The purpose of this section is that of showing the effect of our renegotiation framework on the credit spreads of the two classes of debt. It is intuitive that the opportunity of rescheduling debt, by allocating the continuation surplus of an economically viable firm, allows creditors to improve their payoffs as a group. The appropriation of a share of this surplus gives each creditor the incentive to successfully renegotiate his/her claim vis a vis the equity holders. The opportunity to renegotiate should therefore reduce the credit spreads simply because it increases the claim values with respect to a pure liquidation scenario.

In order to isolate the effect of the renegotiation on credit spreads we first define the spread and then we decompose it into two kinds of premia: a default and a renegotiation premium.

Because our coupon bonds are perpetuity the spread, say CS, can be measured as

CSi = bi Bi −r with i=s, j Bs =S(pt) Bj =J(pt), (61)

where the term bi

Bi represents the yield of the risky bond. By adding and subtracting the liquidation payoff, Li, the credit spread can be rewritten as

rbi/r−Li

Bi

+rLi−Bi

Bi

. (62)

This formulation helps at identifying two kinds of premia. The first term, referred to as LPi, can be interpreted as a pure liquidation premium in that it measures the loss

in liquidation (in terms of difference between the promised contractual coupon payments and the payoff of the claim in liquidation). The second term, shortly denoted asRPi, can

be instead understood as a renegotiation premium, that is, the loss or the gain following from seizing the firm’s assets through liquidation instead of continuing and renegotiating debt according to our optimal restructuring plan.

Similarly as in Mella-Barral, Perraudin, renegotiation premia may be measured in terms of percentage contribution to the spread. Denoting the percentage renegotiation premium as rpi, this can be easily written as47

rpi = RPi CSi = Li −Bi bi/r−Bi , (63)

which applied to the senior and junior claims yields

rps = min{bs/r, VL(pt)} −S(pt) bs/r−S(pt) , rpj = min{bj/r, VL(pt)−Ls} −J(pt) bj/r−J(pt) . (64)

In order to have a first glance at the shape of the risk premia, in Figure 8, we plot the credit spreads, CSi, and the associated percentage renegotiation premia, rpi, for the se-

nior and junior bonds (black and red line respectively). The left and right diagrams in
each row of Figure 8 show the spread and the resulting renegotiation premia respectively.
Moving from the top row to the bottom one, we show the spreads and the renegotiation
premia for different allocations of bargaining power between senior and junior creditor
given the bargaining power of the equity holders and all other parameters48_{. Particularly}

from the top to the bottom of Figure 8, the senior’s bargaining power,ξs, increases from

0 to 0.6 (and, given the equity bargaining power, ξe = 0.3, the junior bargaining power

diminishes correspondingly).

47_{Once determined the percentage contribution of the renegotiation premium, the effect of the liqui-}

dation premium can be written asLP/CS= 1−RP/CS.

48_{The other parameters, which remain unchanged, are:} _{ξ}

e= 0.3,α= 0.2,µ= 0.03,σ= 0.15,r= 0.08,

Looking at the spreads in Figure 8, quite interesting is the fact that the senior credit
spread is not necessarily smaller than the junior one at all level of the state variable.
This could not be the case in a pure liquidation scenario, but it is not surprising within
a renegotiation framework. The senior creditor has a privileged position when the firm is
liquidated and the priority of his/her claim is the crucial and sole factor at determining
the claim value. When instead renegotiation is allowed, holding a senior position plays
still a fundamental role49 but the overall package of concession extracted during renego-
tiation, and hence the claim value, also depends on the bargaining strength. Therefore,
holding a senior but largely under-secured claim50 _{does not imply a smaller credit spread}

compared to a junior claimant. What contributes at reducing the senior credit spread is
therefore the combination of a high liquidation payoff51 _{and strong bargaining power vis}

a vis other players. The credit spreads in Figure 8 captures this second factor: the senior spread decreases (and the junior increases) whenξs increases.

Looking at the percentage renegotiation premia (second column of Figure 8), these ex- clusively determine the overall spread for high levels of the state variable. Precisely, by definition ofrpi, when the state variable is high enough (such thatVL≥bs/r) thenrps = 1

and when pt is even higher (such thatVL−bs/r≥bj/r) then also rpj = 1. Therefore for

high levels of the state variable the spreads are purely due to the renegotiation premia. When pt decreases, by renegotiating, creditors can extract concession depending on

their bargaining power. Therefore the renegotiated claim value becomes greater or at
least equal52 _{to the liquidation value. This is the positive contribution of a renegotiation}

scenario in the sense it contributes at reducing the spreads. Notice in fact that both premia, rps and rpj, become negative when pt and hence the firm’s liquidation value

decrease.

Moreover, looking at 64 it is clear why the renegotiation premia in Figure 8 converge to zero for small level of the state variable. In fact, when pt approaches p, the firm is

liquidated and therefore the option to renegotiate disappears. Trivial to say that at pthe credit spread becomes a pure liquidation premium.

Comparison between senior and junior spread.

49_{In fact, it determines the disagreement payoffs in the Nash bargaining.}

50_{This is the case in Figure 8 where the senior face value is 400, the scrapping value of the firm is only}

200 andα= 0.2.

51_{In fact, the senior spread shift down when}_{α}_{and}_{γ} _{increase.}

So far, little we have said about the difference between the senior and the junior spread. We have briefly argued that the difference is determined by the bargaining power of play- ers in renegotiation and by the fact that the senior creditor is partially secured in the event of liquidation. The comparison between the two spreads is in a way biased by the fact itself that one creditor is partially secured vis a vis the other creditor.

A more meaningful comparison between the two spreads can be achieved by stripping the senior claim for the firm’s scrapping value. By proceeding this way, the junior and the senior claim (stripped forγ) are made more comparable in that they will both receive zero when pt approaches p and the firm goes into liquidation. Stripping the senior claim

for γ also helps at isolating the effect of the secured part of the claim on the spread. Moreover by comparing the junior spread and the spread of the senior ‘unsecured’ claim53 we will highlight an interesting property of our model. This provides a sufficient condition for the senior spread to be smaller than the junior for all levels of the state variable.

We start decomposing the senior spread to isolate the effect of the secured part of the claim. Notice first that by 61) the spread of the senior claim can be rewritten as

CSs = r bs/r−S S = r(bs/r−γ)−(S−γ) S · S−γ S−γ = r ˆ Fs−Sˆ ˆ S · ˆ S S (65)

where ˆS =S−γ and ˆFs=Fs−γ. The two factors in equation 65) can be interpreted as

follows. The first term, denoted as

c CSs=r ˆ Fs−Sˆ ˆ S , (66)

represents the credit spread on the senior claim stripped for the scrapping value γ, i.e. the spread of the unsecured part of the claim.

The second term, ˆS/S, captures the effect of the secured part of the claim on the overall credit spread CSs. In fact, the bigger is the secured part of the claim the smaller

is the ratio ˆS/S and the credit spread CSs.

We can now compare the spreads of the two ‘unsecured’ claims by calculating the

difference CSc_{s}−CS_{j}, which by some simple algebra yields
c
CSs−CSj =r
ˆ
FsJ−SFˆ j
ˆ
SJ . (67)

We can now state the main result of this section in the following Proposition.

Proposition 6.0.3 The difference CScs−CSj is positive if and only if the optimal stop-
ping strategy is {p∗_{j}_{2},p¯}. While, CSc_{s}−CS_{j} is negative if and only if the optimal strategy
is {p∗_{s}_{2},p¯}.54_{.}

Proof See Appendix 8. _{}

The intuition behind this result is clear. As argued in Section 5), point A, the trig- ger strategy i) orders creditors on the ground of their actual bargaining strength, and according to this ii) identifies the claims default regions. Therefore, the bigger the actual bargaining strength of a creditor, the smaller the claim default region and the bigger the claim value, which, in turn, reduces the credit spread.

The relevance of this result is immediate. The above proposition provides a sufficient condition for detecting the case where CSs < CSj for any level of the state variable.

In fact, the inequality CSs < CSj rearranges into CSc_{s}S/S < CSˆ _{j}, which holds if55
c

CSs< CSj.

These results are shown in Figure 9, where, again, we show the credit spreads of Figure
8 and we add the function CSc_{s} (dashed line) which, depending on the parameterisation

and, in turn, the stopping strategy, is above or below the credit spread of the junior
creditor (red line). The differenceCSc_{s}S/Sˆ −CS_{j} appears in the second column of Figure

9. From our analytical result, the difference is positive in the first two plots, where the senior creditor is impaired first, whilst the sign reverts in the last two plots where the junior creditor is the first to be impaired.

54_{The equality holds, and}

d

CSs−CSj = 0 whenp∗j2=p∗s2= ¯p.