Generally speaking, the firm value is equal to the equity value pluses the bond value, no matter what firm has leverage or unleverage under perfect market. When the debt prin-cipal, b/r, is greater than the scrapping value γ, bondholders are the residual claimants, and the debt is risky. Therefore, the value of the firm, Wulwd(ˆpt), which is defined by Vulwd(ˆpt) + Lulwd(ˆpt), will decrease slightly. Leverage generates losses from an ex ante point of view because of the direct bankruptcy costs it entails under perfect market.
From Figure 2, it is easy to see the difference between perfect market and imperfect mar-ket. When b is small, such that b/r is smaller than γ, the debt is riskless, and the payment of the coupon does not affect debtholders’ estimation of the firm value under the unbiased observation price. When b get larger and b/r is slightly larger than γ, leverage becomes costly because it may results in liquidation at pb. The firm may goes to ”liquidation bankruptcy,” which means debtholders will prefer to liquidate the firm than take over at bankruptcy. When b was large enough, we found that there was some trouble with
E[ ˆL(p)|ˆpt, x0, t] > Wulnr(ˆp) as b/r > γ. As a result, debtholders need to modify the debt, firm, and equity value; this is the difference between a levered firm without renegotiation under perfect market and under imperfect market. There is some difference in estimating the value of the debt as pb > px, because the bankruptcy occurs at px. When pb > px, we assert that the firm is liable to ”an operating concern bankruptcy,” since the real price first hits pb; when this happens, bonderholders will take over the firm. Before bonderholders take over the firm, they are unable to obtain all information on the firm, but when they evaluate the value of the debt, they are still concerned with the value between px and pb. As b gets large, such that pb > pt, then the value of the firm will converge to Xulnr, which is still larger than the real firm value, X(p), because of the information asymmetry.
3 Debt and Equity Value with Renegotiation under Imperfect market
3.1 Service Flows with Equityholder Offers
We want to consider how the value of the firm’s security is affected if debtholders and equityholders can renegotiate coupon payments. When equityholders can make take-it-or-leave-it offers to bondholders, since the asymmetric information, they will take advantage of bondholders. We shall assume that possible strategies for equityholders consist of piecewise right-continuous service flow functions of ˆpt, the observation price. First, we notice X(p) satisfy the following PDE:
rX(p) = s(p) + µpX′(p) + σ2
2 p2X′′(p) for p < ps (3.1)
Xulnr(ˆpt) is also assumed to satisfy the same PDE with different µ and σ2, say ˆµ and ˆσ2. Since the parameter µ and σ2 is from the mean and variance of the logarithm of the price under perfect market, we can easily to get the mean and variance of the logarithm of the price under imperfect market and set them be the ˆµ and ˆσ2. Second, Pierre and William (1997) show that s(p) is the optimal debt service flow function under perfect market, so we assume there is another optimal debt service flow function, ˜s(ˆp), under imperfect market.
The intuitive explanation of the service flow function is that debthodlers require a service flow to dissuade them from equityholders need provide debtholders with an income flow whose capitalized value is sufficient to dissuade them from. Therefore, equityholders must to provide enough income flow to capture Xulnr(ˆpt), the value of the firm in the hands of the new owners under imperfect market, when p < ˆps. From above, Xulnr(ˆpt) satisfy the following PDE,
rXulnr(ˆpt) = ˜s(ˆpt) + ˆµˆptXulnr′ (ˆpt) + σˆ2
2 pˆ2tXulnr′′ (ˆpt) for ˆpt< ˆps (3.2) We suppose the following:
Hypothesis 1. If equityholders can make take-it-or-leave-it offers to bondholders regard-ing debt service than there exists trigger levels, ps and pc such that
(1) bankruptcy occurs when pt first hits pc,
(2) for all ˆp < ˆps, ˜s(ˆp) < b and Lwlwd(ˆp) = Xulnr(ˆp) (i.e., when debt service is less than the contracted coupon, the value of debt equals that of debtholder’s observation outside option Xulnr(ˆp)),
(3) for all ˆp ≥ ˆps, ˜s(ˆp) = b(i.e., the contracted coupon is paid).
Now, we can get the service flow function ˜s(ˆp):
From this service flow function, Lwlwre(ˆp)satisfy the following PDE,
rLwlwre(ˆpt) = ˜s(ˆpt) + ˆµˆptL′wlwre(ˆpt) + σˆ2
2 pˆ2tL′′wlwre(ˆpt) (3.4)
The absence of arbitrage implies that Lwlwre(ˆp) = Xulnr(ˆp) for all ˆpt < ˆps. No bubble condition includes limp→∞ˆ Lwlwre(ˆp) = br. Under this situation, equityholders know the real price of the output, so the value of equityholders is the real firm value minus the estimated value of debtholders, i.e. Vwlwre(p, ˆp) = W (p) − Lwlwre(ˆp).
Proposition 3. Under imperfect market, we assume the noisy accounting report of asset is given by ˆpt = pteUt = eXt+Ut = eYt , and Hypothesis 1, equityholders adopt the service
Equity and debt value with equityholder offers is shown in Figure 3. It is clear that renegotiation, even under the imperfect market, still generates the fully efficient outcome.
The main difference between the perfect market and the imperfect market is the point
of the trigger price ps and ˆps. If we set unbiased observation price under imperfect market, debtholders would not agree this contact when equityholders want to renegotiate at the real trigger price, ps. The main reason is debtholders believe that equityholders have some information that debtholders do not know or there is some advantage for equityholders. Therefore, when the observation price hits the real trigger price they still reject the renegotiation. As they reject the renegotiation, even the real price hits the real trigger price, debtholders will not get huge loss since they can be the new owner of the firm. If equityholders still want to own the firm, then they have to renegotiate later and it means debtholders will ask some information premium to make up for information asymmetry.
Now we try to change the variable a, which is the volatility of the noisy accounting, the outcome is shown in Figure 4. It is easy to find that trigger price under the imperfect market becomes smaller when a becomes larger. Basic on the intuition, debtholders will ask more information premium when the market is more imperfect or the information is more asymmetry. If the time, t, between we observe the price of the firm, ˆpt = eYt, and p0 = eX0 is shorter, and the firm does not operate so well that the one might goes to bankruptcy, then it is much difficult to convince debtholders to agree the deal when equityholders want to renegotiate the coupon payments.In order to capture this situation, if we fix the other variable and change the different t, then from penal.1, we can see when t gets smaller then ˆps gets smaller. Therefore, debtholders might ask more information premium because of inferior information. Image that at initial time the firm still operate, what’s the difference between the equityholders want to renegotiate because of financial distress next day and next year? When equityhodlers want to renegotiate next day, debthodlers can keep more full information than one year later, so equityholders need to
release more information to persuade debtholders accept the agreement which will reflect on the value of firm.