Arbitrage (asset) pricing theory

do exercise 1,4,6

Chapter

3

. Arbitrage (asset) pricing theory

Neil Wallace

December

2

6, 20

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1 Introduction

Arbitrage possibilities arise when there are different transactions that achieve
the same final outcome. When there are such possibilities, sharp conclusion
can be drawn about the prices in the different transactions. Consider again
the two-state rainfall model and assume that in addition to making trades
contingent on the outcome of the rainfall, each person can buy or sell plots
of land before rainfall is realized. There are N plots of land and the owner
of plot n gets (wn1,wn2); that is, wn1 is the rice crop on land-n if rainfall is
low and wn2 is that crop if rainfall is high. If wn1 6= wn2, then land-n is what
would ordinarily be called a risky asset. We can use arbitrage to compute
the price of a given plot land in terms of the prices of rice contingent on the
rainfall outcome, the prices we defined above. The general version of doing
that is called arbitrage-pricing theory (or APT).
Suppose p = (p1,p2) is the price of outcome contingent rice. The APT

says that the price-of-land n is p1wn1 +p2wn2. In order to reach that conclu-
sion, assume that any person can either buy land-n or go-short land-n. (To
go-short land-n means promising to pay out wn1 amount of rice if rainfall is
high and wn2 if rainfall is low.) Let v(wn1,wn2) be the pre-outcome price of
land-n (in abstract units of account).

Exercise 1 Suppose contingent rice can be bought or sold at p = (p1,p2).
What purchases and sales would be profitable (without bearing risk) if v(wn1,wn2)
< p1wn1 + p2wn2? What purchases and sales would be profitable (without bearing risk) if v(wn1,wn2) > p1wn1 + p2wn2? (Hint: Answer this question
by assuming, that payment for a plot of land is made by making the follow-
ing promise: If v(wn1,wn2) is the price of land (wn1,wn2), then it can be
purchased for any (x1,x2) that satisfies v(wn1,wn2) = p1x1 +p2x2.)

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Now let’s leave rice, land, and two states behind, and exposit APT more
generally. We assume that there are S states and that p = (p1,p2, …,pS)
denotes the price vector of outcome-contingent goods. We define assets by
their payoffs. In particular, an asset is a vector of outcome-specific payoffs–
say, (a1,a2, …,aS) with the interpretation that the owner of this asset receives
as units of the good if outcome s occurs. (If as < 0, then the owner must pay out as units of the good if outcome s occurs.) The price of the asset is the pre-outcome price at which the asset can be bought or sold. We denote that price v(a1,a2, ...,aS).

Proposition 1 The only price of the asset consistent with no arbitrage prof-
its is

v(a1,a2, …,aS) =

S∑
s=1

psas. (1)

Exercise 2 Argue that proposition 1 is true. (Hint: Show that each person
would profit by buying the asset if v(a1,a2, …,aS) <

∑S
s=1 psas and that no-

one wants to buy it; then show that each person would profit by selling it
short if v(a1,a2, …,aS) >

∑S
s=1 psas.)

Consider two assets: (a1,a2, …,aS) and (b1,b2, …,bS), where bs = zas for
s = 1,2, …,S for some number z > 0. Show that in a CE, v(b1,b2, …,bS) =
zv(a1,a2, …,aS).

2 Limited Liability and the Modigliani-Miller
theorem

We can also use proposition 1 to price the debt and equity of limited-liability
corporations. We define the corporation as an asset in the sense that own-
ership of the entire corporation entitles the owner to a vector of outcome-
specific payoffs– say, (x1,x2, …,xS). As part of limited liability, we now as-
sume that xs ≥ 0 for each s.
We next define debt of the corporation. Debt is a liability with a promised

pay-off that is not contingent. The amount of debt is measured by the
magnitude of that promised pay-off. Let P ≥ 0 be the promised pay-off.
The magnitude P is to be distinguished from what the owners of the debt

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actually get. (After all, junk bonds are called junk because it is understood
that holders of junk bonds will not always receive the promised pay-off.)
Let min{a,b} denote the smaller of a and b. The owners of the debt of

the corporation (x1,x2, …,xS) with promised pay-off P receive the vector
(min{x1,P},min{x2,P}, …,min{xS,P}). We let D denote the pre-outcome
value of this debt. Using proposition 1, we have

D =

S∑
s=1

ps min{xs,P}. (2)

Now we define the equity of the corporation. Consider a corporation
(x1,x2, …,xS) with debt having the promised pay-off P. The owners of the
equity get what is left after the owners of the debt get paid. Therefore,
the owners of the equity of this corporation receive the vector (max{x1 −
P,0},max{x2−P,0}, …,max{xS −P,0}), where max{a,b} means the larger
of a and b. We let E denote the pre-outcome value of this equity. Using
proposition 1, we have

E =
S∑
s=1

ps max{xs −P,0}. (3)

The Modigliani-Miller Theorem concludes that the total value of a cor-
poration, D+E, does not depend on how much debt it has. This conclusion
follows from (2) and (3).

Exercise 3 Show that min{a,b}+max{a− b,0} = a. (Hint: consider each
of the following cases, which together are obviously exhaustive: a > b, a = b,
and a < b.)

Exercise

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(i) Use the result of the last exercise and (2) and (3) to compute
D +E. (You have proved the Modigliani-Miller Theorem.)

Exercise

5

We are describing a corporation by a vector of payoffs, (x1,x2, …,xS).
Suppose S = 2. (i) How would you describe the “riskiness” of a corporation?
(ii) Are less risky corporations worth more than more risky corporations?
Explain.

Because D is the pre-outcome value of the debt and P is the promised
pay-off, the ratio P/D is the (gross) promised yield or (gross) rate-of-return

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on the debt. As we next show, this yield is in general increasing in P, our
measure of the amount of the debt. We draw this conclusion by studying the
ratio D/P. If b > 0, then min{a,b}

b
= min{a

b
,1}. Therefore, from (2), if P > 0,

then
D

P
=

S∑
s=1

ps min{
xs
P
,1}. (4)

Exercise 6 According to (4), higher debt as measured by P either does not
affect the promised yield or increases the promised yield. Describe the cir-
cumstances in which each conclusion holds.

Exercise 7 Competition is sometimes described as a situation in which peo-
ple face prices at which they can buy and sell any amount. Our model is
competitive in the sense that we are assuming that people can trade any
amounts at the price vector of outcome-contingent goods, which we denote
p. Despite that, the last exercise shows that a limited liability corporation
faces a promised yield on debt that can be increasing in P, the amount of
debt it sells. How do you reconcile that result with competition?

Exercise 8 In the US, the base on which the federal corporate income tax is
levied excludes interest payments. In the context of the above model, the base
for the tax in state s is xs − (P −D). Show that a positive tax rate makes
E +D an increasing function of P.

Exercise 9 It is widely reported that hedge-fund operators often charge their
customers “two and twenty”, where two is a percentage that is paid no matter
what the return is and twenty is the share of gains that is paid. (The operators
do not share losses.) For this exercise ignore the two part and focus on the
sharing of gains. Assume that such a hedge-fund operator has customers who
have invested an amount with pre-state value X > 0. Suppose the hedge-fund
operator wants to maximize the pre-state value of their 20% share of gains.
What portfolio of assets with pre-state value X should the hedge-fund operator
buy? (Hint: Start by assuming that there are two states so that the operator
chooses (x1,x2) to maximize its 20% of gains subject to x1 > 0, x2 > 0, and
X = p1x1 +p2x2. Also, if state i occurs, then the gain is max{0,xi −X}.)

Exercise 10 At times, limited liability corporations are given loan guaran-
tees. In a full analysis, we would have to describe how the guarantee is

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financed. Here we will ignore that. If the guarantee is genuine, then the
owners of the debt of the corporation (x1,x2, …,xS) with promised pay-off P
will actually receive P in every outcome. Suppose that is the case. (i) Give
a formula for the pre-outcome value of such debt. (ii) Give a formula for
D + E for such a corporation. (iii) Does D + E depend on P and in what
way? (iv) Suppose the guarantor of the debt imposes some maximum amount
of debt. Given that P is set at the maximum, does the value of the equity of
the corporation depend on the riskiness of the corporation?

Exercise 11 Read a bit about what is called the Volcker Rule, which is about
limited liability corporations called banks whose deposit liabilities are guaran-
teed by the government. Draw an analogy between the setting of the last
exercise and the setting that motivates the Volcker Rule.

These results are the starting point for the study of corporate finance.
That study involves various departures from the above model. After all,
the above model does not even tell us anything about the circumstances in
which the institution of limited liability is desirable. In the setting of the
above model, eliminating that institution would have no consequences. A
good starting point for further study is to read the literature that attempts
to provide a theory of the benefits of limited liability.

3 Concluding remarks

That completes our study of risk-sharing when there is symmetric informa-
tion. We next turn to the study of models of asymmetric information. We
will, however, maintain the expected-utility hypothesis and the assumption
that people agree about the relevent probability distributions.

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