EC9A1, Problem Set 1
G. Trigilia, February 2015
Exercise 1
The example analysed class regarding the Modigliani - Miller Theorem focused on the capital structure for a gas producer. In this question we assume
that the company has decided to raise capital by shares. The company is
considering to hedge the risk of future gas prices by using a forward contract
where the forward contract’s price is $2.75 (per thousand cubic feet).
Based on Modigliani - Miller (MM) should the company hedge the risk
and sign a forward contract? Prove your claim.
Solution. The example discussed in class is the following:
Suppose we found natural gas in the Mediterranean and are interested in extracting the gas. Future natural gas prices (one year
from now) can be either $4 or $2 (per thousand cubic feet) with
equal probabilities. In our field there are 100 Billion cubic feet
and cost of extraction is $100 Millions. For simplicity we assume
that these costs are paid up front. We do not have any cash and
need to raise this amount. Suppose that the risk free interest rate
is 10% and that the discount rate for gas prices is 20%.
Consider first the case of a gas company selling its entire future production
via forward contracts - call it Company A. It will get $275 Millions free of
risk at t = 1, and hence the present discounted value is $275/1.1 = $250
Millions, because the risk free rate is 0.1 and hence the discount factor reads
1/(1 + 0.1).
Now suppose that another, identical gas company - call it Company B
- did not hedge, and has a present discounted value equal to X > $250
Millions. This would contradict the MM proposition, since the structure of
the company’s liabilities should not affect its value.
To prove that we must actually have X = $250 Millions, consider the
following trading strategy: the investor short sells Company B, and use the
proceeds to buy Company A. Further, he sells 100 Millions forward contracts.
At t = 0, the investor gets X from short selling B, and uses $250 Millions
to buy A, making a strict profit of X − $250 > 0 Millions.
At t = 1: he receives $275 Millions from the Company A he purchased,
and pays back the same amount due to the forward contracts he sold. Further, he has to buy the Company B that he did short sell at its market value
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(either $200 or $400 Millions depending on the price of gas). As a result, his
payoff at t = 1 is zero.
Summing up, the trading strategy is strictly profitable at t = 0 and
makes zero profits at t = 1, giving the investor an arbitrage opportunity.
This guarantees that X ≤ $250 Millions.
Finally, the reverse trades allow us to rule out X < $250 Millions, concluding the proof.
Comment: Why the forward contract’s price is $2.75? Assume that there
is a market maker (competitive) that sells forward contracts. This means
that he must provide (at t = 1) 1,000 cubic feet of gas per contract. The
current price of gas (spot) is $2.5 (because it is given by $3/1.2). Therefore,
the market maker should take a loan of $2.5 at t = 0, per contract sold.
For this loan he will have to return $2.75 at t = 1 (because it is given by
$2.5 ∗ 1.1, as the riskless rate is 10%). So, 2.75 is the price required by the
market maker to get the next period for gas.
Exercise 2
A firm wishes to raise money for investment. The firm has existing assets
whose value is £50 Millions or £100 Millions. Only the firm knows what
the value of its assets is, and the market believes that each value is possible
with probability half. To invest, the company needs to raise £100 Millions.
Investment is risky: with probability 0.5 it will succeed and yield a return of
£A Millions, and with prob. 0.5 it will fail and yield zero. The probability
of success is independent of the value of existing asset.
We assume a competitive financial market that is risk neutral. We first
analyse the situation in which the company can only raise equity:
A) What is the percentage of ownership that investors will demand if they
believe that both types of firms raise capital (assuming a competitive
capital market)? For which values of A such an equilibrium exists?
B) What is the percentage of ownership that investors will demand if they
believe that only the lower type raises capital? For which values of A
such an equilibrium exists?
C) Explain why there does not exist an equilibrium in which only higher
type firms raise capital;
Suppose that A = 100 and that the firms only raise funds using debt with a
face value of B.
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D) Show that there does not exist an equilibrium in which both two types
of firms raise capital. (Hint: Investors realise that there is a possibility
that the firm may default);
E) Show that there does not exist equilibrium in which only the lower type
firms raise capital. For what values of A such equilibrium exists?
F) Suppose that firms can offer contracts that consist both on debt repayments, and a probability with which the contract is taken. Construct
a separating equilibrium.
Solution.
(A) If the market believes that the two types of companies raise capital,
then he believes that the average value of a company equals:
0.5[50 + A/2] + 0.5[100 + A/2] = (150 + A)/2
If the competitive market is risk neutral, the rate of ownership it will require,
α, should be given by:
α(150 + A)/2 = 100
⇐⇒
α = 200/(150 + A)
If A < 50, the market does not agree to buy some of the company, hence we
need A ≥ 50. A high type, then, would be willing to invest selling α shares
only if:
(1 − α)[100 + A/2] ≥ 100
Substituting inside the α we calculated previously we get:
A2 − 50A − 40, 000 ≥ 0
The negative root can be ruled out, and the positive root yields A ≥ 226.5.
It is easy to check that whenever the high type is willing to invest, the low
type also agrees to invest. Hence, we conclude that such equilibrium exists
only if A ≥ 226.5.
(B) If the market believes that only the low type is investing, the average
value of the company is equal to:
50 + A/2
If the competitive market is risk neutral, the rate of ownership it will require,
α, should be given by:
α(50 + A/2) = 100
⇐⇒
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α = 200/(100 + A)
If A < 100, the market does not agree to buy some of the company, hence
we need A ≥ 100. A high type, then, would be willing not to invest selling
α shares only if:
(1 − α)[100 + A/2] ≤ 100
Substituting inside the α we calculated previously we get:
A2 − 100A − 40, 000 ≤ 0
The negative root can be ruled out, and the positive root yields A < 256.
Finally, the low type must be willing to invest, that is:
(1 − α)[50 + A/2] ≥ 50
which simplifies to A ≥ 200. Hence, we conclude that such equilibrium exists
only if 200 ≤ A ≤ 256.
Notice that if 226.5 ≤ A ≤ 256, there exist multiple equilibria (the one
derived in part A and the one we just derived in part B). For example, if
A = 250, then there is exists an equilibrium where the investors require 50%
of the company in exchange of an investment of 100, and both types invest.
Further, there exists an equilibrium where the investors require 4/7 of the
company for raising the same amount, and only a low valuation firm agrees
to that.
(C) The condition required for any type to invest is that the fraction
they get to keep of their firm value exceeds (weakly) the value of the firm if
no investment is undertaken. I.e.:
(1 − α)E firm’s value investment ≥ E firm’s value no investment
Notice that the increase in firm’s value due to investment is equal to A/2 for
both firms. Hence, we can write:
A/2 ≥ αE firm’s value investment
Because the RHS is strictly higher for high types, we conclude that whenever
they invest, the low types are investing as well.
(D) First notice that debt is risky, because in the event that the money
went to a low type, and he finds himself in the low state, the value of the
company is 50 < 100. Hence, we conclude that B > 100. If 150 > B, then
the zero profit condition reads:
0.25(50) + 0.25(100) + 0.5B = 100
and indeed B < 150, confirming our conjecture.
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⇐⇒
B = 125
But consider the low type company: the value if it does not invest is 50,
whereas the value if it invests is 0 + (150 − 125)/2 = 12.5 < 50, so it does
not invest and the pooling equilibrium does not exist.
(E) Provided that A = 100, the expected value of the low type of company is 100, so it is clear that the market will require all this amount to
provide the company 100 so as not to lose, so B = 150 (all the company’s
value in case of success). It is clear that the company will not agree to such
a contract because it means giving up all the company’s cash flow in each
state of nature.
We will solve for a general A. First, it must be that the low type company
wants to invest:
50 ≤ (50 + A − B)/2
⇐⇒
A ≥ 50 + B = 200
Finally, the high type company must not invest, i.e.:
100 ≥ (100 + A − B)/2
⇐⇒
A ≤ 100 + B = 250
So if 200 ≤ A ≤ 250, the equilibrium where only the low type invests
exists.
(F) Suppose the market maker believes that the high type is offering a
contract that specifies a face value of debt Bh , and a probability of investment
of ph ∈ [0, 1].1 Moreover, he believes that the low type offers a contract
(Bl , pl = 1) with (Bh , ph ) 6= (Bl , pl ). Does a separating equilibrium exists?
First notice that, at any separating equilibrium, Bh must satisfy the zero
profit condition for the financier investing in the high type company, i.e.:
0.5(100) + 0.5(Bh ) = 100
⇐⇒
Bh = 100
(the high type is issuing a riskless bond), whereas for the low type we already
know Bl = 150.
Second, it must be that at Bh = 100 the high type wants to invest:
100 ≤ ph (100 + A − 100)0.5 + (1 − ph )100
⇐⇒
A ≥ 200
Third, incentive compatibility must hold, i.e. (i) the low type company
should not want to pretend to be a high type; and (ii) viceversa. The (i)
constraint reads:
ph [0.5(50 + A − Bh )] + (1 − ph )50 ≤ [0.5(50 + A − Bl )]
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Off equilibrium, he believes any contract is offered by a low type.
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and plugging the face values of debt:
ph [A − 50] + (1 − ph )100 ≤ A − 100
or, equivalenty:
ph ≤
A − 200
A − 150
The (ii) constraint reads:
ph [0.5(100 + A − Bh )] + (1 − ph )50 ≥ [0.5(100 + A − Bl )]
and plugging the face values of debt:
ph [A] + (1 − ph )200 ≥ A − 50
We get: and
A − 250
A − 200
The two inequalities can be jointly satisfied for every A. However, notice
that the required probability is increasing in A. For instance, if A = 1, 250,
then ph ≥ 0.952, and by the previous inequality ph ≤ 0.955.
ph ≥
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