Prerequisites
Almost essential
Risk
RISK TAKING
MICROECONOMICS
Principles and Analysis
Frank Cowell
July 2015
Frank Cowell: Risk Taking
1
Economics of risk taking
 In the presentation Risk we examined the meaning of
risk comparisons
• in terms of individual utility
• related to people’s wealth or income (ARA, RRA)
 In this presentation we put to this concept to work
 We examine:
• trade under uncertainty
• a model of asset-holding
• the basis of insurance
July 2015
Frank Cowell: Risk Taking
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Overview
Risk Taking
Trade and
equilibrium
Extending the
exchange economy
Individual
optimisation
The portfolio
problem
July 2015
Frank Cowell: Risk Taking
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Trade
 Consider trade in contingent goods
 Requires contracts to be written ex ante
 In principle we can just extend standard GE model
 Use prices piw:
• price of good i to be delivered in state w
 We need to impose restrictions of vNM utility
 An example:
• two persons, with differing subjective probabilities
• two states-of the world
• Alf has all endowment in state BLUE
• Bill has all endowment in state RED
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Contingent goods: equilibrium trade
•
b
xRED
 Certainty line for Alf
 Alf's indifference
curves
Ob
pa
RED
– ____
paBLUE
a
xBLUE
 Certainty line for Bill
 Bill's indifference curves
 Endowment point
 Equilibrium prices & allocation
pbRED
– ____
pbBLUE
Contract
curve
•
b
xBLUE
Oa
July 2015
a
xRED
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Trade: problems
 Do all these markets exist?
• If there are  states-of-the-world…
• …there are n of contingent goods
• Could be a huge number
 Consider introduction of financial assets
 Take a particularly simple form of asset:
• a “contingent security”
• pays $1 if state w occurs
 Can we use this to simplify the problem?
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Financial markets?
 The market for financial assets opens in the morning
 Then the goods market is in the afternoon
 Use standard results to establish that there is a
competitive equilibrium
 Instead of n markets we now have n+
 But there is an informational difficulty
• To do financial shopping you need information about the
afternoon
• This means knowing the prices that there would be in each
possible state of the world
• Has the scale of the problem really been reduced?
July 2015
Frank Cowell: Risk Taking
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Overview
Risk Taking
Trade and
equilibrium
Modelling the demand
for financial assets
Individual
optimisation
The portfolio
problem
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Frank Cowell: Risk Taking
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Individual optimisation
 A convenient way of breaking down the problem
 A model of financial assets
 Crucial feature #1: the timing
• Financial shopping done in the “morning”
• This determines wealth once state w is realised
• Goods shopping done in the “afternoon”
• We will focus on the “morning”
 Crucial feature #2: nature of initial wealth
• Is it risk-free?
• Is it stochastic?
 Examine both cases
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Interpretation 1: portfolio problem

 You have a determinate (non-random) endowment y
 You can keep it in one of two forms:
• Money – perfectly riskless
• Bonds – have rate of return r: you could gain or lose on each
bond
 If there are just two possible states-of-the-world:
• rº < 0 – corresponds to state BLUE
• r' > 0 – corresponds to state RED
 Consider attainable set if you buy an amount b of

bonds where 0 ≤ b ≤ y
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Attainable set: safe and risky assets
 Endowment
 If all resources put into bonds
xBLUE
 All these points belong to A
 Can you sell bonds to others?
 Can you borrow to buy bonds?
unlikely to be
points here
 If loan shark willing to finance you
_
_
y
 P
_
_
y+br′, y+br
_
_
[1+r′ ]y, [1+r]y
_
 P0
[1+rº]y
A
_
y
July 2015
_
[1+r' ]y
Frank Cowell: Risk Taking
unlikely to be
points here
xRED
11
Interpretation 2: insurance problem
 You are endowed with a risky prospect
• Value of wealth ex-ante is y0
• There is a risk of loss L
• If loss occurs then wealth is y0 – L
 You can purchase insurance against this risk of loss
• Cost of insurance is k
• In both states of the world ex-post wealth is y0 – k
 Use the same type of diagram
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Attainable set: insurance
 Endowment
 Full insurance at premium k
xBLUE
 All these points belong to A
 Can you overinsure?
 Can you bet on your loss?
unlikely to be
points here
_
_
y
partial
insurance
 P
L–k
 P0
y0 – L
k
A
_
y
July 2015
unlikely to be
points here
xRED
y0
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A more general model?
 We have considered only two assets
 Take the case where there are m assets (“bonds”)
 Bond j has a rate of return rj
 Stochastic, but with known distribution
 Individual purchases an amount bj
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Consumer choice with a variety of financial
assets
 Payoff if all in cash
 Payoff if all in bond 2
 Payoff if all in bond 3, 4, 5,…
 Possibilities from mixtures
 Attainable set
 The optimum
xBLUE
1
 only bonds 4 and 5
used at the optimum
2
3
A
4

P*
5
6
7
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Frank Cowell: Risk Taking
xRED
15
Simplifying the financial asset problem
 If there is a large number of financial assets many may
be redundant
• which are redundant depends on tastes…
• … and on rates of return
 In the case of #W = 2, a maximum of two assets are
used in the optimum
 So the two-asset model of consumer optimum may be
a useful parable
 Let’s look a little closer
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Frank Cowell: Risk Taking
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Overview
Risk Taking
Trade and
equilibrium
Safe and risky
assets 
comparative statics
Individual
optimisation
The portfolio
problem
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The portfolio problem
 We will look at the equilibrium of an individual risk-taker
 Makes a choice between a safe and a risky asset
• “money” – safe, but return is 0
• “bonds”– return r could be > 0 or < 0
 Diagrammatic approach uses the two-state case
 But in principle could have an arbitrary distribution of r…
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Distribution of returns (general case)
f (r)
 plot density function of r
 loss-making zone
 the mean
r
Er
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Problem and its solution
 Agent has a given initial wealth 𝑦
 If he purchases an amount b of bonds:
• Final wealth then is y = 𝑦 – b + b[1+r]
• This becomes y = 𝑦 + br, a random variable
 The agent chooses b to maximise Eu(𝑦 + br)
 FOC is E(ruy(𝑦 + b*r)) = 0 for an interior solution
• where uy(•) = u(•) / y
• b* is the utility-maximising value of b
 But corner solutions may also make sense…
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Consumer choice: safe and risky assets
Attainable set, portfolio problem
xBLUE
 Equilibrium -- playing safe
 Equilibrium - "plunging"
 Equilibrium - mixed portfolio
_
_
y
 P
P*

P0

A
xRED
_
y
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Results (1)
 Will the agent take a risk?
 Can we rule out playing safe?
 Consider utility in the neighbourhood of b = 0
 Eu(𝑦 + br) |
———— | = uy(𝑦) E r
b
|b=0
 uy is positive
 So, if expected return on bonds is positive, agent will
increase utility by moving away from b = 0
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Results (2)
 Take the FOC for an interior solution
 Examine the effect on b* of changing a parameter
 For example differentiate E(ruy(𝑦 + b*r)) = 0 w.r.t. 𝑦
 E(ruyy(𝑦 + b*r)) + E (r2 uyy(𝑦 + b*r)) b*/𝑦 = 0

b*
– E (ruyy(𝑦 + b*r))
—— = ———————
𝑦
E(r2 uyy(𝑦 + b* r))
 Denominator is unambiguously negative
 What of numerator?
July 2015
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Risk aversion and wealth
 To resolve ambiguity we need more structure
 Assume Decreasing ARA
 Theorem: If an individual has a vNM utility function with
DARA and holds a positive amount of the risky asset then the
amount invested in the risky asset will increase as initial wealth
increases
July 2015
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An increase in endowment
Attainable set, portfolio problem
 DARA Preferences
xBLUE
 Equilibrium
 Increase in endowment
 Locus of constant b
 New equilibrium
_
y+d
_
y


P* o

**
P

A
try same method
with a change in
distribution
_
y
July 2015
xRED
_
y+d
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A rightward shift
 original density function
f (r)
 original mean
 shift distribution by t
 Will this change
increase risk taking?
r
t
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A rightward shift in the distribution
 Attainable set, portfolio problem
xBLUE
 DARA Preferences
 Equilibrium
 Change in distribution
 Locus of constant b
 New equilibrium
_
_
y
 P
Po*

P**

P0


A
What if the
distribution
“spreads out”?
xRED
_
y
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An increase in spread
Attainable set, portfolio problem
 Preferences and equilibrium
 Increase r′, reduce r
xBLUE
 P* stays put
So b must have reduced
You don’t need DARA for this
_
_
y
 P
P*
_
_
y+b*r′, y+b*r

P0

A

xRED
_
y
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Risk-taking results: summary
 If the expected return to risk-taking is positive, then the
individual takes a risk
 If the distribution “spreads out” then risk taking reduces
 Given DARA, if wealth increases then risk-taking increases
 Given DARA, if the distribution “shifts right” then risk-taking
increases
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