Default probabilities of a holding company, with complete and partial information.

Donatien Hainaut† Griselda Deelstra* January 21, 2014

ESC Rennes Business School and CREST, France. Department of Mathematics, Université libre de Bruxelles, Belgium . Email: [email protected]; [email protected] †

*

Abstract This paper studies the valuation of credit risk for rms that own several subsidiaries or business lines. We provide simple analytical approximating expressions for probabilities of default, and for equity-debt market values, both in the case when the information is available in continuous time as well as in the case that it is not instantaneously available. The total rm's asset value being modeled as a sum of lognormal random variables, we use convex upper and lower approximations to infer these analytical approximating expressions. We extend the model to rms nanced by multiple stochastic liabilities and conclude by numerical illustrations. Keywords.

tonicity.

1

default risk, structural model, incomplete information, convex ordering, comono-

Introduction.

The treatment of default is a crucial issue in determining the value of corporate securities and the rm's nancing decisions.

This task is particularly complex when the corporation is itself

a group of subsidiaries that have dependent activities. Structural models such as developed by Merton (1974) and Black and Cox (1976) represent an elegant framework for the valuation of risky debts, when assets are modeled by a single Brownian motion. Since, many alternatives have been developed to replace the Brownian motion by more complex dynamics.

Recently Fiorani

(2010), Ballotta and Fusai (2013) and Hainaut and Colwell (2013) used Lévy, multivariate Lévy and switching Lévy processes.

But, unto our knowledge, there are only very few extensions to

multi-industry rms. Moreover, most of the existing models assume that the dynamics of rm's assets are continuously observed while in practice, the information needed to assess eciently the nancial health is for most of the companies only released at discrete times. As emphasized by Due and Lando (2001), ignoring this aspect leads to an underestimation of short-term credit spreads. The purpose of this work is hence twofold.

Firstly, this paper proposes simple approximating

formulas to appraise default probabilities, risky debts and equity for multi-industry rms. The total rm's asset is a sum of lognormal processes, each one corresponding to a rm's subsidiary. The statistical distribution of rm's total asset value exhibits then more leptokurticity and asymmetry than a single lognormal variable. Secondly, it studies the impact of a lack of information on these quantities.

The framework of our model is partly inspired from papers of Leland (1994, 1998)

and of Leland and Toft (1996). We assume that the default or simply the restructuring events are triggered when the total market value of all subsidiaries falls below a certain threshold. Two cases are considered. In the rst one, this threshold is constant. It can eventually be regulatory

1

imposed, or chosen by the rm's management. In the second case, the threshold is random and the sum of several liabilities. This approach is particularly well adapted for insurance companies that nance their investments by e.g. life or non life provisions. The solution that we propose is based on the concept of convex orders and comonotonicity, which were introduced by Hoefdding (1940) and Frechet (1951) who studied lower and upper bounds for multivariate cumulative distributions.

This theory became popular amongst researchers in

actuarial sciences over the last two decades and has been applied successfully to various elds of research. Dhaene et al. (2002 (a), (b)) proposed in their review comonotone upper and lower approximations in the convex order sense for the sum of a nite number of random variables. The work of Vanduel et al. (2003) reveals that the lower convex bound approximation is extremely accurate for an appropriate choice of parameters. We refer the interested reader to Denuit et al. (2005) for characterizations of convex orders. We further notice that convex orders and stop loss premiums are closely related (see e.g. Dhaene and Goovaerts (1996)). Comonotone bounds have been applied from derivatives pricing (Vanmaele et al. (2006)) to insurance (Ahcan et al. (2006)), including risk management, as in Van Weert et al. (2012). For a recent survey of applications in nance and insurance, we refer e.g. to Deelstra et al. (2011). But we did not encounter any applications of this theory to the valuation of credit risk. Our work tries to ll this gap. The outline of this paper is as follows. The rst section introduces the framework that we adopt to model a multi activity rm. In section 3, we build the convex bounds of the total rm's asset and infer in section 4 approximating formulae for the probabilities of default. In section 5 and 6, we respectively appraise the value of debts with complete and incomplete information. In section 7, the model is adapted to stochastic liabilities. Section 8 contains several numerical applications and we conclude our work in section 9.

2

The model.

We consider a holding company, composed of

N

subsidiaries or various business lines.

Each

subsidiary generates a stream of dividends or cash-ows distributed in its entirety to the parent company. The investment in the subsidiary is assumed irreversible, at least till an eventual re-

(Ω, F, P), where ˜ j for j = 1...M . W i process denoted by Ft

structuring of the holding. Dividends are dened on a ltered probability space

F

is the ltration generated by

The dividend provided by the

M

ith

independent Brownian motions, denoted by

subsidiary is assumed to be a stochastic

and has the following dynamics under the real measure

dFti Fti N ×M

M X

˜ tj σi,j dW

∀i = 1, . . . , N.

(2.1)

j=1

rank(Σ) = M . The > covariance matrix containing the covariances between the ows of dividends is then equal to ΣΣ .

We denote

Σ

= µi dt +

P:

the

matrix of

(σi,j )i=1...N, j=1...M

which are such that

We also assume that there exists a risk free asset, such as a bank account, that provides a constant rate of return

r.

The total ow of dividends paid to the holding company is denoted by

Ft

N X

=

Fti .

i=1 Obviously, cash-ow processes are not tradeable assets. However, as the subsidiary is a separate entity, the entire value of this subsidiary can be seen as a traded asset. In this case, the value of the

ith

subsidiary is equal to the expected sum of cash-ows discounted at the cost of the equity

2

rE

rE > µi :

with

Sti

∞

ˆ

e−rE (s−t) Fsi ds | Ft

= E



t

ˆ



∞

e

=

−rE (s−t)

Fti e

E

µi −

PM

2 σi,j

j=1



2

P j j (s−t)+ M j=1 σi,j (Ws −Wt )

! ds | Ft

t

Fti . rE − µi

=

(2.2)

This last formula is similar to the Gordon-Shapiro formula as rst exposed by Gordon and Myron (1959). The rate

µi

can be seen as the growth rate of dividends distributed by the

ith P,

line. Note that the expectation in equation (2.2) is calculated under the real measure

business and this

is the reason why the discount rate is given by the cost of equity and not by the risk free rate. The dynamics of

Sti

can be rewritten as

dSti Let us denote by

1N

is a

neutral

µ,

M X dFti ˜ tj . = µi Sti dt + Sti σi,j dW rE − µi j=1

=

the vector of

µi

and by

κ,

(2.3)

the vector of market risk premiums

N vector of ones, κ is a solution (not necessary unique) of µ = 1N r + Σκ. Q is dened by the following Radon Nikodym derivative     ˆ ˆ t 1 t 0 dQ 0 ˜ = exp − κ κds − κ dWs . dP t 2 0 0

κj

for

˜ i. W t

If

Then the risk

(2.4)

Under this risk neutral measure are equal to the risk free rate,

Q chosen by market participants, the drifts of the dynamics (2.3), r, to ensure the absence of arbitrage. More precisely, we have that dSti

=

rSti dt + Sti

M X

σi,j dWtj

(2.5)

j=1 where

˜ tj + κj dt dWtj = dW

are here

M

independent Brownian motions dened on

mentioned in e.g. Musiela and Rutkowski (1998) or sure

Q

is unique if

M = N = rank(Σ).

(Ω, F, Q).

As

Dhaene et al. (2013), the risk neutral mea-

Note that all following developments are done under the

PN

i i=1 St and its initial i value is equal to the sum of S0 for i = 1, . . . , N . In our framework, the total market value St of all subsidiaries is a sum of lognormal random variables and is therefore no more distributed as a measure

Q.

The market value of the holding at time

t

is denoted by

St =

lognormal. The investor partly nances his investment by its capital and issues debts to prot from the tax shield oered for interest expenses. The tax rate

θ ∈ (0, 1)

is assumed constant over

time. In the rst part of this work, the debt is modeled as a consol bond. This approach is well suited to t the liabilities structure of most non nancial corporations, that systematically renew their loans for tax purposes. The investor pays continuously and perpetually a constant coupon The tax benet is then

θC .

The debt is issued at time

0

for some amount

D.

C.

In section 7, we will

assume that the holding is nanced by stochastic liabilities. This may be used to model nancial conglomerates such as insurance companies or banks that have several dierent uncertain liabilities. We assume that the equity owner liquidates or restructures the holding when the total value of assets falls below a predetermined value denoted

α,

usually less than the accounting value of

debts or a oor imposed by the regulator. As we will discuss later, management so as to maximize the market value of the equity. of control. The default time is an holding is

Sτ =

PN

i=1

Sτi

Ft

stopping time, denoted by

α

can also be chosen by the

In this case,

τ.

α

is a parameter

The liquidation value of the

and is assigned to the debt holders. In this framework, the market value

3

of the equity, denoted by

E0α ,

is equal to the sum of the expected discounted cash-ows under the

risk neutral measure, decreased by the cost of debts:

E0α

τ

ˆ Q

e

=E

−rt

 (Ft − (1 − θ)C) dt .

(2.6)

0 We assume that in case of bankruptcy, holding shareholders do not receive any income from the sale of the assets. The dierence between the total cash-ow and the coupon paid may be seen as a dividend, that can be positive or negative if the cash-ows are insucient to pay debts. According

Sτ = α,

to the relation (2.2) and given that

ˆ EQ

τ

 X ˆ N e−rt Ft dt = EQ

0

 e−rt Fti dt

0

i=1

=

τ

the rst term in this last expression is equivalent to

N  X

∞

ˆ Q

e

E

−rt



Fti dt

 ˆ − E e−rτ

0

i=1

∞

Q

e

−r(t−τ )



Fti dt

τ


= S0 − EQ e−rτ α .

(2.7)

The market value of the equity can be rewritten as

E0α

 ˆ τ  Q −rt = E e Ft dt − E e (1 − θ)Cdt 0 0   ˆ τ Q −rτ −rt = S0 − E e α+ e (1 − θ)Cdt , ˆ

τ

−rt

Q

(2.8)

0 while the market value of the debt at time

D0α

ˆ

0

is equal to the following expectation:

τ

 e−rt (1 − θ)C dt + e−rτ Sτ ˆ0 τ  Q = E e−rt (1 − θ)C dt + e−rτ α . = EQ

(2.9)

0 To our knowledge, these expressions of equity and debt market values do not admit any analytical solution in this framework, except if there is only one subsidiary. develop lower and upper convex approximations for the sum

PN

In the following sections, we

i i=1 St ,

and infer lower and upper

estimates of default probabilities of the holding company. In reality, the nancial information needed to assess the nancial health of the holding is not necessarily available in continuous time. In particular, if the holding is not listed, the nancial statements that are issued quarterly form the only available information. This remark motivates the second part of this work, in which we explore the impact of this lack of information on the estimates of default probabilities.

3

Convex bounds of

St .

In this section, we briey review results related to comonotone convex upper and lower bounds for a sum of lognormal variables. For details and proofs, we refer to Dhaene et al. (2002 a,b). As mentioned earlier, the sum

St

of market values of subsidiaries at time

t is the sum of N

lognormal

variables:

St

d

=

N X i=1

4

i

S0i eZt .

(3.1)

Under the risk neutral measure,

N (µZ t , ΣΣ> t)

Zt = Zt1 , . . . , ZtN

> r − 12 e> 1 ΣΣ e1 . . .

 µZ =  r− ei

is the

is a Gaussian random vector with distribution

where the mean vector is given by



where




ith

unit root vector of

RN .

1 > > 2 eN ΣΣ eN

 (3.2)

 

To simplify further calculations, the variance of

Zti

is

dened as

V ar(Zti ) = σiZ


2

> t = e> i ΣΣ ei t.

(3.3)

X is said Y in the convex order sense if and only if for all convex functions u(.), we have E(u(X)) ≤ E(u(Y )), provided the expectations exist. This relation is denoted by X ≤cx Y . It has been proven that X ≤cx Y if and only if the stop loss premiums satisfy the relation E (X − d)+ ≤ E (Y − d)+ , for all levels of retention d, and if E[X] = E[Y ]. The following proposition allows us The theory of comonotonicity is closely related to the concept of convex order. A r.v. to precede a r.v.

to build convex bounds for the total market value of subsidiaries. Proposition 3.1.

Zti

Consider the conditioning process Λt dened as the weighted sum of processes Λt

=

N X

γi Zti

(3.4)

i=1

where γi for i = 1, ..., N are constant. Also consider processes dened by Sti,l

=

S0i exp



Sti,c

 
1 2 Z 2 Z l (1 − r ) σ t + r σ W µZ + i i i i t i 2 Z c S0i exp µZ i t + σi Wt

=

(3.5)




(3.6)

and their sums Stl =

N X

Sti,l

Stc =

i=1

N X

Sti,c

i=1

where Wtl and Wtc are independent Brownian motions such that W0l = W0c = 0. The coecients ri are constant and dened as follows ri

=

cov(Zti , Λt ) p p V ar(Zti ) V ar(Λt )

=

> e> i ΣΣ γ p p . > e> γ > ΣΣ> γ i ΣΣ ei

(3.7)

Then, we have the following convex order relations: Stl ≤cx St ≤cx Stc

(3.8)

In most applications, as mentioned in the work of Vanduel et al. (2003), the lower bound approximation

X ≤cx Y

then

Stl

is extremely accurate and can be used as a good proxy for

V ar(X) < V ar(Y )

must hold unless

d

X =Y

St .

Note that if

i . The variances of St , St and

Stc

are

given by the following expressions:

V ar(St )

=

N X N X



S0i S0j e

Z 1 µZ i +µj + 2

i=1 j=1

5



2

(σiZ )

2

+(σjZ )

 t

  j i ecov(Zt ,Zt ) − 1 ,

(3.9)

V ar(Stl )

N X N X

=

S0i S0j e



Z 1 µZ i +µj + 2



(σiZ )

2

2

+(σjZ )

 t



Z

e ri rj σ i

 −1 ,

σjZ t

(3.10)

i=1 j=1

V ar(Stc )

N X N X

=

S0i S0j e



Z 1 µZ i +µj + 2



2

(σiZ )

+(σjZ )

2

 t



Z

eσi

σjZ t

 −1 .

(3.11)

i=1 j=1 Vanduel et al. (2008) recommend weights over

t

 i γi = S0i E eZt

that maximize the variance

V ar(Stl )

γi 's being not adapted to later develStl approximates well St at any time, and according a stop-loss Stl ≤cx St , we know from Kaas et al. (1994, p. 68) that

years. But a nite time horizon and time-dependent

opments, we look for

γi

such that

distance. More precisely, as

ˆ

+∞

−∞

h
i   E (St − k)+ − E Stl − k + dk

=


1 V ar(St ) − V ar(Stl ) . 2

1 l 2 V ar(St ) − V ar(St ) can be interpreted as a measure for the total error made when l approximating the stop-loss premiums of St by those of the convex smaller St . If we adopt this




Then

measure, the best

γi

should then minimize the gap between variances

γi But given that

St

and

Stc

γi

= =


= argmin V ar(St ) − V ar(Stl )

are independent from

= argmin

it is equivalent to


argmin V ar(St ) + V ar(Stc ) − V ar(Stc ) − V ar(Stl )
argmin V ar(Stc ) − V ar(Stl ) ∀t > 0.

By denition (3.10) and (3.11) of

γi

γi ,

∀t > 0.

N N X X

V ar(Stc )

S0i S0j e



and

Z 1 µZ i +µj + 2

V ar(Stl )



(σiZ )

2

2

+(σjZ )

 t



Z

eσi

σjZ t

Z

− e ri rj σ i

σjZ t



∀t > 0,

i=1 j=1 and a rst order Taylor approximation of exponential functions leads to

γi

≈ argmin

N X N X

   Z 2 Z 2 Z Z 1 t i j µi +µj + 2 (σi ) +(σj ) S0 S0 e

(1 − ri rj ) σiZ σjZ t ∀t > 0.

i=1 j=1 As

γi

are time-independent, they ideally should cancel all spreads

use in later developments the

γi

(1 − ri rj ). For this reason, ri rj and 1:

we

that minimizes the quadratic gap between

2 > > > e> ΣΣ γ e ΣΣ γ i j 1 − p  . q γi = argmin > > > ei ΣΣ ei ej ΣΣ> ej γ > ΣΣ> γ i=1 j=1 N X N X



(3.12)

The accuracy of the convex approximations is tested in a numerical applications section concluding this work, see section 8.

4

Approximation of default probabilities.

Before assessing the market value of debt and equity, we build estimates of probabilities of bankruptcy. When the number of subsidiaries is small, the probability that the total asset breaches the oor

α

can always be computed by Monte-Carlo simulations. But once that this number is

6

important, Monte-Carlo simulations require heavy calculations and furthermore, they have not the advantage of analytical tractability of closed or semi-closed formulas. In the following, we will always work (even without further mentioning) under the chosen risk-neutral probability

EQ [.].

we will denote the expectations by for quantile functions

Qp [.],

Q

and

However in order to avoid confusion with the notation

we still will refer to the probability of an event

A

by

P(A)

although

the probability used is the risk-neutral one.

St

In the remainder of this section, lower and upper bounds of functions

g l (., .)

and

g c (., .)

Stl

= =

of time and of Brownian motions

N X

S0i exp



i=1 l

Wtl

are respectively rewritten as

and

Wtc :

 
1 2 Z 2 Z l (1 − r ) σ t + r σ W µZ + i i i i t i 2

g (t, Wtl )

Stc

(4.1)

N X

=

Z c S0i exp µZ i t + σi Wt




i=1

g c (t, Wtc ).

=

(4.2)

g l (t, w) and g c (t, w) are dierentiable with respect to w and then continuous. σiZ > 0 for i = 1...n, the rst order derivative of g c (t, w) with respect to w,

Functions more as

∂g c (t, w) ∂w g c (t, w) is strictly l derivative of g (t, w)

is positive and rst order

∂g l (t, w) ∂w

=

N X

=

Z S0i σiZ exp µZ i t + σi w




i=1

positive increasing in

N X

Further-

S0i ri σiZ exp



i=1

w.

Under the condition that all

ri ≥ 0,

the

  1 2 Z 2 Z µZ + (1 − r )(σ ) t + r σ w i i i i i 2

g l (t, w) is strictly positive increasing. If some ri are negative, a positivity constraint (ri ≥ 0 for i = 1...n) should be added in the optimisation criterion (3.12). If this constraint  c,l  ∂g ∂g c,l is satised, (t, w ) − (t, w ) (w1 − w2 ) ≥ 0, ∀w1 , w2 ∈ R, g l (t, w) and g c (t, w) are also 1 2 ∂w ∂w is also positive and

w. And their inverse functions are well dened. Furthermore, the p quantiles of Stl and c St , respectively denoted by Qp [Stl ] and Qp [Stc ], are given by the following expressions (for a proof see Theorem 1 of Dhaene et al. (2002a)): convex in

If we use the notation

Φ

Qp [Stl ]

= g l (t, Qp [Wtl ])

Qp [Stc ]

= g c (t, Qp [Wtc ]).

for the distribution function of a standard normal random variable, we

can retrieve the distributions of the convex bounds as follows:

FStl (x) FStc (x) Note that the inverses of the functions holding company will go to bankruptcy

=

=

gl

Φ


−1

(g c )

Φ

√ −1

(t, x)

!

t

(t, x) √ t

! .

g k=l,c do not have a simple analytical expression. The when St hits the level α. The distribution of this hitting 7

time is unknown. We would like to estimate the hitting times by using convex lower and upper bounds

Stl

and

Stc : τ l,g

l

 inf t ≥ 0 | Stl ≤ α, Ssl ≥ α ∀s ≤ t n o
−1
−1 inf t ≥ 0 | Wtl ≤ g l (t, α), Wsl ≥ g l (s, α) ∀s ≤ t

(4.3)

inf {t ≥ 0 | Stc ≤ α, Ssc ≥ α ∀s ≤ t} n o −1 −1 = inf t ≥ 0 | Wtc ≤ (g c ) (t, α), Wsc ≥ (g c ) (s, α) ∀s ≤ t

(4.4)

= =

τ c,g

c

=

As shown by Vanduel (2005) in his PhD thesis, the lower comonotonic bound

St

proxy for the original process

Stl

is an excellent

and we expect that default probabilities computed with

Stl

are

also close to real ones. Numerical tests presented in section 8 (right graph of gure 8.1) tend to conrm this. However, even if convex bounds allow us to reduce the dimension, we still face the problem that the hitting time of a non linear function of time by a Brownian motion does not admit any analytical expression, except in very few exceptions. A solution to preserve analytical tractability consists in approximating the function by a linear approximation with respect to time, of functions

gl


−1

(t, α)

and

(g c )

−1

(t, α),

as done by Schmidt and Nokinov (2008). We opt for this approach

and use a rst order Taylor development with respect to time as approximation method. choice is motivated by the fact that if

N = 1, g l


−1

(t, α)

and

−1

(g c )

(t, α)

Our

are precisely linear

function of time e.g. :

c −1

(g )

(t, α)

=

1 ln σ1Z



α S01

 −

µZ 1 t σ1Z

N =1

On another hand, numerical tests presented in section 8 (left graph of gure 8.1) conrm that a linear function of time is close to the original one. To lighten further calculations we denote:

f l (t) = g l Given that


g l t, f l (t) = α,


∂ l g t, f l (t) ∂t

N X

(t, α).

it is easy to calculate that

   

1 1 2 Z 2 Z 2 Z 2 Z l = + (1 − ri ) σi exp µi + (1 − ri ) σi t + ri σi f (t) 2 2 i=1    N X
1 ∂ l 2 Z 2 Z l + S0i ri σiZ exp µZ + (1 − r ) σ t + r σ f (t) f (t) i i i i i 2 ∂t i=1 =

S0i




−1

µZ i

0

f l (t) with respect to time is equal to   

 PN 1 1 i Z 2 Z 2 Z 2 Z 2 Z l S µ + (1 − r ) σ exp µ + (1 − r ) σ t + r σ f (t) i 0 i i i i i i i i=1 2 2 ∂ l   f (t) = − .
 PN 1 ∂t i Z Z 2 Z 2 t + r σ Z f l (t) S r σ exp µ + (1 − r ) σ i i i i i i i i=1 0 2

and therefore we infer that the derivative of

We choose a time

t0 ,

calculate numerically the value

f l (t)

=

f l (t0 ) + (t − t0 )

f l (t0 )

and develop

f (t)

linearly around

t0 :

∂ l f (t)|t=t0 + O(t2 ). ∂t

The accuracy of this approximation is tested in the numerical applications section. We infer from this relation that

gl


−1

(t, α) ≈ β1l − β2l t, 8

(4.5)

β1l and β2l are dened by:   


−1 PN 1 1 2 Z 2 2 Z 2 exp µZ t0 + ri σiZ g l (t0 , α) S0i µZ i + 2 (1 − ri ) σi i + 2 (1 − ri ) σi i=1   , β2l =
 PN Z + 1 (1 − r 2 ) σ Z 2 t + r σ Z (g l )−1 (t , α) i r σ Z exp µ S 0 0 i i i i i i i i=1 0 2

where

(4.6)

β1l

=

g


l −1

(t0 , α) + β2l t0 .

(4.7)

In the same way, we get that

(g c ) c where β1 and

−1

(t, α) ≈ β1c − β2c t,

(4.8)

β2c are dened by:   Z Z c −1 S0i µZ (t0 , α) i exp µi t0 + σi (g )  , β2c = P N i Z Z Z c −1 (t , α) 0 i=1 S0 σi exp µi t0 + σi (g ) PN

i=1

β1c Note that

β1l

β1c

and

(g c )

=

−1

(4.9)

(t0 , α) + β2c t0 .

(4.10)

are negative by construction. Indeed, if this was not the case, the company

0. In most of the cases β2c,l are positive but their signs depend on c,l Z the signs of µi . We will see at the end of this section, that the sign of β2 has a serious impact on the asymptotic probability of ruin. But rst, we dene approximate default times of the convex would necessarily default at time

bounds as follows:

τl

=

τc

=

Given that we

 inf t ≥ 0 | − β1l + β2l t + Wtl ≤ 0 , −β1l + β2l s + Wsl > 0 ∀s ≤ t

(4.11)

inf {t ≥ 0 | − β1c + β2c t + Wtc ≤ 0 , −β1c + β2c s + Wsc > 0 ∀s ≤ t} .
l −1 c −1 replace the frontiers g and (g ) by their linear approximations,

(4.12) we get the

following relations between approximate and exact hitting times of convex estimates: l

P(τ l,g ≤ t) ≤ P(τ l ≤ t) P(τ

c,g c

(4.13)

c

≤ t) ≤ P(τ ≤ t).

As mentioned earlier and in the work of Vanduel et al. (2003), the lower bound approximation

Stl

St . P(τ ≤ t).

is an accurate proxy for

probability of default

We could then expect that

P(τ l ≤ t)

is a good proxy for the real

Numerical applications concluding this work tend to conrm this.

It is well known that the hitting time of a Brownian motion with drift has an Inverse Gaussian (IG) distribution.

From Bielecki and Rutkowski (2004 page 66), the probability that the

t is then approximated by the following

k k P(τ k ≤ t) = Φ hk1 (t) + e2β2 β1 Φ hk2 (t) for k = l or c

holding goes to bankruptcy before time

hk1 (t) =

β1k − β2k t √ t

hk2 (t) =

dP(τ k ≤ t)




3 1 1 1 = ϕ hk1 (t) − β1k t− 2 − β2k t− 2 2 2  
k k 3 1 1 1 +e2β2 β1 ϕ hk2 (t) − β1k t− 2 + β2k t− 2 , 2 2 9

(4.14)

β1k + β2k t √ t

and the density of the default times can consequently by derived as follows for



expressions:

k=l

or

c

ϕ(.) denotes the density function of a standard normal distribution.
2β2k β1k e ϕ hk2 (t) , the density of the default times is rewritten as where

3

= −β1k t− 2 ϕ hk1 (t)

dP(τ k ≤ t)

As mentioned earlier, the sign of to equation (4.14), if

β2l,c

β2l,c


ϕ hk1 (t) =




−β k 1 β1k − β2k t √ 1 exp − 2 t 2πt3

=

Given that


2 ! .

(4.15)

inuences the long term ruin probability. Indeed, according

µZ i

is positive (e.g. when all

are positive), the approximate probability

of default over an innite horizon is given by k

lim P(τ k ≤ t)

k

= e2β2 β1

t→∞

for

k=l

or

c.

This asymptotic probability can be used as a measure of risk, instead of the Value at Risk (VaR) or Tail Value at Risk (TVaR), as done in actuarial sciences for insurance companies. If

β2k=l,c

is

negative, the asymptotic probability of ruin is equal to one and the holding will go to bankruptcy with certainty but the timing is unknown.

5

Valuation of debt and equity.

Approximate market values of debts are provided by the next proposition. These are obtained as the dierence between a default free perpetuity and a payo paid in case of default, weighted by a factor combining discount rate and default probabilities.

Under the assumption that default times are dened as in (4.11) and (4.12), the convex estimates of the market value of the debt are provided by the following expression:

Proposition 5.1.

ˆ D0α,k

τk

! k

e−rs (1 − θ)C ds + e−rτ α

= EQ 0

  q   2 1 β1k β2k + 2r+(β2k ) 1 e (1 − θ)C + α − (1 − θ)C r r

=

Proof.

We prove this result for

k = l.

The market value of the debt is given by the following sum:

ˆ D0α,l

for k = l or c

τl

! e−rs (1 − θ)Cds

= EQ

  l + EQ e−rτ α .

(5.1)

0 Using Fubini's theorem, the rst expectation can be rewritten as follows:

ˆ Q

τl

e

E

ˆ

! −rs

(1 − θ)Cds

+∞

ˆ

t

e−rs (1 − θ)C ds dP(τ l ≤ t)

=

0

0

= =

0

ˆ

+∞


1 1 − e−rt dP(τ l ≤ t) r 0   ˆ +∞ 1 −rt l (1 − θ)C 1− e dP(τ ≤ t) r 0 (1 − θ)C

(5.2)

and the second expectation in (5.1) can be developped as

  l EQ e−rτ α

ˆ 0

10

+∞

e−rt dP(τ l ≤ t)

= α

(5.3)

Equations (5.2) and (5.3) both depend on the integral generating function of

ˆ

τ l: ˆ

+∞

e

´ +∞

−rt

+∞

l

dP(τ ≤ t) = 0

0

0

e−rt dP(τ l ≤ t),

which is the moment

−β l 1 2rt2 + β1l − β2l t √ 1 exp − 2 t 2πt3


2 ! dt

which can be reformulated as

ˆ

+∞

e−rt dP(τ l ≤ t) = e

q  2 −β1l 2r+(β2l ) −β2l

ˆ

0

 +∞

0



 1 −β l  √ 1 exp −  2 2πt3

β1l

  q
2 2 l − 2r + β2 t    dt. t 

But in view of (4.15), the integrand is the density of a hitting time, that we denote by that

P(τ

m,l

≤ ∞) = e ˆ

2β1l

q

2 β2l

2r+(

)

and

P(τ

m,l

+∞

e

−rt

l

dP(τ ≤ t) = e

≤ 0) = 0.

−β1l

τ m,l ,

such

Consequently, we immediately infer that:

 q q 2 2 2r+(β2l ) −β2l 2β1l 2r+(β2l )

e

.

0 

=e

q 2 β1l β2l + 2r+(β2l )



.

Remark that if the holding company is incorporated at time must in theory be equal to its accounting value

D0 ,

t = 0,

the market value of debts

that is the principal lent to the rm. If this

is not the case, the transaction is not free of arbitrage. In this case, the (approximate) fair value of

α

that ensures the equality between market and accounting values of the debt is given by:

αfk air

=

  q   2 1 1 −β1k β2k + 2r+(β2k ) (1 − θ)C + D0 − (1 − θ)C e r r

for

k=l

or

c.

This is the sum of a perpetual annuity and of a kind of option price related to the loss in case of default. In numerical applications ending this work, we illustrate the inuence of the oor

α

on

the debt market value.

Under the assumption that default times are dened as (4.11) and (4.12), the convex estimates of the market value of the equity are approximated by the dierence between market values of assets and debts:

Corollary 5.2.

E0α,k

=

 ˆ S0 − EQ e−rτ α +

τ

 e−rt (1 − θ)Cdt

0

=

  q   2 1 1 β1k β2k + 2r+(β2l ) S0 − (1 − θ)C − α − (1 − θ)C e r r

k=l

or c.

Remark that our results can be used to infer approximations for credit default swap (CDS) premiums. A credit default swap is an insurance protecting the owner of a corporate bond issued by the holding, in case of default.

CDS's are used for hedging and speculation purposes.

In

exchange of regular payments (named the premium leg), the buyer of the CDS receives the part of the bond principal which is not repaid in case of bankruptcy of the bond issuer. The payment done in case of default, is called the default leg. The premium paid for this insurance is usually expressed as a percentage of a unit bond principal. This percentage is called the CDS spread and we denote it by

p.

Premiums are paid at regular intervals of time,

11

∆t,

ranging from

t1

to

tn .

The

premium leg is equal to

P remium leg

= p ∆t

tn X

e−r ti EQ (Iti <τ | Ft0 )

ti =t1

= p ∆t

tn X

e−r ti P(ti < τ )

(5.4)

ti =t1 If the bond issuer goes to bankrupcty, the CDS pays the dierence between the principal and a

R.

constant recovery rate percentage, denoted by

Def ault leg

=

The default leg is then

tn X

(1 − R)

e−r ti EQ Iti−1 <τ



ti =t1

=

tn X

(1 − R)

e−r ti P(ti−1 < τ ≤ ti ).

(5.5)

ti =t1 By equating equations (5.4) and (5.5) and replacing the probabilities of default by their approximations, we get approximate estimates for the CDS spread:

k

p where

P(τ k ≤ ti )

=

(1 − R) ∆t

Ptn

e−r ti P(τ k ≤ ti ) − P(τ k ≤ ti−1 ) Ptn −r ti (1 − P(τ k ≤ t )) i ti =t1 e




ti =t1

k=l

or

c,

(5.6)

is provided by equation (4.14). The weakness of our model is the same as the

one of Merton's structural model. Given that trajectories of assets are continuous and default is the rst hitting time of a barrier, the default is a predictable stopping time.

This leads to an

underestimation of short term probabilities of default. This major aw of the structural model has given rise to other approaches in credit risk modelling. One possibility consists in introducing jumps in the assets dynamics. Another approach is to introduce incompleteness about the available information.

We have explored this alternative in the following section and inferred modied

expressions for the probabilities of default, debts and equity.

6

Valuation of the debt with incomplete information.

If the holding is not listed or if the volume of stocks traded are too low to be considered as reliable, the information about the holding is only available at discrete times. Mainly when the nancial statements are published or when the holding goes to bankruptcy.

Even when the company is

listed, the information is reported to the market with a certain delay. In this section, we assess the impact of the lack of information on default probabilities and on market values of debt and equity. We assume in the remainder of this section that the only information available at time has been communicated at time the ltration by

Ht

(Ft )t

0

is not accessible in continuous time. Let us denote by

the indicator variable

1t<τ ,

t

and that the holding is still active. The information carried by

τ

the time of default and

equal to one if the issuer is still solvent. The information avail-

 Gt = σ S0i , i = 1, ..., N ; Hu , u ≤ t . As before, k we will denote the approximate default times dened in (4.11) and (4.12) by τ with k = l or c, k and then let denote Ht the indicator variable 1t<τ k for k = l or c. We dene the corresponding  i k k ltrations by Gt = σ S0 , i = 1, ..., N ; Hu , u ≤ t .

able to the market is represented by the ltration

We are interested in the probability (still under the risk neutral measure) at time the holding company is still in activity at time

T,

which is given by



P τ > T | S0i , i = 1, ..., N, t < τ = P τ k > T | Gt . 12

t > 0

that

The convex estimates of these survival probabilities are respectively provided by the following expressions:

k

P τ >T

| Gtk




=



k k 1 − Φ hk1 (T ) − e2β2 β1 Φ hk2 (T )

k k 1 − Φ hk1 (t) − e2β2 β1 Φ hk2 (t)

for

k=l

or

c

(6.1)

where

hk1 (s) =

β1k − β2k s √ s

hk2 (s) =

β1k + β2k s √ s

for

k=l

or

c.

(6.2)

The density function of the survival time is then obtained by dierentiating with respect to

dP τ k ≤ T | Gtk




=

=



k k ∂ Φ hk1 (T ) + e2β2 β1 ∂T Φ hk2 (T )

k k 1 − Φ hk1 (t) − e2β2 β1 Φ hk2 (t)   2 −β1k +β2k T ) ( −β1k 1 √ exp − 2 T 2πT 3

for k = l k βk k 2β 1 − Φ h1 (t) − e 2 1 Φ hk2 (t)

T:

∂ ∂T

or

c.

(6.3)

As we will see in numerical applications, the delay in the information disclosure allows us to model

β1k and β2k Z for k = l, c are respectively negative and positive (e.g. if all µi are positive), the approximate probabilities of default over an innite horizon are given by non negligible probabilities of default over a short term period of time. Note that if

k

lim P(τ ≤ T

T →∞

| Gtk )

=



k k k k e2β2 β1 − Φ hk1 (t) − e2β2 β1 Φ hk2 (t)

k k 1 − Φ hk1 (t) − e2β2 β1 Φ hk2 (t)

for

k=l

or

c.

As mentioned earlier, this asymptotic measure of default can be used as substitute to other risk measures such as e.g. VaR or TVaR. The approximate market values of debts are provided by the next proposition and may still be seen as the dierence between a default free perpetuity and a modied cash-ow paid in case of bankruptcy. Proposition 6.1.

k=l

or c :

Dtα,k

The approximate values of the debt are equal to the following expressions, for

  1 1 = (1 − θ)C − (1 − θ)C − α × r r  q     2 2β1k 2r+(β2k ) D,k D,k q  rt e 1 − Φ h1 (t) − e Φ h2 (t) 2 β1k 2r+(β2k ) +β2k

e k k 1 − Φ hk1 (t) − e2β2 β1 Φ hk2 (t)

(6.4)

D,k where hk1 (t) and hk2 (t) are dened by equations (6.2), while hD,k 1 (.) and h2 (.) are given by

q
2 2r + β2k s D,k √ h1 (s) = s q
2 β1k + 2r + β2k s D,k √ h2 (s) = . s β1k −

Proof.

Let us consider the case that

(6.5)

(6.6)

k = l.

If the holding is still active at time t, the value of the debt is equal to the following expectation:

13

Dtα,l



Q

=E

ˆ

e

−r(τ l −t)

α | Gtl



ˆ

τl

Q

+E

! e

−r(s−t)

θ)Cds | Gtl

(1 −

t

  ˆ +∞ 1 e−r(s−t) dP(τ l ≤ s|Gtl ) + (1 − θ)C 1− e−r(s−t) dP(τ l ≤ s|Gtl ) r t t   ˆ +∞ 1 1 e−r(s−t) dP(τ l ≤ s|Gtl ) = (1 − θ)C − (1 − θ)C − α r r t +∞

=α

(6.7)

If we introduce the notation

v(t)

ert
,
l l 1 − Φ hl1 (t) − e2β2 β1 Φ hl2 (t)

=

the integral in (6.7) can be rewritten as follows:

ˆ

+∞

e−r(s−t) dP(τ l ≤ s | Gtl ) t

−β l = v(t) √ 1 2πs3

= v(t)e

−β1l

ˆ

q

+∞

t

!
2 −β1l + β2l s + 2rs2 ds exp − 2s   2

2r+(β2l ) −β2l



ˆ

+∞

t

β1l

 −β l  √ 1 exp − 3  2πs

−

q


l 2

2r + β2 2s

We notice that the integrand is the density of a hitting time, that we denote such that



√ l



2β1 P(τ D,l ≤ s) = Φ hD,l 1 (s) + e

hD,l 1 (s) = l

As

P(τ D,l ≤ ∞) = e2β1 ˆ

q

β1l

p − 2r + β2l 2 s √ s 2

2r+(β2l )

and

+∞

e

−r(s−t)

l

dP(τ ≤

s | Gtl )

=e

−β1l

q

2

τ D,l ,

and which is



 2r+β2l 2 Φ hD,l 2 (s)

hD,l 2 (s) =

P(τ D,l ≤ 0) = 0,

2  s    ds. 

β1l

q
2 + 2r + β2l s √ . s

we immediately infer that:

2r+(β2l ) −β2l

 l

e2β1

q

2r+(β2l )

2

t rt

e ×

 1−Φ





hD,l 1 (t)

2β1l

q

2

2r+(β2l )





hD,l 2 (t)

−e Φ

l l 1 − Φ hl1 (t) − e2β2 β1 Φ hl2 (t)

.

As stated by the following proposition, the market value of equity is the dierence between the conditional expectation of the total value of the assets, conditionally upon the available information, and the value of debts: Proposition 6.2.

k=l

or c :

The approximate values of the equity are equal to the following dierences, for
Etα,k = EQ Stk |Gtk − Dtα,k 14

(6.8)

where Dα,k is provided by proposition 6.1. The conditional expectations of Stl and Stc are given by the following expressions:
l

EQ Stl |Gt

N X

=

    S,l 2β1l (β2l +ri σiZ ) 1 − Φ hS,l (i, t) − e Φ h (i, t) 1 2

l βl l l 2β 2 1 1 − Φ h1 (t) − e Φ h2 (t)

S0i er t

i=1

EQ (Stc |Gtc )

N X

=

S0i er t

(6.9)

    S,c 2β1c (β2c +σiZ ) 1 − Φ hS,c (i, t) − e Φ h (i, t) 1 2 c

(6.10)

c

1 − Φ (hc1 (t)) − e2β2 β1 Φ (hc2 (t))

i=1

where hS,l 1 (i, t)

=

hS,c 2 (i, t)

=

β1l − β2l t − ri σiZ t √ t β1c + β2c t + σiZ t √ t

β1l + β2l t + ri σiZ t √ t c c β + β t + σiZ t 1 2 √ hS,c 2 (i, t) = t hS,l 2 (i, t) =

(6.11)

(6.12)

and where hk1 (t) and hk2 (t) for k = l or c are dened by equations (6.2). Proof.

First, we calculate the expected total value of the assets namely



EQ Stl |Gtl = EQ g l (Wtl )|Gtl .

This expectation can be expressed as

ˆ EQ Stl |Gtl Let us dene

EQ




β1l −β2l t

w0 = w + β2l t ˆ
Stl |Gtl =

g l (t, w)dP(Wtl ≤ w | inf β2l s + Wsl ≥ β1l ). 0≤s≤t

+∞

β1l

Then, by denoting

+∞

=

g l (t, w0 − β2l t)dP(β2l t + Wtl ≤ w0 | inf β2l s + Wsl ≥ β1l ). 0≤s≤t

Xt = β2l t + Wtl

mlt = inf 0≤s≤t Xs

and

P(β2l t + Wtl ≤ w0 | inf β2l s + Wsl ≥ β1l ) 0≤s≤t

(6.13)

, we get that

=

P(Xt ≤ w0 | mlt ≥ β1l )

=

P(Xt ≤ w0 , mlt ≥ β1l ) . P( mlt ≥ β1l )

The denominator is equal to the approximate probability that the company does not go to bankruptcy before time

t: P( mlt ≥ β1l )


= P τl ≥ t =

where

hk1 (t)

and

hk2 (t)



l l 1 − Φ hl1 (t) − e2β2 β1 Φ hl2 (t)

(6.14)

are dened by equations (6.2). Furthermore:

P(Xt ≤ w0 , mlt ≥ β1l )

=

1 − P(mlt ≤ β1l ) − P(Xt ≥ w0 , mlt ≥ β1l )

= P(τ l ≥ t) − P(Xt ≥ w0 , mlt ≥ β1l ) where

P(τ l ≥ t)

is given by (6.14). From Bielecki and Rutkowski (2004 page 68), we have that

0

≥

β1l )

dP(Xt ≥ w0 , mlt ≥ β1l )

=

P(Xt ≥ w ,

mlt

 =

Φ

−w0 + β2l t √ t

 −e

2β2l β1l

 Φ

2β1l − w0 + β2l t √ t



such that

 −ϕ

−w0 + β2l t √ t 15



l l 1 √ + e2β2 β1 ϕ t



2β1l − w0 + β2l t √ t



1 √ t

(6.15)

Hence, the density of

Xt

conditionally to the survival of the holding equals

dP(Xt ≤ w0 | mlt ≥ β1l )

= −

hl1 (t)

1−Φ

1
dP(Xt ≥ w0 , mlt ≥ β1l ) l l − e2β2 β1 Φ hl2 (t)




(6.16)

Introducing the notation

v l (t)

=




hl1 (t)

1−Φ

1
l l − e2β2 β1 Φ hl2 (t)

and combining equations (6.13), (6.15) and (6.16), leads to the following expressions of the conditional expectation of the total value of the assets at t:

ˆ EQ Stl |Gtl

+∞

g l (t, w0 − β2l t)dP(Xt ≥ w0 , mlt ≥ β1l )

= −v l (t)




β1l

1 = v (t) √ t

"ˆ

+∞

g l (t, w0 − β2l t)ϕ

l

−e

2β2l β1l

ˆ

β1l +∞

g l (t, w0 − β2l t)ϕ

β1l

w = w0 − β2l t, "ˆ

By a change of variable

Q

E

Stl |Gtl




=

Given the denition of



l

g (t, w) ϕ β1l −β2l t

g l (w),



−w0 + β2l t √ t



dw0

2β1l − w0 + β2l t √ t



# dw0 .

we get that

+∞

1 v (t) √ t l



w √ t

 dw − e

2β2l β1l

ˆ

+∞ l

g (t, w)ϕ β1l −β2l t



w − 2β1l √ t

Z

e ri σ i

w

 ϕ

w √ t



2

Z 1 = e 2 ( ri σ i ) t ϕ



w − ri σiZ t √ t



and that

e

 ϕ

w − 2β1l √ t

 = e

  2 2 − 1t 2(β1l ) − 12 (2β1l + ri σiZ t)

 ϕ

w − (2β1l + ri σiZ t) √ t



the expectation (6.17) becomes:

"N X

 β1l − β2l t − ri σiZ t √ t i=1   #   N X Z 2 1 −β1l − β2l t − ri σiZ t µZ t+2β1l (β2l +ri σiZ ) i i + 2 (σi ) √ 1−Φ − S0 e t i=1 Q

E

dw .

l




As it is well-known that

ri σiZ w

#

we rewrite the last expression as follows:

"N     ˆ Z 1 X i µZi + 21 (1−ri2 )(σiZ )2 t +∞ w S0 e E = v (t) √ eri σi w ϕ √ dw l l t i=1 t β1 −β2 t #   ˆ +∞   N 2 X Z w − 2β1l µZ + 1 (1−ri2 )(σiZ ) t 2β2l β1l √ S0i e i 2 − dw . e e ri σ i w ϕ t β1l −β2l t i=1 Stl |Gtl

Q



Stl |Gtl




l

= v (t)



S0i e

Z 1 µZ i + 2 (σi )

and we infer the result from this last relation.

16

2

 t





1−Φ

,

(6.17)

To end this paragraph, we say a word about the pricing of CDS in this incomplete framework. If premiums are paid at regular intervals of time,

∆t,

ranging from

t1

to

tn ,

the CDS spread at

time t, given that the last information disclosure has been done at time zero, is provided for or

k=l

k

p (t) where

7

k=c

by:

P(τ k > ti )

(1 − R) ∆t

=

Ptn

ti =t1




e−r ti P τ k > ti−1 | Gtk − P τ k > ti | Gt Gtk ,
Ptn −r ti P τ k > t | G k i t ti =t1 e

(6.18)

is provided by equation (6.1).

Stochastic liabilities.

The model developped in the previous sections is well suited for non nancial corporations that mainly nance their activities by debts and equity.

This model does not t so well nancial

holdings such as banks or insurance companies, which have most of the times random liabilities. To remedy to this problem, we assume in this section that the holding company is nanced by stochastic liabilities

Lit

NL

having the following risk neutral dynamics:

dLit = Lit rdt + Lit

M X

L σi,j dWtj

i = 1, . . . , NL .

(7.1)

j=1

PM

L i i=1 Lt . We denote by Σ the NL × M matrix L of σi,j . The holding invests in NS activities or subsidiaries whose market values are solution of the following stochastic dierential equations:

Lt =

The total value of the liabilities is given by

dSti

=

rSti dt

+

Sti

M X

S σi,j dWtj

i = 1, ..., NS .

(7.2)

j=1

PNS

S i i=1 St . We denote by Σ the NS × M S matrix of σi,j . The market capitalization is the dierence between the total values of investments and liabilities

The total value of the investments is still denoted by

d

Et = St − Lt

=

NS X

St =

i S0i eZt

−

i=1

Zt

where the vector of

>

Z

N (µ t , ΣΣ t)

and the

where

(NS + NL )

NSX +NL

i

Li0 eZt ,

(7.3)

i=NS +1

NS + NL components Zti and is distributed as a multivariate normal the (NS + NL ) × M matrix of assets and liabilities covariances:  S  Σ Σ= (7.4) ΣL

counts

Σ

is

mean vector is given by

Z



µ =

µZS µZL

 (7.5)

with



S S> r − 21 e> e1 1Σ Σ . . .

 µZS =  r−

1 > S S> eNS 2 eNS Σ Σ

  

 and

L L> r − 12 e> e1 1Σ Σ . . .

 µZL =  r−

17

1 > L L> eNL 2 eNL Σ Σ

  

(7.6)

where

µZS

this section

Zti

µZL are respectively drifts of the th N +NL the i unit root vector of R S .

and

assets and the liabilities and where

ei

denotes in

To simplify further calculations, the variance of

is dened as

V ar(Zti ) = σiZS

V ar(Zti ) = σiZL


2


2

> t = e> i ΣΣ ei t

> t = e> i ΣΣ ei t

i = 1, . . . , NS

(7.7)

i = NS + 1, . . . , NL .

(7.8)

PNS +NL

γi Zti that is a weighted sum of processes i=1 i Zt where the constant γi are chosen to minimize the quadratic spread between covariances of upper and lower convex approximations of assets/liabilities: Λt =

As previously, we introduce a process

γi = argmin

NSX +NL NSX +NL i=1

We denote the correlations by

j=1

ri

for

ri

2 > > > e> ΣΣ γ e ΣΣ γ i j  . 1 − p q > > > ei ΣΣ ei ej ΣΣ> ej γ > ΣΣ> γ 

i = 1, ..., NS

(7.9)

as follows:

=

cov(Zti , Λt ) p p V ar(Zti ) V ar(Λt )

=

> e> i ΣΣ γ p p . > e> γ > ΣΣ> γ i ΣΣ ei

(7.10)

We split the vector of correlations in two sub vectors, one related to assets and one to liabilities:

riS = ri

i = 1, ..., Ns ,

riL = rNS +i

i = 1, ..., NL .

The processes dened hereafter are used as lower and upper approximations of asset and liability processes:

Sti,l

Li,l t

=

Li0

 exp

 exp

µZS i

 

1 S 2 ZS 2 S ZS l + (1 − ri ) σi t + ri σi Wt 2

(7.11)

µZL i

 

1 ZL 2 L ZL l L 2 t + ri σi Wt + (1 − ri ) σi 2

(7.12)

Sti,c

ZS = S0i exp µZS Wtc i t + σi

Li,c t

ZL = Li0 exp µZL Wtc i t − σi




(7.13)




(7.14)

Wtc are independent Brownian motions such that W0l = W0c = 0. Note that in the i,c ZL denition of Lt , the sign of σi Wtc is negative because liabilities are substracted from assets (for details, we refer to Dhaene et al. 2002, paragraph 4). Estimates of the market value of equity are

where

Wtl

=

S0i

and

provided by the following dierences:

Etl =

NS X i=1

Sti,l −

NL X

Li,l t

Etc =

i=1

NS X i=1

18

Sti,c −

NL X i=1

Li,c t .

By construction, the following convex order relations are satised

1

Etl ≤cx Etc .

(7.15)

α,

The default is triggered when the equity falls below a certain level denoted by bution of the hitting time is unknown. Convex estimates of

g l (.)

function

and

Etl

g c (.)

=

=

Etc

=

of time and of Brownian motions

NS X

These functions

but the distri-

can be respectively rewritten as

and

Wtc

:



 

1 S 2 ZS 2 S ZS l µZS + (1 − r )) σ t + r σ W i i i i i t 2 i=1    NL X

1 L 2 ZL 2 L ZL l (1 − r Li0 exp µZL + )) σ t + r σ W − i i i i i t 2 i=1 S0i exp

g l (t, Wtl )

NS X

(7.16)

NL
X
ZS c ZL S0i exp µZS t + σ W − Li0 exp µZL Wtl i i t i t − σi

i=1

=

Wtl

Et

i=1

g c (t, Wtc ).

g l (t, w)

(7.17)

g c (t, w)

and

do not admit any simple analytical inverse functions.

gl

in previous sections, we use linear approximations of functions


−1

and

−1

(g c )

As

to keep closed

form expressions for approximate default probabilities. We choose a time t0 , calculate numerically

gl


−1

(t0 , α)

and develop

gl


−1

linearly around

gl where

β1l

and

β2l


−1

t0 :

(t, α) ≈ β1l − β2l t,

(7.18)

are dened by:

   −1 S2 ZS 2 1
 µZS ) t+riS σiZS (gl ) (t0 ,α) 1 S2 ZS 2 i ZS i + 2 (1−ri )(σi (1 − r ) σ e S µ + i i i i=1 0 2   2 −1
2  µZL PNL i  ZL 1 + 12 (1−riL 2 )(σiZL ) t+riL σiZL (g l ) (t0 ,α) e i − i=1 L0 µi + 2 (1 − riL 2 ) σiZL

PNS

β2l =



2



−1

S2 ZS 1 µZS ) t+riS σiZS (gl ) (t0 ,α) S ZS i i + 2 (1−ri )(σi i=1 ri σi S0 e   −1 2 PNL L ZL i µZL + 12 (1−riL 2 )(σiZL ) t+riL σiZL (g l ) (t0 ,α) − i=1 ri σi S0 e i

PNS

.

(7.19)

β1l

=

gl


−1

(t0 , α) + β2l t0

(7.20)

In the same way, we get that,

(g c ) where

β1c

and

true that

=

(t, α) ≈ β1c − β2c t,

ZS i ZS µZS (g c )−1 (t0 ,α) i t+σi i=1 S0 µi e PNS ZS i µZS t+σZS (gc )−1 (t ,α) 0 i i i=1 σi S0 e

ZL

−

PNL

µi Li0 µZL i e

−

PNL

σiZL S0i eµi

i=1

what notations suggest, we don't necessary have

Etl

(7.21)

are dened as follows:

PNS

β2c 1 Despite

β2c

−1

is a comonotonic sum.

19

i=1

Sti,l ≤cx Sti,c

t+σiZL (g c )−1 (t0 ,α)

ZL t+σ ZL i

and

(g c )−1 (t0 ,α)

i,c Li,l t ≤cx Lt .

.

(7.22)

In general it is neither

β1c

(g c )

=

−1

(t0 , α) + β2c t0 .

(7.23)

Linear functions (7.18) and (7.21) delimit the continuation and bankruptcy regions, in function

Wtl and Wtc . But given that g l (t, w) and g c (t, w) are either increasing or decreasing functions c −1 of w , continuation regions can be above or below the linear approximations of (g ) (t0 , α) and
−1 l l c g (t, α). The (approximate) defaults can then occur when Wt and Wt hit an upper or a lower of

boundary. The easiest way to detect if boundaries (7.18) and (7.21) are upper of lower frontiers, is

β1l

to check the sign of

and

β1c .

β1l ≤ 0 or β1c ≤ 0, the continuation region is above them. Indeed, l is in bankruptcy at time 0 given that Wt = 0. The approximate

If

if it is not the case, the holding

default times of convex bounds are then dened as hitting times of

If

β1l > 0

τl

=

 inf t ≥ 0 | − β1l + β2l t + Wtl ≤ 0 , −β1l + β2l s + Wsl > 0 ∀s < t

(7.24)

τc

=

inf {t ≥ 0 | − β1c + β2c t + Wtc ≤ 0 , −β1c + β2c s + Wsc > 0 ∀s < t} .

(7.25)

β1c > 0,

or

the continuation region is below the linear boundaries for the same reason.

The approximate default times are dened then as follows:

τl

=

 inf t ≥ 0 | β1l − β2l t + Wtl ≤ 0 , β1l − β2l s + Wsl > 0 ∀s < t

(7.26)

τc

=

inf {t ≥ 0 | β1c − β2c t + Wtc ≤ 0 , β1c − β2c s + Wsc > 0 ∀s < t} .

(7.27)

As these are all hitting times of a Brownian motion with drift, the approximate probabilities of default are then provided by similar expressions to those obtained in section 4, except that the sign of

β1l,c

plays now an important role:



k k P(τ k ≤ t) = Φ hk1 (t) + e2β2 β1 Φ hk2 (t) hk1 (t) = −sign(β1k )



β1k − β2k t √ t

The asymptotic probabilities of ruin when



for

k=l

or

hk2 (t) = −sign(β1k )

β1k β2k > 0

or

β1k β2k < 0

c

(7.28)

β1k + β2k t √ . t

are respectively

1

and

k

k

e2β2 β1 .

The CDS premium in this setting can be easily computed by formula (5.6) in which we substitute the probabilities of default by these obtained for random liabilities.

We now consider that the

information about the holding is not continuously but as in section 6. We assume that the only information available at time

t

has been disclosed at time

The information carried by the ltration

(Ft )t

0

and that the holding is still active.

is not accessible in continuous time. As previously

the information available to the market is represented by the ltration with the ltrations

Gtk

for

k=l

or

c

Gt ,

and we will work

for the approximations. Given the similarities between the

approximate default times for models without and with stochastic liabilities, we can easily infer that the approximate default probabilities are still provided by the formula (6.1):

k

P τ >T

| Gtk






k k 1 − Φ hk1 (T ) − e2β2 β1 Φ hk2 (T )

= k k 1 − Φ hk1 (t) − e2β2 β1 Φ hk2 (t)

for

k=l

or

c.

(7.29)

Furthermore, the estimates of the equity, provided in the next corollary, are obtained in the same way as in proposition 6.2, except that a particular care must be granted to the sign of Corollary 7.1.


l

EQ Stl |Gt

β1l,c .

The expected value of Stl given the information Gtl , equals =

    S,l S,l 2β1l (β2l +riS σiZS ) 1 − Φ h (i, t) − e Φ h (i, t) 1 2 ( ) t

S0i e l l 1 − Φ hl1 (t) − e2β2 β1 Φ hl2 (t) i=1     l l L ZL   1 − Φ hL,l (i, t) − e2β1 (β2 +ri σi ) Φ hL,l (i, t) NL 2 X 1 2 µZL + 21 (σiZL ) t

− Li0 e i (7.30) l βl l l 2β 2 1 1 − Φ h1 (t) − e Φ h2 (t) i=1

NS X



1 µZS i +2

2 σiZS



20

where hS,l 1 (i, t)

=

hL,l 1 (i, t)

=


−sign(β1l ) β1l − β2l t − riS σiZS t √ t
−sign(β1l ) β1l + β2l t − riL σiZL t √ t


−sign(β1l ) β1l + β2l t + riS σiZS t √ = t
l l −sign(β ) β + β2l t + riL σiZL t 1 1 L,l √ h2 (i, t) = t hS,l 2 (i, t)

The expected value of Stc given the information Gtl is obtained by replacing β1l , β2l , riS and riL respectively by β1c , β2c , 1 and -1. CDS premiums can also be calculated by substituting the right approximate probabilities of default in the formula (6.18).

8

Numerical Applications.

In a rst example, we compare the exact and approximate default probabilities of a holding composed of ve business lines that deliver correlated dividends. The initial value of investments, their volatilities and correlations are respectively set to:

S0 = (20, 20, 20, 20, 20)0

(8.1)


σiZ = (10%, 20%, 30%, 40%, 50%)0    ρ=   The risk free rate is equal to equal to

α = 90.

1.0 −0.3 −0.6 −0.2 −0.1

r = 2%

−0.3 1 0.5 0.3 0.1

−0.6 0.5 1 0.7 0.2

−0.2 0.3 0.7 1 0.3

−0.1 0.1 0.2 0.3 1

(8.2)

   .  

(8.3)

and the oor triggering the default of the company is set

In a following series of tests, multiple levels of

minimization of the criterion (3.12) yields the following

α

are considered. The numerical

γi : 0

(γi )i = (1.6497, 0.5774, 0.3840, 0.2427, 0.2318) . The coecients

ri

that dene the lower convex bound of

St

are 0

(ri )i = (0.1050, 0.5714, 0.5742, 0.7448, 0.5673) . r1 is the smallest. The −1 (t, α), (g c ) (t, α) and their

As the rst business line is negatively correlated with others, the coecient left graph of gure (8.1) presents the exact inverse functions linear approximations (4.5) and (4.8), calculated with oer a good t.

t0 = 0 .

gl


−1

In this case, linear approximations

The right graph of the same gure shows the approximate default probabili-

ties obtained with the lower and upper linear bounds (dotted and dot dash curves).

The real

probabilities of default have also been computed by Monte Carlo simulations (5000 runs) with discretization step

∆t = 0.0005

(continuous curve). Using the same method, we also calculate the

lower convex estimate of default probabilities, dened by (4.3) calculated with the exact inverse boundary

gl


−1

(t, α)

(see dashed line). Probabilities of default computed with the lower convex

bound are not far from the real one. The upper convex estimate of default probabilities is relatively less accurate. As the driving risk processes used are continuous Brownian motions, default probabilities vanish over a short period. This feature is not necessary observed in reality.

21

0.2

1

0.1

0.9

0.8 0 0.7 −0.1 0.6 −0.2 gl −1

Monte Carlo Monte Carlo Lower Bound Lower Bound linear proxy Upper Bound linear proxy

0.5

linear proxy for gl −1 −0.3

gc −1 linear proxy for gc −1

0.4

−0.4 0.3 −0.5 0.2

−0.6

0.1

0

1

2

3

4

0

5

g l,c

Figure 8.1: Example of function


−1

0

(t, α)

1

2

3

4

5

and default probabilities.

Figure 8.2 exhibits estimates of the market value of debts, in function of the oor

α.

The

accounting value of the debt is set to 90 and the tax rate is null. Two scenarios are considered. On the left side, the coupon rate is 2% while on the right side, the coupon rate is 2.5%.

We

remark that the market value of debts and equity can respectively be minimized and maximized by a judicious choice of

α.

The lower approximation being the most relevant, the holding's owner

optimizes the equity market value by closing his activities when the market value of assets reaches 62.50 for a cost of debts of 2% or 70.00 for a cost of debts of 2.5% .

110

120

105 c=2.5% Lower Approx c=2.5% Upper Approx

110

c=2.0% Lower Approx c=2.0% Upper Approx

100

95 100 90

85

90

80 80 75

70 70 65

60

0

20

40

Value of α

60

80

60

100

0

20

40

Value of α

60

Figure 8.2: Market Value of debts as a function of

22

80

α.

100

α = 70

α = 90

0.35

0.8

0.7

0.3

0.6

delay 0y delay 1y delay 2y

0.25

0.5 0.2 0.4 0.15 0.3

0.1

delay 0y delay 1y delay 2y

0.2

0.05

0

0.1

0

1

2

3

4

0

5

0

1

2

3

4

5

Figure 8.3: Inuence of a delay on probabilities of default (lower approximation).

Figure 8.3 illustrates the inuence of a delay in the disclosure of information on the probabilities of default for maturities, estimated with the lower comonotone bound. Two scenarios are considered. In the rst one, the oor is set to value value of assets (S0

= 100).

α = 70

and is then relatively far from the initial

In this case, the higher is the time lag after the last publication

of information, the higher are the estimated probabilities of default. In the second scenario, the oor triggering the default is close to the initial value of assets,

α = 90.

Here, the inuence of

the delay is exactly the opposite and decreases estimated probabilities of default. We explain this as follows: if the holding is still alive after one or two years, the probability that the total assets value is far above

α is high.

After one or two years, the simple fact that the company did not go to

bankruptcy, is sucient to infer that the company is now in a less critical situation than in the past. The lack of information also inuences the appraisal of debts and equity. trated in gure 8.4.

This point is illus-

It exhibits on the left side, the market value of debts (coupon 2.5%, debt

accounting value 90) for dierent levels of the oor

α,

and for a delay of one or two years. The

right graph presents the expected value of the total of assets. We observe that the higher is the delay, the lower is the debt market value and the higher is the expected value of assets, whatsoever the level of the oor. The dierence between the expected market value of assets and debts gives the market value of the equity, which can also be maximized by a judicious choice of

23

α.

130

130

125 125 120 120 Debts delay 1y Debts delay 2y

115

E(St) delay 1y 115

E(St) delay 2y

110

105

110

100 105 95 100 90 95 85

80

0

20

40

Value of α

60

80

90

100

0

20

40

Value of α

60

80

Figure 8.4: Market values of debts and equity, in function of

100

α.

To end this section, we test the model of holdings nanced by stochastic liabilities. We have considered a rm that invests in two business lines and nanced by two random liabilities. The parameters chosen (initial values, volatilities and correlations) are the following:

S0 = (50, 50)0
ZS

σi

L0 = (40, 40)0
σiZL = (20%, 20%)0

= (10%, 10%)0

 1 0.3 0.6 0.2  0.3 1 0.5 0.3   ρ=  0.6 0.5 1 0.7  0.2 0.3 0.7 1 

The initial market value of the equity is then 20. The risk free rate is set to triggering the default of the company is yields the following

α = 16.

r = 2%

and the oor

The numerical minimization of the criterion (7.9)

γi :

0

(γi )i = (0.7724, 0.7621, 0.2560, 0.3756) . The coecients

ri

that dene the lower convex bound of

St

are 0

(ri )i = (0.7004, 0.7084, 0.9040, 0.7169) . The left gure of exhibit 8.5 presents convex estimates of probabilities of default. Compared to the results that we get in previous examples, the lower bound convex estimates are relatively far from the real probabilities of default, obtained by Monte Carlo simulations. Our approximation seems less reliable. In this case, as done in Vyncke et al. (2004), we can still work with a weighted average of estimates to price the debt or the equity.

The right graph of gure 8.5 reveals the

inuence of a delay in the disclosure of information on the expected market value of the equity, for dierent levels

α.

For the chosen parameters, the higher is the time lag, the lower is the price.

24

1

20

0.9 18 0.8

0.7 16 0.6

0.5

14

Lower Bound Monte−Carlo Upper Bound

0.4

12

0.3

0.2

E(Et) delay 1y Lower Approx

10

E(Et) delay 1y Upper Approx E(Et) delay 2y Lower Approx

0.1

E(Et) delay 2y Upper Approx 0

0

1

2

3

4

8

5

5

10

Value of α

15

20

Figure 8.5: Probabilities of default and expected equity.

9

Conclusions.

This paper investigates an approximating method to assess the credit risk related to a multiindustry rm.

If the market values of subsidiaries are lognormal, the distribution of the total

asset's rm is not know. In this case, equity and debts cannot be appraised analytically. Convex bounds allow us to circumvent this drawback. It leads to simple analytical expressions for approximate probabilities of default, equity and debts, of a holding company, both when the complete information about the holding is released in a continuous way and when only incomplete information is available. Numerical applications tend to conrm that probabilities of default estimated from the lower convex bounds are close to these obtained by Monte Carlo simulations. The main drawback of our approach is that as the driving risk processes used are Brownian motions, default probabilities vanish over a short period. This feature is not necessarily observed in reality. It is possible to remedy to this by using jump processes, but in this case, analytical tractability is lost. Our approach has however other advantages. It duplicates dependent business lines. And as the total rm's asset is a sum of lognormally distributed variables, its distribution presents more asymmetry and leptokurticity than a single lognormal variable. Furthermore the managerial implications of our model are multiple. Asymptotic ruin probabilities may be used as a risk indicator. The management can also use it to determine a threshold triggering the holding's default, so as to maximize the shareholder's interests.

And an operator on CDS markets can

eventually use our approach for pricing purposes. Finally, the model can be extended to holdings nanced by stochastic liabilities. However, numerical results seem less persuasive in the case of stochastic liabilities.

Acknowledgement. The authors thank Steven Vanduel, of the Vrije Universiteit Brussel, for his helpful comments. They further are grateful to the referee for his/her constructing remarks which helped improving the paper. Griselda Deelstra acknowledges support of the ARC grant IAPAS "Interaction between Analysis, Probability and Actuarial Sciences" 2012-2017.

25

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