Topic 3. Introduction to Portfolio Selection
under Uncertainty
ECON30024 Economics of Finanical Markets
Shuyun May Li
Department of Economics
The University of Melbourne
Semester 1, 2025
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Outline
1. Risk and uncertainty.
2. The expected utility hypothesis (EUH)
• Expected utility
• Risk aversion
• Portfolio selection with expected utility
3. The Mean-Variance approach
Required reading: Chap. 4 of Bailey
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1. Risk and Uncertainty
• As discussed in Topic 1, a theory of asset price determination
requires modeling portfolio selection under uncertainty.
• Such decision also involves specifying a utility function and a
budget constraint for investors.
• How people incorporate uncertainty in their preference or
utility function?
• The expected utility hypothesis (EUH) remains the most
popular approach to modeling uncertainty in economics.
• The mean-variance objective is a special case of expected
utility.
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• In economics, risk refers to those unknown events for which
‘objective probabilities’ can be assigned.
• Uncertainty applies to events for which probabilities cannot
be assigned without being subjective.
• Unknown events in financial markets are typically located
somewhere between the two extremes.
• We’ll assume probabilities for unknown events can be assigned
(objectively or subjectively), and use the two terms
interchangeably.
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2. The Expected Utility Hypothesis (EUH)
• Basic ingredients in portfolio selection problem under
uncertainty
1) States: uncertainty is represented by states of the world:
S = {s1, s2, ..., sK},
where each sk denotes a state that could occur.
2) Actions: a portfolio choice made prior to the state of the
world being revealed:
(x1, x2, .., xn)
where each xj denotes units of asset j chosen to hold.
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3) Consequences: terminal wealth from the portfolio in each
state of the world
Wk = vk1x1 + · · ·+ vknxn =
nX
j=1
vkjxj , k = 1, . . . ,K (1)
where vkj is the unit payoff of asset j in state k.
4) Preferences: consequences are valued according to a
utility function:
U(W1,W2, ..,WK)
5) Constraints: we focus on budget constraint
nX
j=1
pjxj = A (2)
where A is initial wealth, pj is unit price of asset j.
• An investor’s portfolio selection problem: choose (x1, . . . , xn)
to maximise U , subject to the budget constraint.
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2.1 Expected utility
• Under the expected utility hypothesis, preference U is specified
as an expected utility:
E(u(W )) = pi1u(W1) + pi2u(W2) + . . .+ piKu(WK) (3)
– pik is the probability the investor assigns to state sk,
reflecting her belief about which state will occur.
– u(·) is a function of a single possible consequence that is
the same for all states, reflecting the investor’s preferences
over the consequences.
– u is known as the von Neumann-Morgenstern (vNM)
utility function.
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• The expected utility form seems quite natural, however, it was
mathematically derived under some strong assumptions on
preferences.
– This is established in the Expected Utility Theorem.
– For instance, one assumption is that preferences are
independent of beliefs.
– Further discussion on the underlying assumptions are in
Tutorial 3.
– Despite evidence on individual behaviour that contradicts
its assumptions (Topic 11), EUH remains the mainstream
approach in modeling decision making under uncertainty.
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2.2 Risk aversion
• At the center of the expected utility form in (3) is the vNM
utility function u.
• First, u reflects investors’ attitude toward wealth.
– It’s natural to assume more wealth is preferred to less.
– We assume u(W ) is strictly increasing in W , or
u′(W ) > 0.
• Second, u reflects investors’ risk preference.
– Do you like variability of outcomes across states?
· No: Risk averse
· Indifferent: Risk neutral
· Yes: Risk loving
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– An example: Consider a bet that pays $0 with probability
0.5 and $100 with probability 0.5.
Would you choose the bet or $50 for sure?
What can you infer about your utility function u?
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– Formalise risk preference by the curvature of u
Risk averse : E[u(W )] < u(E[W ])
Risk neutral : E[u(W )] = u(E[W ])
Risk loving : E[u(W )] > u(E[W ])
Equivalently,
Risk averse : u is strictly concave, or u′′(W ) < 0
Risk neutral : u is linear, or u′′(W ) = 0
Risk loving : u is strictly convex, or u′′(W ) > 0
– There is considerable empirical evidence that individuals are
risk averse, so u is assumed to be strictly concave.
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Figure 3.1: A concave vNM utility function u(W )
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• Third, further assumptions on u to reflect the degree of risk
aversion
– Absolute risk aversion (ARA): would you increase or
decrease the absolute amount of wealth invested in risky
assets if your wealth increases?
· increase: decreasing ARA
· no change: constant ARA
· decrease: increasing ARA
– Relative risk aversion (RRA): would you increase or
decrease the percentage of wealth invested in risky assets if
your wealth increases?
· increase: decreasing RRA
· no change: constant RRA
· decrease: increasing RRA
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– Formalise the degree of absolute risk aversion:
· Measure of ARA:
RA(W ) = −
u′′(W )
u′(W )
> 0
· Examples of u(W ) that exhibit increasing, constant, or
decreasing ARA:
IARA R′A(W ) > 0 u(W ) =W − bW
2, b > 0, 0 ≤W < 12b
CARA R′A(W ) = 0 u(W ) = − exp(−cW ), c > 0
DARA R′A(W ) < 0 u(W ) =
W 1−γ
1−γ , γ > 0, γ 6= 1
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– Formalise the degree of relative risk aversion:
· Measure of RRA:
RR(W ) =WRA(W ) = −
Wu′′(W )
u′(W )
> 0
· Examples of u(W ) that exhibit increasing, constant, or
decreasing RRA:
IRRA R′R(W ) > 0 u(W ) = − exp(−cW ), c > 0
CRRA R′R(W ) = 0 u(W ) =
W 1−γ
1−γ , γ > 0, γ 6= 1
DRRA R′R(W ) < 0 u(W ) = −
1
W−1
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– Negative exponential utility (− exp(−cW )) and power
utility (W
1−γ
1−γ ) are often used in the literature.
· u(w) = − exp(−cW ) exhibits CARA:
RA(W ) ≡ −
u′′(W )
u′(W )
= c
· u(w) = W
1−γ
1−γ exhibits CRRA:
RR(W ) ≡ −
Wu′′(W )
u′(W )
= γ
Verify these results and check your work with the solution in
Exercise Topic3.
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– Relationship between ARA and RRA (see Exercise Topic3)
· If an investor has CRRA, then she has increasing,
decreasing or constant ARA?
– There is empirical evidence of CRRA and DARA (Blume
and Friend, 1975, AER).
• In summary, typical assumptions imposed on the vNM utility
function u(W )
– u is strictly increasing
– u is strictly concave
– u exhibits CRRA (often assumed, not always)
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2.3 Portfolio selection with expected utility
• The static portfolio selection problem for an investor:
max
(x1,...,xn)
E(u(W ))
s.t.
nX
j=1
pjxj = A
– Recall that E(u(W )) ≡
∑K
k=1 piku(Wk), where
Wk = vk1x1 + · · ·+ vknxn =
n∑
j=1
vkjxj , k = 1, . . . ,K (1)
– Since (1) holds for all k, we have
W =
n∑
j=1
vjxj (4)
where vj is asset j’s unit payoff (a random variable).
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• Alternative formulation of the portfolio selection problem
– Let aj ≡
pjxj
A
be the proportion of initial wealth invested in
asset j, then the budget constraint is re-written as
nX
j=1
aj = 1
– Let rj denote the rate of return on asset j, j = 1, . . . , n, then
rj =
vj − pj
pj
⇒ vj = (1 + rj)pj
– Then (4) becomes
W =
nX
j=1
vjxj =
nX
j=1
(1 + rj)pjxj =
nX
j=1
(1 + rj)ajA
=

nX
j=1
aj +
nX
j=1
ajrj
!
A =

1 +
nX
j=1
ajrj
!
A
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– Define
rP ≡
n∑
j=1
ajrj (5)
then
W = (1 + rP )A
Note that rP is the rate of return on the portfolio.
– The portfolio selection problem can be re-written as:
max
(a1,...,an)
E
[
u [(1 + rP )A]
]
, s.t.
n∑
j=1
aj = 1
– Given u and the distributions of rj ’s, the constrained
maximisation problem can be solved (though not easy).
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3. The Mean-Variance Approach (MVA)
• In the MVA, the preference or objective of the investor is
assumed to be
G(µP , σ
2
P )
– µP : the expected rate of return on the portfolio
– σ2P : the variance of the rate of return on the portfolio,
representing the ‘risk’ of the portfolio.
– G is increasing in µP and decreasing in σP , i.e., µP is a
‘good’, and σP is a ‘bad’.
• Example: G(µP , σ
2
P ) = µP − ασ
2
P ,
what is the investor’s risk preference if α > 0, or α = 0?
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• The mean-variance objective is a special case of expected
utility.
– If u is a quadratic function, the expected utility E(u(W ))
can be written as a mean-variance objective.
(see Exercise Topic3)
• Mean-variance objective is a reasonable approximation to
investors’ preference.
• However, it could ignore other things about the portfolio which
investors may view as important.
(Tutorial 3 discussion)
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• Indifference curves of G(µP , σ
2
P )
– Indifference curves are drawn in (σP , µP ) space.
– Each indifference curve represents the combinations of σP
and µP that give the same level of utility.
An example: Let G(µP , σ
2
P ) = µP − ασ
2
P , where α > 0, plot
an indifference curve of G.
G(µP , σ
2
P ) = g0, i.e., µP − ασ
2
P = g0
⇒ µP = ασ
2
P + g0.
That is, µP is an increasing convex function of σP .
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Figure 3.2: Indifference curves of a mean-variance objective
Why are indifference curves of a mean-variance objective
upward slopping and convex? (Tutorial 3)
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Review Questions
1. What is the difference between ‘risk’ and ‘uncertainty’?
2. Understand the definitions of states, actions, consequences and
preferences.
3. Under the EUH, how can you express the preferences of investors?
4. Understand the concepts of risk averse, risk neutral, and risk loving.
5. What are the mathematical properties of the vNM utility function
for it to exhibit risk aversion, risk neutrality, and risk loving?
6. Why do we use utility functions that exhibit risk aversion in our
analysis?
7. Understand the concept of absolute risk aversion and relative risk
aversion.
8. What are the relationships between ARA and RRA?
9. How to measure ARA and RRA?
10. Give a utility function that exhibits constant RRA.
11. How to formulate the portfolio selection problem with expected
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utility in two alternative ways? One way is to choose the quantity of
each asset to hold, another is to choose the proportion of initial
wealth invested in each asset.
12. How is the rate of return on a portfolio defined? Why it is a
weighted sum of the rate of returns on the individual assets? Derive
this expression mathematically.
13. What is the mean-variance objective function? Why is it increasing
in µP and decreasing in σP ?
14. Understand why the indifference curves of a mean-variance objective
are upward sloping and convex.
15. For G(µP , σ
2
P ) = µP − ασ
2
P , be able to draw the indifference curves.
16. Roughly understand why the mean-variance objective is a special
case of expected utility.
17. Think about other things that you care about other than expected
return and variance of return on the portfolio when choosing a
portfolio of assets to invest in.
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