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\begin_body

\begin_layout Title
Finance
\end_layout

\begin_layout Author
Michael Peters
\end_layout

\begin_layout Thanks
\begin_inset ERT
status open

\begin_layout Plain Layout


\backslash
today{}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The edgeworth box provides a nice way to think about trade in markets.
 The kind of market for which that theory probably best applies is financial
 markets.
 This note explains how to use the edgeworth box to understand asset pricing.
\end_layout

\begin_layout Standard
The approach begins with two traders and two goods.
 The first step to understanding all this is to think about income in different
 states as representing two distinct goods.
 Insurance is an example in which this is true.
 An insurance policy only pays you money if you have an accident.
 Money means a lot more to you when you have an accident than when you don't,
 which is why you might be willing to give up money when you don't have
 an accident by paying an insurance premium in order to get it when you
 do have an accident.
\end_layout

\begin_layout Standard
Lets start with a simple Walrasian equilibrium in a two good exchange economy
 similar described with an Edgeworth box.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename /home/peters/files/docs/book/finance/finance_fig1.eps

\end_inset


\end_layout

\begin_layout Standard
The figure describes a Walrasian equilibrium for an economy with two goods.
 Trader 1 starts at the origin, and has quantity 
\begin_inset Formula $y$
\end_inset

 of good 1 and 
\begin_inset Formula $y-d$
\end_inset

 of good 2.
 He makes a trade with trader to in which he exchanges 
\begin_inset Formula $z_{1}$
\end_inset

 units of good 1 for 
\begin_inset Formula $z_{2}$
\end_inset

 units of good 2.
 This takes both of them to the point where the indifference curves are
 tangent so that the allocation is pareto optimal.
\end_layout

\begin_layout Standard
In finance, we want to interpret the goods by imagining that there is an
 as yet unrealized event, say a recession.
 Trader 1 has income 
\begin_inset Formula $y$
\end_inset

 if there is no recession, but has much lower income 
\begin_inset Formula $y-d$
\end_inset

 if there is.
\end_layout

\begin_layout Standard
Before it is known whether or not the recession occurs, there is a market
 in which two 'securities' can be traded.
 The first security, 
\begin_inset Formula $a$
\end_inset

, pays 
\begin_inset Formula $1$
\end_inset

 dollar if and only if there is no recession.
 Security 
\begin_inset Formula $b$
\end_inset

 pays 
\begin_inset Formula $1$
\end_inset

 dollar if and only if there 
\size normal
\emph on
is 
\emph default
a recession.
 We imagine that the way the event contingent transfer occurs is that trader
 1 sells 
\begin_inset Formula $z_{1}$
\end_inset

 unit of security 
\begin_inset Formula $a$
\end_inset

, for which he receives total revenue 
\begin_inset Formula $qz_{1}$
\end_inset

.
 He uses the revenue to buy 
\begin_inset Formula $z_{2}=qz_{1}$
\end_inset

 units of security 
\begin_inset Formula $b$
\end_inset

 from individual 2.
\end_layout

\begin_layout Standard
The, once the event is realized, when a recession occurs, trader 2, since
 she has sold 
\begin_inset Formula $z_{2}$
\end_inset

 units of security 
\begin_inset Formula $b$
\end_inset

, is obliged to pay 
\begin_inset Formula $1$
\end_inset

 
\begin_inset Formula $z_{2}$
\end_inset

 dollars.
 If the recession doesn't occur, then 
\begin_inset Formula $1$
\end_inset

 is on the hook since he is the one who sold off 
\begin_inset Formula $z_{1}$
\end_inset

units of security 
\begin_inset Formula $a$
\end_inset

.
\end_layout

\begin_layout Standard
There is a second way to look at this tradeoff.
 From 
\begin_inset Formula $1$
\end_inset

's perspective, what he does is to plan how much income he would like to
 have in each of the two events.
 His plan is to arrange it so that he has income 
\begin_inset Formula $c_{n}$
\end_inset

 if there is no recession and 
\begin_inset Formula $c_{a}$
\end_inset

 when there is a recession.
 To accomplish this, he realizes that he has to purchase and sell assets
 that will force him to pay 
\begin_inset Formula $z_{1}$
\end_inset

 if there is no recession, but will leave him with a payment of 
\begin_inset Formula $z_{2}$
\end_inset

 when a recession occurs.
\end_layout

\begin_layout Standard
The way he might do this is to take the matrix of asset returns, given by
 
\begin_inset Formula 
\[
\left[\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right]
\]

\end_inset

 and post multiply it by his portfolio 
\begin_inset Formula $\left(z_{1},z_{2}\right)$
\end_inset

 viewed as a column vector to get the payments that he wants.
 In other words, he would solve the equation
\begin_inset Formula 
\[
\left[\begin{array}{c}
c_{n}-y\\
c_{a}-y+d
\end{array}\right]=\left[\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}\right]\cdot\left[\begin{array}{c}
z_{1}\\
z_{2}
\end{array}\right]
\]

\end_inset

 which obviously has the solution described above.
\end_layout

\begin_layout Standard
As we are working here with a Walrasian equilibrium, trader 2 is happy with
 the trade she makes, which is to receive 
\begin_inset Formula $c_{n}-y$
\end_inset

 when there is no recession, but to pay 
\begin_inset Formula $c_{a}-y+d$
\end_inset

 when there is.
\end_layout

\begin_layout Standard
At this point, we could ask what the prices of the securities have to be
 so that the market for them clears.
 To see this, we just consider the Walrasian equilibrium first and imagine
 that the price of good 1 is 
\begin_inset Formula $q$
\end_inset

 while the price of good 2 is 
\begin_inset Formula $1.$
\end_inset

 The bundle 
\begin_inset Formula $\left(c_{n},c_{a}\right)$
\end_inset

 has cost 
\begin_inset Formula $qc_{n}+c_{a}$
\end_inset

which is equal to 
\begin_inset Formula $qy+y-d$
\end_inset

 since it is a Walrasian equilibrium.
 Then of course, 
\begin_inset Formula $q\left(c_{n}-y\right)+\left(c_{a}-y+d\right)=0$
\end_inset

, from which it is obvious that the price of asset 
\begin_inset Formula $a$
\end_inset

 has to be 
\begin_inset Formula $q$
\end_inset

 to get this to work.
\end_layout

\begin_layout Standard
We could express this as a problem is computing a security price 
\begin_inset Formula $\rho$
\end_inset

 for asset 1 such that
\begin_inset Formula 
\[
\rho z_{1}+z_{2}=\rho\left(c_{n}-y\right)+\left(c_{a}-y+d\right)=0
\]

\end_inset

or
\begin_inset Formula 
\[
\rho=-\frac{c_{a}-y+d}{c_{n}-y}=q.
\]

\end_inset


\end_layout

\begin_layout Standard
So far this is pretty obvious, so lets enrich the model to make it look
 more like a stock market.
 Suppose that our two traders are entrepreneurs.
 Each of them has an inheritance 
\begin_inset Formula $\omega$
\end_inset

 that they have no matter what.
 But each has a start up venture they are running.
 The start up of trader 1 gives profits 
\begin_inset Formula $\pi_{a1}>0$
\end_inset

 if there is no recession and 
\begin_inset Formula $\pi_{a2}<0$
\end_inset

 if there is.
 Trader 2 has another start up that does well in a recession.
 Her company earns 
\begin_inset Formula $\pi_{b1}<0$
\end_inset

 if there is no recession, but make a profit 
\begin_inset Formula $\pi_{b2}$
\end_inset

 if there is a recession.
\end_layout

\begin_layout Standard
Now lets take the first figure above, and just relabel it so that it coincides
 with this new information.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename /home/peters/files/docs/book/finance/finance_fig2.eps

\end_inset


\end_layout

\begin_layout Standard
Here is the same diagram with the trades labeled.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename /home/peters/files/docs/book/finance/finance_fig3.eps

\end_inset


\end_layout

\begin_layout Standard
Now we can imitate what we did before.
 We'll let 
\begin_inset Formula $q=\frac{c_{a}-\omega-\pi_{a2}}{c_{n}-\omega-\pi_{a1}}$
\end_inset

 be the price ratio that seems to support the Walrasian equilibrium.
\end_layout

\begin_layout Standard
To support this outcome, trader 1 has to buy a portfolio 
\begin_inset Formula $\left(z_{1},z_{2}\right).$
\end_inset

 As the assets are now profits in start up companies, the portfolio has
 to satisfy the following matrix equation:
\begin_inset Formula 
\[
\left[\begin{array}{c}
c_{n}-\omega-\pi_{a1}\\
c_{a}-\omega-\pi_{a2}
\end{array}\right]=\left[\begin{array}{cc}
\pi_{a_{1}} & \pi_{b1}\\
\pi_{a2} & \pi_{b_{2}}
\end{array}\right]\cdot\left[\begin{array}{c}
z_{1}\\
z_{2}
\end{array}\right]
\]

\end_inset

It is straightforward how we should do this, pre multiply both sides of
 the equation by the inverse matrix
\begin_inset Formula 
\[
\left[\begin{array}{cc}
\pi_{a_{1}} & \pi_{b1}\\
\pi_{a2} & \pi_{b_{2}}
\end{array}\right]^{-1}\left[\begin{array}{c}
c_{n}-\omega-\pi_{a1}\\
c_{a}-\omega-\pi_{a2}
\end{array}\right]=
\]

\end_inset


\begin_inset Formula 
\[
\frac{1}{\pi_{a1}\pi_{b2}-\pi_{b1}\pi_{a_{2}}}\left[\begin{array}{cc}
\pi_{b2} & -\pi_{b1}\\
-\pi_{a2} & \pi_{a_{1}}
\end{array}\right]\left[\begin{array}{c}
c_{n}-\omega-\pi_{a1}\\
c_{a}-\omega-\pi_{a2}
\end{array}\right]=\left[\begin{array}{c}
z_{1}\\
z_{2}
\end{array}\right]
\]

\end_inset

 So we can compute the portfolio rather easily.
 The interpretation is just as before.
 Trader 1 is going to sell off 
\begin_inset Formula $z_{1}$
\end_inset

 of the shares in his venture and use the proceeds to purchase shares in
 the venture being run by trader 2.
\end_layout

\begin_layout Standard
All we have left to do at this point is to try to figure out how the shares
 of the two ventures should be priced.
 Lets use the convention that the shares in venture 2 should have a price
 1, so that all we need to figure out is what the corresponding relative
 price 
\begin_inset Formula $\rho$
\end_inset

 should be of the shares in venture 1.
\end_layout

\begin_layout Standard
Now it is just computation.
 We have to have 
\begin_inset Formula $\rho z_{1}+z_{2}=0$
\end_inset

, which means
\begin_inset Formula 
\[
\frac{1}{\pi_{a1}\pi_{b2}-\pi_{b1}\pi_{a_{2}}}\left[\begin{array}{cc}
\rho & 1\end{array}\right]\left[\begin{array}{cc}
\pi_{b2} & -\pi_{b1}\\
-\pi_{a2} & \pi_{a_{1}}
\end{array}\right]\left[\begin{array}{c}
c_{n}-\omega-\pi_{a1}\\
c_{a}-\omega-\pi_{a2}
\end{array}\right]=0
\]

\end_inset

If we want to solve this, we can obviously forget about the constant 
\begin_inset Formula $\frac{1}{\pi_{a1}\pi_{b2}-\pi_{b1}\pi_{a_{2}}}$
\end_inset

 and solve
\begin_inset Formula 
\[
\left[\begin{array}{cc}
\rho & 1\end{array}\right]\left[\begin{array}{cc}
\pi_{b2} & -\pi_{b1}\\
-\pi_{a2} & \pi_{a_{1}}
\end{array}\right]\left[\begin{array}{c}
c_{n}-\omega-\pi_{a1}\\
c_{a}-\omega-\pi_{a2}
\end{array}\right]=0
\]

\end_inset

instead.
 This gives first
\begin_inset Formula 
\[
\left[\begin{array}{cc}
\rho\pi_{b2}-\pi_{a2} & \pi_{a1}-\rho\pi_{b1}\end{array}\right]\left[\begin{array}{c}
c_{n}-\omega-\pi_{a1}\\
c_{a}-\omega-\pi_{a2}
\end{array}\right]=0
\]

\end_inset

or
\begin_inset Formula 
\[
\left(\rho\pi_{b2}-\pi_{a2}\right)\left(c_{n}-\omega-\pi_{a1}\right)+\left(\pi_{a1}-\rho\pi_{b1}\right)\left(c_{a}-\omega-\pi_{a2}\right)=0.
\]

\end_inset

From the Walrasian equilibrium, this is
\begin_inset Formula 
\[
\left(\rho\pi_{b2}-\pi_{a2}\right)=-q\left(\pi_{a1}-\rho\pi_{b1}\right)
\]

\end_inset

or
\begin_inset Formula 
\[
\rho=-\frac{q\pi_{a1}-\pi_{a_{2}}}{\pi_{b2}-q\pi_{b1}}.
\]

\end_inset


\end_layout

\begin_layout Standard
So this formula describes the basics of asset pricing.
 To compute the value of an asset, you need to know a couple of things.
 The first of which is the returns on other assets - even though we are
 trying to find the relative price of asset 
\begin_inset Formula $a$
\end_inset

 we have to know 
\begin_inset Formula $\pi_{b1}$
\end_inset

 and 
\begin_inset Formula $\pi_{b2}$
\end_inset

 to do that.
 Second, we need to know the state price 
\begin_inset Formula $q$
\end_inset

, which we can compute by solving for a Walrasian equilibrium.
\end_layout

\end_body
\end_document