Liquidity and Credit Risk

THE GRADUATE INSTITUTE OF INTERNATIONAL STUDIES 40, Bd. du Pont d’Arve PO Box, 1211 Geneva 4 Switzerland Tel (++4122) 312 09 61 Fax (++4122) 312 10 26 http: //www.fame.ch E-mail: admin@fame.ch

Liquidity and Credit Risk

FAME - International Center for Financial Asset Management and Engineering McGill University

Reseach Paper N° 42 August 2001

Jan ERICSSON

London School of Economics Olivier RENAULT

INTERNATIONAL CENTER FOR FINANCIAL ASSET MANAGEMENT AND ENGINEERING 40 bd du Pont d'Arve • P.O. Box 3 • CH-1211 Geneva 4 • tel +41 22 / 312 0961 • fax +41 22 / 312 1026

http://www.fame.ch • e-mail: admin@fame.ch

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ESEARCH

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APER

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ERIES

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LIQUIDITY AND CREDIT RISK

Jan ERICSSON

Olivier RENAULT

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Abstract

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Liquidity and Credit Risk

Jan Ericsson and Olivier Renault

Executive summary

This paper focuses on the interaction between credit risk and liquidity spreads in the

setup of a firm- value based model of credit risk.

Motivation

Credit and liquidity risks have been put forward as the two main justifications for the

existence of yield spreads above benchmark treasury notes or bonds. While a rapidly

growing body of literature has focused on credit risk since Merton (1974), liquidity

has remained a relatively unexplored topic, in particular for defaultable securities. The

purpose of this paper is to propose a simple structural model of liquidity and credit

risk in an attempt to better understand the interaction between these two sources of

risk.

Structural credit risk models along the lines of Merton (1974) are often criticized on

the following two counts. First it is argued that the levels of yield spreads generated

by the models are too low to be consistent with observed spreads. However it should

be noted that empirical tests typically fail to take into account that these models only

price credit risk. If prices on corporate bonds reflect compensation for other sources

of risk such as illiquidity then one should expect to find that these models overprice

bonds.

A second criticism often brought up is that the levels of credit spreads obtained with

most structural models are negligible for very short maturities which is inconsistent

with empirical evidence. Our model implies non trivial liquidity premia for short

maturities and can thus resist this line of criticism.

In recent empirical work on credit spreads, Anderson and Sundaresan (2000) have

shown that although sophisticated structural models are often able to capture large

parts of the variation in spread indices, during some subsamples they fail to do so.

This suggests that another factor exogenous to their model is at play. We posit that a

natural candidate for an additional factor should be one that captures the liquidity of

corporate debt markets.

The Model

The paper models credit risk in a modified Merton (1974) framework and introduces

liquidity risk. When the firm is solvent, the bearer of a bond is subjected to random

liquidity shocks. Such shocks can for example reflect cash constraints or rebalancing

of the investor's portfolio because the specific bond is no longer appropriate for

hedging or diversification purposes. With a certain probability he may have to sell his

bond immediately. The price he would have to sell at is assumed to be a random

fraction of the price in a perfectly liquid market. The distribution of this fraction is

modelled as a function of the number of traders active in the market for a particular

bond.

The supply side of the market is an endogenous function of the state of the firm and

the probability of liquidity shocks. When there is no liquidity shock, the bondholder

still has the option to sell if the price he can obtain is good enough. Although if a

bondholder could hold the bond until maturity he would avoid accepting a discount

altogether, he will sell if the price is better than the expected value of waiting. By

waiting he exposes himself to the risk of being forced to sell at a less favourable price.

Numerical results and empirical tests

Our model implies that liquidity spreads are decreasing and convex functions of time

to maturity. This is consistent with empirical evidence on markets for Treasury

securities both in the US and Germany. Our model thus allows to reconcile structural

models of default with observed non- zero spreads for very short maturities.

In addition to the shape of the term structure our model implies that the level of

liquidity premia is correlated to the probability of financial distress. The premia will

be higher when default is likely as bondholders will have less time to look for

attractive offers and may have to liquidate their positions on unfavourable terms.

Furthermore, we show that the optimal trading behaviour of the bondholder is a

function of time to maturity, firm risk and leverage. The bondholder's behaviour is

summarized by the discount that he is willing to sell his security at. When time to

maturity is short there is a relatively small chance that he will be forced to sell his

bond at an unfavourable price. He will thus be unwilling to accept a large discount.

However when the bond maturity is distant he will be willing to sell at a relatively

lower price. Hence our model implies that the market for recent vintages should be

more active than seasoned ones. Furthermore our model predicts that trading should

be more active the riskier the issue. When the firm is near default the probability of a

sudden shortening of a bond's effective maturity may be substantial and the

willingness of bondholders to sell will increase.

Two of the predictions of our model are that young issues should be more liquid (i.e.

carry a lower liquidity spread) than seasoned ones and that liquidity spreads should be

correlated with the level of credit risk. Taking credit ratings as measures of default

risk and choosing two proxies of liquidity (time elapsed since issuance and an

on-the-run / off- the-on-the-run spread), we test these hypotheses on a sample of over 500 US

corporate bonds. We find that, on average, a recently issued corporate bond can be

expected to trade at around 14 basis points discount compared to an identical (same

remaining maturity, rating etc.) but older bond. There is evidence in our sample of

larger liquidity spreads for shorter bonds as predicted by our model. We also find

support for the correlation between credit risk and liquidity spreads which is also a

prediction of the model. The latter result is compatible with the intuition that being

able to liquidate a position quicky is more valuable when default is near than when

the firm is in good financial health.

1

Introduction

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Wzr ri rxu prgho*v hpslulfdo lpsolfdwlrqv 0 wkh srvlwlyh fruuhodwlrq ehwzhhq ghidxow dqg oltxlglw| ulvn dqg wkh qhjdwlyh vorsh ri wkh whup vwuxfwxuh ri oltxlg0 lw| vsuhdgv 0 duh whvwhg rq d sdqho ri dssur{lpdwho| 833 XV frusrudwh erqgv1 Frqwuroolqj iru fuhglw ulvn e| d vhw ri frpprqo| xvhg yduldeohv/ zh h{dplqh wkh lpsdfw ri wzr sur{lhv iru oltxlglw| ulvn= d phdvxuh ri oltxlglw| ulvn lq Wuhd0

vxu| pdunhwv dqg d phdvxuh ri erqg djh1 D frpsdulvrq ri sdudphwhu hvwlpdwhv dfurvv vxevdpsohv frqvwuxfwhg dorqj fuhglw udwlqjv dqg erqg pdwxulwlhv uhyhdov vxssruw iru rxu prgho*v suhglfwlrqv1

Wkh vwuxfwxuh ri wkh sdshu lv wkh iroorzlqj1 Vhfwlrq 5 suhvhqwv wkh prgho dqg ghvfulehv wkh ghidxow jhqhudwlqj phfkdqlvp dqg wkh vrxufhv ri looltxlglw|1 Vhfwlrq 6 dqdo|}hv oltxlglw| vsuhdgv lq wkh ghidxow0iuhh fdvh dqg uhsruwv wkh jhqhudo vkdsh ri wkh whup vwuxfwxuh ri oltxlglw| vsuhdgv lq wklv frqwh{w1 Vhfwlrqv 7 dqg 8 ghvfuleh wkh ghidxow ulvn| fdvh dqg lwv lpsolfdwlrqv iru erqg wudglqj uhvshfwlyho|1 Lq vhfwlrq 9 zh h{whqg rxu dqdo|vlv wr d pruh uhdolvwlf prgho ri ghdohu ehkdylru1 Vhfwlrq : uhsruwv rq rxu hpslulfdo whvwv rq wkh prgho*v suhglfwlrqv dqg vhfwlrq ; frqfoxghv1

2

The Model

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wkh txdqwlwdwlyh rxwsxwv ri rxu prgho wkh| duh xqolnho| wr fkdqjh wkh txdolwdwlyh uhvxowv1

Wkh xqfhuwdlqw| uhodwlqj wr wkhup ydoxhiywjw_w@_@3W lv prghohg xvlqj d vwdq0 gdug pxowlsolfdwlyh elqrpldo irupxodwlrq: _{lq zklfk ghqrwhv wkh yrodwlolw| ri}

9_{Ihz sdshuv kdyh dgguhvvhg wkh sureohp ri ghidxow ulvn lq lqfrpsohwh pdunhwv1 Orw}}

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:_{Wkxv wkh suredelolw| ri dq xs pryh lq d jlyhq wlph lqwhuydo{|lv jlyhq e|} R' eo{|3_

3_ c

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Dw pdwxulw| wkh kroghu uhfhlyhv wkh sulqflsdo uhsd|phqw zkhq wkh up lv vroyhqw ru d iudfwlrq ri wkh ydoxh ri wkhup zkhq lw lv lq ghidxow1

EO+W, @S LyTS. pd{ +yW N>3,LyT?S=>

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rwkhuzlvh dqg N uhsuhvhqwv wkh frvwv ri qdqfldo glvwuhvv1< _{Wkhvh duh wdnhq}

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Zh dvvxph wkdw dw pdwxulw| wkh up lv oltxlgdwhg dqg surfhhgv duh glv0 wulexwhg wr wkh uhvshfwlyh fodlpdqwv1 Wkhuh lv khqfh qr pdunhw oltxlglw| sureohp

zkhuh

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_'ej I

{|

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edqnuxswf| frvwv duh lqyhvwru vshflf1

dw wklv gdwh dqg wkh looltxlg erqg sulfh lv

EL+W, @EO+W,=

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O@j h{s+uW,S=

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wkhup lv vroyhqw1 Li qrw/ wkh erqgv duh zruwk=

EO+w, @ pd{ +ON>3,> EL+w, @ pd{ +ON>3,>

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f

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Hhw i @ 4 X q@3 hq q$ q q.4=

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jlyhq pdunhw dw wkh vdph wlph4:₁

FIGURE 1

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i

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Vr idu/ zh kdyh wdnhq wkh suhydlolqj pdunhw frqglwlrqv wr eh {hg1 Wkh dyhudjh qxpehu ri wudghuv zloolqj wr wudgh d jlyhq erqg kdv ehhq dvvxphg frqvwdqw1 Wklv grhv qrw doorz iru dgyhuvh vkrfnv wr wkh hfrqrp| dv d zkroh/

53_{Ghwdlov ri wkh fdofxodwlrqv ri} h_B|:BW | i dqg.|3 h B|U_Bt:B t i fdq eh irxqg lq wkh dsshqgl{1 46

dffrpsdqlhg e| zruvhqlqj fuhglw0pdunhw frqglwlrqv1 Wklv dvvxpswlrq frxog eh uhod{hg dv iroorzv1 Zh frxog dvvxph wkdw qrupdo pdunhw frqglwlrqv duh uhsuhvhqwhg e| dq dyhudjh qxpehu ri wudghuvQ _{dqg wkdw zlwk suredelolw|} F

wkhuh lv d ghwhulrudwlrq lq wkh zloolqjqhvv ri wudghuv wr ghdo zlwk frusrudwh erqgv wr wkh h{whqw wkdw wkh Srlvvrq sdudphwhu iru wkh h{shfwhg qxpehu ri rhuv gursv wr F _? Q₁ 54 _{Zh gr qrw suhvhqw wklv h{whqvlrq dv lw grhv qrw}

dowhu wkh txdolwdwlyh uhvxowv ri wkh prgho1

Qrz wkdw zh kdyh glvfxvvhg rxu dvvxpswlrqv dqg wkh uhvxowlqj prgholqj iudphzrun/ zh surfhhg wr suhvhqw rxu qxphulfdo uhvxowv1 Zh ehjlq zlwk wkh fdvh ri fuhglw ulvn0iuhh ghew dqg wkhq lqwurgxfh wkh fdvh ri frusrudwh ghidxowdeoh ghew1

3

Results in the Default Risk-free Case

Zh vwxg| wkh fdvh ri d ghidxow ulvn0iuhh erqguvw1 Wklv vhuyhv d gxdo sxusrvh1 Iluvw/ zh ghqh d ehqfkpdun fdvh qhfhvvdu| wr dqdo|}h wkh lqwhudfwlrq ri fuhglw dqg oltxlglw| ulvn1 Zlwkrxw nqrzlqj zkdw wkh whup vwuxfwxuh ri oltxlglw| vsuhdgv orrnv olnh lq wkh fuhglw ulvn0iuhh fdvh/ lw zrxog eh kdug wr xqghuvwdqg wkh ixoo lpsdfw ri ghidxow ulvn1 Vhfrqg/ zh doorz wkh lpsolfdwlrqv ri wkh prgho wr eh uhodwhg wr hpslulfdo uhvxowv zklfk duh dydlodeoh iru wkh Wuhdvxu| pdunhwv1

Iru doo vlpxodwlrqv lq wkh iroorzlqj vhfwlrqv/ zh ghqh d vhw ri sdudphwhuv zklfk zloo eh xvhg xqohvv vwdwhg rwkhuzlvh1 Wklv lv wr idflolwdwh frpsdulvrqv ehwzhhq wkh ydulrxv judskv dqg wdeohv1 Wklv edvh fdvh kdv wkh iroorzlqj sdudp0

54_{Wklv h{whqvlrq zrxog qrw dhfw wkh vwuxfwxuh ri rxu erqg sulflqj irupxod exw zrxog}

fkdqjh wkh txdqwlwdwlyh rxwsxw1 Wkh suredelolw| ri rewdlqlqj d jlyhq qxpehu ri rhuv zrxog eh E'? 'ó3w´_e3N¡¢? ?- nwe3 C¡¢? ? -!

dqg wklv zrxog gluhfwo| dhfw wkh h{shfwhg iudfwlrq wkdw d erqgkroghu frxog h{shfw wr rewdlq erwk lq wkh hyhqw ri d irufhg vdoh dqg d suhhpswlyh wudgh1

hwhuv = wkh wlph vwhs lv w @ 4@45 +rqh prqwk,/ wkh ulvn0iuhh lqwhuhvw udwh lv

u@ :(/ wkh dyhudjh qxpehu ri pdunhw0pdnhuv dfwlyh iru wkh vhfxulw| lv@ :>

wkh suredelolw| ri d oltxlglw| vkrfn lv@ 3=75:(zklfk fruuhvsrqgv wr d |hduo| suredelolw| ri8(ri kdylqj dw ohdvw rqh vkrfn1 Lq wkh qh{w vhfwlrq +ghidxow ulvn| fdvh,/ zh zloo vxssohphqw wkh edvh fdvh zlwk ydoxhv ri edqnuxswf| frvwv dqg ri wkh ghidxow sdudphwhuv1

FIGURE 2

Iljxuh 5 sorwv wzr vshflfdwlrqv ri wkh oltxlglw| vsuhdg1 Wkh gdvkhg olqh/ zklfk lv doprvw vwudljkw/ lv wkh oltxlglw| vsuhdg zkhq wkh erqgkroghu fdqqrw vhoo klv srvlwlrq ehiruh pdwxulw| xqohvv kh lv irufhg wr gr vr e| d oltxlglw| vkrfn1 Wkh vrolg olqh lv wkh oltxlglw| vsuhdg zkhq zh oliw wkh frqvwudlqw ri qr hduo| vdoh1 Lw lv wkhq d ghfuhdvlqj dqg frqyh{ ixqfwlrq ri wlph wr pdwxulw|1 Wklv looxvwudwhv wkdw lw lv wkh delolw| wr vhoo yroxqwdulo| sulru wr pdwxulw| zklfk |lhogv wkh vkdsh ri wkh oltxlglw| fxuyh1 Wklv sduwlfxodu vkdsh lv frqvlvwhqw zlwk hpslulfdo hylghqfh ri Dplkxg ) Phqghovrq +4<<4, dqg Nhpsi ) Xkulj +4<<:, iru wkh XV dqg Jhupdq jryhuqphqw ghew pdunhwv uhvshfwlyho|1

Dplkxg ) Phqghovrq +4<<4, vwxg| wkh pdunhwv iru XV Wuhdvxu| qrwhv dqg eloov ri htxdo pdwxulw|/ lqvwuxphqwv zlwk lghqwlfdo lqwhuhvw udwh ulvn h{srvxuh dqg uhpdlqlqj sd|rv1 Wkh glhuhqfh ehwzhhq wkhvh lqvwuxphqwv olhv lq wkh oltxlglw| ri wkhlu vhfrqgdu| pdunhw1 Wkh|qg wkdw wkh |lhog vsuhdgv rq wkhvh vhfxulwlhv glhu rq dyhudjh e| derxw 75 edvlv srlqwv dqg wkdw wkh glhuhqwldo lv d frqyh{ dqg ghfuhdvlqj ixqfwlrq ri wlph wr pdwxulw|1

Nhpsi ) Xkulj +4<<:, whvw iru wkh h{lvwhqfh ri d oltxlglw| vsuhdg lq orqjhu jryhuqphqw erqgv lq wkh Jhupdq pdunhw155 _{Wkh dxwkruv qg d vwdwlvwlfdoo|}

55_{Wkh| frpsduh wkh |lhogv ri d vdpsoh ri odujh lvvxhv +lvvxh vl}h lv xvhg dv d sur{| iru}

oltxlglw|, djdlqvw wkrvh ri vpdoohu lvvxhv ryhu wkh shulrg 4<<50<71 Doo 476 erqgv lq wkhlu vwxg| zhuh lvvxhg ehwzhhq 4<;5 dqg 4<<7 zlwk lqlwldoo| 43|hduv wr pdwxulw| dqg kdg d uhvlgxdo pdwxulw| ehwzhhq 318 dqg 43 |hduv1

vljqlfdqw dyhudjh vsuhdg ri 4: edvlv srlqwv1 Wklv ohyho dqg wkdw irxqg e| Dpl0 kxg ) Phqghovrq +4<<4, fdq uhdglo| eh uhsolfdwhg e| rxu prgho iru uhdvrqdeoh sdudphwhu ydoxhv1

Qrwh wkdw d ghfuhdvlqj ixqfwlrq iru|lhog vsuhdgvgrhv qrw phdq wkdw oltxlglw| kdv d vpdoohu lpsdfw rqsulfhviru orqj erqgv1 Rq wkh frqwudu|/ dv zloo eh vkrzq lq wkh qh{w vhfwlrq/ shufhqwdjh sulfh glhuhqfhv lqfuhdvh lq pdwxulw|1 Udwkhu lw lv wkh suhvhqfh ri dq xsshu erxqg rq groodu orvvhv gxh wr looltxlglw| wkdw jhqhudwhv ghfuhdvlqj vsuhdgv1

Zh qrz wxuq wr wkh fdvh ri ghidxowdeoh ghew1

4

Results in the Default-risky Case

Lq wklv vhfwlrq zh dgg fuhglw ulvn lqwr rxu iudphzrun lq rughu wr vwxg| wkh lqwhudfwlrq ehwzhhq oltxlglw| ulvn dqg wkh srvvlelolw| riqdqfldo glvwuhvv1

Zhuvw suhvhqw wkh vkdsh ri oltxlglw| vsuhdg fxuyhv lq wkh wzr fdvhv zkhq yroxqwdu| vdohv ehiruh pdwxulw| duh suhfoxghg ru doorzhg1 Zh wkhq dqdo|}h wkh uhvshfwlyh vkduh ri |lhog vsuhdgv h{sodlqhg e| oltxlglw| dqg fuhglw ulvn dqg vkrz wkdw oltxlglw| suhpld fdq h{sodlq d vljqlfdqw sursruwlrq ri vsuhdgv lq rxu prgho1 Erxqgv duh wkhq rewdlqhg iru sulfh glvfrxqwv +wkh glhuhqfh ehwzhhq wkh sulfh ri d oltxlg dqg dq rwkhuzlvh lghqwlfdo looltxlg erqg, dqg vrph frpsdudwlyh vwdwlfv ri wkh oltxlglw| vsuhdgv duh surylghg lq vxevhfwlrq 7171

4.1

The Term Structure of Liquidity Spreads

FIGURE 3

Ohw xvuvw uhylhz vrph qrwdwlrq1 Uhfdoo wkdwghqrwhv wkh suredelolw| ri d oltxlglw| vkrfn lq d jlyhq wlph lqwhuydow1 Wkh sdudphwhufdq eh wkrxjkw ri dv wkh qxpehu ri wudghuv dfwlyh lq wkh pdunhw iru d sduwlfxodu frusrudwh erqg1 Ohyhudjh lv phdvxuhg e| wkh txdvl0ghew udwlrtzklfk uhodwhv wkh ulvn0iuhh suhvhqw

ydoxh ri ghew frpplwphqwv wr wkh fxuuhqw dvvhw ydoxh/ zklohlv d uhgxfhg irup phdvxuh iru wkh frvw ri looltxlglw| lq wkh pdunhw iru glvwuhvvhg ghew1

Dv ehiruh/ wkh edvh fdvh iru vlpxodwlrqv zloo eh w @4@45> u @ :(/ @ :> @ 3=75:(1 Zh qrz dgg sdudphwhuv vshflf wr ghidxow ulvn= wkh txdvl0ghew udwlrt@ 3=;/ wkh yrodwlolw| ri dvvhwv lv@ 3=49>edqnuxswf| frvwv duh N@ 63

dqg wkh vxssohphqwdu| frvwv iru looltxlg vhfxulwlhv duh@ 53=Ilqdoo|/ wkh ghidxow erxqgdu| lvO@j h{s+uW,S>zkhuhj@41

Zh qrwh wkdw vsuhdgv duh vwloo ghfuhdvlqj lq ghew pdwxulw| vr wkdw wkh txdolwd0 wlyh vkdsh ri wkh oltxlglw| whup vwuxfwxuhv grhv qrw fkdqjh zlwk wkh lqwurgxfwlrq ri fuhglw ulvn lqwr wkh dqdo|vlv +Iljxuh 6,156 _{Wklv lv lq olqh zlwk wkh uhvxowv ri}

Orqjvwd +4<<7, zkr vwxglhg wkh Mdsdqhvh pdunhw dqg irxqg vlplodu sdwwhuqv iru fuhglw ulvn| erqgv lvvxhg e| wkh Mdsdq Ilqdqfh Frusrudwlrq ri Pxqlflsdo Hqwhusulvh/ wkrvh ri wkh Wrn|r Phwursrolwdq Jryhuqphqw dqg ghehqwxuhv ri wkh Lqgxvwuldo Edqn ri Mdsdq157

Wkh ghfuhdvlqj whup vwuxfwxuh ri oltxlglw| vsuhdgv fdq khos wr h{sodlq d vw|ol}hg idfw iru kljk judgh frusrudwh erqg vsuhdgv1 Vwuxfwxudo prghov fdqqrw uhsurgxfh wkh dw vkdsh ri whup vwuxfwxuhv ri vsuhdgv deryh Wuhdvxu| ehqfk0 pdunv iru kljk judgh erqgv/ dv uhsruwhg e| Gxhh +4<<<,1 Wkhvh prghov surgxfh lqfuhdvlqj whup vwuxfwxuhv ri fuhglw vsuhdgv iru orz ulvn vhfxulwlhv1 Krzhyhu/ li zh dgg wkh oltxlglw| frpsrqhqw/ lw lv fohduo| srvvleoh wr uhfrqfloh d vwuxfwxudo prgho zlwk d dw whup vwuxfwxuh ri vsuhdgv/ wkh lqfuhdvh lq fuhglw vsuhdgv ehlqj rvhw e| wkh ghfuhdvh lq oltxlglw| vsuhdgv158

56_{Wklv uhvxow lv urexvw wr fkdqjhv lq wkh sdudphwhuv1 Lq vrph h{wuhph fdvhv/ wkhuh lv d}

kxps wkdw shdnv dw yhu| vkruw pdwxuwlhv exw wkhq prvw ri wkh fxuyh vwloo uhwdlqv wkh ghvfulehg vkdsh1

57_{Wkh dxwkru dovrqgv mxpsv lq vsuhdgv durxqg vl{ wr vhyhq |hduv wr pdwxulw| exw wkhvh}

duh gxh wr vshflflwlhv ri wkh Mdsdqhvh pdunhw1

58_{Wkh odfn ri vprrwkqhvv ri wkh fxuyhv lv gxh wr wkh suhvhqfh ri d eduulhu lq wkh wuhh1}

4.2

The Components of Yield Spreads

FIGURE 4

Pdunhwdelolw| suhpld duh dq lpsruwdqw sduw ri wkh wrwdo |lhog vsuhdg iru vkruw pdwxulwlhv/ ohvv vr iru orqjhu erqgv +vhh Iljxuhv 7 dqg 8/ rewdlqhg xvlqj wkh edvh fdvh sdudphwhuv59_{,1 Wklv iroorzv iurp rxu prgho ri fuhglw ulvn zklfk}

suhglfwv wkdw ghidxow lv xqolnho| iru yhu| vkruw pdwxulwlhv dv zhoo dv iurp rxu prgho ri pdunhwdelolw| ulvn zklfk suhglfwv kljkhu oltxlglw| vsuhdgv dw wkh vkruw hqg ri wkh whup vwuxfwxuh1 Wklv txdolwdwlyh uhvxow lv frqvlvwhqw zlwk wkh uhvxowv ri Orqjvwd+4<<7, zkrqgv d vlplodu vsolw iru Mdsdqhvh gdwd1

FIGURE 5

4.3

Bounds on price discounts due to illiquidity

Lq rughu wr xqghuvwdqg wkh suhglfwhg vkdsh ri wkh whup vwuxfwxuh lw zloo eh xvhixo wr jdlq vrph lqwxlwlrq iru wkh ehkdylru ri sulfh glvfrxqwv1 Frqvlghu Iljxuh 9 zklfk sorwv wkhvh glvfrxqwv lq wkh fdvh zkhq qr hduo| vdohv duh doorzhg/ wkrvh lq wkh xquhvwulfwhg fdvh dqgqdoo|/ wkhlu xsshu dqg orzhu erxqgv1

Sulfh glvfrxqwv fdqqrw eh juhdwhu wkdq wkh lpphgldwh sd|phqw ri wkh dggl0 wlrqdo edqnuxswf| glvfrxqwli wkh erqg lv vxemhfw wr ghidxow ulvn1 Lq wkh gh0 idxow ulvn0iuhh fdvh/ lw fdqqrw h{fhhg wkh h{shfwhg glvfrxqw luuhvshfwlyh ri wudghuv txrwhv4=Wkh sulfh glvfrxqw lv wkxv erxqghg deryh e|pd{¡>4¢

Wkh glvfrxqw lv erxqghg ehorz e|4_{>rwkhuzlvh wkh erqgkroghu zrxog eh}

zloolqj wr lpphgldwho| vhoo rklv srvlwlrq1 Zlwklq wkhvh erxqgv/ wkh glvfrxqw lqfuhdvhv dqg lv frqfdyh/ uh hfwlqj wkh kljkhu suredelolw| ri dq dgyhuvh oltxlglw| vkrfn1 Qdwxudoo|/ wkh suredelolw| ri d oltxlglw| vkrfn gxulqj wkh olih ri d yhu| orqj erqg whqgv wr4dqg li qr hduo| vdoh lv doorzhg/ wkh vsuhdg frqyhujhv wr lwv

59_{{|'*2c o' .I/' .cw' fe2.I/^' fHc j' fSc g' fcV' 2fc }'1}

xsshu dv|pswrwh1 Krzhyhu wklv lv qrw wkh fdvh zkhq zh fdq vhoo gxulqj wkh olih ri wkh erqg ehfdxvh wkh suredelolw| ri d yroxqwdu| vdoh dovr lqfuhdvhv zlwk wlph wr pdwxulw| dqg wkxv rvhwv vrph ri wkh lpsdfw ri wkh lqfuhdvlqj olnholkrrg ri d vkrfn1 Lq idfw/ lw frqyhujhv wr wkh orzhu erxqg/ ehfdxvh iru orqjhu pdwxulwlhv wkh suredelolw| wkdw d erqgkroghu vhoov lq dqwlflsdwlrq ri d oltxlglw| vkrfn dssurdfkhv rqh15: _{Wkh frqyhujhqfh wr wkh orzhu dv|pswrwh gxh wr yroxqwdu| vdohv hqirufhv d}

vpdoohu fds rq wkh groodu orvvhv dulvlqj iurp looltxlglw|1 Wkh suhvhqfh ri dq xsshu erxqg rq sulfh glvfrxqwv e| lwvhoi |lhogv d grzqzdug vorslqj whup vwuxfwxuh ri oltxlglw| vsuhdgv1 Wkh srvvlelolw| ri suhhpswlyh wudghv e| wkh erqgkroghu wr zdug rixwxuh oltxlglw| vkrfnv ixuwkhu orzhuv wkh orqj hqg ri wklv whup vwuxfwxuh dqg e| grlqj vr/ whqgv wr jlyh lw d frqyh{ vkdsh1

FIGURE 6

4.4

Comparative Statics

Rxu pdlqqglqj vr idu lv wkh ghfuhdvlqj vkdsh ri wkh whup vwuxfwxuh ri oltxlglw| vsuhdgv1 Zh qrz surfhhg wr vwxg| lq juhdwhu ghwdlo krz wkh frpshqvdwlrq iru oltxlglw| ulvn ghshqgv rq rwkhu prgho sdudphwhuv1

Wkh jxuhv ehorz vkrz krz wkuhh ri wkh prvw lpsruwdqw sdudphwhuv ri rxu prgho lpsdfw rq oltxlglw| vsuhdgv iru glhuhqw pdwxulwlhv1 Zh uvw frq0 vlghu zklfk lv wkh suredelolw| ri idflqj d oltxlglw| vkrfn1 Iljxuh : uhsruwv wkh whup vwuxfwxuhv ri oltxlglw| vsuhdgv iru ydulrxv ydoxhv ri fruuhvsrqg0 lqj uhvshfwlyho| wr dq dqqxdo suredelolw| ri dw ohdvw rqh oltxlglw| vkrfn ri

8( +@ 3=75:(,>43( +@ 3=;:7(, dqg 53( +@4=;75(,1 Dv h{shfwhg/ wkh juhdwhu wklv suredelolw|/ wkh juhdwhu wkh oltxlglw| vsuhdg ehfdxvh wkh pruh olnho| wkh erqgkroghu lv wr eh irufhg wr vhoo dw dq xqidyrudeoh sulfh1

5:_{Wkh uhdvrq wkdw wkhuh lv qr glvfrxqwlqj hhfw iroorzv iurp rxu vshflfdwlrq ri txrwhv lq}

looltxlg pdunhwv dv iudfwlrqv ri wkhlu ydoxh lq shuihfwo| oltxlg pdunhwv1

Iljxuh ; sorwv oltxlglw| vsuhdgv iru ydulrxv ohyhov ri dup*v dvvhw yrodwlolw|1 Lw lv fohdu iurp wklv sorw wkdw oltxlglw| vsuhdgv whqg wr lqfuhdvh zlwk fuhglw ulvn1 Wklv kdv lpsruwdqw lpsolfdwlrqv iru hpslulfdo zrun rq frusrudwh ghew ydoxdwlrq prghov1 Wkhvh uhvxowv vkrz wkdw fuhglw dqg oltxlglw| ulvn fdqqrw eh wuhdwhg lqghshqghqwo|1 Rqh fdqqrw vlpso| dgg d oltxlglw| suhplxp deryh fuhglw vsuhdgv1 Vsuhdgv deryh ehqfkpdunv zloo eh zlghu iru vshfxodwlyh0judgh ghew qrw rqo| wr frpshqvdwh iru fuhglw ulvn exw dovr ehfdxvh ghidxow ulvn lpsdfwv rq oltxlglw| vsuhdgv1

Krzhyhu/ fuhglw ulvn lv qrw rqo| d pdwwhu ri ghidxow suredelolw| +pdlqo| lq0 xhqfhg e| ohyhudjh dqg yrodwlolw|,/ exw lv dovr ghwhuplqhg e| wkh orvv uhdol}hg lq ghidxow1 Wkh odvwjxuh +Iljxuh <, sorwv wkh uhodwlrqvkls ehwzhhq wkh dggl0 wlrqdo orvv lqfxuuhg e| looltxlg vhfxulwlhv xsrq ghidxow dqg wkh oltxlglw| suhplxp1 Djdlq d srvlwlyh uhodwlrqvkls lv irxqg1

FIGURES 7,8,9

5

Bondholder Trading Behavior

Lq wklv vhfwlrq zh dqdo|}h wkh zloolqjqhvv ri d erqgkroghu wr vhoo klv vhfxulw| dw d glvfrxqw wr wkh sulfh wkdw zrxog suhydlo lq d shuihfwo| oltxlg pdunhw1 Zkhq qdqfldo glvwuhvv dssurdfkhv/ wkh erqgkroghu uhdol}hv wkdw wkhuh lv dq lqfuhdvhg olnholkrrg wkdw klv rswlrq wr vhhn rxw idyrudeoh rhuv sulru wr pdwxulw| zloo ehfrph zruwkohvv1 Khqfh kh zloo eh zloolqj wr vhoo dw d orzhu sulfh1 Djdlq wklv zrxog vxjjhvw wkdw wudglqj yroxph lqfuhdvhv dv ghidxow ehfrphv pruh olnho|1 Dq hpslulfdo vwxg| rq kljk0|lhog ghew e| Vfkxopdq hw do1 +4<<6, vxssruwv wkh lghd wkdw dqwlflsdwlrq ri ghidxow uhvxowv lq juhdwhu wudglqj dfwlylw|1

FIGURE 10

Iljxuh 43looxvwudwhv lq d elqrpldo wuhh wkh shufhqwdjh sulfh glvfrxqwv5; _wkdw

duh dffhswdeoh wr wkh erqgkroghu dv d ixqfwlrq ri wlph wr pdwxulw| dqg sur{lplw| ri ghidxow1 Zh vhh wkdw wkh forvhu zh duh wr pdwxulw|/ wkh orzhu wkh dffhswdeoh glvfrxqw ehfdxvh wkh erqgkroghu rqo| kdv d vkruw wlph wr zdlw ehiruh wkh erqg lv uhghhphg dqg wkh olnholkrrg ri d irufhg vdoh dw dq xqidyrudeoh sulfh lv vpdoohu1 Rq wkh rwkhu kdqg/ zkhq wkh lvvxhu ri wkh erqgv dssurdfkhvqdqfldo glvwuhvv/ wkh erqgkroghu zloo eh pruh zloolqj wr vhoo klv vhfxulw| dw d juhdwhu glvfrxqw wr dyrlg wkh ulvn ri idflqj uhrujdql}dwlrq frvwv1

Wdeoh 4 irfxvhv pruh vshflfdoo| rq wkh lpsdfw ri ghidxow ulvn rq oltxlglw| glv0 frxqwv1 Ghidxow ulvn lv phdvxuhg e| ohyhudjh +txdvl0ghew udwlr= Sh{s+uW,@y3,

dqg wkh yrodwlolw| ri dvvhwv = =Wkh glvfrxqw wkh erqgkroghu lv uhdg| wr vhoo klv erqg dw fohduo| lqfuhdvhv lq erwk ohyhudjh dqg yrodwlolw| dqg fdq uhdfk frqvlghu0 deoh ohyhov1 Wkhvh uhvxowv duh urexvw wr wkh fkrlfh ri wkh rwkhu prgho sdudphwhuv

+>> N>> u,=

TABLE 1

6

Dealer Trading Behavior

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FIGURE 11

6.1

Bond pricing with state dependent dealer behavior

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FIGURE 12

7

Testing the Model’s Predictions

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TABLE 2

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7.1

The data

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7.2

Results

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Wkh ohyho ri wkh ulvn iuhh lqwhuhvw udwh lv dozd|v qhjdwlyho| uhodwhg wr wkh vsuhdg ohyhov1 Lw lv vwurqjo| vljqlfdqw h{fhsw iru rqh sduwlfxodu vshflfdwlrq zlwk orqj whup erqgv1 Wkh uhodwlrqvkls lv pruh pdunhg iru upv zlwk orz fuhglw udwlqj ru erqgv zlwk vkruw pdwxulwlhv1

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66_{Wdeoh 7 surylghv d fruuhodwlrq pdwul{ iru wkh lqghshqghqw yduldeohv1 Iru yduldeohv wkdw}

duh furvv0vhfwlrq vshflf/ wkh wdeoh uhsruwv wkh phdq ri fruuhodwlrqv frpsxwhg iru hdfk furvv0 vhfwlrqdo xqlw1

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Lq rughu wr vhh li rxu wzr suhglfwlrqv duh eruqh rxw e| wkh gdwd/ zh frqvlghu wzr vxevdpsohv= rqh edvhg rq fuhglw udwlqj/ wkh rwkhu rq erqg pdwxulw|1 Zh zloo uvw glvfxvv zkhwkhu zh qg hylghqfh ri d srvlwlyh fruuhodwlrq ehwzhhq fuhglw dqg oltxlglw| vsuhdgv1

Zhqg wkdw wkh sdudphwhu hvwlpdwh iru wkhRW Ugxpp| lv dssur{lpdwho| wkuhh wlphv odujhu lq wkh vxevdpsoh ri erqgv zlwk orz fuhglw udwlqj1 Wklv vxj0 jhvwv wkdw erqgv wkdw duh r0wkh0uxq dqg fuhglw ulvn| zloo kdyh wr uhzdug wkhlu kroghuv zlwk dq dgglwlrqdo |lhog zklfk fdq eh wkuhh wlphv kljkhu lq edvlv srlqwv wkdq wkh fruuhvsrqglqj h{wud |lhog iru kljk fuhglw txdolw| erqgv1 Vlploduo|/ wkh lpsdfw ri W OLT lv doprvw irxu wlphv odujhu lq wkh orz udwlqj vdpsoh1 Wklv lv frqvlvwhqw zlwk d ghfuhdvh lq oltxlglw| lq Wuhdvxu| pdunhwv wkdw sursdjdwhv lqwr frusrudwh ghew pdunhwv zlwk d pdunhgo| vwurqjhu lpsdfw rq wkh vshfxodwlyh judgh vhjphqw1 Erwk ri wkhvhqglqjv duh vxssruwlyh ri wkh srvlwlyh uhodwlrqvkls ehwzhhq fuhglw dqg oltxlglw| ulvn lpsolhg e| rxu prgho1

Wxuqlqj wr wkh vkdsh ri wkh whup vwuxfwxuh ri oltxlglw| vsuhdgv/ zhqg wkdw wkh lpsdfw ri wkh RWU gxpp| glhuv dfurvv erqg pdwxulwlhv1 Wkh ehqhw wr ehlqj rq0wkh0uxq iru vkruw whup erqgv lv derxw 83( kljkhu wkdq iru orqj erqgv 0 vxjjhvwlqj wkdw wkh oltxlglw| frpsrqhqw ri |lhog vsuhdgv glplqlvkhv zlwk pdwxulw|1 Wkh glhuhqfh ehwzhhq wkh sdudphwhu hvwlpdwhv iru WOLT lv qrw vwdwlvwlfdoo| vljqlfdqw dqg grhv khqfh qrw dgg ixuwkhu vxssruw wr rxu k|srwk0

hvlv1

Ryhudoo/ zh eholhyh wkdw wkh hpslulfdo uhvxowv uhsruwhg lq wklv vhfwlrq duh vxssruwlyh ri wzr hpslulfdo lpsolfdwlrqv ghulyhg iurp rxu prgho ri fuhglw dqg oltxlglw| ulvn1

8

Concluding Remarks

Zh kdyh ghyhorshg d vlpsoh prgho wr looxvwudwh wkh lpsdfw ri oltxlglw| ulvn rq wkh |lhog vsuhdgv ri frusrudwh erqgv1 Ghvslwh lwv vlpsolflw|/ wkh prgho kdv d qxpehu ri lqwhuhvwlqj ihdwxuhv1 Rxu pdlq txdolwdwlyhqglqj lv wkdw wkh ohyho ri oltxlglw| vsuhdgv vkrxog eh srvlwlyho| fruuhodwhg zlwk fuhglw ulvn dqg wkdw wkh| vkrxog eh ghfuhdvlqj ixqfwlrqv ri wlph wr pdwxulw|1

Dqrwkhuqglqj lv wkdw iru uhdvrqdeoh sdudphwhu lqsxwv wkh prgho lv deoh wr jhqhudwh qrq0qhjoljleoh |lhog vsuhdgv hyhq iru vkruw pdwxulwlhv1 Wklv dgguhvvhv d frpprq fulwlflvp ri vwuxfwxudo erqg sulflqj prghov dqg khosv wr uhfrqfloh wkhp zlwk hpslulfdo hylghqfh1

Rxu uhvxowv duh frqvlvwhqw zlwk suhylrxv hpslulfdo uhvhdufk qrw rqo| dv idu dv wkh vkdsh ri wkh whup vwuxfwxuh +ghfuhdvlqj dqg frqyh{, lv frqfhuqhg exw dovr lq whupv ri wkh ohyhov1 Wkh uhodwlyh vkduhv ri vsuhdgv h{sodlqhg e| fuhglw ulvn dqg oltxlglw| ulvn lq rxu prgho duh frpsdudeoh zlwk hduolhu hpslulfdoqglqjv1 Lq dgglwlrq/ rxu prgho lv doljqhg zlwk d qxpehu ri vw|ol}hg idfwv derxw ghew vhfxulwlhv1 Iru h{dpsoh/ lw surylghv vrph mxvwlfdwlrq iru wkh juhdwhu oltxlglw| ri |rxqj lvvxhv dqg wkh kljkhu iuhtxhqf| ri wudghv qhduhu ghidxow uhsruwhg lq wkh olwhudwxuh1

Ilqdoo|/ zhqg wkdw XV frusrudwh erqg gdwd vxssruw wkh suhglfwlrq ri rxu prgho wkdw oltxlglw| vsuhdgv vkrxog eh srvlwlyho| fruuhodwhg zlwk wkh olnholkrrg ri ghidxow dqg wkdw vsuhdgv vkrxog ghfuhdvh zlwk wlph wr pdwxulw|1

Appendix A : Expected Discounts

Wkh h{shfwhg ehvw iudfwlrq ri wkh oltxlg sulfh wkdw wkh vhoohu zloo eh rhuhg lv

Hhw i @X4 q@3 S+Q @q,Z 4 3 i q_+,g>

zkhuh wkh ghqvlw| iq_{+{, lv wkh suredelolw| wkdw { lv wkh ehvw sulfh iudfwlrq}

rewdlqhg jlyhq q rhuv1 Jlyhq rqo| rqh rhu/ iru d xqlirup glvwulexwlrq wkh suredelolw| ri jhwwlqj d iudfwlrq ri ohvv wkdq{lv

I+{, @{>

zkhuhI lv wkh fxpxodwlyh glvwulexwlrq1 Wkh suredelolw| ri jhwwlqj qrqh kljkhu wkdq{zlwkqlqghshqghqw rhuv lv wkxv

+I+{,,q@{q_>

dqg vr wkh ghvluhg ghqvlw| ixqfwlrqiq _lv iq_{+{, @} C+I+{,,q

C{ @q{q4>

dqg wkxv/ jlyhq wkdw wkh qxpehu ri rhuv lv Srlvvrq zlwk sdudphwhu>zh jhw

Hhw i @ X4 q@3 hq q$ Z 4 3 q q_g> @ 4 X q@3 hq q$ qq.4=

Qrz/ lq rughu wr frpsxwh wkh ydoxh ri dq looltxlg erqg dw d jlyhq qrgh lq

wkh elqrpldo wuhh zh qhhg wr vroyh Hw h  wLtAt i dqg S³wAw ´ @HhLtAt i Uhfdoo wkdw S+Q @q, @hq q$>

dqg wkdw frqglwlrqdo rqqrhuv rxu dvvxpswlrq ri xqlirupo| glvwulexwhg rhuv |lhogv wkh iroorzlqj ghqvlw| iru wkh sulfh iudfwlrq rhuhg

iq_{+, @q}q4₌ Wkhq lw iroorzv wkdw HhLtAt i @ 4 X q@3 S+Q@q,HhLtAt mQ @q i @ X4 q@3 S+Q@q,Z 4 t qq4_g @ 4 X q@3 hq q$ ¡ 4+w,q¢= 67

Hw h  wLtAt i @ X4 q@3 S+Q@q,HhwLtAtmQ @q i @ 4 X q@3 S+Q@q,Z 4 t qqg @ X4 q@3 hq q$qq.4 ³ 4+ w,q.4 ´ = 68

Appendix B : Spread Constructions

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rq d erqg zlwk pdwxulw|W lv jlyhq e|

U+w> W, @3. +4.5,4h{s +_W@W@6,

6 5h{s +W@6,

Xvlqj ulvn0iuhh }hur0frxsrq erqgv +pdlqo| vwulsv, wr ghulyh wkh ehqfkpdun fxuyhv hqdeohv xv wr rewdlq d qhduo| shuihfw w ri revhuyhg ulvnohvv udwhv e| pd{lpxp olnholkrrg +zh xvh wkh FPO wrro lq Jdxvv,1 Krzhyhu zhqg wkdw wkh Qhovrq0Vlhjho surfhgxuh lv ryhu0sdudphwul}hg iru }hur0frxsrq erqgv dqg ohdgv wr zlgh glhuhqfhv lq wkh sdudphwhu hvwlpdwhv lq vslwh ri rqo| plog yduldwlrqv lq wkhlu lqlwldo ydoxhv1 Zh wkxv lpsrvh d uhvwulfwlrq rq wkh uvw sdudphwhu zklfk lv wkh rqo| rqh zlwk d fohdu hfrqrplf lqwhusuhwdwlrq1 Pruh suhflvho|/ wkh uvw sdudphwhu uhsuhvhqwv wkh |lhog ri d shushwxdo ulvn0iuhh erqgU+w>4,= Zh dssur{lpdwh lw e| wkh 63|hdu X1V1 Wuhdvxu| udwh dqg wkxv rewdlq d frqvlvwhqw dqg urexvw vhw ri rswlpdo sdudphwhuv1 Wkh frqvwudlqw dovr wxuqv rxw wr |lhog srvlwlyh iruzdug udwhv iru doo pdwxulwlhv dqg doo revhuydwlrq shulrgv/ wkhuhe| dyrlglqj rqh ri wkh pdlq fulwlflvpv ri wkh dojrulwkp1 Iru hdfk prqwk/ zh h{foxgh

ulvn| erqgv zkrvh pdwxulw| idoov rxwvlgh wkh udqjh vsdqqhg e| wkh ulvn0iuhh erqgv wr dyrlg wkh lpsuhflvlrqv ri wkh lqwhusrodwlrq surfhgxuh rxwvlgh wklv udqjh1

Appendix C: Dealer behavior

Lq wklv dsshqgl{/ zhuvw ghulyh wkhuhsruwhg lq vhfwlrq 91 Zh wkhq fdofxodwh wkh suredelolw| ri d yroxqwdu| vdoh lq wkh lqyhqwru| prgho/ dv zhoo dv wkh h{shfwhg glvfrxqw lq fdvh ri d irufhg vdoh ru d yroxqwdu| vdoh1

Required discount

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