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1、Credit Risk Analysis and Security DesignRoman InderstHolger M. MllerNovember 2002We thank Andres Almazan, Jacqes Cremer, Darrell Duffie, Colin Mayer, Enrico Perotti, Patrick Rey, TonySaunders, Raghu Sundaram, Jean Tirole, Ivo Welch, JeffWurgler, and seminar participants at NYU, Yale, Kellog,Rocheste
2、r, UNC Chapel Hill, Wisconsin-Madison, LBS, LSE, Toulouse, Amsterdam,Frankfurt, Humboldt, FUBerlin, WZB Berlin, the Oxford Finance Summer Symposium (2002), and the European Summer Symposiumin Financial Markets (ESSFM) in Gerzensee (2002) for helpful comments and suggestions. Inderst acknowledgesfina
3、ncial support from the Financial Markets Group (FMG).London School of Economics our assumptions and his assumptions do not conflict.) Theoutcome is less clear ifin addition to the ex-ante incentive problem considered herethereis ex-post moral hazard both on the part of the borrower and on the part o
4、f the lender. Suchdouble-sided moral hazard is frequently assumed in models of venture capital contracting. There,the conclusion is usually that some equity-linked security such as convertible debt is optimal.Exploring the tension between double-sided ex-post moral hazard (which appears to favor equ
5、ity-linked securities) and ex-ante lender moral hazard (which appears to favor debt) in the samemodel is an interesting avenue for future research.6AppendixProof of Proposition 2. By Assumptions 1-3 the lenders optimal credit policy involves aunique cutoffsignaleven if a menu is offered. (See Lemma
6、2.) By the argument in the maintext we then have that sSB sFBif and only if V 0. The optimal menu minimizes theeffi ciency loss from excessive rejection. In a slight deviation from what we specified in Section3.1 we now assume that in case of indifference the lender accepts. This allows us to determ
7、inea contract tSBthat is implemented at the lowest acceptable signal s = sSB.26As s = sSBis azero-probability event this change in assumptions is without consequences.Suppose a menu T = tiiIis offered from which the lender selects contracts other than tSBwith positive probability. We show that such
8、a menu is not optimal. Consider the degeneratemenu T0:=tSB, which is constructed from T by deleting all contracts but tSB. Note that T0implements the same cutoffsignal as T. As the deletion restricts the lenders choice for highersignals s sSBthe borrower cannot be worse off. We now distinguish betwe
9、en the two cases:one where tSBis a debt contract and one where tSBis not a debt contract. Suppose first that tSBis not a debt contract. We can then construct a new degenerate menu T00:=tSBwhere thedebt contracttSBis constructed from tSBas in the Proof of Proposition 1. By the same logic asin that pr
10、oof the cutoffsignal implemented by T00is strictly lower than sSB, which immediately26In case of randomization tSBlies in the support of the lenders distribution if s = sSB.29implies that T cannot be optimal. Suppose next that tSBis a debt contract. We show thatby switching from T to T0the borrowers
11、 participation constraint becomes slack. This followsimmediately if the lender strictly prefers some contract ti T to tSBafter observing s sSB.In what follows we show that this is indeed true if contracts other than tSBare implementedwith positive probability. Precisely, we prove that if the lender
12、is indifferent between tSBandsome contractbt 6= tSBat some signal b s b s. This is implied by MLRP of G(x | s). The proof uses an argument similar to that in theProof of Proposition 1.By the definition ofbt we haveRxxz(x)g(x | b s)dx = 0, where z(x) := tSB(s)bt(x). Moreover,by Assumption 3 and since
13、 tSBis a debt contract there exists some x (x,x) such that z(x) 0for all x x, where the inequalities are strict over sets of positivemeasure. Using MLRP of G(x | s) we can then rewriteRxxz(x)g(x | s)dx for s b s asZ xxz(x)g(x | b s)g(x | s)g(x | b s)dx +Zx xz(x)g(x | b s)g(x | s)g(x | b s)dx0 we arg
14、ue again to a contradiction.We assume that some contract t that is not debt is optimal. We next construct a debt contractt that would leave both the borrowers and lenders expected payoffs unchanged if the cutoffsignal sSB(t) was used. In a slight abuse of the notation used in the Proof of Propositio
15、n 1 wedefine z() :=RxXt(x) t(x)g(x)dx. By the definition oft we haveXh()1 F (sSB(t)z() = 0.(8)We now need to introduce some additional notation. If all that was known is that the signal isgreater than some value s the probability the conditional distribution would put on type ish( | s) :=h()1 F( s)P
16、 0h(0)1 F0( s).With this definition at hand (8) transforms toXz()h( | sSB(t) = 0.(9)30Claim 1. There exists a type with 1 2 it holds that 1, z() 0, and z() 0 for all 0 with = 1.ii) For all it holds that z() 0. Moreover, by Assumption 3 and the construction oft there exists a value x in the interior of X such thatt(x) t(x) for x x, where theinequalities hol
