Investment Analysis and
Portfolio Management
Lecture 5
Gareth Myles
Risk
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An investment is made at time 0
The return is realised at time 1
Only in very special circumstances is the
return to be obtained at time 1 known at time 0
In general the return is risky
The choice of portfolio must be made taking
this risk into account
The concept of states of the world can be used
Choice with Risk

State Preference

The standard analysis of choice in risky situations
applies the state preference approach

Consider time periods t = 1, 2, 3, 4, ...
 At each time t there is a set of possible events
(or "states of the world")
et  1t ,2t ,3t ,4t ,
Choice with Risk

When time t is reached, one of these states is
realized
 At the decision point (t = 0), it is not known
which
 Decision maker places a probability on each
event pt  pt1 , pt2 , pt3 , p 4t ,
 The probabilities satisfy


E
p
i 1
i
t
1
Choice with Risk
t=0
t=1
t=2
12
11
22
10
32
42
21
Event tree
52
Choice with Risk

Each event is a complete description of the
world
i
 Let re = return on asset i at time t in state e
t
then
et  re1t , re2t ,



This information will determine the payoff in
each state
 Investors have preferences over these returns
and this determines preferences over states
Choice with Risk

Expected Utility

Assume the investor has preferences over wealth in
each state described by the utility function U  U W 
U
U=U(W )
W

Preferences can be defined over different sets of
probabilities over the states
Choice with Risk

Assume 1 time period and 2 states
 Let wealth in state 1 be W1 and in state 2 W2
 Let p denote the lottery {p, 1-p} in which state
1 occurs with probability p
 Lottery q is defined in the same way
Example
 Let W1  10 , W  5, p  0.9,0.1 , q  0.1,0.9
2
 Then any investor who prefers a higher return to a
lower return must rank p strictly preferable to q
Choice with Risk

We now assume that an investor can rank
lotteries

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1. Preferences are a complete ordering
2. If p is preferred to q, then a mixture of p and r is
preferred to the same mixture of r and q
3. If p is preferred to q and q preferred to r, then
there is a mixture of p and r which is preferred to q
and a different mixture of p and r which is strictly
worse then q
The investor will act as if they maximize the
expected utility function
EU  p1U W1   p2U W2 
Choice with Risk

This approach can be extended to the general
state-preference model described above
 For example, with two assets in each state
  

  
EU  p1U a1 1  r11  a2 1  r12  p2U a1 1  r21  a2 1  r22

where ai is the investment in asset i
 Summary

Preferences over random payoffs can be described
by the expected utility function
Risk Aversion

Consider receiving either
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A fixed income M
A random income M[1 + r] or M[1 – r], each
possibility occurring with probability ½
An investor is risk averse if
U(M) > ½ U(M[1 + r]) + ½ U(M[1 – r])
The certain income is preferred to the
random income
This holds if the utility function is concave
Risk Aversion
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A risk averse investor
will pay to avoid risk
The amount the will pay
is defined as the solution
to
U(M - r) = ½U(M[1 + r])
+ ½U(M[1 – r])
r is the risk premium
The more risk averse is
the investor, the more
they will pay
Utility
U W0  h2 
UW0  r  EU
U W0  h1 
W0  h1
W0  r
W0  h2
Wealth
Portfolio Choice
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Assume a safe asset with return rf = 0
 Assume a risky asset
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Return rg > 0 in “good” state
Return rb < 0 in “bad” state
Investor has amount W to invest
 How should it be allocated between the
assets?
Portfolio Choice
Let amount a be placed in risky asset, so W – a
in safe asset
 After one period
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Wealth is W - a + a[1 + rg] in good state
Wealth is W - a + a[1 + rb] in bad state

A portfolio choice is a value of a
 High value of a

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More wealth if good state occurs
Less wealth if bad states occurs
Portfolio Choice
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Possible wealth levels are illustrated on a
“state-preference” diagram
Wealth in
bad state
a=0
W
a=W
W[1+rb]
W
W[1+rg]
Wealth in
good state
Portfolio Choice

Adding indifference curves shows the choice
 Indifference curves from expected utility function
EU = pU(W - a + a[1 + rg]) + (1-p)U(W - a + a[1 + rb])
 The investor chooses a to make expected utility
as large as possible
 Attains the highest indifference curve given the
wealth to be invested
Portfolio Choice
Wealth in
bad state
W
a=0
a*
W[1+rb]
a=W
W
W[1+rg]
Wealth in
good state
Portfolio Choice
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Effect of an increase in risk aversion
What happens if rb > 0 or if rg < 0?
When will some of the risky asset be
purchased?
When will only safe asset be purchased?
Effect of an increase in wealth to be invested?
Mean-Variance Preferences

There is a special case of this analysis that is
of great significance in finance
 The general expected utility function
constructed above is dependent upon the
entire distribution of returns
 The analysis is much simpler if it depends on
only the mean and variance of the distribution.
 When does this hold?
Mean-Variance Preferences
~
 Denote the level of wealth by W . Taking a
Taylor's series expansion of utility around
expected wealth
 
  
  
 
~
~
~ ~
~
U W U E W U' E W W  E W
  

~ ~
~
1
 U '' E W W  E W
2

2
 R3
 
~
EW
~
W
Here R3 is the error that depends on terms
~
~ 3
involving W  EW  and higher
Mean-Variance Preferences

Taking the expectation of the expansion
     
    
1
~
~
~
~2
E U W  U E W  U ' ' E W  W  ER3 
2
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The expected error is

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    
1 n  ~ n ~
E R3    U E W m W
n  3 n!
The expectation involves moments (mn) of all orders
(first = mean, second = variance, third)
The problem is to discover when it involves only the
mean and variance
Mean-Variance Preferences
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Expected utility depends on just the mean and
variance if either
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  
~
E W = 0 for n > 2. This holds if utility is
1. U
quadratic
Or
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
n 
2. The distribution of returns is normal since then all
moments depend on the mean and variance
In either case
  
   
~
~
~ 2

E U W  U  E W , W 


Risk Aversion
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With mean-variance
preferences
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Risk aversion implies
the indifference
curves slope upwards
Increased risk
aversion means they
get steeper
rp
Less risk
averse
More risk
averse
p
Markowitz Model

The Markowitz model is the basic model of
portfolio choice
 Assumes
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A single period horizon
Mean-variance preferences
Risk aversion
Investor can construct portfolio frontier
Markowitz Model
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Confront the portfolio frontier with meanvariance preference
 Optimal portfolio is on the highest indifference
curve
 An increase in risk aversion changes the
gradient of the indifference curve
 Moves choice around the frontier
Markowitz Model
Expected
return
Less risk
averse
Optimal
portfolio
Xa = 1, Xb = 0
rMVP
More risk
averse
Xa = 0, Xb = 1
 MVP
Choice with risky assets
Standard
deviation
Markowitz Model
Expected
return
Optimal
portfolio
Less risk
averse
Borrowing
Xa = 1, Xb = 0
More risk
averse
Lending
Xa = 0, Xb = 1
Choice with a risk-free asset
Standard
deviation
Markowitz Model

Note the role of the tangency portfolio
 Only two assets need be available to achieve
an optimal portfolio
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Model makes predictions about
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Riskfree asset
Tangency portfolio (mutual fund)
The effect of an increase in risk aversion
Which assets will be short sold
Which investors will buy on the margin
Markowitz model is the basis of CAPM