Question

Consider equity holders’ choice between two mutually exclusive projects: Project A and Project
B. Both have upfront cost at Year 0 of $50 million and equity plans to finance entirely with debt.
Cash flows in the year following the investment are partially state-contingent. Project A has a
certain cash flow of $60 million regardless of the state of the world in Year 1, but Project B has a

cash flow of $90 in the Good state, and a cash flow of $20 million in the Bad state. The risk-
neutral probability of the Good state is 1/2.

Assume that the risk-free rate is 0% and that equity holders are considering the choice of two
mutually exclusive debt markets to tap: Naïve and Sophisticated. In the Naïve market, potential
lenders are not aware of agency costs of debt and always believe that equity holders will take
actions that maximize firm value regardless of capital structure. In the Sophisticated market,

potential lenders are aware of agency costs of debt and the incentives of equity to engage in ex-
post appropriation of wealth from bondholders. In particular, Sophisticated market lenders

understand the risk-shifting incentive of equity that arises from the convexity of their claim in
the firm’s asset volatility.

Answer the following questions:
1. Which project maximizes firm value?
2. Assume equity taps the Naïve market
a. What promised face value of debt must equity issue to raise the $50 million
upfront cost?
b. Based on the face value you calculated in (a), calculate the payoff to equity in
each project and find which project they will take.
c. What is the expected payoff to the Naïve investors given the promised face value
from (a) and project choice from (b)?
3. Assume equity taps the Sophisticated market
a. What promised face value of debt must equity issue to raise the $50 million
upfront cost?
b. Assume that equity cannot commit to taking a certain project ex-ante. Which
project do they select in this case?

Answer 1

1. Risk free rate, r = 0%

NPV (project A) = -50 + 60/(1+r) = +10

NPV (project B) = -50 + 0.5*[90/(1+r) + 20/(1+r)] = +5

NPV (project A) > NPV (project B)

Project A maximizes the firm value.

2. (a) NAIVE MARKET - The equity will raise the $50 million by issuing a bond with face value $50 to naive lenders.

(b) In project A, the equity holders will have to pay back $50 million to the bondholders from the payoff of the project. Hence, Payoff to equity holders = $10 million.

In project B, there are two scenarios with equal likelihood (probability = 0.5):

1. If the payoff from project B is $90 million, the debt holders will get back their $50 million, and equity holders' payoff is $40 million from the project.

2. If the payoff from project B is only $20 million, debt holders will receive only $20 million and lose $30 million. Equity holders will not receive anything and their payoff will be $0 million.

Average payoff for equity holders from project B = 0.5*$40 million + 0.5*$0 million = $20 million.

Payoff to equity holders (t=1)
Project	Poor	Good
A	10	10
B	0	40
Difference (B-A)	-10	30
Probability	0.5	0.5
Average payoff	10

Therefore, equity holders are better off with project B than project A.

(c) Payoff to debt holders:

Project B :  -$50 million + 0.5*($50 million + $20 million) = -$15 million

Payoff to bondholders (t=1)
Project	Poor	Good
A	50	50
B	20	50
Difference (B-A)	-30	0
Probability	0.5	0.5
Average payoff	-15

3. (a) As long as the sophisticated market lenders have lent only up to $30 million, the equity holders would have no incentive to go for project B (riskier project). In this case, the interests of the debt holders and equity holders would be aligned.

Therefore, the sophisticated market lenders would lend only $30 million.

Payoff to bondholders (t=1)
Project	Poor	Good
A	30	30
B	20	30
Difference (B-A)	-10	0
Probability	0.5	0.5
Average payoff	-5

Payoff to equity holders (t=1)
Project	Poor	Good
A	30	30
B	0	60
Difference (B-A)	-30	30
Probability	0.5	0.5
Average payoff	0

(b) In case they can not select a project ex ante, even equity holders would then prefer project A.