Pricing Recovery Risk in Bonds and Swaps · Pricing Recovery Risk in Bonds and Swaps Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and

Pricing Recovery Risk in Bonds and Swaps

Albert Cohen

Actuarial Sciences ProgramDepartment of Mathematics

Department of Statistics and ProbabilityMichigan State University

East Lansing [email protected]

http://actuarialscience.natsci.msu.edu

Travelers Research SymposiaApril 26, 2019

The Peter J. Tobin College of BusinessSt.John’s University

Queens, NY

Albert Cohen (MSU) Pricing Recovery Risk in Bonds and Swaps Travelers Research Symposia 1 / 41

Outline

1 Review of Some Structural ModelsFundamentals of Structural ModelsMerton ModelBlack-Cox Model

2 Stochastic RecoveryCorrelated Asset-Recovery ModelBond Price - SRBC ModelCDS - SRBC Model

3 Connection with Change of MeasureRecovery Risk and Default Risk under Change of Measure

4 Numerical AlgorithmMatching Stochastic Recovery Model with Input ValuesBenchmark Example


Single-Name Default Models

Single-name default models typically fall into one of three main categories:

Structural Models. Attempts to explain default in terms offundamental properties, such as the firms balance sheet and economicconditions (Merton 1974, Black-Cox 1976, Leland 1994, etc..)

An excellent resource on structural models in corporate finance is thelecture series by Hayne Leland, including ”Lecture 1: Pros and Consof Structural Models: An Introduction.”

Reduced Form (Intensity) Models. Directly postulates a model forthe instantaneous probability of default via an exogenous process λt(Jarrow-Turnbul 1995, Duffie-Singleton 1999, etc..) via

P[τ ∈ [t, t + dt)|Ft ] = λtdt. (1)

Hybrid Models. Incorporates features from structural andreduced-form models by postulating that the default intensity is afunction of the stock or of firm value (Madan-Unal 2000,Atlan-Leblanc 2005, Carr-Linetsky 2006, etc..)


Simplest Framework for Structural Models

The idea pioneered by Merton 1 is to model equity Et,T at time t as acall option on the assets A of the firm at expiry T of thezero-coupon, notional N bond issued by the firm :

Assume Modigliani-Miller Theorem holds: The value of the firm isinvariant to its capital structure (debt B to equity E :)

At = Et,T + Bt,T . (2)

Equity is a call option on assets with notional N as strike:

EMertont,T = E

[e−r(T−t)(AT − N)+ | At = A

]. (3)

With a tractable debt structure, investors are able to compute defaultprobability, bond price, credit spreads, recovery rates, etc..

1R. Merton, On the Pricing of Corporate Debt: the Risk Structure of Interest Rates,Journal of Finance 29, 1974, pp. 449-470.


Merton Model Asset Evolution

Assume a probability space (Ω,F ,P):

Underlying asset is modeled as GBM under a physical measure P:

dAt = µAAtdt + σAAtdWAt . (4)

Default is implicitly assumed to coincide with the event AT ≥ Nc :

τMerton = T1AT<N +∞1AT≥N. (5)

This results in a turnover of the company’s assets to bondholders ifassets are worth less than the total value of bond outstanding.

At maturity, the bond value (payoff) is

BMertonT ,T = AT1AT<N + N1AT≥N. (6)

For pricing purposes, assume a risk-neutral probability measure P.


Merton Model Bond Price

Consequently, using the Feynmann-Kac approach to solution viaexpectation, we obtain

BMertont,T = e−r(T−t)Et [B

MertonT ,T ] = e−r(T−t)Et [AT1AT<N + N1AT≥N]

= e−r(T−t)

[∫ N

0A · Pt [AT ∈ dA] + N

∫ ∞N

Pt [AT ∈ dA]

]

= e−r(T−t)

[Et [AT | AT < N] · Pt [AT < N] + N · Pt [AT ≥ N]

]

= Ne−r(T−t)

[1− Et

[(N − AT

N

) ∣∣∣AT < N

]· Pt [AT < N]

]:= Ne−r(T−t)

[1− Et [Loss | Default] · Pt [Default]

].

(7)


Merton Model PD and LGD

BMertont,T = Ne−r(T−t)Φ(d0) + AtΦ(−d1)

PDMertont,T = Pt [AT < N] = Φ(−d0)

LGDMertont,T = Et

[(N − AT

N

) ∣∣∣AT < N

]= 1− er(T−t)

At

N

Φ(−d1)

Φ(−d0)

d1 =ln(At/N) + (r + 1

2σ2A)(T − t)

σA√T − t

= d+

d0 =ln(At/N) + (r − 1

2σ2A)(T − t)

σA√T − t

= d−

Φ(x) :=1√2π

∫ x

−∞e−

u2

2 du.

(8)


Merton Model Credit Spread

The yield-to-maturity credit spread Yt,T is defined as the spread overthe risk free rate r which prices the bond as

Bt,T = Ne−(r+Yt,T )(T−t). (9)

Solving for Y yields

Yt,T =1

T − tln

(N

Bt,T

)− r . (10)

For the Merton Model,

YMertont,T = − 1

T − tln

[Φ(d0) +

At

Ner(T−t)Φ(−d1)

]. (11)

Assumes only one possible time for default.


Black-Cox Model (Bondholder Covenant)

Same asset dynamics as in Merton model:

dAt = µAAtdt + σAAtdWAt .

Default can happen at times other than maturity, in particular whenfirm assets fall below a prescribed default point (barrier) K :

τBC > T = τK > T , τMerton > TτK = inft ≥ 0 : At ≤ K

τMerton = T1AT≥Nc +∞1AT≥N.

(12)


Black-Cox Model (Bondholder Covenant)

In the original Black-Cox paper 2, K was taken to beK (t) = Ke−α(T−t). We assume that α = 0.

Stochastic Approach: At times t ≤ T , we define and solve

BBCT ,T = N1τK>T ,AT≥N + AT1τK>T ,AT≥Nc

BBCt,T = e−r(T−t)Et [B

BCT ,T ].

(13)

2F. Black & Cox, J.C., Valuing Corporate Securities: Some Effects of BondIndenture Provisions, Journal of Finance 31, 1976, 351-367.


Merton / Black-Cox Model Criticisms

Information gap arises from estimating (A, σA), leading to surprises indefault time (Default Risk) and recovery amount (Recovery Risk.)

Single factor model combines Recovery Risk with Default Risk.

As there is only one source of risk (A) for both debt and equity, thismodel assumes that debt and equity are perfectly correlated. Samefor PD-LGD correlation.

Credit Risk is often misestimated 3.

Would be nice to decouple the default and recovery drivers 4.

What do empirical recovery rate-to-time plots look like?

3V.V. Acharya, T.S. Bharath, & A. Srinivasan, Does industry-wide distress affectdefaulted firms? Evidence from creditor recoveries, Journal of Financial Economics 85(2007) 787-821.

4G. Giese, The Impact of PD/LGD Correlations on Credit Risk Capital, Risk, April2005, pp 79-85.


http://www.nber.org/papers/w16151

Bond Price under Correlated A-R Model

Define the quantities:

At , the asset value at time t > 0

Rt , the recovery amount at time t > 0. Recovery amount evolves in amanner that is correlated to asset dynamics.

The dynamics for the asset and recovery 5 are, under a risk neutralmeasure P, modeled to be

dAt

At= rdt + σAdW

At

dRt

Rt= rdt + σRdW

Rt

dW At dW R

t = ρA,Rdt.

(14)

5A. Levy & A. Hu, Incorporating Systematic Risk in Recovery: Theory andEvidence, Modeling Methodology, Moody’s KMV, 2007.


Stochastic Recovery Black-Cox Model (SRBC)

Let K define our (level) default point written into the bondholdercovenant on the asset A, and define τK as the first time A reachesthis default point.

The default time τ can also be defined, as in the Black-Cox model,via the event Default ∈ FT :

Default = τK > T ,AT ≥ Nc . (15)

An important quantity is γ := ρA,RσRσA

, which factors heavily into ourSRBC bond price. Beyond the usual measure of sensitivity ofrecovery relative to the underlying asset, γ reflects the extra riskinherent in decoupling loss given default from probability of default.


Stochastic Recovery Black-Cox Model (SRBC)

This leads to a barrier-option interpretation of debt, one that carriesover into a decoupled Stochastic Recovery Black-Cox (SRBC) Modelthat allows for partial information to play a role in pricing andestimating credit risk while allowing for default before expiry.

Stochastic Approach: At times t ≤ T , we define

BSRBCT ,T = N1τK>T ,AT≥N + RT1τK>T ,AT≥Nc

BSRBCt,T = e−r(T−t)Et [B

SRBCT ,T ].

(16)


Main Theorem: Bond Price under SRBC model.

Theorem

(Bond Price under Stochastic Recovery Black Cox Model) a

I. Weak Covenant Case. If K ≤ N the price of a zero-coupon bond isgiven by

BSRBCt,T (K ) = Ne−r(T−t)

[Φ(dw

0 )−(K

At

) 2r

σ2A

−1Φ(xw0 )

]

+ Rt

[Φ(−dw

γ ) +

(K

At

) 2r

σ2A

+2γ−1Φ(xwγ )

]BSRMt,T = BSRBC

t,T (0) = Ne−r(T−t)Φ(dw0 ) + RtΦ(−dw

γ ).

(17)

aA. C. & N. Costanzino Bond and CDS Pricing via the StochasticRecovery Black-Cox Model, Risks. Vol. 5, No. 26. 2017.


Distance-to-Default

In the above closed form solutions for the SRBC bond prices, we use

dw0 =

ln(AtN

)+ (r − 1

2σ2A)(T − t)

σA√T − t

d s0 =

ln(AtK

)+ (r − 1

2σ2A)(T − t)

σA√T − t

xw0 =ln(

K2

NAt

)+ (r − 1

2σ2A)(T − t)

σA√T − t

x s0 =ln(

KAt

)+ (r − 1

2σ2A)(T − t)

σA√T − t

.

(18)


Distance-to-Default Under Partial Information

Adjusting for recovery, we also have

dwγ = dw

0 + γσA√T − t

d sγ = d s

0 + γσA√T − t

xwγ = xw0 + γσA√T − t

x sγ = x s0 + γσA√T − t.

(19)


Distance-to-Default Under Partial Information

The risk-adjusted SRBC distances-to-default dγ , xγ in (19) reduce tothe standard distances-to-default (d0, x0) and (d1, x1) of the BCmodel if γ = 0 or γ = 1, respectively.

This reflects the uncertainty of the firm manager in the partialinformation setting of what the recoverable value of the firm’s assetstruly are, and affects only the recovery term.

The probability of default in the SRBC model is the same as in theBC model.


CDS: SRBC Model

Extending the BC CDS 6 premium to include stochastic recovery results in

PSRBCt,T =

Et

[D(t, τBC)

(1− RτBC

N

)1τBC≤T

]∫ Tt D(t, s)Pt [τBC > s]ds + 1

2

∫ Tt D(t, s)Pt [τBC ∈ ds]

. (20)

The quantities in (20) are also computed in closed form 7

6I.H. Gokgoz, O. Ugur, & Y. Okur, On the single name CDS price under structuralmodeling, Journal of Computational and Applied Mathematics, Volume 259, Part B, 15March 2014, pp 406-412.

7A. C. & N. Costanzino Bond and CDS Pricing via the Stochastic RecoveryBlack-Cox Model, Risks. Vol. 5, No. 2. 2017.


Recovery Risk and Default Risk under Change of Measure

In order to incorporate recovery risk into structural models andunderstand it’s effect on the credit spreads, a coupled stochasticrecovery risk driver R was added to the classical one-dimensionalstructural models of Merton and Black-Cox.

To enable this extension, the asset process A becomes the jointprocess of (A,R) which lives in a filtered probability space(Ω,F , Ftt≥0 ,P) satisfying the usual conditions.

In this setting, we assume that P is a risk-neutral measure.



If we further define the default event as a set D ∈ FAT ⊆ FT ⊆ F , then

the price BNRt,T of a bond is the exchange of asset for notional in the

traditional structural models without stochastic recovery (NR):

BNRt,T = e−r(T−t)EP[AT1D + N1Dc | FA

t ]. (21)



Under the stochastic recovery framework (SR), specifically for Merton andBlack-Cox models, the price of a defaultable bond was shown to be similarin nature to the traditional structural models without stochastic recovery:

BSRt,T = e−r(T−t)EP[RT1D + N1Dc | Ft ]. (22)

By appealing to barrier option theory, Ishizaka & Takaoka 8 define a newmeasure Q with Radon-Nikodym density process defined by

Z :=dQdP

= e−r(T−t)AT

At. (23)

8M. Ishizaka & K. Takaoka, On the Pricing of Defaultable Bonds Using theFramework of Barrier Options, Asia-Pacific Financial Markets, Volume 10, 2003, pp151-162.



We set the notation above as

Pt [·] := P[· | FAt ]

Qt [·] := Q[· | FAt ].

(24)

It follows that the bond price without stochastic recovery can be written as

BNRt,T = e−r(T−t)N · Pt [D

c ] + At ·Qt [D]. (25)



This line of reasoning, that of changing numeraire, can be carried over intothe case where recovery is used as the numeraire, to build a new measureQ∗ to estimate probability of default, with Radon-Nikodym density processand conditional probability defined by

Z ∗ :=dQ∗

dP= e−r(T−t)

RT

Rt

Pt [·] := P[· | Ft ]

Q∗t [·] = Q∗[· | Ft ].

(26)



Armed with this measure, we use recovery R to estimate the probability ofdefault triggered by asset value A. We presented closed form solutions inthe Merton and Black-Cox cases, which can now be shown to equal

BSRt,T = e−r(T−t)N · Pt [D

c ] + Rt ·Q∗t [D].

= e−r(T−t)N · Pt [Dc ] + Rt ·Q∗t [D].

(27)


Connection with Distance-to-Default

Note that recovery R is only partially informed by asset level A, andso using R as the numeraire reflects a risk-adjustment in estimatingthe probability of default.

We also point out that since the default event (trigger) remains in thesmaller filtration FA

t ⊆ Ft , the default-free part of the bond price(27) remains the same as it was in the structural model withoutrecovery (22).

The effect of incorporating stochastic recovery into the model isreflected in the new estimate Q∗t [D] for probability of default.


Partial Information Transform

Definition

We define the Partial Information Transform (PIT) of Pt [τ ≤ T ] for amodel where assets A evolve via a continuous Geometric Brownian Motion(GBM) and default is a set D ∈ FA

T , to be the transform which mapsPt [τ ≤ T ]→ Q∗t [τ ≤ T ] a .

aA. C. and Nick Costanzino (2017) A General Framework forIncorporating Stochastic Recovery in Structural Models of Credit Risk, Risks.Vol. 5, No. 26. 2017.


Partial Information Transform

For the two-factor Asset Recovery model, we can compute the PIT via thefollowing Lemma:

Lemma

Q∗t [D] = Q∗t [τ ≤ T ] :=e [

12γ(1−γ)σ2

A−γr ](T−t)

AγtEP[AγT1τ≤T | FA

t

]. (28)

The quantity (28) also represents the solution to a two-factorbond-pricing PDE, and is highly dependent on the boundaryconditions representing default covenants (i.e. the set D ∈ FA

T .)

The parameter γ, to be estimated from market data, is the only termthat relates to post-default recovery R in the Partial InformationTransform.Note that when

γ = 1, we return the value Q∗t [τ ≤ T ] = Qt [τ ≤ T ].

γ = 0, we return the value Q∗t [τ ≤ T ] = Pt [τ ≤ T ].


Merton Revisited

For example, in the Merton Case the default set D := AT < N and sousing the result e−r(T−t)Et

[RT1AT<N

]= RtΦ(−dγ), we can compute

the asset and recovery numeraire defined probability of default as(PMerton,rt [D],QMerton,r

t [D])

= (Φ(−d0),Φ(−d1))

Q∗t [D] = Q∗t [AT < N] = e−r(T−t)Et

[RT

Rt1AT<N

]=

1

Rte−r(T−t)Et

[RT1AT<N

]=

1

RtRtΦ(−dγ) = Φ(−dγ).

(29)

In terms of transformations, we can write

Q∗t [D] = Φ(

Φ−1(PMerton,rt [D]

)− γσA

√T − t

)= Φ

(Φ−1

(QMerton,r

t [D])

+ (1− γ)σA√T − t

).

(30)


Matching Equity

In the Stochastic Recovery Merton framework, we assume thatequity is a call on recovery and not assets:

ESRMt,T := e−r(T−t)Et [(RT − N)+] . (31)

The next step is to calibrate Rt and σR . Since in the model these aremarket defined quantities, they must be calibrated using market data.

Hence, we use the equity and equity volatility to set up two equationsfor the two unknowns:

EMarkett = ESRM

t (R, σR) = RtΦ(dR1 )− Ne−r(T−t)Φ(dR

0 )

σMarketE EMarket

t = σRRtΦ(dR1 )

(32)

where σMarketE and EMarket

t are the equity and volatility observeddirectly from the market. This procedure yields calibrated values Rt

and σR .


Matching Bonds

Finally, we set up two equations for the two remaining unknowns(d0, ρA,R). The two coupled equations are,

BMarkett,T = BSRM

t,T (A, σA, ρA,R ,Rt , σR)

= Ne−r(T−t)Φ(d0) + RtΦ(−dγ)(σMarketB

)2=(σSRMB (A, σA, ρA,R ,Rt , σR)

)2= Ω2

Aσ2A + Ω2

Rσ2R + 2ΩAΩRρA,RσAσR .

(33)

Our flow in the algorithm is thus

(N)→ (R, σR)→ (d0, ρA,R). (34)


Physical vs Risk-Neutral Quantities

As equity is modeled as a call option on assets, we have that the Sharperatios of the equity and recovery are the same. It can be shown that

µB − r = ΩA(µA − r) + ΩR(µR − r) (35)

and so we can estimate directly from market data:

µR − r

σR=µE − r

σE

µA − r

σA=

(µB − r)− ΩRσRµE−rσE

ΩAσA.

(36)


Physical vs Risk-Neutral Quantities

The Sharpe ratio for assets A also enables us to convert from risk-neutraldistance-to-default to physical distance-to-default via

dµA0 = d0 +µA − r

σA

√T − t

dµAγ = dγ +µA − r

σA

√T − t

= d0 + ρA,R σR√T − t +

µA − r

σA

√T − t.

(37)

With all of this information, we can also calculate the physical probabilityof default and expected recovery-given-default:

P[τ ≤ T ] = Φ (−dµA0 )

E[RT

N| AT < N

]= eµR(T−t)

Rt

N

Φ (−dµAγ )

Φ (−dµA0 ).

(38)


Challenges

In the following analysis, using our stochastic recovery model, we canobtain data from balance sheet information and use this data to calibrateour model. However, there are many difficulties that our two-dimensionalmodel encounters in such a calibration that the original Merton modeldoesn’t:

First, in using bond data we are faced with many different bondissues. In the original Merton model, we must only concern ourselfwith the single estimate of equity and its volatility.

Second, each bond issue usually has coupons attached to it, whereasour model is for zero-coupon bonds.


Challenges

To tackle these two issues, we input a corresponding zero-couponnotional N and an average time-to-maturity of more than one bondissue

The calibration that follows is computed by Matlab code, using thebuilt in fsolve command, designed by Dr. Aditya Viswanathan.


Market Inputs

Some more notes regarding inputs to our calibration:

EMktt is the market capital obtained from balance sheet information.

σMktE is the historical volatility of the stock returns estimated from

historical prices. µMktE is also calculated using this historical data.

BMktt,T is the total debt obtained from balance sheet information.

σMktB is the estimated historical volatility of bond returns.

µMktB is also calculated using this historical data.


Market Inputs

We begin with a benchmark test of the algorithm.

Consider the following inputs to benchmark the algorithm:

Table: Market-based Input Values

EMarkett 19.03931396852958

σMarketE 0.6109279750612985

BMarkett,T 82.47066247641656

σMarketB 0.04713145769720459

µMarketE 0.10

µMarketB 0.05

T − t 7.84

r 0.0156

N 100


Model Outputs

The algorithm returns calibrated values

Table: Algorithm Final Values

R 80.000000

σR 0.25000

d0 1.000000

ρAR 0.400000.

Using this data, our default and recovery metrics are calculated to be

Table: Metrics

Et

[RTN

∣∣∣D] Et

[RTN

∣∣∣D] PAnnualizedt [D] PAnnualized

t [D]

0.5713913 0.3755446 0.0217939 0.0000120703.


Notional Sensitivity

Subsequent runs of the algorithm show that solution of this nonlinearsystem (32) - (33) is indeed sensitive to variation of the input parameters.We concentrate on the shifting of the input notional and the effect on theexpected recovery and asset-recovery correlation:

if the notional is lowered to 99.9, then the algorithm produces ahigher physical and risk-neutral expected recovery pair of(59.1%, 39.7%), but asset-recovery correlation decreases to 37.7%.

if the notional is increased to 100.1, then the algorithm produces alower physical and risk-neutral expected recovery pair of(54.5%, 34.7%), but asset-recovery correlation increases to 43.2%.

if the notional is further increased to 100.2, then the algorithmproduces a much lower physical and risk-neutral expected recoverypair of (48.7%, 28.6%), but asset-recovery correlation increasessignificantly to 50.5%.


Conclusions and Next Papers?

Have extended one-factor models to decouple recovery risk fromdefault risk, with closed form solutions (both Merton and Black-Cox.)

Connections between default risk and recovery risk exist, via thenotion of partial information.

Can use with other products, such as Coupon Bonds.

Recovered assets can exceed pre-default estimated assets. Historicallypossible (foreclosure / bank fees.)

Can also include bounded recovery in this framework via PartialInformation Transform.

Calibration to market data for both CDS and Bond prices.


Acknowledgments

Joe Campolieti (WLU)

Peter Carr (NYU)

Nick Costanzino (co-author, Barclays Capital)

Stephen J. Mildenhall (St. John’s)

J. Austin Murphy (Oakland)

Harvey Stein (Bloomberg)

Aditya Viswanathan (UM - Dearborn)


Documents

Pricing Recovery Risk in Bonds and Swaps · Pricing Recovery Risk in Bonds and Swaps Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and