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Chapter 4Hedging Market and Credit Risk in CorporateBond Portfolios

Patrizia Beraldi, Giorgio Consigli, Francesco De Simone, Gaetano Iaquinta,and Antonio Violi

Abstract The European market for corporate bonds has grown significantly overthe last two decades to become a preferable financing channel for large corporationsin the local and Eurobond markets. The 2008 credit crisis has, however, dramati-cally changed corporations funding opportunities with similar effects on borrowingpolicies of sovereigns as well. Accordingly institutional and individual investorshave progressively reduced the share of credit risky instruments in their portfolios.This chapter investigates the potential of multistage stochastic programming to pro-vide the desired market and credit risk control for such portfolios over the recent,unprecedented financial turmoil. We consider a Eurobond portfolio, traded in thesecondary market, subject to interest and credit risk and analyse whether a jump-to-default risk model and a dynamic control policy would have reduced the impactof severe market shocks on the portfolios during the crisis, to limit the systemicimpact of investment strategies. The methodology is shown to provide an effectivealternative to popular hedging techniques based on credit derivatives at a time inwhich such markets became extremely illiquid during the Fall of 2008.

Keywords Multistage stochastic programming · Bond portfolio optimization ·Credit risk

4.1 Introduction

The 2007–2009 financial turmoil witnessed an unprecedented market downturn forfinancial instruments carrying credit risk (Abaffy et al. 2007; Berndt 2004; Crouhyet al. 2000; Duffie and Singleton 1999) spanning both the secondary markets forcorporate and sovereign securities and the markets for derivatives based on suchinstruments. The crisis propagated in US markets from mortgage-backed securities(MBS) and collateralized debt obligations (CDO) to credit instruments traded over-the-counter and thus into international portfolios. Widespread lack of liquidity inthe secondary market induced first the Federal Reserve, then the European Cen-tral Bank, to adopt an expansive monetary policy through a sequence of base rate

P. Beraldi (B)Department of Electronics, Informatics and Systems, University of Calabria, Rende, Italye-mail: [email protected]

M. Bertocchi et al. (eds.), Stochastic Optimization Methods in Finance and Energy,International Series in Operations Research & Management Science 163,DOI 10.1007/978-1-4419-9586-5_4, C© Springer Science+Business Media, LLC 2011

73

74 P. Beraldi et al.

reductions. Such policy partially limited the fall of bond prices but could do verylittle against instability in the corporate equity and fixed income markets.

The debate over the causes and possible remedies of such a prolonged financialcrisis involved, from different perspectives, policy makers, financial intermediaries,and economists (European Central Bank 2010), all interested in analyzing the equi-librium recovery conditions, possibly within a quite different market architecture.

Financial investors, on the other hand, suffered dramatic portfolio losses withinincreasingly illiquid money, secondary stock and bond and credit derivative markets.In this chapter we present a stochastic model for interest rate and credit risk appliedto a portfolio of corporate bonds traded in the Eurobond market with portfolio strate-gies tested over the 2008–2009 crisis. The portfolio management problem is formu-lated as a dynamic stochastic program with recourse (Consigli and Dempster 1998;Pflug and Römisch 2007; Zenios and Ziemba 2007). This chapter provides evidenceof the potential offered by dynamic policies during a dramatic market crisis. Key tothe results presented are the definitions of

• a statistical model capturing common and bond-specific credit risk factors thatwill determine jointly with the yield curve the defaultable bonds price behavior(Dai and Singleton 2003; Das and Tufano 1996; Duffie and Singleton 2000;Jarrow and Turnbull 2000; Kijima and Muromachi 2000; Longstaff et al. 2005)and

• a multistage strategy determined by a risk-reward objective function explicitlyconsidering an extreme risk measure (Bertocchi et al. 2007; Consigli et al. 2010;Dempster et al. 2003; Jobst et al. 2006; Jobst and Zenios 2005; Rockafellar andUryasev 2002). The problem considers a constant investment universe excludingcredit derivatives and exogenous hedging strategies.

The stochastic model for bond returns and the optimization model are strictly relatedand their accuracy will determine the quality of the resulting portfolio strategy.The return model leads to the definition of bond price scenarios used as inputs toa stochastic programming model over a 1-year time horizon with monthly rebal-ancing decisions. The stochastic program includes buying, holding, and sellingdecisions regarding an extended universe of corporate bonds, spanning several eco-nomic sectors and all rating classes under Standard and Poor’s classification (Abaffyet al. 2007; Moody’s Investors Service 2009). Dynamic bond portfolio manage-ment approaches, since the seminal work by (Bradley and Crane 1972), have beenwidely adopted in the past for Treasury portfolios, while limited experience hasbeen reported for corporate bond portfolios (Consigli et al. 2010; Jobst and Zenios2005). This motivates our study. We show that, under the given market conditions,a dynamic policy would have provided an effective contingency plan without theneed to hedge against credit losses with derivative contracts.

The chapter evolves as follows. Section 4.2 describes the market and credit riskmodel which represents the bond dynamics. In Section 4.3 a mathematical modelfor the optimal bond portfolio management is introduced, while Section 4.4 reportson the scenario generation procedure for the simulation of input data for the decisionmodel. Computational experiments for the validation of the proposed approach arereported in Section 4.5. Some concluding remarks end the chapter.

4 Hedging Market and Credit Risk in Corporate Bond Portfolios 75

4.2 Corporate Bonds Risk Exposure

The recent credit crisis witnessed an unprecedented credit spread increase acrossall maturities in the Eurobond market while, first in the USD monetary area andthen in the UK and Europe, interest rates were rapidly, though ineffectively, drivendown to try to facilitate a liquidity recovery in the markets. In this work we assumea risk model incorporating both common and specific risk factors and allow theinvestor to determine her optimal policy by exploiting a scenario representationof the credit spreads evolution which may be affected at random times by severemarket shocks. We consider a universe of Euro-denominated bonds with ratingsfrom AAA to CCC−C (see (Abaffy et al. 2007)) plus one default-free governmentbond. Ratings are alphanumeric indicators specified by international rating agenciessuch as standard and poor, moody and fitch, defining the credit merit of the bondissuer and thus the likelihood of possible defaults over a given risk horizon. Ratingrevisions can be induced by market pressures or expert opinions and only occurinfrequently. Bonds trading in the secondary market on the other hand generate acontinuous information flow on expected interest rates and yield movements.

Bonds belonging to the same rating class may be distinguished according totheir maturity and more importantly the activity sector to which the issuer belongs.Bond issuers within the same rating class and industry sector may finally be dis-tinguished according to their market position and financial strength: the riskier theissuer, according to the above classification, the higher the credit spread that will berequested by investors to include a specific fixed income security in the portfolio.

Corporate prices are thus assumed to be driven by

• a common factor affecting every interest-sensitive security in the market (Coxet al. 1985), related to movements of the yield curve,

• a credit risk factor, related to movements of the credit curves (Abaffy et al. 2007;Duffie and Singleton 1999; Jobst and Zenios 2005), one for each rating class, and

• a specific bond factor, related to the issuer’s economic sector and its generalfinancial health.

A degree of complexity may be added to this framework when trying to take intoaccount the possible transition over the portfolio lifetime of bond issuers across dif-ferent rating classes (Jarrow et al. 1997) and bond-specific liquidity features result-ing in the bid-ask spread widening in the secondary market. In this work we willfocus on the solution of a 1-year corporate bond portfolio management problemwith monthly portfolio revision, not considering either transition risk or liquidityrisk explicitly.

4.2.1 Market and Credit Risk Model

We consider a credit-risky bond market with a set I of securities. For each bondi ∈ I , we denote by Ti its maturity. Let K be the set of rating classes. Assuming


a canonical S&P risk partition from AAA to CCC-C and D (the default state), wewill denote by k = 1, 2, . . . , 8 the credit risk indicator associated, respectively, withdefault-free sovereign and corporate AAA, AA, A, BBB, BB, B, CCC-C, and Dratings. A difference is thus postulated between a sovereign and a corporate AAArating: the first one will be associated with the prevailing market yield for default-free securities with null credit spread over its entire market life, while the secondmay be associated with a positive, though negligible, credit spread. The security icredit risk class is denoted by ki ,while Ik is the set of bonds belonging to rating classk. The price of security i at time t will be denoted by vi

t,Ti. For given security price

and payment structure the yield yit,Ti

can be inferred from the classical price–yieldrelationship:

vit,Ti=

∑t<m≤Ti

cime−yi

t,Ti(m−t)

, (4.1)

where the cash payments over the security residual life are denoted by cim up to and

including the maturity date. In (4.1) the yield yit,Ti

will in general reflect the currentterm structure of risky interest rates for securities belonging to class ki . We assumeyi

t,Tigenerated by a bond-relevant credit spread π i

t,Tiand a default-free interest rate

rt,Ti , i.e. yit,Ti= rt,Ti + π i

t,Ti.

A one-factor model is considered for both the default-free interest rate curveand the security-specific credit spread. The latter is assumed to be generated by arating-specific factor πk

t and an idyosincratic factor ηit .

The three state variables of the model are assumed to follow the s.d.e.’s:

drt (ω) = μr dt + σr (t, r)dW rt (ω), (4.2)

dπkt (ω) = μkdt + σk(t, π

k)∑l∈K

qkldW lt (ω) ∀k, (4.3)

dηit (ω) = μηdt + σi dW i

t (ω)+ β i (ω)d� it

(λi)∀i, (4.4)

where ω is used to identify a generic random variable. The first equation describesthe short interest rate evolution as a diffusion process with constant drift and pos-sibly state- and time-dependent volatility. dW r

t ∼ N (0, dt) is a normal Wienerincrement. The k = 1, 2, . . . , 7 credit spread processes in (4.3) are also modeled asdiffusion processes with time- and state-varying volatilities. Credit spreads and therisk-free rate are correlated: the coefficients qkl denote the elements of the correla-tion matrix lower triangular Choleski factor linking the eight risk factors together,given the independent Wiener increments dW k

t , k = 1, 2, . . . , 7.Equation (4.4) assumes an independent security spread process with a diffusion

component generated by a Wiener process W it and a jump-to-default component

generated by a default process � it with intensity λi and jump size β i. The bond

price will either follow a continuous path up to maturity if no default occurs or will


default before maturity generating a price shock. All security prices in the fixed-income market are assumed to be driven by the short rate rt . The credit spreadsπ

kit in (4.3) will determine the price behavior of bonds i belonging to class ki and

thus the relative behavior of bonds in different rating classes. The bond-specificmarginal spread processes ηi

t will finally determine the relative price dynamics ofbonds within the same rating class.

From (4.1), given the current time 0 price vi0,Ti

, price dynamics for t ∈ [0, T ] arederived from the yield scenarios by introducing the canonical duration-convexityupdate in the real-world, market, probability space:

dvit,Ti= vi

t,Ti

[yi

t dt − δit dyi

t + 0.5γ it (dyi

t )2], (4.5)

where yit = rt + πki

t + ηit . Given the current price and duration-convexity pair

(δit , γ

it ), each yield increment will determine an associated price transition. Then an

update of the duration and convexity parameters leads to a new price transition forthe given yield increment, and so on up to maturity, unless default occurs.

To specify the spread process in (4.4) and employ the pricing update in (4.5),the marginal spread ηi

t must be identified. Below in the case study we present amodel assuming a pure jump process for the marginal spread. In the general case,for a given price history vi

l,Ti, l ≤ t , at time t and cash flow structure, we imply the

marginal spread as

ηil = π i

l − πkil ,

π il = − ln

(vi

l,Ti

vf

l,Ti

)· (δi

l )−1. (4.6)

vf

l,Tiin (4.6) denotes the price of a default-free security with the same payment

structure as bond i and δil is the duration of bond i at time l. Such a price is computed

from the interest rate term structure history rl,T . The spread πkil denotes instead the

CDS spread for the rating class introduced in (4.3). We use the above approximationto differentiate risky price dynamics within a rating class.

Scenario generation for the corporate bond portfolio problem relies on the set ofequations (4.5) for all i ∈ I and the underlying risk factor processes (4.2), (4.3),and (4.4).

4.2.2 Default Model

Rating agencies regularly estimate default frequencies observed in the economyfor borrowers belonging to a certain rating class by year cohorts leading to thedefinition of real-world default probabilities at the rating class level. Moody’s 2009report (Moody’s Investors Service 2009) updates previous studies presenting default


statistics for the 1982–2008 period that we consider in our case study. We refer toactual default frequencies and expected losses per unit credit published by the ratingagency as the natural or real-world, or empirical, default probabilities and recoveryrates considered in the model. These probabilities need to be distinguished frommarket default probabilities implied from bond prices (Berndt et al. 2005; Consigli2004; Duffie and Singleton 1999; Elton et al. 2001).

Market implied default probabilities are recognized to over estimate default fre-quencies actually observed in the economy (Berndt et al. 2005). Defaultable spreadsdo price additional risk factors such as liquidity risk, company evolving finan-cial conditions, and the stability of the default intensities in the economy. Ratingassessments moreover occur at discrete times with occasional rating revisions: toovercome this drawback we propose below a simple and practical heuristic for thedefault intensity coefficient λi in the model.

In particular, let Ψ it be the default process affecting security i . Prior to default

the process is at 0 and moves to 1 upon default. In a discrete setting for everytime stage t = 1, 2, . . . , T the process transition from 0 to 1 is determined byan intensity λi with two components: the 1-year rating-specific default frequencyλki and a user-defined multiplier ξ i to capture the security-specific perspectives asestimated through available market and company information by credit analysts.The security-specific risk coefficient ξ i is assumed to depend on the company geo-graphic exposure, its economic sector, and the security default premium (Berndtet al. 2005). We assume that credit analysts map the above analysis into eight riskpartitions ξ i = κ ∗ 0.125, κ = 4, 5, . . . , 12, where

λi = λki × ξ i . (4.7)

This heuristic provides a user-based update of the historical default frequencies. Theloss-given default for a bond investor is determined by the market price loss sufferedby the investor upon default announcement. Moody’s report in (Moody’s InvestorsService 2009) the historic recovery rates on a large sample of Eurobonds in whichthe canonical inverse relationship between default frequencies and recovery rates isconsistently confirmed across time and economic sectors.

4.2.3 Price–Yield Relationship

Following the intensity specification in (4.7) an individual security can at any timebe affected by a jump-to-default event attributed by the arrival of new informationthat will drive the security into a default state. The relationship between the incre-mental spread behavior and the security price is as follows. Let the value of thedefault process dΨ i

τ = 1 at a random time τ . Then, given a recovery value β i , wehave


viτ,Ti=

∑τ<m≤Ti

cime−βi

e−(rτ+πkiτ )(m−τ). (4.8)

At the default announcement, the bond price will reflect the values of all pay-ments over the bond residual life discounted at a flat risky discount rate includingthe risk-free interest rate and the class ki credit spread after default. Widening creditspreads across all the rating classes will push default rates upward without discrim-inating between corporate bonds within each class. On the other hand, once defaultis triggered at τ then the marginal spread upward movement will be absorbed by abond price drop which is consistent with the postulated recovery rate.

Following (4.5) the price movement will depend on the yield jump induced by ηit .

This is assumed to be consistent in the mean with rating agencies recovery estimates(Moody’s Investors Service 2009) and carries a user-specified uncertainty reflectedin a lognormal distribution with stochastic jump size β i ∼ ln N

(μβi , σβi

). Then

ln(β i

) ∼ N(μβi , σβi

)so that β i

t ∈ (0,+∞) and both e−βi(the recovery rate) and

1− e−βi(the loss rate) belong to the interval (0, 1).

In summary the modeling framework has the following characteristic features:

• default arrivals and loss upon default are defined taking into account rating-specific estimates by Moody’s (2009) adjusted to account for a borrower’s sectorof activity and market-default premium;

• over a 1-year investment horizon default intensities are assumed constant and willdetermine the random arrival of default information to the market; upon defaultthe marginal spread will suffer a positive shock immediately absorbed by a price-negative movement reflecting the constant partial payments of the security overits residual life;

• the Poisson process and the Wiener processes drive the behavior of an indepen-dent ideosyncratic spread process which is assumed to determine the securityprice dynamics jointly with a short rate and a credit spread process;

• the credit spreads for each rating class and the short interest rate are correlated.

The model accommodates the assumption of correlated risk factors whose behaviorwill drive all securities across the different rating categories, but a specific defaultrisk factor is included to differentiate securities behavior within each class. Nocontagion phenomena are considered and the exogenous spread dynamics impactspecific bond returns depending on their duration and convexity structure.

4.2.4 The Portfolio Value Process

We now extend the statistical model to describe the key elements affecting the port-folio value dynamics over a finite time horizon T . Assuming an investment uni-verse of fixed income securities with annual coupon frequency we will have onlyone possible coupon payment and capital reimbursement over a 1-year horizon.Securities may default and generate a credit loss only at those times. We assume


a wealth equation defined at time 0 by an investment portfolio value and a cashaccount W0 = ∑

k∈K∑

i∈IkXi0 + C0, where Xi0 is the time-0 market value of

the position in security i , generated by a given nominal investment xi0 and by thecurrent security price vi

0,Ti. For t = 0,�t, . . . , T −�t , we have

Wt+�t,ω =∑k∈K

∑i∈Ik

xitvit,Ti(1+ ρi

t+�t (ω))+ C(t +�t, ω). (4.9)

In (4.9) the price return ρit+�t is generated for each security according to (4.5) as

vi (t+�t,ω)vi (t)

− 1 = [−δit dyi

t + yit dt + 0.5γ i

t (dyit )

2], where yit represents the yield

return at time t for security i . For k = 0 we have the default-free class and the yieldwill coincide with the risk-free yield. As time evolves an evaluation of credit lossesis made. Over the annual time span as before, we have

C(t +�t, ω) = C(t)ert (ω)�t +∑k∈K

∑i∈Ik

ci,t+�t +

−∑k∈K

∑i∈Ik

ci,t+�t

(1− e

[−βit+�t (ω)d�

it+�t (ω)

]). (4.10)

The cash balance C(t) evolution will depend on the expected cash inflows ci,t

and the associated credit losses for each position i , the third factor of (4.10). Couponand capital payments in (4.10) are generated for given nominal investment by thesecurity-specific coupon rates and expiry date. If a default occurs then, for a givenrecovery rate, over the planning horizon the investor will face both a cash loss in(4.10) and the market value loss of (4.9) reflecting an assumption of constant partialpayments over the security residual time span within the default state.

Equation (4.10) considers payment defaults explicitly. However, the mathemat-ical model introduced below considers an implicit default loss definition. This isobtained by introducing the coefficient specification

fi,t+�t (ω) = ci

(e−

[βi

t+�t (ω)d�it+�t (ω)

])(4.11)

and, thus, ci,t+�t = xit fi,t+�t , where ci denotes the coupon rate on security i .In this framework the corporate bond manager will seek an optimal dynamic

strategy looking for a maximum price and cash return on his portfolio and at thesame time trying to avoid possible default losses. The model is quite general andalternative specifications of the stochastic differential equations (4.2), (4.3), and(4.4) will lead to different market and loss scenarios accommodating a wide rangeof possible wealth distributions at the horizon.


4.3 Dynamic Portfolio Model

In this section we present a multistage portfolio model adapted to formulate andsolve the market and credit risk control problem. It is widely accepted that stochasticprogramming provides a very powerful paradigm in modeling all those applicationscharacterized by the joint presence of uncertainty and dynamics. In the area of port-folio management, stochastic programming has been successfully applied as wit-nessed by the large number of scientific contributions appearing in recent decades(Bertocchi et al. 2007; Dempster et al. 2003; Hochreiter and Pflug 2007; Zeniosand Ziemba 2007). However, few contributions deal with the credit and market riskportfolio optimization problem within a stochastic programming framework. Herewe mention (Jobst et al. 2006) (see also the references therein) where the authorspresent a stochastic programming index tracking model for fixed income securities.

Our model is along the same lines, however, introducing a risk-rewardtrade-off in the objective function to explicitly account for the downside generatedby corporate defaults.

We consider a 1-year planning horizon divided in the periods t = 0, . . . , T cor-responding to the trading dates. For each time t , we denote by Nt the set of nodes atstage t . The root node is labeled with 0 and corresponds to the initial state. For t ≥ 1every n ∈ Nt has a unique ancestor an ∈ Nt−1, and for t ≤ T − 1 a non-empty setof child nodes Hn ∈ Nt+1. We denote by N the whole set of nodes in the scenariotree. In addition, we refer by tn the time stage of node n. A scenario is a path fromthe root to a leaf node and represents a joint realization of the random variablesalong the path to the planning horizon. We shall denote by S the set of scenarios.Figure 4.1 depicts an example scenario tree.

The scenario probability distribution P is defined on the leaf nodes of thescenario tree so that

∑n∈NT

pn = 1 and for each non-terminal node pn =

Fig. 4.1 A sample scenario tree


∑m∈Hn

pm,∀n ∈ Nt , t = T − 1, .., 0, that is each node receives a conditionalprobability mass equal to the combined mass of all its descendant nodes. In our casestudy portfolio revisions imply a transition from the previous time t − 1 portfolioallocation at the ancestor node to a new allocation through holding, buying, andselling decisions on individual securities, for t = 1, . . . , T − 1. The last possiblerevision is at stage T − 1 with one period to go. Consistent with the fixed-incomeportfolio problem, decisions are described by nominal, face-value, positions in theindividual security i ∈ I of rating class ki ∈ K .

4.3.1 Parameters

Let xi be the initial holding in bond i ∈ I and let C0 and g denote the initial cashand debt, respectively. The random coefficients in the optimization problem include

rn the one period interest rate for cash deposits available at node n;πk

n the credit spread of rating class k at node n;ηi

n the bond-specific incremental spread of bond i at node n;bn the one period interest rate on borrowing decisions taken at node n;vin the market price of bond i at node n, generated given the initial price vi0 as

a function of the security-specific yield dynamics (cf. (4.5));fin the cash flow generated by bond i at node n following (4.11) which takes

into account possible cash defaults over the planning horizon.

With rn , πkn , and ηi

n as state variables in the statistical model, for given initial valuesvi0, r0, π

k0 , η

i0, i ∈ I , for t = 1, 2, .., T and n ∈ Nt we have

vin = vian

[1− δi

andyi

an+ yi

andt + 0.5γ i

an(dyi

an)2], (4.12)

where dyian= yi

n − yian=

[(rn − ran

)+ (π

kin − πki

an

)+ (ηi

n − ηian

)]is defined up

to the end of the last stage. A default event in node n will simultaneously affectthe dηi

anincrement and thus the bond return in the given subperiod, and the cash

account through the coefficient fin specified in the previous section.For a given tree structure, following the corresponding set of conditional sample

paths, an update of the duration and convexity coefficients is needed at each nodebefore the next price transition is generated according to

δin =

∑tn≤m≤Ti

((m − tn) cine−yi

n(m−tn))

(vi

n

) , (4.13)

γ in =

∑tn≤m≤Ti

((m − tn)2 cine−yi

n(m−tn))

(vi

n

) , (4.14)


where tn denotes the time stage of node n and m defines the expected payment datesfor bond i up to maturity Ti .

A set of stage-independent upper bounds is considered to reflect exogenouspolicy constraints imposed by the investment manager to limit the portfolio riskexposure. The maximum investment-grade portfolio fraction is denoted by φ, whileζ identifies the maximum speculative-grade portfolio fraction. Parameter νk rep-resents the maximum portfolio fraction for bonds belonging to rating class k. Anupper bound for the debt level with the bank is also introduced in the model by theparameter γ . Finally, χ+ and χ− are proportional buying and selling transactioncosts.

Decision Variables

For each node n of the scenario tree and each bond i we define

xin nominal amount (in euros) held in bond i at node n;x+in buying decision (nominal amount) on bond i at node n;x−in selling decision on bond i at node n.

In addition, we denote by gn and zn the nominal debt and the amount invested incash at node n, respectively. All the decision variables are constrained to be non-negative.

4.3.2 Constraints

The portfolio composition is optimized under various restrictions. The modelincludes the classical inventory balance constraints on the nominal amount investedin each bond, for each node of the scenario tree:

xi0 = xi + x+i0 − x−i0 ∀i ∈ I, (4.15)

xin = xian + x+in − x−in ∀i ∈ I, ∀n ∈ N . (4.16)

The cash balance constraint is imposed for the first and later stages:

∑i∈Ivi0x+i0(1+ χ+)+ z0 − g0

= C0 − g +∑i∈Ivi0x−i0(1− χ−), (4.17)

∑i∈Ivin x+in(1+ χ+)+ zn − gn

= ∑i∈Ivin x−in(1− χ−)+ zan eran

−gan eban +∑i∈I

fin xian ∀n ∈ N . (4.18)


The model also includes constraints bounding the amount invested in each rat-ing class (4.19) as well as in investment grade (4.20) and speculative grade (4.21)classes, respectively, as fractions of the current portfolio value:

∑i∈Ik

vin xin ≤ νk∑i∈Ivin xin ∀k ∈ K , ∀n ∈ N , (4.19)

4∑k=0

∑i∈Ik

vin xin ≤ φ∑i∈Ivin xin ∀n ∈ N , (4.20)

7∑k=5

∑i∈Ik

vin xin ≤ ζ ∑i∈Ivin xin ∀n ∈ N . (4.21)

Finally, a limit on the debt level for each node of the scenario tree is imposed:

gn ≤ γ (C0 +∑i∈I vi0 xi − g) ∀n ∈ N . (4.22)

4.3.3 Objective Function

The goal of financial planning is twofold: maximize the expected wealth generatedby the investment strategy while controlling the market and credit risk exposureof the portfolio. This trade-off can be mathematically represented by adopting arisk-reward objective function:

max(1− α)E[Wn] − αθε[Wn], (4.23)

where α is a user-defined parameter accounting for the risk aversion attitude andn are the leaf nodes (n ∈ NT ) of the scenario tree. The higher the α the moreconservative, but also the less profitable, the suggested financial plan. The first termof (4.23) denotes the expected value of terminal wealth, computed as

E[Wn] =∑

n∈NT

pnWn, (4.24)

where the wealth at each node n is

Wn =∑i∈I

vin xin + zn − gn . (4.25)

The second term in (4.23) accounts for risk. In particular, we have considered theconditional value at risk (CVaR) at a given confidence level ε (usually 95%). CVaRmeasures the expected value of losses exceeding the value at risk (VaR). It is a“coherent” risk measure, suitable for asymmetric distributions and thus able to con-trol the downside risk exposure. In addition, it enjoys nice computational properties(Andersson et al. 2001; Artzner et al. 1999; Rockafellar and Uryasev 2002) and


admits a simple linear reformulation. In tree notation the CVaR of the portfolioterminal wealth can be defined as

θε = ξε + 1

1− ε∑

n∈NT

pn [Ln − ξε]+ , (4.26)

where ξε denotes the VaR at the same confidence level. Here Ln represents theloss at node n, measured as the negative deviation from a given target value of theportfolio terminal wealth:

Ln = max[0, Wn −Wn

], (4.27)

where Wn represents a reference value, computed on the initial wealth 1-year com-pounded value for given current (known) risk-free rate:

W0 = (C0 +∑i∈I vi0 xi − g), (4.28)

Wn =W0er0tn ∀n �= 0. (4.29)

The overall objective function can be linearized through a set of auxiliary vari-ables (ςn) and the constraints as follows:

θε = ξε + 1

1− ε∑

n∈NT

pnςn, (4.30)

ςn ≥ Ln − ξε ∀n ∈ NT , (4.31)

ςn ≥ 0 ∀n ∈ NT . (4.32)

This model specification leads to an easily solvable large-scale linear multi-stagestochastic programming problem. Decisions at any node explicitly depend on thecorresponding postulated realization for the random variables and have a directimpact on the decisions at descendant nodes. Depending on the number of nodes inthe scenario tree, the model can become very large calling for the use of specializedsolution methods.

4.4 Scenario Generation

The bond portfolio model implementation requires the definition of an extended setof random coefficients (Dupacová et al. 2001; Heitsch and Römisch 2003; Pflug2001). We focus in this section on the relationship between the statistical model


actually implemented to develop the case study and the set of scenario-dependentcoefficients included in the stochastic programming formulation.

The statistical model drives the bond returns over the planning horizon 0, . . . , T .At time 0 all coefficients in (4.2), (4.3), (4.4), (4.5), (4.7), and (4.8) have beenestimated and, for a given initial portfolio, Monte Carlo simulation can be usedto estimate the credit risk exposure of the current portfolio. In this applicationa simple random sampling algorithm for a fixed, pre-defined, tree structure isadopted to define the event tree structure underlying the stochastic programmingformulation.

A method of moments (Campbell et al. 1997) estimation is first performedon (4.2), (4.3), and (4.4) from historical data, then Moody’s statistics (Moody’sInvestors Service 2009) are used to estimate the natural default probability and therecovery rates, which are calibrated following (4.7), to account for recent marketevidence and economic activity, and (4.8), allowing a limited dispersion from theaverage class-specific recovery rates.

In the case study implementation the risk-free rate and the credit spread processesare modeled as correlated square root processes according to the tree specification,for s ∈ S, t = 1, . . . , T, n ∈ Nt , where hn denotes the child node in the givenscenario s. For each k ∈ K and initial states r0 and πk

0 , we have

�rn = μr (thn − tn)+ σ r√rn√

thn − tnen,

�πkn = μk(thn − tn)+ σ k

√πk

n

√thn − tn

∑l∈K

qkln el

n . (4.33)

In (4.33), eln ∼ N (0, 1), for l = 0, 1, .., 7, independently and qkl denote the

Choleski coefficients in the lower triangular decomposition of the correlationmatrix. The nodal realizations of the risk-free rate and the credit spreads are alsoused to identify the investor’s borrowing rate bn = rn + π k

n in the dynamic modelimplementation, where k denotes the investors specific rating class.

The incremental spread ηin for security i has been implemented in the case study

as a pure jump-to-default process with null mean and volatility. The associatedidlosyncratic tree processes, all independent from each other, will in this case forall i follow the dynamic

dηin = β i

nd� i (λi , n) (4.34)

with constant default intensity and random market impact on the security-specificspread. For given initial security prices and any specified tree processes it is possibleto derive the input coefficients to the portfolio problem.

The procedure can be sketched as follows:


Corporate Bonds Scenario Generation

given vi0, yk

0 = r0 + πk0 , yi

0 = yk0 + ηi

0,∀i, kfor t = 1, ..., T − 1

for n ∈ Nt

generate rn, πkn correlated

beginfor i = 1, ..., I

sample d� in ∈ Poisson(λi )

if d� in = 0, dηi

n = 0, yin = yk

nderive dyi

n, vin, δ

in, γ

in , fin

elseif d� in = 1

sample β in ∈ Ln(β i , σβi )

generate dηin = β i

n, dyin = dyk

n + dηin

derive vin, δ

in, γ

in , fin

end ifend for i

end for nend for t

Following the above assumptions we can summarize the key steps supporting thescenario generation algorithm for the credit risk management problem. Inputs to theprocedure are

• the investment universe prices at time 0 and the associated duration and convexitypairs;

• the value at time 0 of 1-year risk-free rate and credit spreads, one for every ratingclass k ∈ K together with the set of statistical coefficients in (4.33);

• the time 0 security-specific marginal spreads and the shock distributions (jumpmagnitude distribution and intensity).

4.5 Computational Experiments

In this section, we present the results obtained by considering a 1-year planning hori-zon (starting from June 2, 2008) with monthly rebalancing. The universe of financialinstruments contains a risk-free treasury security and 36 corporate Eurobonds span-ning the Standard and Poor’s rating classes. Details are reported in Table 4.1. Inparticular, for each bond i ∈ I , we report the ISIN code, the issuer rating class, thesector of activity and geographic area, the coupon rate, its frequency, and maturity.

The implementation relies on two main phases strictly interconnected: scenariogeneration, which has been performed by using MATLAB R2009B,1 and the for-

1 www.mathworks.com

www.mathworks.com


Tabl

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rate

(%)

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urity

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man

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Tabl

e4.

1(c

ontin

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ISIN

Gua

rant

orSe

ctor

Cou

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ing

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rate

(%)

CPN

frq

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mulation and solution of the stochastic programming problem, which has been per-formed by GAMS 23.42 with CPLEX 123 as its solution kernel. All the computa-tional experiments have been carried out on a DELL workstation with an INTELXEON quadcore 2 GHz processor and 8 GB RAM. The final aim of the extensivetesting phase has been to investigate the effectiveness of the stochastic programmingformulation as a decision tool for simultaneously controlling market and credit risk,in comparison with other strategies.

In the case study we have considered a maximum debt level γ = 20%, amaximum investment grade fraction φ = 70%, a maximum speculative gradefraction ζ = 70%, and a maximum fraction of investment in each rating classνk = 30%, k ∈ K . Transaction costs for both buying and selling are assumed con-stant and equal to 0.5% of the bond value. The initial holdings consist of a portfoliowith an equal allocation in each bond, with no initial cash or debt.

4.5.1 Scenario Analysis

Scenario generation has been performed by implementing the procedures describedin Section 4.4. The following tables specify the parameters adopted in the simulationprocess. In particular, Table 4.2 reports the values of the correlation matrix amongthe risk-free rate and the spread for each rating class, together with the adopteddefault intensities.

Further, in Table 4.3, we report the weights used to compute the default intensityof each bond (last column of Table 4.1) as a function of the guarantor economic sec-tor and geographic region, with ξ i in (4.7) determined by multiplying the specifiedcoefficients for each sector–region pair.

Figure 4.2 shows the scenario trees generated for a low-risk (AA) and high-risk(CCC−C) bond, respectively. Unlike the AA bond, the CCC−C security is affectedby a default event on several scenarios.

Scenario generation represents a critical issue in any stochastic programmingformulation since the quality of the generated scenarios heavily affects the quality of

Table 4.2 Correlation matrix and Moody’s rating homogeneous default intensities

RF AAA AA A BBB BB B CCC−CDefaultintensity

RF 1 0.0005 0.3044 0.2865 −0.0022 −0.0937 −0.16 −0.153 0AAA 1 0.903 0.904 0.923 0.835 0.919 0.896 0AA 1 0.991 0.899 0.833 0.845 0.83 0.0002A 1 0.915 0.829 0.853 0.844 0.0004BBB 1 0.863 0.936 0.945 0.0048BB 1 0.936 0.896 0.0243B 1 0.977 0.134CCC−C 1 0.29

2 www.gams.com3 http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/

www.gams.com

http://www-01.ibm.com/software/integration/optimization/cplex-optimizer/


Table 4.3 Heuristic default intensity weights for economic sector and geographic region

Sector Weight Region Weight

Automotive 1.25 Europe 1Telecommunications 1 UK 1.25Utility 0.75 USA 1.25Manufacture 1 Emerging markets 1.25Financial 1.5

Fig. 4.2 AA and CCC−C bond scenario tree

the recommendations provided by the model. With the aim of assessing the robust-ness of the stochastic programming approach, additional tests have been performedto measure the so-called in-sample stability. This is evaluated by generating severalscenario trees and solving the corresponding stochastic programming problem foreach tree.

Figure 4.3 reports, for an increasing set of scenarios, the values of the expectedfinal wealth and CVaR generated by each problem solution. The center of the circlerepresents the average value, whereas the radius measures the standard deviation.


Fig. 4.3 Solutions stability

As expected the bigger the tree, the lower the radius and thus the solution pairdispersion. The analysis clearly shows that robust solutions may be obtained withscenario trees of limited size. Following these preliminary results, we consider inour experiments a 1024 scenario tree with 2071 nodes. The resulting mathematicalproblem contains 227,810 variables and 97,374 constraints.

4.5.2 Portfolio Strategies

We analyze the optimization output for different levels of the risk aversionparameter α. Figure 4.4 shows the trade-off between the expected terminal wealthand the CVaR risk measure as α increases from 0.01 to 0.99. Each point on the fron-tier is generated by a stochastic program solution with an underlying 1024 scenariotree over the 12 stages.

In Table 4.4 we show how the portfolio composition determined at the first stage(referred in the following as implementable or here-and-now solution) changes fordifferent values of the risk aversion parameter.

As α increases, the model solution reflects the agent’s increasing risk aversion.For α = 0.99 in particular the fraction invested in investment grade securities is60%, whereas for α = 0.1, speculative grade securities span 46% of the optimalportfolio value.

Figure 4.5 reports the wealth evolution along specific scenarios prior and afteroptimization to clarify the theoretical impact of the optimal multistage strategy onthe investor wealth evolution. For the given statistical model, the do nothing against


Fig. 4.4 Risk-return trade-off

Table 4.4 Implementable solution for different α values

Rating class α = 0.99 α = 0.5 α = 0.1

AAA 20% 22% 0%AA 0% 0% 0%A 20% 24% 20%BBB 20% 18% 34%BB 0% 18% 29%B 2% 0% 0%CCC−C 4% 0% 17%Cash 34% 18% 0%

the dynamic strategy outputs are compared in order to clarify the financial improve-ment expected to be brought about by the dynamic strategy.

Figures 4.3, 4.4, 4.5, and 4.6 and Table 4.4 present a set of results associatedwith multistage stochastic program defined over a 1-year planning horizon startingon June 2, 2008. Next we consider the results of adopting a 12-month rolling win-dow with monthly forward updates of the scenario tree and the associated stochasticcoefficients definition to analyze the risk control performance induced by a sequenceof H&N solutions over the most severe period of the crisis. An average risk-averseinvestor with α = 0.5 was considered. Figure 4.7 shows how the optimal H&Nportfolio composition changed during the test period. Without imposing specificconstraints on the portfolio policy, highly volatile financial markets lead to substan-tial portfolio revisions from one stage to the other.

Figure 4.8, finally, shows the out-of-sample market performance of the opti-mal H&N strategies evaluated on realized historical prices. For comparative pur-poses, the results have been computed for different levels of risk aversion α on a


Fig. 4.5 Wealth evolution along the scenario tree

Fig. 4.6 Distribution of the final wealth


Fig. 4.7 Portfolio evolution

normalized portfolio value of 100 at the beginning of June 2008. In the same Fig. 4.8we report, for relative valuation purposes, the performance of the initial, perfectlydiversified, portfolio and the IBRCPAL corporate bond index, a benchmark for cor-porate bond traders in the Euro market.4

Fig. 4.8 Backtesting analysis

4 http://kbc-pdf.kbc.be/ng/feed/am/funds/star/fa/BE0168961846.pdf

http://kbc-pdf.kbc.be/ng/feed/am/funds/star/fa/BE0168961846.pdf


The best- and worst-case possible portfolio outcomes are also displayed for thegiven investment universe considering ex post the bonds price evolution: a maxi-mum 70% loss of the initial portfolio value would have been possible between June2008 and March 2009 adopting the worst possible portfolio policy and, conversely,under perfect foresight a maximum 31% profit might have been achieved. For differ-ent degrees of risk aversion the figure shows that the set of H&N optimal decisionswould have protected the portfolio performance for a highly risk-averse investor byoverperforming the market index, while as the risk coefficient decreases the backtest shows that a risk-seeking investor optimal strategy would have led to roughly a20% loss over the period.

4.6 Conclusions

The chapter focuses on the potential of a multistage stochastic program formulationof a bond portfolio management problem including securities exposed to marketand credit risk factors. The recent credit crisis has lead to unprecedented portfoliolosses in the market of corporate liabilities and represents an ideal stress period foradvanced modeling paradigms at a time in which not only the market suffered asevere crisis but also the credit derivative market became very illiquid and madeclassical hedging impractical.

The formulation of an optimal dynamic strategy in the corporate bond market hasbeen evaluated over the June 2008 to March 2009 period relying on a risk model inwhich the specific credit risk carried by individual securities has been separatedfrom the rating-based credit risk factors affecting all securities within a rating class.No transition risk has been considered in the analysis and the bond risk exposurehas been modeled through a jump-to-default model calibrated on available Moody’sdefault statistics (Moody’s Investors Service 2009). Consistent with industry stan-dards, the final assessment of the individual borrower’s default risk has been basedon available sector and rating-specific default data enriched with credit analysts’views, here reflected in a simple heuristic applied to the available default estimates.

The evidence suggests that a multistage strategy for a sufficiently large invest-ment universe would have protected the investors from market and credit lossesduring the worst part of the crisis.

Acknowledgments The authors acknowledge the support given by the research grant PRIN2007“Optimization of stochastic dynamic systems with applications to finance,” no. 20073BZ5A5,sci.resp. G. Consigli.

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