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```									          Principles of Finance

Grzegorz Trojanowski

Lecture 2:
Theory of investor’s choice

October 12, 2004                  Principles of Finance - Lecture 2   1

Lecture 2 material
Elton et al., Chapters 1, 10
Sharpe et al., Chapter 6
textbook

October 12, 2004                  Principles of Finance - Lecture 2   2

1
Lecture 2: Checklist
• By the end of this lecture you should:
Be familiar with the notions of preference relation,
utility function, certainty equivalent, and risk
Be familiar with different measures of risk aversion
Be able to compute risk premium coefficients for a
given utility function
Be familiar with the properties of the most
commonly used utility functions
Be aware of the corresponding empirical evidence

October 12, 2004              Principles of Finance - Lecture 2      3

Example:
Choice under certainty (1)
• Two periods: Year 1 and Year 2
• Investor receives an income of ₤ 10,000 each
period (with certainty)
• Interest rate for borrowing/lending: 5%

• Questions:
What are the options open to an investor?
Which of them should be chosen?

October 12, 2004              Principles of Finance - Lecture 2      4

2
Example:
Choice under certainty (2)
• Scenario A: Save all the income until Year 2; the
amounts consumed equal to ₤ 0 in Year 1 and ₤ 20,500
(i.e. ₤ 10,000 + ₤ 10,000 · 1.05) in Year 2
• Scenario B: Consume the Year 1 income in Year 1 and
the Year 2 income in Year 2; the amounts consumed
equal to ₤ 10,000 in each period
• Scenario C: Consume as much as possible in Year 1 and
borrow against Year 2 income; the amounts consumed
in Year 1 and 2 equal to ₤ 19,524 and ₤ 0, respectively
10,000
X + 0.05 X = 10,000 ⇒              X=             = 9,524
1.05
October 12, 2004                 Principles of Finance - Lecture 2             5

Example:
Choice under certainty (3)
• Intermediate scenarios also possible, e.g. the investor
can consume just ₤ 5,000 in Year 1 and save the
remainder of the Year 1 income until Year 2
• All the available options (the opportunity set) can be
described by the following equation:

⎡Year 2       ⎤ ⎡Year 2 ⎤ ⎡Amount           ⎤
⎢consumption ⎥ = ⎢income⎥ + ⎢saved in Year 1⎥ (1 + 0.05)
⎣             ⎦ ⎣          ⎦ ⎣              ⎦
C2 = £ 10,000 + (£ 10 ,000-C1 ) ⋅1.05
C2 = £ 20,500 + 1.05 ⋅ C1
October 12, 2004                 Principles of Finance - Lecture 2             6

3
Example:
Choice under certainty (4)

October 12, 2004     Principles of Finance - Lecture 2   7

Example:
Choice under certainty (5)

• Investor is likely to have some preferences
over the possible options (consumption
bundles)

• A useful tool for the analysis of preferences
are the indifference curves

October 12, 2004     Principles of Finance - Lecture 2   8

4
Example:
Choice under certainty (6)

October 12, 2004   Principles of Finance - Lecture 2   9

Example:
Choice under certainty (7)

October 12, 2004   Principles of Finance - Lecture 2   10

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Preference relation
• Consider a consumption bundle
x ∈ {x1, x2, …, xn}
from some feasible set, where the elements of x
are sure things
• For pairs (x, y):
If x ≥ y, then x is weakly preferred to y
Strict preference: x > y if x ≥ y but not y ≥ x
Indifference: x ~ y, if x ≥ y and y ≥ x
• A preference relation is a binary relation with the
properties of completeness and transitivity
October 12, 2004              Principles of Finance - Lecture 2   11

Preference relation (aside)

• Completeness (or comparability) means that
given any two bundles x and y, the agent is able to
state whether x ≥ y, y ≥ x, or x ~ y

• Transitivity means that the choices are consistent,
i.e. if x ≥ y and y ≥ z, then x ≥ z

October 12, 2004              Principles of Finance - Lecture 2   12

6
Ordinal utility functions
• A preference relation ≥ can be represented by an
ordinal utility function Φ:
Φ(x) > Φ(y) ⇔ x > y
Φ(x) = Φ(y) ⇔ x = y

• The function Φ is not unique
• It can only be used to rank bundles
• It does not quantify exactly how much better
one bundle is compared to another
October 12, 2004             Principles of Finance - Lecture 2    13

Risk and preference
• Suppose x is not a sure thing, but a lottery
Example: The agent can receive £ 20, £ 12, or £ 4
with equal probability (i.e. ⅓ for each outcome)

• How much is the lottery worth?
• The certainty equivalent of a lottery (i.e. the certain
payment which makes the agent indifferent between
this payment and the lottery) depends on the agent’s
preference towards risk
• Certainty equivalent for such a lottery would be very
small if the agent is very risk averse
October 12, 2004             Principles of Finance - Lecture 2    14

7
State-contingent payoffs (1)
• Let’s think of x as an asset with payoffs
depending on the state of the world next period

State          ω1                ω2          …   ωn
Payoff          x1                x2         …   xn

• Suppose that the probability for state ω1 is π1,
the probability for state ω2 is π2, etc.
• Then, x is like a lottery
October 12, 2004            Principles of Finance - Lecture 2            15

State-contingent payoffs (2)
• Can we define a preference relation on lotteries?
• We can use ordinal utility function to rank these
lotteries
• However, if there are many possible states (sometimes
infinitely many), x may be a large vector (sometimes
of an infinite length)
• We need a more convenient solution
• Let’s construct a measure to replace ordinal ranking
of states with cardinal ranking of certain prospects

October 12, 2004            Principles of Finance - Lecture 2            16

8
Expected utility

• Let U(·) be a cardinal utility measure on sure things
n
with the expected value Q = ∑ π iU ( xi )
i =1

• U(·) has the property that
n                          n
x ≥ y ⇔ ∑ π iU ( xi ) ≥ ∑ π iU ( yi )
i =1                       i =1

• Q can be thought of as an expected utility function

October 12, 2004              Principles of Finance - Lecture 2     17

Risk aversion
• At a given wealth level, an investor is risk averse if and
only if he refuses every fair and immediately resolved
gamble which has consequences only for wealth
• An investor who is risk averse at all relevant wealth
levels is globally risk averse
• An investor is risk averse if and only if his utility
function is strictly concave
• Let w be the wealth level. For risk averse investors
Ε[U ( w)] < U [Ε( w)]

October 12, 2004              Principles of Finance - Lecture 2     18

9
• Let ε be a lottery where E(ε) = 0; ε is a fair gamble
• Let U(·) be the utility function of an investor who is
only concerned about his wealth level
• The Markowitz risk premium is the solution πM to
the equation Ε[U ( w + ε )] = U ( w − π M )
• It is the difference between the investor’s expected
wealth (given the lottery) and the certainty equivalent
wealth
• Interpretation: It is the maximum the investor would
pay to avoid the gamble
October 12, 2004                    Principles of Finance - Lecture 2         19

• For small and fair gambles ε, expand U(w+ε) as a Taylor series to
U (w + ε ) =

= U ( w) + U ' ( w)[( w + ε ) − w] + U ' ' ( w)[( w + ε ) 2 − w] + K
1
2
• Take expectation on both sides and ignore higher order terms

Ε[U ( w + ε )] ≈ Ε[U ( w)] + U ' ( w)Ε[ε ] + U ' ' ( w)Ε[ε 2 ]
1
2
• Note that U(w) is constant, E[ε] = 0, and Ε[ε 2 ] = σ ε2
• So                                      1
Ε[U ( w + ε )] ≈ U ( w) + U ' ' ( w)σ ε2
2

October 12, 2004                    Principles of Finance - Lecture 2         20

10

• The Taylor series expansion for U(w - π ) is

U ( w − π ) = U ( w) − πU ' ( w) + K

• Using equation for E[U(w+ε)] we have
1
U (ε ) + U ' ' ( w)σ ε2 = U ( w) − πU ' ( w)
2
• Solving for π renders the Pratt-Arrow risk premium
1 U ' ' ( w) 2
π =−               σε
2 U ' ( w)
October 12, 2004                    Principles of Finance - Lecture 2           21

Absolute risk aversion
• The Pratt-Arrow coefficient of absolute risk aversion
U ' ' ( w)
(ARA) is defined as A( w) = −
U ' ( w)

• Sometimes, risk tolerance function is used instead

1
• Risk tolerance function is defined as T ( w) =
A( w)

October 12, 2004                    Principles of Finance - Lecture 2           22

11
Absolute vs. relative risk aversion

• ARA is a measure of local absolute risk aversion
• The coefficient of relative risk aversion (RRA)
measures aversion to gambles that are proportional
to wealth levels
R(w) = w · A(w)
• The more risk averse the investor, the higher the
risk premium required from the risky asset in order
to persuade the investor all of his wealth in it

October 12, 2004                Principles of Finance - Lecture 2   23

Risk aversion and wealth (1)
• How does the investor preference change when
his wealth increases?
If the investor increases the amount of invested
in risky assets as his wealth increases, he has a
decreasing absolute risk aversion: A’(w) < 0
If the investor holds the same amount of money
in risky assets as his wealth increases, he has a
constant absolute risk aversion: A’(w) = 0
If the investor invests more money in risky
assets as his wealth increases, he has an
increasing absolute risk aversion: A’(w) > 0
October 12, 2004                Principles of Finance - Lecture 2   24

12
Risk aversion and wealth (2)
• How does the percentage of wealth invested in
risky assets change as the wealth increases?
If the investor increases the percentage of
wealth invested in risky assets, he has a
decreasing relative risk aversion: R’(w) < 0
If the investor invests the same percentage of
wealth in risky assets, he has a constant relative
risk aversion: R’(w) = 0
If the investor decreases the percentage of
wealth invested in risky assets, he has an
increasing relative risk aversion: R’(w) > 0
October 12, 2004                Principles of Finance - Lecture 2          25

Preferences and risk aversion:
Empirical evidence
• Sources of evidence:
Experiments from simple choice situations
Survey data on investor’s asset choices
• Major findings:
Investors prefer more to less
Vast majority of individuals can be considered risk
averse
Investors are usually found to have decreasing absolute
risk aversion
The evidence on the coefficient of relative risk aversion
appears inconclusive
October 12, 2004                Principles of Finance - Lecture 2          26

13
b 2
• Quadratic utility function is given by U ( w) = w − w
2
• The first derivative: U’(w) = 1 – bw

• The second derivative: U’’(w) = -b
b
• Absolute risk aversion is A( w) =
1 − bw
b2
• Thus, A' ( w) =            >0
(1 − bw) 2
• The quadratic utility function exhibits increasing
absolute risk aversion
October 12, 2004           Principles of Finance - Lecture 2   27

• The relative risk aversion coefficient equals to
bw
R ( w) = wA( w) =
1 − bw
b
• Thus, R ' ( w) =            >0
(1 − bw) 2
• The quadratic utility function exhibits increasing
relative risk aversion
mean variance analysis being optimum
• However, if the investor is assumed to prefer more to
less, the quadratic utility function is valid only for a
restricted range of wealth
October 12, 2004           Principles of Finance - Lecture 2   28

14
Log utility function (1)

• The log utility function is given by U(w) = ln(w)

1
• The first derivative: U ' ( w) =
w
1
• The second derivative: U ' ' ( w) = −
w2
• An investor with a log utility function prefers
more to less and is risk averse

October 12, 2004           Principles of Finance - Lecture 2       29

Log utility function (2)
1
• Absolute risk aversion: A( w) =
w
1
• Hence, A' ( w) = −
w2
•     The log utility function exhibits decreasing
absolute risk aversion
•     The relative risk aversion coefficient equals
R(w) = wA(w) = 1
•     Therefore, R’(w) = 0
•     The log utility function exhibits constant relative
risk aversion
October 12, 2004           Principles of Finance - Lecture 2       30

15
HARA utility function (1)

• The         hyperbolic           absolute                risk-aversion

(HARA) or linear risk-tolerance (LRT)
γ
1 − γ ⎛ αw   ⎞
utility is given by: U ( w) =       ⎜    +β⎟
γ ⎝1 − γ   ⎠
αw
• It is properly defined for                         +β >0
1− γ

October 12, 2004            Principles of Finance - Lecture 2                   31

HARA utility function (2)

• For γ < 1 there is a lower bound on the domain

• For γ > 1 there is an upper bound on the domain

• The risk tolerance is linear in w:
1      w    β
T ( w) =        =     +
A( w) 1 − γ α

October 12, 2004            Principles of Finance - Lecture 2                   32

16
HARA utility function (3)
• For γ → 1, HARA reduces to a linear function
(characterised by risk neutrality)
• For γ = 2, one obtains quadratic utility function
• For γ → -∞ and β = 1 we obtain (negative)
exponential utility (with constant absolute risk
aversion)
• For γ < 1 and β = 0, we obtain power utility
• For γ → 0 and β = 0, we obtain log utility
October 12, 2004       Principles of Finance - Lecture 2   33

17
Principles of Finance
Week 3: October 19, 2004

Tutorial problems

Problem 1
•   ECBG Chapter 1, Exercise 4, p. 9

Problem 2
•   EGBG Chapter 1, Exercise 12, p. 10

Problem 3
•   EGBG Chapter 10, Exercises 3 and 4, p. 229

Problem 4
•   EGBG Chapter 10, Exercise 7, p. 229

Problem 5
•   EGBG Chapter 10, Exercise 10-11, p. 229-230

Problem 6 (optional)
HARA utility function is defined as:
γ
1 − γ ⎛ αw   ⎞
U ( w) =       ⎜    +β⎟
γ ⎝1 − γ   ⎠
where α, β, and γ are the parameters, while w is the level of wealth. As discussed during the
lecture, this representation renders many commonly used utility functions as the special
cases. In particular:
•    For γ → 1, HARA reduces to a linear function (characterised by risk
neutrality)
•    For γ = 2, one obtains quadratic utility function
•    For γ → -∞ and β = 1 we obtain (negative) exponential utility (with constant
absolute risk aversion)
•   For γ < 1 and β = 0, we obtain power utility
•   For γ → 0 and β = 0, we obtain log utility
A. Prove that you can indeed obtain linear, quadratic, exponential, power, and log utility
functions, respectively, for the values of parameters α, β, and γ given above.
B. Derive the measures of ARA and RRA for each of those special cases.

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