investment analysis and portfolio management lecture 5 gareth myles
TRANSCRIPT
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Investment Analysis and Portfolio Management
Lecture 5
Gareth Myles
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Risk
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
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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")
,4,3,2,1 ttttte
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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
The probabilities satisfy
,,,, 4321ttttt ppppp
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i
itp
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Choice with Risk
t = 0 t = 1 t = 2
10
11
21
12
22
32
42
52
Event tree
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Choice with Risk
Each event is a complete description of the world
Let = return on asset i at time t in state e then
This information will determine the payoff in
each state Investors have preferences over these returns
and this determines preferences over states
,, 21tt eet rre
iet
r
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Choice with Risk
Expected Utility Assume the investor has preferences over wealth in
each state described by the utility function
Preferences can be defined over different sets of probabilities over the states
WUU
W
U
U=U ( W )
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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 wayExample Let , , , Then any investor who prefers a higher return to a
lower return must rank p strictly preferable to q
101 W 52 W 1.0,9.0p 9.0,1.0q
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Choice with Risk
We now assume that an investor can rank lotteries 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
2211 WUpWUpEU
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Choice with Risk
This approach can be extended to the general state-preference model described above
For example, with two assets in each state
where ai is the investment in asset i Summary
Preferences over random payoffs can be described by the expected utility function
222
1212
212
1111 1111 raraUpraraUpEU
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Risk Aversion
Consider receiving either 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
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Risk Aversion
A risk averse investor will pay to avoid risk
The amount the will pay is defined as the solution to
U(M - ) = ½U(M[1 + r]) + ½U(M[1 – r]) is the risk premium The more risk averse is
the investor, the more they will pay
Wealth
Utility
10 hW 20 hW
20 hWU
10 hWU
EUWU 0
0W
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Portfolio Choice
Assume a safe asset with return rf = 0
Assume a risky asset 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?
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Portfolio Choice
Let amount a be placed in risky asset, so W – a in safe asset
After one period 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
More wealth if good state occurs Less wealth if bad states occurs
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Portfolio Choice
Possible wealth levels are illustrated on a “state-preference” diagram
W
W
W[1+rg]
W[1+rb] a = W
a = 0
Wealth ingood state
Wealth inbad state
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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
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Portfolio Choice
W
W
W[1+rg]
W[1+rb] a = W
a = 0
Wealth ingood state
Wealth inbad state
a*
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Portfolio Choice
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?
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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?
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Mean-Variance Preferences
Denote the level of wealth by . Taking a Taylor's series expansion of utility around expected wealth
Here R3 is the error that depends on terms involving and higher
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W~
3 ~~WEW
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Mean-Variance Preferences
Taking the expectation of the expansion
The expected error is
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
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2
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WmWEUn
RE nn
n
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Mean-Variance Preferences
Expected utility depends on just the mean and variance if either 1. = 0 for n > 2. This holds if utility is
quadratic
Or 2. The distribution of returns is normal since then all
moments depend on the mean and variance
In either case
WEU n ~
2~,
~~WWEUWUE
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Risk Aversion
With mean-variance preferences Risk aversion implies
the indifference curves slope upwards
Increased risk aversion means they get steeper
pr
p
More risk averse
Less risk averse
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Markowitz Model
The Markowitz model is the basic model of portfolio choice
Assumes A single period horizon Mean-variance preferences Risk aversion Investor can construct portfolio frontier
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Markowitz Model
Confront the portfolio frontier with mean-variance 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
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Markowitz Model
MVP
MVPr
Standarddeviation
Expectedreturn
Optimalportfolio
Less riskaverse
More riskaverse
Xa = 1, Xb = 0
Xa = 0, Xb = 1
Choice with risky assets
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Markowitz Model
Standarddeviation
Expectedreturn
Optimalportfolio
Less riskaverse
More riskaverse
Xa = 1, Xb = 0
Xa = 0, Xb = 1
Borrowing
Lending
Choice with a risk-free asset
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Markowitz Model
Note the role of the tangency portfolio Only two assets need be available to achieve
an optimal portfolio Riskfree asset Tangency portfolio (mutual fund)
Model makes predictions about 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