Transcript The securities market economy -

The securities market
economy -- theory
Abstracting again to the twoperiod analysis - - but to different
states of payoff
The securities market
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We again assume an initial period, o,
and some date in the future, 1
This is a two-period analysis as an
abstraction from the securities market
to point out some basic foundations,
now that we have laid out basic
intertemporal utility issues, expected
utility, and risk aversion
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Again, consumption at date o is Co
Consumption at date 1 is C1
But C1 can come in different states, S
C1 is actually an S-dimensional vector
C1 = (C11, C12, . . . . , C1s) for states
S = 1, . . ., s states of the world
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C1s can be Cs if there is no confusion
that we are in the future date 1
Co and C1 (or C1s) may be restricted
to be positive
We will assume there is utility over
this intertemporal consumption as
U(Co, C1), like, for example U =
Ln(Co, C1) as we have seen before
Utility and Consumption
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U = Ln(Co, C1) restricts Co, C1 > 0
We assume there are I agents in the
economy, i = 1, 2, . . . , I
Utility of the ith agent is given by
Ui(Co, C1) --- which gives a
consumption plan for the agent
The agents endowment
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Agent i’s endowment of wealth or
previous payoff from securities that
adds to wealth is given by Woi at date
o -- Wo could mean i starts with some
portfolio of securities
The agents endowment of wealth or
previous payoff from securities at date
1 is W1i
Properties of the utility
function
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U is increasing at date o if
U(Co΄, C1) ≥ U(Co, C1) when Co΄≥Co
for every C1
U is increasing at date 1 when
U(Co, C1΄) ≥ U(Co, C1), for C1΄ ≥ C1
for every Co
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U is strictly increasing for U(Co΄, C1)
> U(Co, C1) for Co΄> Co for every C1
U(Co, C1΄) > U(Co, C1) for every
C1΄> C1 for every Co
If U is strictly increasing over all dates,
Co, C1, then Utility is strictly
increasing
The agents problem in
this securities market
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Let P = price
Let h = holding, or holdings, like a
portfolio of securities (h is not payoff
or payment as we discuss later in
connection with expected utility)
Let X be the payoff matrix from the
holdings of securities
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The agent’s problem is to:
MaxCo,C1,h U(Co, C1)
subject to:
Co  Wo – Ph
C1  W1 + hX
and this is the two-period securities
problem
Now let’s go back to the
agent’s problem
MaxCo,C1,h U(Co, C1)
subject to:
Co  Wo – Ph
C1  W1 + hX
Notice, Wo – Ph is exactly how much we can consume in period 0, or
this is just wealth minus how much we are investing
Notice also, that C1 is just added wealth + what we get from our
holdings as a return
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Now we have an optimization problem
to maximize U(Co, C1) subject to two
constraints
One constraint says that current
consumption is less than or equal to
the initial period wealth endowment
minus how much the agent pays for
the portfolio
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The second constraint indicates that
future consumption has to be less
than future period endowment plus
the payoff of the holdings
We also have a hidden constraint in
here from our first discussions of this
security world problem that Co, C1 ≥
0
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So, we now have what is called a
Kuhn-Tucker optimization problem (as
opposed to the Lagrangian problem
we dealt with previously in our
economic foundations discussions)
We now have inequalities to deal with
The Kuhn-Tucker problem
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Max U(Co,C1) + λ[Co  Wo-Ph] +
[C1  W1 + hX], with two constraint
multipliers, λ,  to deal with
But we can solve these using some
optimization algorithm ---- Excel,
Matlab, etc
The constrained optimization conditions
(Kuhn-Tucker conditions)
(*) ∂oU(Co,C1) – λ  0 and
[∂oU(Co,C1) – λ]Co = 0
(**) ∂S U(Co,C1) – S  0 and
[∂S U(Co,C1) – S]CS = 0
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(s, being some state of the payoffs )
λP = X {Well, duh!!}
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All that ∂oU(Co,C1) means is that we
have taken the derivative of the utility
function with respect to Co
All that ∂oU(Co,C1) – λ means is that
we have taken the derivative of the
whole constrained optimization
problem with respect to Co --similarly for ∂S U(Co,C1) - S
The solution?
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If utility, U, is quasi concave, meaning
U΄> 0 and U΄΄ 0, the Kuhn-Tucker
conditions derived from the
constrained optimization are sufficient
and necessary for the maximization
(and we will not give the mathematics
of that proof here) --- meaning we have at least
quasi-concave utility
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And, if the solution is interior (no
border or axis solutions) and
∂oU(Co,C1) > 0, ∂S U(Co,C1) > 0, then
inequalities in (*) and (**) are
satisfied with equality
So λP = X becomes,
X(/λ) = P = X(∂S U(Co,C1) /
∂oU(Co,C1) ) = X/r = JUST PRESENT
VALUE
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So the price of security j ( the cost in
units of date o consumption of a unit
increase in the holding of the jth
security) = the sum over states of its
payoff in each state multiplied by the
marginal rate of substitution between
consumption in that state and
consumption at date 0
Intertemporal marginal
rate of substitution
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∂S U(Co,C1) / ∂oU(Co,C1) is the marginal
rate of substitution between consumption in
that state s and consumption at date o ---MRSCo,C1
Notice, that P = X/r is just SIMPLE PRESENT
VALUE
Present value P = X/r = (the
earnings)/discounted
We have basic finance!
We can also get NPV
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Notice also, that net present value is
just NPV = -P + X/r, and if X/r > P,
then NPV > 0 , which is a concept with
which we are all familiar
What are we doing?
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Security j is identified by its payoff, xj
Security j comes in payoffs of different
states, s, at date 1, xjs --- these
payoffs are in terms of consumption
good, C, since the payoff is converted
to consumption
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Securities are claims on cash flows
coming from the production and sale
of goods and services
The payoffs xjs can be positive,
negative or equal to zero
There are a finite number, j, of
securities with payoffs x1, . . , xj
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Therefore, X is a j by s matrix of payoffs of
all securities in the economy
 x1 
 
X  x2 
x j 
 
Payoff matrix
The portfolio
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A portfolio is the holdings of J
securities, j = 1, 2, . . . , J
Holdings, h, can be positive, negative
or equal to zero
h > 0 is a long position in that
particular set of securities
h < 0 is a short position
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So the portfolio is the J-dimensional vector,
h, where hj is the holdings of security j
The portfolio payoff is hX for all the holdings
The set of payoffs available via trades in
securities markets is the Asset Span, but we
will not worry too much about this definition
Price of the security
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P = the price of the security
P = (P1, P2, . . ., PJ)
Ph = the sum of the prices multiplied
by the holdings over J prices and
securities, and this is the price (or
value) of the portfolio h at securities
prices P
Return
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Return, rj, on security j is its payoff
divided by its price --- xj / Pj, for Pj
being nonzero
rj = xj/Pj = gross return
Net return is gross return - 1
Example:
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S = 1, 2, 3 --- three states
x1 = (1, 1, 1), x2 = (1, 2, 2) --- two
securities
1 1 1
X 
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1 2 2
Payoff matrix
Payoff states and risk
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Let’s look at security 1 over all its
states, x1 = (1, 1, 1)
The payoff is 1 in each state!
Security 1 is risk free, that is the
payoff is 1 in each state
E (x) = 1, VAR (x) = 0 = 2
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Now look at security 2
x2 = (1, 2, 2)
Security 2 is risky, with a payoff of 1 in
state 1, and then a payoff of 2 in
states 2 and 3
E (x) = mean = 1.67, VAR (x) = 0.33
= 2
St. dev (x) = √2 = 0.58
The excel commands for
mean and variance
mean
variance
st. dev
1
1
1
1
0
1
2
2
1.666667
0.333333
0.57735
0.57735
The excel formulae
mean
variance
1
1
1
=AVERAGE(C3:E3)
=VAR(C3:E3)
1
2
2
=AVERAGE(C5:E5)
=VAR(C5:E5)
st. dev
=SQRT(G5)
=STDEV(C5:E5)
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We let P1 = 0.8 and P2 = 1.25
Then, r1 = X/P = (1/0.8, 1/0.8, 1/0.8),
which is (1.25, 1.25, 1.25)
And r2 = X/P = (1/1.25, 2/1.25,
2/1.25) = (0.8, 1.6, 1.6)
Notice what these are? C1/Co – 1
= r, or C1/Co = 1 + r