,1. Utility Theory
Jing Yao F71AH&PT 1 / 29
,Summary
In economics, utility is a measure of the satisfaction or happiness from
consumption of goods or service. A higher (lower) level of utility means a
higher (lower) degree of satisfaction or happiness.
Mathematically, utility can be expressed in terms of a function, namely, an
utility function, which assigns a real number to a given level of consumption or
wealth. Utility functions are basic or fundamental objects of economic analysis.
Many theories in economics and finance, in particular, in modern portfolio
theory and financial economics, are established under the assumption that
(rational) economic agents maximize their utilities, expressed in terms of utility
functions, subject to some (budget) constraints. So, utility functions do play an
indispensable role in economic and financial analysis.
In this chapter, we discuss some important characteristics of utility functions
and go through some typical examples of utility functions. We also discuss how
utility functions can be used to make decisions or choices in an uncertain world,
in which future financial outcomes are uncertain or not anticipated. We
introduce an expected utility theorem or criterion, which describes the
utility-maximization behavior of an economic agent. We also describe an
important concept, called risk aversion, which describes the typical risk
avoiding behavior of an economic agent.
Jing Yao F71AH&PT 2 / 29
,Chapter outline
1.1: General features of utility functions
1.2: Examples
1.3: The expected utility criterion
1.4: The certainty equivalent
1.5: Risk aversion
Reading: Joshi and Paterson, chapters 7-9.
Jing Yao F71AH&PT 3 / 29
,1.1 General Features of Utility Functions
Utility function: u(x) is a real-valued function of
a real variable x
x : Wealth of an economic agent
Intuition:
u(x) measures how happy or satisfied an
economic agent is if they have a wealth of x
units
Jing Yao F71AH&PT 4 / 29
,General Features of Utility Functions
Two main features:
Non-satiation:
u(x) is increasing
u′(x) ≥ 0
We prefer more wealth to less wealth
Risk-averse:
u(x) is concave
u′′(x) ≤ 0
Given that we have £x , gaining £∆ > 0 is
good, but losing £∆ is very much worse
Jing Yao F71AH&PT 5 / 29
,1.2 Examples
Exponential Utility:
u(x) = 1− e−αx , where α > 0 is a fixed
parameter
x ∈ (−∞,∞)
u′(x) = αe−αx > 0
u′′(x) = −α2e−αx < 0
Quadratic Utility:
u(x) = x − αx2, where α > 0 is fixed
x ∈ (−∞, 1/(2α))
u′(x) = 1− 2αx > 0, for x < 1/(2α)
u′′(x) = −2α < 0
Jing Yao F71AH&PT 6 / 29
,1.2 Examples (cont.)
Logarithmic Utility:
u(x) = ln x
x ∈ (0,∞)
u′(x) = 1/x > 0
u′′(x) = −1/x2 < 0
Power Utility:
u(x) = x
γ
γ , where γ < 1, γ 6= 0 is fixed
x ∈ (0,∞)
u′(x) = xγ−1 > 0
u′′(x) = (γ − 1)xγ−2 < 0
Alternative Power Utility: u(x) = (xγ − 1) /γ
Jing Yao F71AH&PT 7 / 29
,1.2 Examples (cont.)
Linear Utility:
u(x) = x
x ∈ (−∞,∞)
u′(x) = 1 > 0
u′′(x) = 0
Jing Yao F71AH&PT 8 / 29
,1.3 The Expected Utility Criterion
Suppose
1 the attitude to wealth of an economic agent (a
person or a corporation) is represented as a
utility function u(x);
2 the agent has current wealth w ;
3 the agent has to choose 1 out of n possible
actions, where Xi represents an increase in
wealth if action i is chosen and Xi is a random
variable.
Jing Yao F71AH&PT 9 / 29
,1.3 The Expected Utility Criterion (cont.)
Then, the expected utility criterion says that the
agent should choose action k , where
E [u(w + Xk)] = max
1≤i≤n
E [u(w + Xi)]
If
E [u(w + Xk)] = E [u(w + Xj)]
= max
1≤i≤n
E [u(w + Xi)] ,
then the agent is indifferent between action k and
action j
Jing Yao F71AH&PT 10 / 29
,1.3 The Expected Utility Criterion (cont.)
1 Rationale: Action i ⇒ wealth w + Xi
The value of this to the agent is u(w + Xi).
This is a random variable. Simple comparison of
random variables is by comparing their means.
So, we choose the action with the largest value
of E [u(w + Xi)].
2 Reasonable postulates
⇒ Easily accepted
⇒ Decisions made on the basis of the EUC with
respect to some utility function
Jing Yao F71AH&PT 11 / 29
,1.3 The Expected Utility Criterion (cont.)
3 Positive linear transformations:
The utility functions u(x) and a + bu(x), where
b > 0 ⇒ Identical decisions
4 In the textbook, the authors do not include
initial wealth explicitly, but, it is implicit in their
treatment.
5 For an exponential utility function, initial wealth
does NOT affect decisions (using the expected
utility criterion)
Jing Yao F71AH&PT 12 / 29
,1.3 The Expected Utility Criterion (cont.)
Proof of (5):
Suppose u(x) = 1− e−αx , where α > 0. Then,
E [u(w + Xi)] = E [1− e−α(w+Xi )]
= 1− e−αwE [e−αXi ].
So, E [u(w + Xk)] = max
1≤i≤n
E [u(w + Xi)]
⇐⇒ 1− e−αwE [e−αXk ] = max
1≤i≤n
{1− e−αwE [e−αXi ]}
⇐⇒ −e−αwE [e−αXk ] = max
1≤i≤n
{−e−αwE [e−αXi ]}
⇐⇒ E [e−αXk ] = min
1≤i≤n
E [e−αXi ]
Jing Yao F71AH&PT 13 / 29
,1.4 The Certainty Equivalent
Suppose an agent has a utility function u(x) and
current wealth level w .
The agent COULD choose an action, which will
increase wealth to w + X .
Then, the CERTAINTY EQUIVALENT of this
uncertain outcome X (given w and u(x)) is the
constant K such that
u(w + K ) = E [u(w + X )]
Jing Yao F71AH&PT 14 / 29
,1.4 The Certainty Equivalent (cont.)
Remarks:
1 For a well behaved utility function u(x) and
random variable X , K exists and is unique
2 Intuitively, the agent is indifferent between
receiving the certain amount K and the random
amount X (i.e. u(w + K ) = E [u(w + X )])
Jing Yao F71AH&PT 15 / 29
,1.4 The Certainty Equivalent (cont.)
3 Examples:
(a) X ≤ 0 represents losses from damage to your car in
one year
X ≤ 0⇒ K < 0.
−K =the maximum premium you are willing to pay to
cover the losses
(b) X > 0 represents the net payoff from a lottery
K > 0 represents the fixed amount you are prepared to
receive to be indifferent to participating in this lottery
Jing Yao F71AH&PT 16 / 29
,1.5 Risk Aversion
Definition 1.5.1: Absolute Risk Aversion
For a given utility function u(x), at a given level of
wealth x ,
ARA(x) = −u
′′(x)
u′(x)
Definition 1.5.2: Relative Risk Aversion
RRA(x) = −x u
′′(x)
u′(x)
Jing Yao F71AH&PT 17 / 29
,1.5 Risk Aversion (cont.)
Interpretation:
u′(x) measures how much you would like to have
more wealth (given current wealth x)
u′′(x) measures how much more you dislike
losing wealth than gaining wealth
⇒ −u′′(x)u′(x) will be higher for someone who is not
particularly keen to gain more wealth (low u′(x))
and does not like losing wealth (high u′′(x))
⇒ −u′′(x)u′(x) is high for RISK AVERSE people
(ARA)
Jing Yao F71AH&PT 18 / 29
,1.5 Risk Aversion (cont.)
ARA = −u′′(x)u′(x) is a function of current wealth x
Question: How does ARA behave as a function
of wealth x?
Answer: It depends on the agent’s utility
function
General features:
−u′′(x) usually decreases as x increases while
u′(x) decreases as x increases since u′′(x) < 0.
So, ARA might increase or decrease as x
increases.
Jing Yao F71AH&PT 19 / 29
,1.5 Risk Aversion (cont.)
RRA= −x u′′(x)u′(x) is an attempt to measure a risk
aversion in a way independent of current wealth x
Intuition: If −u′′(x)u′(x) decreases with wealth x , then
−x u′′(x)u′(x) MIGHT be reasonably constant
One major study: RRA is reasonably constant
Another major study: RRA decreases with wealth
Jing Yao F71AH&PT 20 / 29
,1.5 Risk Aversion (cont.)
Invariance under linear transformation:
ARA and RRA remain unchanged if we replace u(x)
by a + bu(x), where b > 0
Jing Yao F71AH&PT 21 / 29
,1.5 Risk Aversion (cont.)
Example: Insurance
An economic agent has a power utility function
u(x) = xγ/γ, 0 < γ < 1
(a) Determine the ARA and RRA as functions of x
(b) The agent faces an uncertain loss X over the coming year,
where X ∼ U(0, 4000). The agent can insure against this loss
by paying a premium of P . Calculate the maximum premium
the agent is prepared to pay given:
(A) γ = 0.1
(B) γ = 0.01
1. Current wealth = 10,000
2. Current wealth = 100,000
Jing Yao F71AH&PT 22 / 29
,1.5 Risk Aversion (cont.)
Solution:
(a) u(x) = x
γ
γ
⇒ u′(x) = xγ−1 > 0 and u′′(x) = (γ − 1)xγ−2 < 0
⇒ ARA= 1−γ
x
(decreasing as a function of x)
⇒ RRA = 1− γ (constant as a function of x)
(b) From Part (a), the utility function behaves as we might
expect
A: γ = 0.1 B: γ = 0.01
(1) w = 10,000 P1,A P1,B
(2) w = 100,000 P2,A P2,B
Note that ARA and RRA are decreasing functions of γ.
Jing Yao F71AH&PT 23 / 29
,1.5 Risk Aversion (cont.)
What do we expect?
P1,A < P1,B (B is more risk averse than A)
P1,A > P2,A (loss from (1) is more disastrous than (2))
General case: Given wealth = w , parameter γ, we want to find
the maximum premium = P
Two choices:
Insure: wealth = w − P with certainty
Do not insure: wealth = w − X ∼ U(w − 4000,w)
Compare expected utilities:
E [u(w − P)] = u(w − P)
E [u(w − X )]
Jing Yao F71AH&PT 24 / 29
,1.5 Risk Aversion (cont.)
Expected utility criterion:
Buy insurance if E [u(w − P)] > E [u(w − X )]
Do not buy if E [u(w − P)] < E [u(w − X )]
The maximum premium is P such that
u(w − P) = E [u(w − X )]
⇒ u(w − P) = ∫ w
w−4000
1
4000
( x
γ
γ
)dx
⇒ (w−P)γ
γ
=
∫ w
w−4000
1
4000
( x
γ
γ
)dx
⇒ (w − P)γ = 1
4000
[
xγ+1
γ+1
]w
w−4000
Jing Yao F71AH&PT 25 / 29
,1.5 Risk Aversion (cont.)
Hence
P1,A = 2, 076.00
P2,A = 2, 006.12
P1,B = 2, 083.65
P2,B = 2, 006.84
Jing Yao F71AH&PT 26 / 29
,1.5 Risk Aversion (cont.)
Example: Choosing between two investments A and B.
Given a quadratic utility function u(w) = w − 0.05w 2
u′(w) = 1− 0.1w > 0 (non-satiation) if w < 10
u′′(x) = −0.1
⇒ ARA = 0.1
1−0.1w =
1
10−w (increasing with w , unusual !)⇒ RRA = w
10−w (increasing with w)
Investment A
Outcome Utility Probability
5 3.75 0.2
7 4.55 0.5
10 5 0.3
Investment B
Outcome Utility Probability
6 4.2 0.3
8 4.8 0.6
9 4.95 0.1
Jing Yao F71AH&PT 27 / 29
,1.5 Risk Aversion (cont.)
Expected utility:
Investment A: EU = 4.525
Investment B : EU = 4.635
The investor would prefer Investment B
Jing Yao F71AH&PT 28 / 29
,1.5 Risk Aversion (cont.)
Note that E [u(W )] = E [W ]− 0.05E [W 2]
Also
E [A] = 5× 0.2 + 7× 0.5 + 10× 0.3 = 7.5
E [B] = 7.5 = E [A]
E [A2] = 59.5 and E [B2] = 57.3
The investor is RISK AVERSE
Quadratic utility + same expected value
⇒ Prefer lower variance
Hence Variance is often used as a measure of risk.
Jing Yao F71AH&PT 29 / 29