Endogenous Credit Spreads and Optimal Debt Financing Structure in the Presence of Liquidity Risk

• Author: Eva Lütkebohmert,Daniel Oeltz,Yajun Xiao
• Date: 01 January 2017
• DOI: http://doi.org/10.1111/eufm.12089
• Published date: 01 January 2017

Endogenous Credit Spreads and

Optimal Debt Financing Structure in

the Presence of Liquidity Risk

Eva L€

utkebohmert

Department of Quantitative Finance, University of Freiburg, Platz der Alten Synagoge, 79098

Freiburg, Germany

E-mail: eva.luetkebohmert@finance.uni-freiburg.de

Daniel Oeltz

RIVACON GmbH, Im Apfelgrund 4, 61381 Friedrichsdorf, Germany

E-mail: d.oeltz@rivacon.com

Yajun Xiao

Business School, University of Technology Sydney, PO Box 123 Broadway, NSW 2007 Sydney,

Australia

E-mail: Yajun.Xiao@uts.edu.au

Abstract

We present a structural model that allows a firm to effectively manage its exposure

to both insolvency and illiquidity risk inherent in its financing structure. Besides

insolvency risk, the firm is exposed to rollover risk through possible runs by short-

term creditors. Moreover, asset price volatilities are subject to macroeconomic

This work was supported by the Excellence Initiative through the project ‘Pricing of Risk in

Incomplete Markets’within the Institutional Strategy of the University of Freiburg as well

as by the German Research Foundation through the project ‘Modelling of Market, Credit-

and Liquidity Risks in Fixed-Income Markets’. The financial support is gratefully

acknowledged by the first and the third author. Several helpful comments and suggestions

from Rama Cont, Damir Filipovic, Paul Glasserman, Xue-Zhong He, Gechun Liang and

Lakshithe Wagalath are very much appreciated. The authors also thank the participants at

the IMA Conference on Mathematics in Finance in Edinburgh 2013, at the Conference on

Current Topics in Mathematical Finance in Vienna 2013, at the Conference of the French

Finance Association in Lyon 2013, and at the 8

th

World Congress of the Bachelier Finance

Society 2014 in Brussels for several valuable remarks. Moreover, the authors are grateful to

Frank Koster for sharing his knowledge on the discretisation scheme of the Heston process

and on least square Monte Carlo methods in general. His suggestions and remarks were

highly valuable for the implementation of an efficient scheme to simulate the proposed

model. Finally, the authors would like to thank the Editor John Doukas and two anonymous

referees for valuable comments and suggestions that helped to significantly improve the

paper. Correspondence: Eva L€

utkebohmert.

European Financial Management, Vol. 23, No. 1, 2017, 55–86

doi: 10.1111/eufm.12089

© 2016 John Wiley & Sons, Ltd.

shocks and influence creditors’risk attitudes and margin requirements. Credit

spreads are derived endogenously depending on the firm’s total default risk. Equity

holders have to bear the rollover losses. An optimal debt structure that maximises

the firm’s equity value is determined by trading off lower financing costs and

higher rollover risk.

Keywords: funding liquidity, optimal capital structure, rollover risk, structural

credit risk models

JEL classification: G01, G32, G33

1. Introduction

Heavy reliance on short-term financing, liquidity dry-ups in the short-term debt market

and creditors’panic runs were among the major reasons of the various firm failures

during the 2007/2008 financial crisis.

1

Since then there has been extensive research into

how liquidity dry-ups in the short-term lending market can lead to spiking credit spreads

and ultimately can cause firm failures. The question of how a firm can quantify and

practically cope with the risks inherent in short-term financing has been discussed much

less. This, however, is a prevailing issue, especially as, in the aftermath of the crisis,

financial institutions surprisingly again increase reliance on short-term financing

sources.

This article proposes a comprehensive and tractable model that takes into account

insolvency risk, illiquidity risk associated with a firm’s debt structure, and potential

interactions between both sources of risk. The paper has four major contributions.

Firstly, we study the effect of both insolvency and illiquidity risk on short-term credit

spreads which are determined endogenously in our setting. When the firm is in distress,

investors will be more pessimistic and the firm needs to offer higher coupon rates to

induce them to roll over their funding. In this way, our model reflects the empirically

observed fact that spreads on short-term debt increase when funding liquidity tightens.

Moreover, by investigating the effect of market deteriorations on a firm’s credit spread,

our paper contributes to explaining the flight-to-quality observable during the financial

crisis of 2007/2008. Secondly, our approach allows us to study how a firm’sfinancing

structure affects its default probability. The more the firm relies on short-term funding,

the higher is its default probability as rolling over short-term debt becomes more

sensitive to fluctuations of the firm’s asset value and its volatility.

In this way, our model can pr ovide valuable insights in to a firm’s internal risk

management, especially as feedback effects of changing cred it spreads on the firm’s

solvency situation are taken into ac count. Thirdly, the model allows us to det ermine a

firm’s optimal debt financing struct ure, i.e., its optimal rat io of short- to long-term

debt. When short-term rate s increase, equity holders have to bear the rollover losses

subject to their limited li ability. Short-term finan cing is attractive as it is usually

1

In fact, it has been shown in Adrian and Shin (2008, 2010) that the use of very short-term

financing dramatically increased during the financial crisis of 2007/2008 and the period

immediately before that. Gorton and Metrick (2012) and Brunnermeier (2009) provide

further evidence that debt runs were one of the main reasons for the crisis.

© 2016 John Wiley & Sons, Ltd.

56 Eva L€

utkebohmert, Daniel Oeltz and Yajun Xiao

cheaper than long-term financing. However, the more a firm relies on short-term

funding, the higher is its exposu re to rollover risk. Our results sh ow that equity value

can be maximised by trading of f benefits and costs associat ed with short-term

financing. Thereby the ratio of sh ort- to long-term debt is optimi sed. Finally, our

model allows us to study the feedbac k effects of external macroe conomic shocks on

afirm’s internal funding liquidity. I n this way, our approach can contrib ute to the

construction of efficient rules on firms’balance sheets that can help to prevent

the emergence of liquidity spir als and hence yields valuable impl ications for the

construction of regulato ry guidelines.

Our paper is related to several st reams of literature. In the red uced-form approaches

of Longstaff et al. (2005) and Filipovic and Trol le (2013), the default rate and the

liquidity rate are exogenous processes which are independent of each other. Thes e are

converted into individual credit spread components co rresponding to default a nd

liquidity risk, respective ly. In stark contrast to this ap proach, our paper studies the

intertwined effect between insol vency and illiquidity risk. Th erefore, we build on

the structural credit risk mod el of Black and Cox (1976) where a default happens when

the firm fundamental value falls bel ow an exogenous default barr ier. We incorporate

funding liquidity risk in thi s setting by modelling creditors’rollover decisio ns at short-

term debt maturities. In our model the decision faced by short-term creditors of

whether to roll over or to withd raw their funding depends o n both future insolvency

risk and creditors’beliefs about future debt runs. Liquidit y risk resulting from the

binary rollover decision f ollowing creditors’coordi nation is determined in par tial

equilibrium. In this regar d our approach differs consi derably from Ericsson and

Renault (2006), He and Xion g (2012b), and He and Milbrad t (2014), who employ the

framework of Leland and Toft (199 6) and introduce an exogenous liquidity shock. The

assumed staggered debt stru cture which implies stabl e coupon rates over time as wel l

as the assumption that liqu idity shocks are exogenous, allows the authors to

endogenously determine the d efault barrier which is not possi ble in our setting. It has

been documented, however , that the spread on many short- term financing instruments

climbed dramatically duri ng the financial crisis, whic h makes the assumption of

constant coupon rates disputab le when the average debt maturity is not lar ge.

Therefore, we allow the coupo n rate on short-term debt to vary over tim e. Thus,

instead of endogenising the defau lt barrier, in this paper we endoge nously determine

the coupon rate such that the firm can react to ch anging market conditions and p revent

creditors from running by a ppropriately adjusting t he coupon rate on short-term debt.

In He and Milbradt (2014) an d He and Xiong (2012b) rollo ver costs are exogenous

and have to be absorbed by equity hol ders. In our model, equity hol ders also need to

absorb rollover costs, howev er, these are endogenous as t hey are caused by increasing

short-term risk premia. Thi s allows us to analyse the optim al ratio of short-term debt

to long-term debt that max imises the firm’s equity value.

Further, our model is relevant to the bank run literature. The classical bank run models

of Bryant (1980), Diamond and Dybvig (1983) and Rochet and Vives (2004) provide

evidence for the fact that bank runs can occur as a result of coordination problems among

short-duration depositors’rollover decisions. These static models have been extended to

dynamic coordination problems in Guimaraes (2006) and He and Xiong (2012a). In

Morris and Shin (2010) the authors focus on a two-period setting where short-term

creditors face a binary decision in terms of global games at an interim time point. Liang

et al. (2014, 2015) combine ideas of global games with the dynamic coordination

© 2016 John Wiley & Sons, Ltd.

Endogenous Credit Spreads and Optimal Debt Financing 57