ECO359: Assignment 2
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ECO359: Assignment 2
Due 10 AM on March 14.
1. Consider the following variation of the trade-off model discussed in class. There are three dates t = 0, 1/2, 1. At date t = 0, the firm has F0 = 0.4 of debt outstanding and chooses a new debt amount F . The outstanding debt comes with the covenant that F < 0.7. At date t = 1/2, the firm potentially learns news about its future cash flows and can potentially change the project (see the specification of each question below). At date t = 1, cash flows Y are realized. Initially, cash flows are uniformly distributed on [0, 1]. The corporate tax rate is 25% and there is zero recovery upon default. Suppose that all risk is idiosyncratic and rf = 0.
(a) Suppose first that nothing happens at date t = 1/2. Compute the optimal debt F* that the equity holders choose at t = 0. Is the optimal debt level higher/lower/equal to the case without the outstanding debt analyzed in class (i.e., when F0 = 0)? What is the logic behind this result? Hint: This part is similar to the case of additional debt issuance that we covered in class when we talked about the leverage ratchet effect. Note that the equity holders payoff equals the equity value of E(F) plus the issuance revenue (F - F0 )P (F) if F > F0 (minus repurchase costs (F0 - F)P (F) if F < F0) .
Solution: The equity holders’ payoff equals
E (F) + (F - F0 ) P (F) = 3 (1 - F)2 + (F - F0 ) (1 - F) - max,
where we used the formula for E(F) derived in class.
FOC: optimal F* satisfies
- (1 - F* ) + 1 + F0 - 2F* = 0 =号 - F* + F0 = 0 =号 F* = = 0.52.
F* = 0.52 is larger than the optimum when F0 = 0 (which equals 0.2 as we derived in class) . This is due to the leverage ratchet effect. Equity holders get benefits of new debt issuance in the form of issuance revenue, but share higher expected default costs with existing debt holders. In particular, this implies that the equity holders have no incentives to reduce debt to 0.2, which maximizes the firm value E(F) + D(F) .
(b) For the rest of the problem, suppose that at date t = 1/2, the equity holders can change the project at no cost, which changes the distribution of cash flows to
Y ≠〈 1
(à
w.p. 1/4.
w.p. 3/4贮
(1)
For any fixed F ∈ [0, 1] chosen at t = 0 (not necessarily F* derived in the previous part!), compute the equity value, call it Enew (F), from changing the project. Compare it with the equity value, call it Eold (F), from keeping the original project (i.e., the original distribution of cash flows). For which values of F will the shareholders prefer to change the project? Hint: You need to generalize the argument presented in class when we talked about asset substitution. You can show that there
is a threshold such that
E (F) =.,Eold (F) ,
F < ,
F > .
Solution: This is a straightforward generalization of the argument in class. For new distribution (1), the equity value equals
Enew (F) = P(Y = 1) (1 - F) = (1 - F).
For the old distribution (i. e., uniform on [0,1]), the equity value equals
Eold (F) = (1 - F)2 .
The equity holders prefer to switch if and only if
Enew (F) > Eold (F)
年号 (1 - F) > (1 - F)2
年号 F > 0.5
Thus, equity holders switch the project whenever F > 0.5 . Thus,
E (F) =., (1 - F)2 , F < 0.5,
(c) For any fixed F chosen at date t = 0, compute the debt price P (F). Hint: Similarly to the previous part, you can show that there is a threshold such that
P (F) =.,Pold (F) ,
F < ,
F > .
Solution: From part b,
P (F) = .1(1) - F,
( 4 ,
F < 0.5,
F > 0.5.
(d) Compute the optimal debt level chosen by the equity holders at t = 0. Recall that the outstanding debt comes with the covenant that F < 0.7 so you only need to consider F ∈ [0, 0.7]. Is it higher/lower/equal to the optimal F you found in part (a)? Provide intuition for your answer. Hint: The equity holders maximize E(F)+(F -F0 )P (F), where functions E and F are determined in parts (b) and (c) . You need to maximize this function separately on the interval F < 0.5 and on F > 0.5 and then compare which maximum gives a higher payoff to the equity holders. If you prefer, you can use Python, WolframAlpha, or any other program to draw a graph of E(F) + (F - F0 )P (F) and find the maximum of this function numerically.
Solution: Consider first F < 0.5 . Then, the equity holders maximize
(1 - F)2 + (F - F0 ) (1 - F) .
The derivative of this function is
- (1 - F) + 1 - 2F + F0 = - F + F0 > 0 for F < 0.5.
Thus, this function is maximized at F = 0.5 with the payoff to equity holders of *0.25+0.1*0.5 =
0.14375 .
Consider next F ∈ (0.5, 0.7] . Then, the equity holders get
(A - F) + (F - F0 ) = ╱ - F0 + F、= 0.0875 + F/16,
which is strictly increasing in F . Thus, it is maximized at F = 0.7 with a value of 0.1313 which is below 0.14375 . Thus, the equity holders optimal choice of F** = 0.5 . This is below F* = 0.52 . This is because of the asset substitution problem. Namely, the equity holders will change the project if the debt level is too high (above 0. 5) . This lowers the debt price, and hence, the equity holders payoff from issuing debt above 0. 5. Thus, it’s optimal for equity holders to choose a lower debt F** = 0.5 that avoids the asset substitution problem. The graphical illustration for this part is
below.
(e) For the rest of the problem, suppose that at t = 1/2 news about future cash flows are realized. Specifically, the equity holder learn that cash flows are either distributed uniformly on [0, 1/2] (call it “bad news”) or uniformly on [1/2, 1] (call it “good news”). Both news arrive with probability 1/2. Suppose first that the equity holders cannot change the project. For any fixed F ∈ [0, 0.7], compute how will the debt-to-equity ratio change after bad news. Does it decrease/increase/stays the same compared to the case before the news? Provide your intuition for this result. Hint1: You need to compute equity and debt values for the new distribution of cash flows after bad news. Hint2: The probability density function for a uniform distribution on [a, b] is given by
f(y) =.,1/(b - a),
.0,
for y ∈ [a, b],
otherwise.
Hint3: If both equity and debt values are zero, then suppose that the leverage ratio is infinity.
Solution: Consider the case of bad news. Let’s compute the equity value. For F > 1/2, the firm
defaults and Eo(b)ld(ad)(F) = 0 and D(F) = 0 so D/Ebar = e . For F < 1/2,
Eo(b)ld(ad)(F) = F1/2(Y - F)p(Y)dY
3 1/2
= d(Y2 - 2FY )
4 F
= /1/4 - F + F2、
= (1 - 2F)2
and
1/2
D(F) = Fp(Y)dY
F
= F (1 - 2F)
Thus, the leverage ratio is
D/Ebad = =
D/E0 = =
D/Ebad > D/E0
16 F 8 F
年号 >
年号 >
年号 1 > 0
which always holds. The debt-to- equity ratio increases even though the debt amount doesn’t change. This happens because both the debt and equity values decrease after bad news, but the latter de- creases by a larger amount.
(f) Suppose that after learning the news, the equity holders can choose whether to change the project or not. For the case of bad news, compute the range of F’s for which the equity holders make a change. Hint: This exercise essentially asks you to redo part (b) but for the distribution corre- sponding to the case of bad news.
Solution: Consider the case of bad news. Let’s compute the equity value. For F > 1/2, the firm
defaults and Eo(b)ld(ad)(F) = 0 . For F < 1/2,
Eo(b)ld(ad)(F) = F1/2(Y - F)p(Y)dY
3 1/2
= d(Y2 - 2FY )
4 F
= /1/4 - F + F2、
= (1 - 2F)2
The equity holders prefer to switch the project if and only if
Enew (F) > Eo(b)ld(ad) (F)
年号 (1 - F) > (1 - 2F)2
年号 3/4 > F.
This means that the equity holders always prefer to switch the project after bad news.
(g) (Harder. Not for credit!) Redo parts (e) and (f) for the case of good news. Find the optimal debt level at t = 0.
2. Read the article “CFO ratings” by Matt Levine posted on Quercus. Respond to the following questions.
(a) The author writes “Why would the sliced-up claims (senior bonds, junior bonds, shares) be worth more ($105) than a single unitary claim ($100)? <...> You could tell psychological stories, stories about conservative investors wanting safe assets and aggressive investors wanting risky assets and nobody wanting stuff in the middle.” This argument suggest that differences in risk aversion can explain why issuing a combination of relatively safe claims (senior bonds) and riskier claims (junior bonds, shares) allows you to raise more money than issuing just one claim (unlevered equity). Use the CAPM to either prove or disprove this argument. Hint: Assume that even senior bonds have some market risk, and in particular, they differ from risk-free bonds in the CAPM.
Solution: According to the CAPM, investors choose the same risky portfolio and the risk aver- sion only affects their leverage/borrowing. Hence, both more risk averse and more risk tolerant investors would hold the safer and riskier claims in the same proportion – in the same proportion as they are in the capital structure of the firm. Thus, there is no value in catering to the risk aversion of investors within the CAPM framework.
(b) The author then provides an alternative explanation based on ratings and regulation. The article talks about institutional investors in general, but for clarity, we will talk here about banks (you can equally use pension funds or insurance companies as examples). There are two departures from the Modigliani-Miller framework here. First, banks have different preferences (rather than mean-variance preferences): Other things equal, they prefer to minimize their equity financing and maximize their expected returns. Second, by banking regulation, different assets imply different capital requirements for banks (i.e., the minimal amount of equity financing that a bank should raise to hold a specific asset on its balance sheet).
Suppose that each bank is owned by a risk-averse individual (say, Jamie Dimon or David Solomon)
who must contribute her own wealth as the equity capital in the bank. Based on the material covered in the lectures, provide a justification for why bank owners with standard mean-variance preferences might prefer to minimize her equity in the firm. Hint: Think of the diversification preferences of bank owners.
Solution: Owning a large equity stake in the firm exposes the owner to the idiosyncratic risk associated with the firm’s assets. The owner prefers to diversify this risk and ideally sell all the equity and invest in the diversified portfolio of assets.
(c) Suppose you want to start a firm that invests in a risky asset and raises financing from banks by selling them debt and equity claims. Suppose that bank owners are willing to contribute at most X% to equity financing of their banks, that is, if a bank buys $100 worth of claims then the equity financing can constitute at most 100*X%, otherwise, bank owners are not willing to contribute equity capital. Modify the example on pages 3-4 of the article to argue that in this case, the Modigliani-Miller’s Proposition 1 can fail: the value of your levered firm can exceed the value of the unlevered firm. Hint: You will need to choose appropriate value of X that makes the example work as well as specify risk weights of different claims (equity and debt), base capital requirement, value of asset, etc. You can specify them arbitrarily to make the example work, but please try stay as close as possible to the example in the article so that it’s easier to grade your response. Try to construct an example in which the value of the unlevered firm is 0, while the value of the levered firm is positive (you don’t have to follow this hint, but this might be the easiest example) . Solution: An example solution is as follows. Suppose the capital requirement is 8%. Suppose the risky asset is worth $100 and has risk weight of 50%. Suppose that X=3.8. If you only issue unlevered equity, then the minimal equity requirement for banks for investing in your equity is $4, which is above X. Thus, banks will not finance your firm and the value of your firm is 0. Now, if alternatively you issue $80 of senior debt with risk weight 0 and $20 of risky equity with risk weight of 200%, then the minimal equity requirement for banks for investing in both is only $20*16%=$3.2. This is less than X and so you will be able to finance your asset, and the value of your firm will be $100.
(d) In the example constructed in the previous part, assuming that the senior claim has risk weight of zero, find a proper calculation of the risk weight of the equity claim (wE ), issued by your firm as a function of the firm’s leverage (D/E) and the risk weight of the asset (wA ) that restores the Modigliani-Miller’s Proposition 1. Hint: You need to ensure that the capital requirements for investing in both debt and equity of your firm is the same as the capital requirement for investing the unlevered equity of your firm.
Solution: You essentially need to derive MM’s Proposition 2.
D E D + E
E + D E + D E
2023-04-13