2
versus
LB = (500; 0; 1)
In addition to whatever you own, you have been given $2, 000.
You are now asked to choose between
LC = (−1, 000; 0; 0.5)
versus
LD = (−500; 0; 1)
A majority of subjects chose B in case 1 and C in case 2, but this is
inconsistent with any preference relation over wealth gambles.
The difference between the two cases is that the outcomes are framed
as gains relative to the reference wealth level in case 1, but as losses in
case 2.
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Risk Aversion
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
Introduction
Investors want to avoid risk, i.e., they want to smooth their
consumption across states of nature
Hence, restrictions on the NM expected utility representation
must be imposed to guarantee such behaviour
Since probabilities are objective and independent of investor
preferences, we must restrict the utility function to capture risk
aversion
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Risk Aversion
How To Measure Risk Aversion?
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
How To Measure Risk Aversion?
Definition of Risk Aversion
Definition (Risk aversion)
An investor is said to be (weakly) risk averse if
e)
u (w) ≥ E u (w
e with mean w. Strict risk aversion is
for any random wealth w
represented by strict inequality.
An equivalent definition is:
Definition (Risk aversion)
An investor is said to be (weakly) risk averse if
u (a) ≥ E u (a + e
ε) ,
where e
ε is a zero-mean random variable and a is a constant.
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Risk Aversion
How To Measure Risk Aversion?
Concavity and Risk Aversion
A risk-averse investor would prefer to avoid a fair bet
The above inequalities are known as Jensen’s inequality and are
equivalent to concavity of the utility function See Appendix
Concavity is preserved by monotone affine transformations, so,
for given preferences, either all utility functions are concave or
none are
For a differentiable function u, concavity is equivalent to
non-increasing marginal utility:
u0 (w1 ) ≤ u0 (w0 )
if w1 > w0
Strict concavity is equivalent to decreasing marginal utility (strict
inequality above)
For a twice differentiable function u, concavity is equivalent to
u00 (w) ≤ 0 for all w (strict concavity implies strict inequality)
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Risk Aversion
How To Measure Risk Aversion?
Concavity and Risk Aversion – Intuition
Consider a financial contract where the investor either receives an
amount h with probability 1/2, or must pay an amount h with
probability 1/2 (i.e., a lottery (h, −h, 1/2))
Intuition tells us that a risk-averse investor would prefer to avoid
such a security, for any level of personal income Y
Formally,
u (Y ) >
1
1
u (Y + h) + u (Y − h) = E (u) = U (Y )
2
2
This inequality can only be satisfied for all income levels Y if the
utility function is (strictly) concave
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Risk Aversion
How To Measure Risk Aversion?
A Strictly Concave Utility Function
Source: Danthine and Donaldson (2005), Figure 4.1
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Risk Aversion
How To Measure Risk Aversion?
Indifferent Curves
Source: Danthine and Donaldson (2005), Figure 4.2
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Risk Aversion
How To Measure Risk Aversion?
Measure Risk Aversion
Given that u00 (·) < 0, why not say that investor A is more risk
00 (w)| ≥ |u00 (w)| for all wealth levels?
averse than investor B iff |uA
B
Let uA (w) ≡ auA (w) + b with a > 0
Since the utility function is invariant to affine transformations, uA
and uA must describe the identical ordering and must display
identical risk aversion
However, if a > 1, we have that
00
uA (w) > u00 (w)
A
This a contradiction
We need a measure of risk aversion that is invariant to affine
transformations
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Risk Aversion
How To Measure Risk Aversion?
Coefficients of Risk Aversion
Absolute risk aversion:
α (w) = −
u00 (w)
u0 (w )
It depends on the preferences and not on the particular utility
function representing the preferences
For any risk-averse investor, α (w) ≥ 0
Relative risk aversion:
ρ (w) = wα (w) = −
wu00 (w)
u0 (w )
Risk tolerance:
τ (w) =
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1
u0 (w )
= − 00
α (w)
u (w)
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Risk Aversion
Interpreting Risk Aversion
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
Interpreting Risk Aversion
Absolute Risk Aversion and the Odds of a Bet
Consider an investor with wealth w who is offered the lottery
L = (h, −h, π )
The investor will accept the bet if π is high enough and reject if π
is low enough
The willingness to accept the bet must also be related to her
current level of wealth
Proposition
Let π = π (w, h) the probability at which the investor is indifferent between
accepting and rejecting the bet. Then we have that
1 1
π (w, h) ∼
= + h α (w) .
2 4
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Risk Aversion
Interpreting Risk Aversion
Proof.
By definition, π (w, h) must satisfy
u (w) = π (w, h) u (w + h) + 1 − π (w, h) u (w − h)
Taylor approximation yields
1
u (w + h) = u (w) + hu0 (w) + h2 u00 (w) + R1
2
1
u (w − h) = u (w) − hu0 (w) + h2 u00 (w) + R2
2
After substitution we have
1 2 00
0
u (w) = π (w, h) u (w) + hu (w) + h u (w) + R1 +
2
1
+ 1 − π (w, h) u (w) − hu0 (w) + h2 u00 (w) + R2
2
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Risk Aversion
Interpreting Risk Aversion
Proof (cont’d).
Collecting terms yields
1 2 00
u (w) = u (w) + 2π (w, h) − 1 hu (w) + h u (w) +
2
+ π (w, h) R1 + 1 − π (w, h) R2
{z
}
|
0
≡R
Solving for π (w, h) results in
1 1
R
u00 (w)
1 1
∼
π (w, h) = + h − 0
−
= + h α (w)
2 4
u (w)
2hu0 (w)
2 4
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Risk Aversion
Interpreting Risk Aversion
Example: Exponential Utility
Example
Consider the utility function of the form
1
u (w) = − e−νw
ν
It is easy to show that α (w) = ν.
Hence, we have that
1 1
π (w, h) ∼
= + hν
2 4
Now the odds demanded are independent of the level of initial wealth,
but depend on the amount of wealth at risk.
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Risk Aversion
Interpreting Risk Aversion
Relative Risk Aversion and the Odds of a Bet
Proposition
Consider an investment opportunity similar to the one above, but now the
amount at risk is proportional to the investor’s wealth, i.e., h = θw. Then we
have that
1 1
π (w, θ ) ∼
= + θρ (w) .
2 4
Proof.
Very similar to the case of absolute risk aversion.
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Risk Aversion
Interpreting Risk Aversion
Example: Power Utility
Example
Consider the utility function of the form
 1− γ
 w
if γ > 0, γ 6= 1
u (w) =
1−γ

ln w
if γ = 1
It is easy to show that ρ (w) = γ.
Hence, we have that
1 1
π (w, h) ∼
= + θγ
2 4
Now the odds demanded are independent of the level of initial wealth.
but depend on the fraction of wealth that is at risk.
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Risk Aversion
Interpreting Risk Aversion
Risk-Neutral Investors
Risk-neutral investors are identified with an affine utility function
u (w) = c w + d
with c > 0
It is easy to see that both α (w) = 0 and ρ (w) = 0
Such investors do not demand better than even odds when
considering risky investments (π (w, h) = π (w, θ ) = 1/2)
The are indifferent to risk and are concerned only with an asset’s
expected payoff
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Risk Aversion
Risk Premium and Certainty Equivalent
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
Risk Premium and Certainty Equivalent
Definitions
Consider an investor, with current wealth w, evaluating an
e
uncertain risky pay-off Z
Relying on the definition of risk aversion, we have
h
i
e ≥ E u w+Z
e
u w+E Z
If an uncertain pay-off is available for sale, a risk-averse agent will
only be willing to buy it at a price less than its expected pay-off
Certainty equivalent (CE): the maximal certain amount of money
an investor is willing to pay for a lottery
Risk premium (π): the maximum amount the agent is willing to
pay to avoid the gamble
The risk premium is the difference between the expected value of
the bet and the certainty equivalent
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Risk Aversion
Risk Premium and Certainty Equivalent
Mathematical Formulation
According to the above definition, the certainty equivalent and
the risk premium are the solutions of the following equations:
h
i
e = u w + CE w, Z
e =
E u w+Z
e − π w, Z
e
= u w+E Z
It follows that
e =E Z
e − π w, Z
e
CE w, Z
or
e =E Z
e − CE w, Z
e
π w, Z
Note that from the definition
of CE we have that
e
u (w + CE) < u w + E Z
e
Since u is increasing (recall that u0 > 0), we must have CE < E Z
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Risk Aversion
Risk Premium and Certainty Equivalent
CE and Risk Premium: An Illustration
Source: Danthine and Donaldson (2005), Figure 4.3
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Risk Aversion
Risk Premium and Certainty Equivalent
Example: Certainty Equivalent
Example
Consider an investor with log utility and initial wealth w0 = 1, 000.
e = (200, 0, 0.5). What is the certainty
She is offered the lottery Z
equivalent of this gamble?
Solution
The equation that yields the CE is
h
i
e = u w + CE
E u w+Z
Substitution results in
0.5 · ln (1, 200) + 0.5 · ln (1, 000) = ln (1, 000 + CE)
from which CE = 95.45
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Risk Aversion
Risk Premium and Certainty Equivalent
Relation Between Risk Premium and Risk Aversion
e = 0, u is twice continuously differentiable and
Assume that E Z
the gamble is a bounded random variable
It can be shown that
1
e ∼
π w, Z
= σz2 α (w)
2
The amount one would pay to avoid the gamble is approximately
proportional to the coefficient of absolute risk aversion
e then
Let π = θw be the risk premium of w + wZ;
1
θ∼
= σz2 ρ (w)
2
The proportion θ of initial wealth that one would pay to avoid a
e of initial wealth depends on
gamble equal to the proportion Z
e
relative risk aversion and the variance of Z
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Risk Aversion
Risk Premium and Certainty Equivalent
Example: Power Utility
Example
1− γ
Consider again the utility function u (w) = w1−γ with γ = 3,
e = ($100, 000; −$100, 000; 0.5).
w = $500, 000, and Z
Then the risk premium will be
1 γ
1
3
e ∼
π w, Z
= $30, 000
= σz2 = · 100, 0002 ·
2 w
2
500, 000
To double-check the approximation, calculate
e = u (500, 000 − 30, 000) = −2.26347−12
u w − π w, Z
h
i
e = 1 u (600, 000) + 1 u (400, 000) = −2.25694−12
E u w+Z
2
2
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Risk Aversion
Risk Premium and Certainty Equivalent
Certainty Equivalent in Terms of Returns
Let the equivalent risk-free return be defined as
e
u w 1 + rf
= u w + CE w, Z
e can also be converted into a return
The random pay-off Z
e
e
distribution via Z = erw, or, er = Z/w
Hence, rf is defined by the equation
u w 1 + rf
= E u w 1 + er
The return risk premium, π r , is defined as
π r = E er − rf or E er = rf + π r
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Risk Aversion
Risk Premium and Certainty Equivalent
Example: Log Utility
Example
Consider the utility function u (w) = ln w, and assume w = $500, 000,
e = ($100, 000; −$50, 000; 0.5).
and Z
Then the risky return is 20% with probability 0.5 and
−10% with
probability 0.5, and with an expected return of E er = 5%.
The certainty equivalent must satisfy
ln (500, 000 + CE) =
1
1
ln (600, 000) + ln (450, 000)
2
2
which implies that CE = 19, 615
It follows then that the risk-free return is
1 + rf =
519, 615
= 1.0392
500, 000
The return risk premium is π r = 5% − 3.92% = 1.08%
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Risk Aversion
Constant Absolute Risk Aversion
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
Constant Absolute Risk Aversion
CARA Utility
Constant Absolute Risk Aversion (CARA): absolute risk aversion
is the same at every wealth level
Every CARA utility function is a monotone affine transform of the
exponential utility function
u (w) = −e−αw
where α is a constant and equal to the absolute risk aversion
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Risk Aversion
Constant Absolute Risk Aversion
No Wealth Effects With CARA Utility
With CARA utility there are no wealth effects
h
i
e = 0, we have that u (w − π ) = E u w + Z
e
Assuming that E Z
Calculating the two sides yields
u (w − π ) = −e−αw eαπ
and
e = −e−αw e−αZe
u w+Z
e
Taking expectations and equating gives eαπ = E e−αZ , from
which it follows that
π=
1
e
ln E e−αZ
α
Hence, an investor with CARA utility will pay the same to avoid a
fair gamble no matter what her initial wealth might be
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Risk Aversion
Constant Absolute Risk Aversion
Normally Distributed Bet
e is Gaussian, then the risk premium can be
If the gamble Z
calculated more explicitly
Recall that if e
x is Gaussian with mean µ and variance σ2 , then
1 2
E eex = eµ+ 2 σ
e which has zero mean and variance α2 σz2 ;
Now we have e
x = − αZ
thus the risk premium becomes
π=
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1 2
ασ
2 z
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Risk Aversion
Constant Relative Risk Aversion
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
Constant Relative Risk Aversion
CRRA Utility
Constant Relative Risk Aversion (CRRA): relative risk aversion
is the same at every wealth level
Any CRRA utility function has decreasing absolute risk aversion
as α (w) = ρ (w) /w
Any monotone CRRA utility function is a monotone affine
transform of one of the following functions:
Log utility: u (w) = log w
Power utility:
u (w) =
1 γ
w
γ
γ < 1, γ 6= 0
A more convenient formulation of power utility is
u (w) =
w1−ρ
1−ρ
ρ = 1 − γ > 0, ρ 6= 1
It is easy to show that ρ is the coefficient of relative risk aversion
of the power utility function
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Risk Aversion
Constant Relative Risk Aversion
Properties of CRRA Utility Functions
Log utility has constant relative risk aversion equal to 1
An investor with power utility is said to be more risk averse than
a log-utility investor if ρ > 1 and to be less risk averse if ρ < 1
The fraction of wealth a CRRA-utility investor would pay to avoid
a gamble that is proportional to initial wealth is independent of
her wealth
Log utility is a limiting case of power utility obtained by taking
ρ −→ 1 (by l’Hôpital’s rule)
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Risk Aversion
Quadratic Utility
Outline
1
An Introduction to Choice Theory
Dominance
Choice Theory Under Certainty
Choice Theory Under Uncertainty
How Restrictive Is Expected Utility Theory? The Allais Paradox
2
Risk Aversion
How To Measure Risk Aversion?
Interpreting Risk Aversion
Risk Premium and Certainty Equivalent
Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
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Risk Aversion
Quadratic Utility
Definition and Properties
Quadratic utility takes the form
b
u ( w ) = w − w2
2
b>0
The marginal utility of wealth is u0 (w) = 1 − bw, and is positive
only when b < 1/w
Thus, this utility function makes sense only when w < 1/b, and
the point of maximum utility, 1/b, is called the “bliss point”
The absolute and relative risk aversion coefficients are
α (w) =
b
1 − bw
and
ρ (w) =
bw
1 − bw
It has increasing absolute risk aversion, an unattractive property
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Risk Aversion
Quadratic Utility
Quadratic Utility and Mean-Variance Preferences
Quadratic utility has a special importance in finance as it implies
mean-variance preferences
Since all derivatives of order higher than 2 are equal to zero, the
investor’s expected utility becomes See Appendix
1
e ) = u E (w
e ) + u00 E (w
e ) Var (w
e)
E u (w
2
Specifically,
b
e ) = µ − σ 2 + µ2
E u (w
2
For any probability distribution of wealth, the expected utility
depends only on the mean and variance of wealth
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A PPENDIX
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Some Important Definitions and Results
Concavity and Jensen’s Inequality
Definition (Concavity)
A function f is concave if, for any x and y and any λ ∈ (0, 1),
f λx + (1 − λ) y ≥ λf (x) + (1 − λ) f (y) .
If the inequality is strict, the function is strictly concave.
Theorem (Jensen’s inequality)
Let f be a concave function on the interval (a, b), and e
x be a random variable
such that Pr [e
x ∈ (a, b)] = 1. Provided that the expectations E (e
x) and
x)] exist,
E [f (e
E f (e
x) ≤ f E (e
x) .
Moreover, if f is strictly concave and Pr [e
x = E (e
x)] 6= 1, then the inequality
is strict.
Return
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Some Important Definitions and Results
Taylor Expansion of a Function
Definition (Taylor Expansion of a Function)
Let f (x) be a real- or complex-valued function, which is infinitely
differentiable at a point x0 . Then the function can be written as a
power series, called the Taylor expansion of the function, as
f ( x ) = f ( x0 ) +
+
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f 0 ( x0 )
f 00 (x0 )
( x − x0 ) +
( x − x0 ) 2 +
1!
2!
f 000 (x0 )
( x − x0 ) 3 + · · ·
3!
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Some Important Definitions and Results
Taylor Expansion for the Mean of a Random Variable
Theorem (Taylor Expansion for the Mean of a Random Variable)
Let e
x be a random variable with mean µ and variance σ2 (both finite),
and let
f be a twice differentiable function. Then the expected value of f e
x can be
approximated as
∼ f (µ) + 1 f 00 (µ) σ2 .
E f (e
x) =
2
Proof.
The Taylor expansion of the expected value of f (e
x) is
1 00
2
0
∼
E f (e
x) = E f (µ) + f (µ) (e
x − µ) + f (µ) (e
x − µ) .
2
The expectation of the term containing the first derivative is zero, thus
the expression simplifies to the one presented in the theorem.
Return
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