Valuation in Discrete – Time Recovery
Li
Wang
Graduate School of Math and Stats
McMaster University
1. Introduction
Bond investors loose all of their investment in the event of a default. In
practice, however, investors frequently receive some recovery payment
upon default. With the (in homogeneous) Poisson process we now have a
first mathematical model to model the arrival time of a default event.
Assuming constant interest rates r > 0, for a defaultable zero bond
maturing at T with zero recovery we get
d 0T  E Q [e rT 1{ T }]  e rT P[  T ]  e ( r  )
That is, in the intensity based framework we can value a defaultable bond
as if it were default free by simply adjusting the discounting rate. Instead
of discounting with the risk-free interest rate r, we know discount with the
default-adjusted rate r+ λ where λ is the risk-neutral intensity. For
modeling recovery purpose, we hope to get the similar form by adjusting
the discounting rate. In this paper, we try to figure out this intuition by
valuation in discrete-time recovery. We consider two conventions:
fractional recovery of face value and fractional recovery of market value.
2. Reduced -Form Default Models
2.1 Poisson Process
Let T1......Tn denote the arrival times of some physical event. We call
the sequence (Ti) (homogeneous) Poisson process with intensity λ if the
inter-arrival times Ti+1 - Ti are independent and exponentially distributed
with parameter, equivalently, letting N (t )  i 1{Ti  t} count the number
of event arrivals in the time interval [0, t], we say that N  ( N (t )) t 0 is a
(homogeneous) Poisson process with intensity λ if the increments
N (t )  N ( s ) are independent and have a Poisson distribution with
parameter λ(t-s) for s < t , i.e.,
P[ N (t )  N ( s )  k ] 
1
( (t  s )) k e  (t  s )
k!
Poisson process has a number of important properties making it
ubiquitous for modeling discrete events. Being Markovian, the
occurrence of its next k jumps during any interval after time t is
independent from its history up to t. The probability of one jump during a
small interval of length ∆t is approximately λ∆t and that the probability
for two or more jumps occurring at the same time is zero. In Practice, the
default time is set equal to the first jump time of the Poisson process N.
Thus   T1 is exponentially distribution with (intensity) parameter and
the default probability is given by
F (t )  p[  t ]  1  e  t
The intensity is the conditional default arrival rate given no default:
lim
h0
1
p[  (t , t  h) |   t ]  
h
Letting ƒ denote the density of F we can also write

f (t )
1  F (t )
In the structural approach a default is predictable, i.e. it can be anticipated.
Since the jumps of a Poisson process are totally unpredictable, in the
intensity based approach the default is unpredictable as well. This has
important consequence for the term structure of credit spreads.
We plot default probabilities F(T) as a function of horizon T for
varying degree of intensities λ = 0.005, 0.01, and 0.02.
Clearly, F(T) is for fixed T increasing in the default intensity λ.
2.2 Reduced form models
“Reduced-form” defaultable term-structure models typically take as
primitives the behavior of default free interest rates, the fractional recovery
of defaultable bonds at default, as well as a stochastic intensity process for
default. The intensity λt may be viewed as the conditional rate of arrival of
default. For example, with constant, default is a Poisson arrival.
These models are distinguished somewhat by the manner in which the
recovery at default is parameterized. Jarrow and Turnbull (1996) stipulated
that, at default, a bond would have a market value equal to an exogenously
specified fraction of an otherwise equivalent default-free bond. Duffie and
Singelton (1997) followed with a model that, when specialized to
exogenous fractional recovery of market value at default, allows for closeform solutions in a wider range of cases, by showing that cash flows can
be discounted simply at the short-term default-free rate plus the riskneutral rate of expected loss of market value due to default.
Here, we propose models with two recovery assumptions: fractional
recovery of face value and fractional recovery of market value. In order to
see the basic idea of valuation with the models, we suppose that recovery
payments are made at the time of default and the default occurs only at
discrete time intervals.
3. Valuation in discrete-time recovery
The following sections specialize to obtain explicit results.
3.1 Ingredients and Assumptions
The model has several basic ingredients:
 A default time τ for default of the given issuer. The default time is
assumed to have an intensity process λ with a constant intensity λ,
for example, default has a Poisson arrival at intensity λ. More
generally, for t before τ, we may view λt as the conditional rate of
arrival of default at time t, given all information available up to that
time. In other words, for a small time interval of length ∆, the
conditional probability at time t that default occurs between t and
t+∆, given survival to t, is approximately λ∆t.
 A bounded short-rate process r and equivalent martingale measure Q.
 Assuming that the risk-neutral probability r and default risk λ are
independent with a known constant fraction for the face value or
market value recovered at default.
3.2 Fractional Recovery of face value
Fractional Recovery of face value is based on a legalistic interpretation of
bond covenants that would have defaulting firm liquidating their assets and
returning to bondholders some fraction of the face values of their bonds
according to the priority of their holdings.
Recovery at default is given by a bonded random variable W, per unit
of face value. With discrete-time recovery, W is measured and received as
of the first date after default among a pre-specified list T (1), T (2),...T (n)  T
of times, with Ti  Ti 1 , where T is maturity. The discrete –time recovery
assumption may also be viewed simply as an approximation, with the
virtue of explicit pricing, of the pricing that would apply with continual
recovery.
Suppose that recovery payments are made at the time of default and that
default occurs only at discrete time intervals of length ∆. For example,

1
, recovery is measured as of the end of the day of default. We
365
assume that the number of periods before maturity, n  (T  t ) /  , is an
integer. We let Z (t , i ) denote the market value at time t of any default
recoveries to be received between times t  (i  1) and t  i . We let
d (t , T ) denote the price of a defaultable zero-coupon bond maturing at
time T, We have
n
d (t , T )  d 0 (t , T )   Z (t , i )
i 1
Since the risk-neutral time t conditional probability of default during any
time interval ∆ is p * (t , t  (i  1))  p * (t , t  i) (the difference in the
survival probabilities to the beginning and the end of the period), we have
z (t , i )   (t , t  i) ( p * (t , t  (i  1))  p * (t , t  i))
Where δ(0, s) is the price of a default-free zero-coupon bond maturing at
time s.
3.3 Fractional Recovery of market value
An alternative recovery assumption is that, at each time t, conditional on
all information available up to but not including time t, a specified riskneutral mean fraction Lt of market value is lost if default occurs at time t.
With a risk-neutral default-intensity process λ*, the risk-neutral conditional
expected rate of loss of market value owing to default is St  t * Lt . The
T
pre-default market value is B . Then the price of a zero-coupon bond is
d (0, T )  E Q [e  rT 1{ T }  e  r (1  s)dT 1{ T } ]  e  ( r  s )T
which is the value of a zero recovery bond with thinned (risk-neutral)
default intensity S  L *  .
An attraction feature of this recovery convention, and the associated
pricing relation showed in above formula, is that one may use a model for
the default-adjusted short-rate process R = r + s of a type that admits an
explicit discount d(0,T) . Then, we can value a default bond on the same
computational platform used for default-free bond pricing.
In order to promote intuition for this pricing result, we will price a 3year defaultable zero-coupon bond in a event-tree setting. We suppose r
the default-free annual interest rate and upward and downward changes in r
are assumed to have equal risk-neutral probabilities. The bond may default
in any year with 8% (risk-neutral) probability. The bond loses 60% of its
market value if and when it defaults, i.e., L = 0.6.
λ* = 0.08 annual risk Neutral Default probability
L=0.06 Loss Rate
95.20
85.57
76.57
r=11.25%
95.20
r=7.5%
87.34
70.57
r=9%
r=5%
95.20
79.10
r=6%
88.81
r=7.2%
95.20
Figure1. Valuation of a 3-year zero-coupon bond with
default risk
We calculate the bond Price at each point as follows:
The bond price at the third year:
V  100  (1  *)  100   * (1  L)
V  100  (1  0.08)  100  0.08  0.6  95.2
If the default-free interest rate moves upwards to 11.25%, the value of the
bond at year 2, assuming that it has no defaulted, will be the risk-neutral
probability of surviving another year without default, multiplied by the
payoff given survival, plus the risk-neutral default probability multiplied
by the payoff given default. Then, at the second year nodes, assuming no
default before then, the bond prices are therefore
100
(1  L)  100
  *
(1  r )
(1  r )
100
(1  0.6)  100
0.92 
 0.08 
85.57
1.1125
1.1125
100
(1  0.6)  100
0.92 
 0.08 
 88.81
1.072
1.072
100
(1  0.6)  100
0.92 
 0.08 
 87.34
1.09
1.09
V  (1   *) 
In the same way, we can get the bond prices at the first year nodes:
85.57
(1  0.06)  85.57
 0.08 
 75.78
1.075
1.075
87.34
(1  0.06)  87.34
0.92 
 0.08 
 77.35
1.075
1.075
75.78  77.35
 76.57
2
0.92 
Since there are two prices at the second nodes each with 50% probability,
so we have to take the average of the two bond prices.
All bond prices at each node are shown in Figure 1.
From the event-tree setting, we can get a hint that the impact of default can
be captured by an effective discount rate at any node of
100  85.57
 16.86%
85.57
In general, the effective discount rate at any node in the tree is R, where
1
1

[ * (1  L)  (1  *)]
1 R 1 r
r *L
R
1  * L
with λ* us the risk-neutral probability of default at that node and L the
risk-neutral expected loss in market value, as a fraction of the market value
at that node, conditional on default in the next period.
For time periods of length ∆, if we substitute r, R and λ* in annualized
form, as
R 
r  L  * 
1  L * 
Dividing through by ∆ and allowing ∆ to coverage to zero, leaving the
continuous-time default-risk-adjusted short –rate process:
R  r  L *
We show the all default-adjusted short rates as following tree sitting.
16.86%
R= (r+λ*)/(1- λ*L)
P=100
85.57
12.92%
P=100
76.57
14.5%
10.29%
87.34
70.57
P=100
11.34%
79.10
12.6%
88.81
P=100
Figure2. Valuation at default-adjusted short rates
If we use the formula of R, we can get the same rate as the discount rate
R

r   * L 0.1125  0.08  0.6

 16.86%
1  * L
1  0.08  0.6
In the same way, we can get the default-adjusted short rate at the beginning:
79.10  70.57
 12.09%
70.57
76.57  70.57
 8.5%
70.57
12.09%  8.5%
 10.29%
2
0.05  0.08  0.6
also ,
 10.29%
1  0.08  0.6
We can confirm the bond price at default-adjusted rate R=10.29% node
76.57
79.10
 0.5 
 0.5  70.57
1.1029
1.1029
Therefore, when we use as a specification for a default-risk-adjusted short
rate process R, there is no loss of generality, when pricing a defaultable
claim, in treating the claim as if it is default-free, once the short-term
default –free discounting rate r is replaced by the default-adjusted short
rate R. In other words, the simplified valuation tree in Figure 2 is sufficient
for pricing defaultable zero-coupon bonds.
References:
[1] D. Duffie, M. Schroder, and C. Schroder, and C. Skiadas (1996),
“Recursive Valuation of Defaultable Securities and the Timing of
Resolution of Uncertainty,” Annals of Applied Probability, 6: 1075-1090
[2] D. Duffie and K. Singleton (1997), “Modeling Term Structures of
Defaultable Bond,” Working paper, Graduate School of Business, Stanford
University.
[3] M. R. Grasselli, “Math 772 – Credit Risk and Interest Rate Modeling,”
Class notes, Graduate School of Math and Stats, McMaster University,4445