AN ANALYTICAL MODEL OF BOND RISK DIFFERENTIALS: A COMMENT. Avery B. Cohan*

Transcription

1 JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS June 1978 AN ANALYTICAL MODEL OF BOND RISK DIFFERENTIALS: A COMMENT Avery B. Chan* In a paper published in the December 1975 issue f this Jurnal, Bierman 2 and Hass (BH) cnstruct a steam rller fr the purpse f cracking a nut. BH's paper is essentially an attempt t use subjective prbabilities t set yields n new bnd issues. I am cncerned primarily with the first tw-thirds f their paper (pp ), which, in my view, cntains a number f statements that are seriusly misleading. The first sectin f this cmment will briefly summarize thse prtins f pp f their paper. The secnd sectin cntains the cmment itself, plus a few bservatins n the final prtin f their paper. I. (1) Assuming a risk-neutral buyer f debt issues and given what they call the "prbability f survival" (P), BH shw hw t btain the "required" yield n a new risky perpetuity (their equatin 4, p. 759). They use a perpetuity in rder t avid, at the utset, the cmplicatins created by the fact that (risky) brrwers must als, in every case, make nt nly interest payments but als payments n principal. They then state as their cnclusins t this initial sectin f their paper that: Deceased, frmerly University f Nrth Carlina, Chapel Hill. The authr was indebted t Richard W. McEnally and t tw annymus referees fr a number f helpful suggestins n an earlier draft f this paper. " Harld Bierman, Jr. and Jerme E. Hass, "An Analytical Mdel f Bnd Risk Differentials," Jurnal f Financial and Quantitative Analysis (December 1975), pp Which in fact had already been cracked. See [2, pp ] and [3, pp ]. BH state that they are building n Fisher's 1957 article. Fisher des talk briefly abut the prbability f default but his article runs essentially in terms f risk premiums, nt prbabilities. BH are actually "building," althugh they d nt realize it, n Macaulay [8] and [2] and [3]. Macaulay called this the prbability f payment f interest and principal in full and n time. See [8, p. 61]. 371

2 (a) If P is 1 ("the interest payments are certain"), the "required" rate n the new issue wuld be the riskless rate (!). (b) As the "prbability f survival" increases, the required rate declines (curve M-M, Figure 1), and vice versa. (2) They then discuss "the relatinship between the (dllar) amunt f interest that has t be paid and the prbability f survival" (p. 759). Given sme prbability distributin ver EBIT, the prbability f survival can be related t the dllar vlume f interest payments which, f curse, is the prduct f r (the required rate) and B. But BH d nt make clear whether B is the amunt f a new issue r the ttal amunt f debt the issuer has utstanding. (3) BH cnclude this sectin f the paper by drawing curve Q-Q (Figure III) which they call "the interest rate-survival prbability feasible set." The difference between M-M and Q-Q is unclear, except that M-M is the investr's curve and Q-Q, the firm's. Bth curves wuld have t have been derived in much the same way. (4) BH then suggest that the curve M-M (Figure 1) which relates P and the lender's required rate f return and seems entirely subjective (a kind f * indifference curve) can be juxtapsed n curve Q-Q t btain r which is defined as "the minimum interest rate required by the; market fr a fixed amunt f debt," B. II. My cmments fllw: (1) What are BH saying, esssntially? They are saying essentially that lenders wh are risk-neutral can fix the "apprpriate" yield n a new issue by dividing the return n a riskless security by the firm's "prbability f survival" (their equatin 4). At bttm, this is what the paraphernalia summarized in their Figure 2 means. But given their definitin f P (prbability f survival), it (P) cannt be determined by reference t a prbability distributin ver EBIT, unless B is the ttal amunt f debt the firm will have utstanding after the new issue has been sld. In any event, their statement that "frm the prbability distributin f earnings... fr al] pssible levels f interest payments" must assume ne state f nature ver the life f the bnd(s?), fr therwise, we wuld need families f distributins fr each cupn perid. The statement 4 P BH, pp

3 als assumes that the same distributin wuld apply t each cupn perid. Bth assumptins are dubius. One referee pints ut that "we nrmally will nt take issue with a theretical treatment (mdel) if... it predicts r explains," hwever unrealistic the assumptins n which it rests may be. Quite s. But, he adds, BH ffer n empirical evidence whatever in supprt f their mdel. What might they have dne? They might, fr example, have used their equatin 4 (p. 759) t "predict" the yields n a sample f new issues. But this wuld have required a P fr each issue, a P they did nt have. T btain a P they might have used a variatin n their equatin 4, thus: P =, (see [2]) but then they culd nt have used P, which wuld have been derived in part frm r, t predict r. (2) As indicated abve, their P is nt cnceptually clear unless their B is the ttal amunt f bnds the firm will have utstanding after the new bnd is issued. But their final paragraph, p. 758, suggests virtually unmistakably, that bth B and P refer t the new bnd, nt t the ttal amunt f bnds the firm has utstanding. (3) The derivatin f Q-Q (Figures 2 and 3) is far frm clear. All ne can say is that, given the rate f interest at which the firm wishes t brrw and the amunt it wishes t brrw, ne can find the ttal amunt f interest (I) (-r B ) the firm wuld be required t pay (Figure 2, IV). Then, given a curve relating I and P (Figures 2 and 3), ne can find P (=P ). One can then relate P t r via Figure 2, quadrants II and I (=r ). r then becmes the first pint n the "feasible set" Q-Q. Nw change r, hld B cnstant, and anther "feasible" pint can be btained, and s frth until the entire curve Q-Q (Figure 3) is develped. Nte that the relatinship between P and I cannt be derived withut an implicit (r explicit) prbability distributin ver EBIT. Nte als that the shapes f the curves in quadrants I and III and in Figure 3 are unexplained and are nt bvius. (4) The statement in paragraph (4) f Sectin I abve assumes that every investr wuld apply the same P t every B r cmbinatin, r in ther wrds that the P's are "bjective." Mrever, t the naked eye, the real difference between M-M (Figure 1) and Q-Q (Figure 3) is far frm clear: bth relate P t r. And their juxtapsitin, t btain an equilibrium pint, is pssible nly because f the way the curves are drawn: M-M is cnvex thrughut while Q-Q fluctuates between being cncave and cnvex. In fact, bth shuld have the same general shape and the same elasticity, because bth relate p and r and therefre reflect 373

4 the same prbability distributins ver EBIT. There is n technical reasn why M-M and Q-Q shuld intersect r be tangent t ne anther. (5) BH then give us a numerical illustratin the utcme f which is simply implicit in the assumptin that the prbability distributin ver EBIT is rectangular. Further, BH make the illustratin heavier ging than need be. If the prbability distributin is given, P can be btained directly frm it. If it is.983, r wuld simply be 1.04/.983 = , r precisely their result. And r 5.80 percent wuld be pints n bth M-M and Q-Q. But we shuld nte again that the critical input is the distributin ver EBIT, withut which n result wuld be frthcming. At wrst, n such distributin can be btainedr at best it wuld be s subjective, ver any reasnable perid (15-20 years) as t be virtually wrthless. (6) In the remainder f the paper, BH allw fr principal repayment and fr revisin f the prbabilities, cnditinal upn past survival. By and large, all the cmments made abve apply but several additinal pints are wrth making. BH say (p. 768) that "equatin 4, in fact, hlds fr a bnd f any... maturity, if the prbability f survival is cnstant ver time." This statement is incrrect. Equatin 4 hlds whether P is a cnstant r nt prvided nly that it is the n rt f whatever the prduct f the n P's may be in ther wrds, that it is a gemetric average f the n P's. This is very different frm saying that P must be a cnstant. The individual P's can take n any time shape whatsever. The prf that P need nt be a cnstant is as fllws: Start with the mst general frm f the relevant present value equatin: r U+r 1 P... P l P l r 2 P l P 2 (1+r n )P l P 2 + ^ ^ 2 d+g 1 ) 2 n where r is the ne-perid yield n a risky bnd sld at par; r is the twperid yield n a risky bnd sld at par, etc. r =f= r, etc. P is the prbability that r will be paid in full and n time. V is the prbability that r will be paid in full and n time, and s frth P \ P, etc. (1+g ) is the ne-perid rate n riskless bnds; (l+g_) is the tw-perid rate n riskless bnds, and s frth 1+g, = = l+g_» etc. As nted abve, we wuld need a separate jint distributin ver every pssible state f nature fr each cupn perid. P. 768, underlining supplied. 374

5 Nw extract gemetric averages f the P's, the r's, and the (1+g)'s. these averages P, r, and (1+g). Then rewrite equatin (1) as fllws: Call (2) 1 = -^- -f (1+g) (1+i) (1+i) Nw slve (2) fr P: (3) P = 1+r P is identical t BG's P. We nw knw that if we divide the return (1 + yield) n a risky bnd int the return n a riskless security f the same maturity, we will btain the average prbability that each payment will be made in full and n time, prvided the lender is risk neutral and prvided als that the riskless bnd and the risky bnd are the same in every ther way marketability, callability, and s 9 frth. But we knw als that if we can, in sme way, btain the prbability that payment f interest and principal will be made in full and n time f a risky bnd, we can estimate an apprpriate return and yield n that risky bnd, thus: (4) This equatin is identical t BH's equatin 4. (7) BH say als that "the risk differential is invariant with the life f the bnd" when investrs are risk neutral. This statement is als incrrect. Q As prf: = 1. (l+g) r (1+r) d+g), 1+r (1+r) 2 _ d+i) 2 (1+r) 2 (l+i) n 1+r 11 d+g?,, (1+r). (l+g)" 9 BH d nt tuch at all n these "minr" difficulties. P. 769, underlining supplied. 375

6 As indicated abve, r is a gemetric average f the varius n perid rates frm 1 t n, given by the term structure(s) f risky rates. f their equatin 4 is in fact this average. And what falls ut If the r's are different, then the "risk differentials" wuld be "cnstant" nly by accident. (8) BH btain a prbability by assumptin, a methd which is smewhat less than fully satisfactry. Mre satisfactry methds are available. One such methd is illustrated in [2] and [4]. In [2], an explratry paper published in 1971, P's fr sme 3000 direct placements were derived by using equatin (3) abve. Each f the 3000 P's was adjusted fr callability and fr demand and supply cnditins in the market fr lwer grade risky debt issues. In (4), these P's s adjusted were regressed, in varius ways, n the variables life insurance cmpanies take int accunt in fixing the yield n a risky issue tw f which, by the way, were (a) a five-year average, adjusted fr trend, f EBIT and (b) the cefficient f variatin f that average. In any event, the result was a regressin equatin which culd be used t predict P. And as we have seen, given P and the rate n a (cmparable) riskless security f the same 12 maturity, an estimate f the yield n sme given risky security can be btained. (9) The BH mdel, as with s many ther mdels being cnstructed these days in finance, requires inputs which are simply nt available, and except in the mst tenuus way, cannt be cnstructed. The BH mdel requires a prbability distributin ver EBIT. BH imply that they mean ne distributin. But what they really need wuld be a jint distributin ver all pssible states f nature separately fr each cupn perid. T put the requirement in these terms is, bviusly, t suggest that it cannt be met. l:l See [2], pp The cnceptual basis f equatin (4) abve, was discussed in [3, pp ]. 13 See [3, p. 14]. 376

7 REFERENCES [1] Chan, Avery B. Yields n Crprate Debt Directly Placed. Natinal Bureau f Ecnmic Research. New Yrk (1967), Chapter 2. [2]. "The Ex-Ante Quality f Direct Placements." In Essays n Interest Rates, J. M. Guttentag, ed. Vl. 2. Natinal Bureau f Ecnmic Research. New Yrk (1971). [3]. The Risk Structure f Interest Rates. General Learning Press. Mrristwn, N.J. (1973). [4]. "Estimating Pint f Departure Yields n New Direct Placements." Frthcming. [5] Fisher, Lawrence. "Determinants f Risk Premiums n Crprate Bnds." The Jurnal f Plitical Ecnmy (June 1959). [6] Hickman, Braddck. Crprate Bnd Quality and Investr Experience. Natinal Bureau f Ecnmic Research (1956). [7] Jhnsn, Ramn E. "Term Structures f Crprate Bnd Yields as a Functin f Risk f Default." Jurnal f Finance (May 1967). [8] Macaulay, Frederick R. Sme Theretical Prblems Suggested by the Mvements f Interest Rates, Bnd Yields and Stck Prices in the United States since Natinal Bureau f Ecnmic Research. New Yrk (1938), p. 60. [9] Malkiel. "Review f Essays n Interest Rates, Vl. 2." In The Jurnal f Finance (June 1972). [10] Silvers, J. B. "An Alternative t the Yield Spread as a Measure f Risk." The Jurnal f Finance (September 1973). 377