Transcript Risk

Lecture 3: Arrow-Debreu Economy
• The following topics are covered:
– Arrow-Debreu securities
– Optimal portfolios of Arrow Debreu securities
– How Arrow-Debreu securities differ framework differs from the
standard utility maximization?
– Implications in asset pricing
– Example
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Arrow-Debreu Assets
• There are S possible states indexed by s=0, 1, …, S-1.
• A pure security (or say an Arrow-Debreu asset) stays in each state, paying
$1 if a given state occurs and nothing if any other state occurs at the end of
the period
• Let Пs denote the price of the Arrow-Debreu security associated with s, i.e.,
the price to be paid to obtain one monetary unit if state s occurs
• State price Пs can be decomposed into the probability of the state, ps, and
the price of an expected dollar contingent on state s occurring (or say the
state price per unit of probability of associated contingent claims), πs.
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Complete Market
• The market is considered to be complete when investors can structure any
set of state-contingent claims by investing in the appropriate portfolio of
Arrow-Debreu securities
• In other words, (1) there are enough independent assets to “span” the entire
set of all possible risk exposures; (2) The market will be complete if there
are at least as many assets who vectors of state-contingent payoffs are
linearly independent as there are number of states
• Can we say the market is complete?
• Applying this idea, we have the following example
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An Example of Two State
• Two assets, (a) risk free bond: with r in both states; (b) risky asset: final
value is Ps, s= 0, 1. initial price of the risky asset is 1
• P0<1+r<P1
• Replicating the Arrow-Debreu security associated with state s=1 by
purchasing alpha units of the risky asset and by borrowing B at the risk-free
rate, in such a way that
0 = α 0 – (1+r)B
1 = αP1 – (1+r)B
P
P0
1
and B 
P1  P0
( P1  P0 )(1  r )
Let  s denote the price of Arrow-Debreu security associated with s, we have
We have:  
1    B 
1  P0 /(1  r )
0
P1  P0
Applying the same logic, we have  0    B 
1  P1 /(1  r )
0
P0  P1
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Option and AD Securities
– Assuming P0 < P1 – 1, then the AD asset associated with state s=1 is a
call option with strike price P1 – 1
– In other words, buying a call option with an exercise equal to P1 – 1 has
the same payoff as an AD security
– Why?
– Exercise 5.4 (b)
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Пs vs Security Price P
•
•
•
Pure assets can be replicated by market security
On the other hand, each market security may be considered as a specific set of
payoff combination of AD assets. In other words, it represents a particular
investment choice in AD assets.
A particular example is the risk-free asset
– It has a payoff of 1 in each state of nature at the end of the period
– We have the following:
PB  (1  r ) 1 -- the present value of $1 received at the end of period.
Apply the no-arbitrage argument, we have the following expression for the state price,
S 1

s 0
s
 PB  (1  r ) 1
S 1
Alternatively,
p
s 0
s
s
 PB  (1  r ) 1
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Risk Neutral Probability and P
^
^
Define p s   s (1  r ) . p s is summed to 1 and often referred to as the risk-neutral
probability for state s.
S 1 ^
S 1
p
ys
S 1
Ey
The price of a market asset: P    s y s 

  s ps ys
1

r
1

r
s 0
s 0
Where  s is the state price per unit of probability. We often use c for y.
s 0
s
Solving exercise 5.1
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Optimal Portfolio of AD Assets
• Go through the example in CW
• Let cs denote the investment in AD asset with state s
S 1
max  p s u (c s ) subject to
c0 ,..., cs
0
S 1
 c
s s
w
0
where cs is the investors’ initial wealth. ps is the probability of state s. Пs is the price for
AD asset s.

*
FOC: u ' (c s )   s for all s = 0, 1, …, S-1. (please go through CW example)
ps

*
Assuming  s  s , we have u' (cs )   s .
ps
Where  s is the state price per unit of probability (the probability refers one in any state).
 s is referred to in the finance literature as the pricing kernel.
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When πs stay constant across states
• If πs stays the same across all states, then it is optimal for
the agent to purchase the same quantity of these claims.
^
• It can be shown that p s  p s
• Thus the above condition is equivalent to the case that a
risk is actuarially priced, we need to purchase full
insurance on it
• The asset is a risk free asset: cs=w(1+r)
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When πs differs across states
^
• When πs differs across states, then ps  ps
• We have different level of consumption of each value of πs
• Note cs is what we want to solve for
Write c s*  C ( s ) for all s. u'[C ( s )]   ; We have C ( s )  u ' 1 ( )
u’ is decreasing due to risk aversion since u is concave, thus C ( s ) is non-increasing.
In other words, one consumes less in more expensive states.
• Figure 5.1
• Exercise 5.2
– Key: the consumption curve changes from 2 to 1
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Graphic Illustration of the s=2 Case
Objective function:
max p0 u(c 0 )  p1u(c1 ) subject to  0 c0  1c1  w
c0 ,c1
Note that  s   s p s . A special case is that when πs = 1 (state price per unit of
probability stays the same and interest rate = 0), we have p0 c0  p1c1  w as the
constraint.
Figure 5.3 does the plot. The optimal consumption set is determined by equating the
slope of indifference curve and iso-expected value locas.
An application of the separation theorem: shown in the next two slides.
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Analogy to Intertemporal Consumption Decisions
The idea here is consistent with the choice between consumption and
investment discussed in CW Chapter 1. It incorporates:
(1) Utility function
(2) Indifference curve
(3) Maximization under constraint – a decreasing return investment function
only; i.e., consumption and investment without capital market
(1) Consumption is about the choice between consuming now and the
future
(2) Investment is about choosing optimal investment return, which
affects consumption pattern
(3) Here investment and consumption decisions are not separatable
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Then with the Capital Market
•
Fisher separation theorem: Given perfect and complete capital markets,
the production decision is governed solely by an objective market
criterion (represented by maximizing attained wealth) without regard to
individuals’ subjective preferences which enter into their consumption
decisions
–
Choose optimal production first,
–
Choose optimal consumption pattern (C0, C1) based on each
individual’s utility function (indifference curve)
–
Less risk averse individual will consume more today
–
Transaction costs break down the separation theorem
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Examples: Exercise 5.4(a)
E5.4: Three assets: asset A (2, 5, 7); asset B (2, 4, 4); asset C (1, 0, 2)
How to construct AD in state 1?
How to construct a risk-free asset?
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Implications of Arrow Debreu Securities
• A means to model uncertainty
• About consumption in different states
• The same idea can be applied to the consumption over time
– Allocate wealth over time versus allocate wealth across states
• Easy to achieve any payoff, such as option payoffs
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Exercises
• EGS, 5.2
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