Commitment problems, reputation, and randomized strategies The question below shows that not being able to commit to an action can lead to suboptimal outcomes. It also shows that randomized strategies can be more efficient than fixed strategies. Finally, it shows how reputation can play an important role in determining optimal actions.

11) Today is t = 0. All investors are risk neutral and the risk free rate is 0%. You are an entrepreneur and have an investment opportunity that costs 100 at t = 0. The opportunity can have a good outcome (payout = 160) and a bad outcome (payout = 80) at t = 1. If you exert full effort (work), then the probability of each outcome is 50%. If you dont work (shirk), then the outcome will be bad (80) with certainty, but you will derive a private benefit of 18. You are trying to finance the project by issuing a 1-year zero coupon bond with price = 100 and face value = F. In case of default the assets are liquidated and the proceeds are turned over to the debtholders. There are no bankruptcy costs. Note that if you shirk you still derive the $18 private benefit, even if there is a default.

a) Would it be possible to issue debt if debtholders thought you would shirk?

b) Would it be possible to issue debt if debtholders though you would work? If so, what face value would they require? (Hint: You must also confirm that given this face value, it is in fact optimal for you to work.) Now assume that in case of default, you can propose to the bondholders a renegotiation instead of liquidation. Under renegotiation the debt would be worth 85 (instead of 80 under the liquidation scenario), and equity would be worth 10 (instead of 0 under the liquidation scenario). Comment: The above scenario could happen if, for example, a business is worth more as a going concern than in liquidation. Another example could be a homeowner who has defaulted on their mortgage. The bank could foreclose on the property (liquidate), or they could renegotiate the loan. It could be that renegotiating the mortgage has a higher NPV than foreclosure for the bank, and is also beneficial to the homeowner relative to foreclosure.

c) Given these new assumptions, what is the optimal strategy for the bondholders if ever the bad outcome occurs? (The options are liquidate or accept renegotiation.)

d) Suppose you anticipate that the bank will take their optimal action in case of default. What is your optimal action (work or shirk) given the face value you found in part b?

e) Suppose that bondholders anticipate your answer to part d. Would they still be willing to buy for 100 the bond with the face value you found in part b? Is there any face value that would enable you to raise the required 100?

f) Suppose bondholders could commit in advance to always liquidating in case of default. Would you be able to issue the bond with the face value you found in part b? Comment: Would this commitment be credible? Perhaps it could be if the lender has reputational concerns and wants to show other borrowers that this is the type of action they take when there is a default. While committing to liquidate allows the project to be financed with debt, it is not an efficient solution since there is a 50% chance that the firm will be liquidated (you work, but 50% of the time the outcome ends up being bad anyways). From a total welfare point of view, we would say that the most efficient outcome (first best) would be for you to work and for the bondholders to always accept renegotiation in case of default. Unfortunately, this does not work in this example since if that were the lenders policy, it would actually be optimal for you to shirk, not work. So the first best solution cannot be implemented. What is the second best solution? It turns out that always liquidating is not the second-best solution. The second best solution involves a randomized strategy by the lender: in case of default, liquidate with probability x and renegotiate with probability (1-x), for the lowest value of x that ensures that it is optimal for you to work. The face value of the bond also has to satisfy the condition that the value of the bond is in fact equal to 100. The lower the probability of liquidation, the more efficient, since the firm is worth more under renegotiation than under liquidation.

g) Find x and the face value F that implements the second best solution described above. (Hint: you will have two equations and 2 unknowns. First equation: given x and F you will be indifferent between work and shirk. Second equation: given that you work and given x and F, the bond is in fact worth 100.)

Comment: The above model can explain why lenders will sometimes renegotiate and sometimes liquidate. For reputational purposes lenders need to show that they are tough (liquidate sometimes). They need to show that they are just tough enough to make it worthwhile for borrowers to try their hardest to pay back their loans. Indeed, this is supported by some empirical findings and explains how reputational concerns can explain some seemingly counterintuitive liquidations. The above model is based on an influential paper by Bolten and Scharfstein (they derive many other results as well).