Corporate Financing – Understanding Payoffs and Risk Preference This problem is less direct than the previous ones, and
Posted: Wed Mar 30, 2022 3:43 pm
Corporate Financing – Understanding Payoffs and Risk
Preference
This problem is less direct than the previous ones, and you may
have to work on it
more slowly. It will help you think and apply the theory in more
economic and graphic
terms.
Consider a company that has only one investment with a net return
of Y (Y is a random
variable, positive and smaller than infinite). The company is
financed by both debt and
equity; the investors of debt and equity are called lender and
shareholder,
respectively, and the shareholder exercises full control on the
company (no debt
covenants in place) and may ultimately decide on the company’s risk
level.
The debt contract is standard; in other words, the company agrees
to repay the lender
a pre-determined amount D, as long as Y > D. If Y < D, then
the lender gets the whole
Y. Therefore, the shareholder acts as the residual claimer on Y;
the firm’s payoff Y
is first used to repay the lender, then the shareholder.
1. Describe algebraically the payoff functions of both the
lender and the
shareholder as a function of Y and D. Tips:
• A payoff is the amount received by a type of investor depending
on
what occurs to Y. Algebraically, it means “If Y > D → payoff
shareholder
= […], & payoff lender = […]” and so on for all cases and
players.
• There are two cases to consider: Y < D and Y > D. You may
check
that it does not matter whether you consider the equality Y = D at
the
lower or upper case.
2. If you think of financial options, what is the payoff of the
shareholder akin to
(suggestions: owing a call option, issuing a call option, owing a
put option or
issuing a put option)? On the other hand, what is the payoff of the
lender akin
to? Remember to consider what the sum of the payoffs of the lender
and the
shareholder must equal to.
3. Considering the payoffs of each investor (lender and
shareholder), what will be
their risk preference vis-à-vis the average risk undertaken by the
company?
Explain why.
4. Explain how the lender could use debt covenants to align the
incentives of the
shareholder to its own. Can you give some examples of financial
covenants that
the lender would like to have in place?
5. As leverage increases, how does the lender’s appetite for
risk change?
Why?
6. As leverage decreases, how does the shareholder’s appetite
for risk change?
Why?
7.Consider an extension of the problem above by including a
subordinated debt, which
is supplied by a subordinated lender. Consider the lender on the
first part of the
problem was the senior lender, or the first one to be paid in case
Y is not high enough.
The subordinated debt contract is standard; in other words, the
company agrees to
repay the subordinated lender a pre-determined amount d as long as
Y > D + d. If Y <
D + d, then the proceeds first flow to the senior lender, then to
the subordinated lender.
The shareholder therefore still acts as the residual claimer on
Y.
8. Describe algebraically the payoff functions of each
investor
(senior lender, subordinated lender and shareholder) as a function
of Y, D and
d. There are now three cases to consider: Y < D; D < Y < D
+ d; and Y > D + d.
9. Explain how the inclusion of the subordinated lender altered
the shareholder’s
risk preference in comparison to the previous case. How does the
change in
risk assumed by the company (the shareholder remains in control)
affect the
senior lender’s expected payoff?
10. Explain why the senior lender would have liked to prevent
the firm from
issuing subordinated debt if it could have control over this
decision
(covenants).
11. Explain why the subordinated lender’s risk preference has
characteristics of
both equity and senior debt; specifically, resembling equity at low
values of Y
and resembling senior debt at high values of Y. Furthermore, show
that her
payoff may be represented by the combination of owning (being long
on) a call
option (with strike price K1) and writing (being short on) another
call option (with
strike price K2); for that, determine both K1 and K2 in terms of D
and d.
12. Based on your response on the previous question, explain how
the
subordinated lender’s risk preference changes for different values
of D and
d. More specifically, explain how its risk preference changes as
(i) d increases,
and as (ii) D decreases.
Preference
This problem is less direct than the previous ones, and you may
have to work on it
more slowly. It will help you think and apply the theory in more
economic and graphic
terms.
Consider a company that has only one investment with a net return
of Y (Y is a random
variable, positive and smaller than infinite). The company is
financed by both debt and
equity; the investors of debt and equity are called lender and
shareholder,
respectively, and the shareholder exercises full control on the
company (no debt
covenants in place) and may ultimately decide on the company’s risk
level.
The debt contract is standard; in other words, the company agrees
to repay the lender
a pre-determined amount D, as long as Y > D. If Y < D, then
the lender gets the whole
Y. Therefore, the shareholder acts as the residual claimer on Y;
the firm’s payoff Y
is first used to repay the lender, then the shareholder.
1. Describe algebraically the payoff functions of both the
lender and the
shareholder as a function of Y and D. Tips:
• A payoff is the amount received by a type of investor depending
on
what occurs to Y. Algebraically, it means “If Y > D → payoff
shareholder
= […], & payoff lender = […]” and so on for all cases and
players.
• There are two cases to consider: Y < D and Y > D. You may
check
that it does not matter whether you consider the equality Y = D at
the
lower or upper case.
2. If you think of financial options, what is the payoff of the
shareholder akin to
(suggestions: owing a call option, issuing a call option, owing a
put option or
issuing a put option)? On the other hand, what is the payoff of the
lender akin
to? Remember to consider what the sum of the payoffs of the lender
and the
shareholder must equal to.
3. Considering the payoffs of each investor (lender and
shareholder), what will be
their risk preference vis-à-vis the average risk undertaken by the
company?
Explain why.
4. Explain how the lender could use debt covenants to align the
incentives of the
shareholder to its own. Can you give some examples of financial
covenants that
the lender would like to have in place?
5. As leverage increases, how does the lender’s appetite for
risk change?
Why?
6. As leverage decreases, how does the shareholder’s appetite
for risk change?
Why?
7.Consider an extension of the problem above by including a
subordinated debt, which
is supplied by a subordinated lender. Consider the lender on the
first part of the
problem was the senior lender, or the first one to be paid in case
Y is not high enough.
The subordinated debt contract is standard; in other words, the
company agrees to
repay the subordinated lender a pre-determined amount d as long as
Y > D + d. If Y <
D + d, then the proceeds first flow to the senior lender, then to
the subordinated lender.
The shareholder therefore still acts as the residual claimer on
Y.
8. Describe algebraically the payoff functions of each
investor
(senior lender, subordinated lender and shareholder) as a function
of Y, D and
d. There are now three cases to consider: Y < D; D < Y < D
+ d; and Y > D + d.
9. Explain how the inclusion of the subordinated lender altered
the shareholder’s
risk preference in comparison to the previous case. How does the
change in
risk assumed by the company (the shareholder remains in control)
affect the
senior lender’s expected payoff?
10. Explain why the senior lender would have liked to prevent
the firm from
issuing subordinated debt if it could have control over this
decision
(covenants).
11. Explain why the subordinated lender’s risk preference has
characteristics of
both equity and senior debt; specifically, resembling equity at low
values of Y
and resembling senior debt at high values of Y. Furthermore, show
that her
payoff may be represented by the combination of owning (being long
on) a call
option (with strike price K1) and writing (being short on) another
call option (with
strike price K2); for that, determine both K1 and K2 in terms of D
and d.
12. Based on your response on the previous question, explain how
the
subordinated lender’s risk preference changes for different values
of D and
d. More specifically, explain how its risk preference changes as
(i) d increases,
and as (ii) D decreases.