Intensity Based Models
Advanced Methods of Risk Management
Umberto Cherubini
Learning Objectives
• In this lecture you will learn
1. Extract default intensity from market
data
2. Model the dynamics of the intensity of
default and the credit spread
3. Calibrate the recovery rate from the
CDS and the bond markets
4. Extend the model to portfolios
Reduced form models
• In the reduced form models, default probability and
recovery rate (LGD) are modelled based on statistical
assumptions instead of an economic model of the
firm.
• The modelling techniques that are used are very
close to those applied in insurance mathematics,
specifying the frequency of occurrence of the event
(default probability) and the severity of loss in case
the event takes place (loss given default)
• These models use then the concept of intensity and
for this reason are also called intensity based
Credit spread and survival analysis
• Denote, in a structural model, Q the probability of
survival of the obligor after the maturity of the
obligation, (the default probability is then DP = 1 – Q)
and LGD the loss given default figure. Then, the
credit spread is given by
Credit Spread = – ln[1 – (1 – Q )LGD]/(T – t)
• Assume now the most extreme case in which all the
exposure is lost (LGD = 1). We have
Credit Spread = – ln[Q]/(T – t)
• Models from survival analysis (actuarial science) can
help design the credit spread.
The simplest model
• Assume the default event to be drawn from a Poisson
distribution (remember that it describes the
probability of a countable set of events in a period of
time).
• The Poisson distribution is characterized by a single
parameter, called intensity. The probability that no
event takes place before time T (in our case meaning
survival probability beyond that)is given by the
formula
Prob( > T) = exp (–  (T - t))
Constant intensity model
• Applying the survival probability function
Q = Prob( > T) = exp (–  (T - t))
to the credit spread formula (again
under the assumption LGD = 1)
Credit Spread = – ln[Q]/(T – t)
we get
Credit Spread = 
Intensity vs structural
• Intensity  denotes the probability of an event
in an infinitesimal interval of time. The
expected time before occurrence of the event
is 1/.
• Differently from structural models, the default
event comes as a “surprise”. Technically, it is
said that default is an inaccessible time.
• The intensity corresponds to the concept of
instantaneous forward rate in interest rate
models.
Cox models
• If the intensity parameter is not fixed, but changes
stochastically with time, the model is called Cox
model (or double stochastic models)
• For every maturity we can consider an average
intensity (t,T) and the credit spread curve will be
Credit Spread(t,T) = (t,T)
• Notice that the relationship between , that is the
instantaneous intensity, and the average intensity is
the same as that between instantaneous spot rate
and yield to maturity in term structure models
Affine intensity models
• Assume the dynamics of default intensity is
described by a diffusive process like
d (t) = k( – (t))dt + dz(t)
where setting  = 0, 0.5 deliver standard
affine term structure models for the credit
spread
Debt(t,T) = v(t,T)exp(A(T-t) - B(T -t) (t))
with A and B the affine functions in Vasicek (
= 0) or Cox Ingersoll Ross ( = 0.5) models
Positive recovery rate
• If we assume positive recovery rate (and so LGD < 1)
and independence between interest rate in default
intensity we can easily extend the analysis. Denote 
the recovery rate and compute
Debt(t,T; )=v(t,T)[Prob( > T)+ Prob(  T)]
Debt(t,T; )=  v(t,T) +(1-) Prob( >T)v(t,T)
Debt(t,T; 0)= Prob( >T)v(t,T), from which...
Debt(t,T; )=  v(t,T) +(1-) D(t,T; 0)
• The price is obtained as a portfolio of the risk free
asset and a defaultable exposure with recovery rate
zero.
Default probabilities
• The spread of a BBB 10 exposure over the risk-free
yield curve is 45 basis points.
• Assuming zero recovery rate we get
Prob( >T) = exp (– .0045 10) = 0.955997
and the probability of default is
1 - 0.955997 = 4.4003%
• Assuming a 50% recovery rate we have
Prob( >T) = [exp (– .0045 10) - ]/(1- ) =
0.911995
and default probability is 1 - 0.911995 = 8.8005%
Recovery rate
• The other point involved in modelling the
expected loss refers to the recovery rate.
• This topic is particularly involved, and the
research on the subject is not very
developed.
• In particular, the point is how to recover
– the value of the recovery rate
– the dependence of recovery rate and default
– whether the recovery rate is computed with
respect to face value or market value, or other
Recovery rate and default
• In case the recovery rate is independent of
the default probability expected loss can be
computed using whatever distribution defined
on the support between 0 and 1
• Typical example is the beta distribution that is
very often used to study the recovery from
samples of defaults.
• Altman,Resti and Sironi find that the number
of defaults and the amount of recovery are
negatively correlated across business cycles
Recovery of what?
1. Recovery of face value: it is the legal
concept of recovery. Once in default,
payments of interest is stopped and a
fraction of principal is allocated to creditors
2. Recovery of treasury: it refers to a fraction
of a risk free bond (Treasury!!) with the
same financial structure of the defaultable
bond
3. Recovery of market value: it refers to the
market value of the bond prior to default
Recovery of face value
• It is based on the idea of legal default. It is useful for
corporate bonds.
• Moody’s compute the recovery rate as the ratio of the
value of the bond on the secondary market one
month after default and the face value.
• Technically, default should be considered at every
time, assuming that the bond is substituted with a
percentage of par.
• In practice, it can be approximated assuming
payment of the recovery at maturity of the bond, as in
the previous example.
Recovery of treasury
• It is based on the assumption that in case of
default there is rescheduling.
• It is typical of sovereign risk applications in
which case there is no formal bankruptcy
procedure leading to recovery of fraction of
par
• Typical example is the concept of “haircut”
that is currently used in the current discussion
on the sovereign debt crisis in the Euro area.
Recovery of market value
• It is considered as an approximation that
makes the model easier to handle.
• In fact, it turns out that intensity can be
modified by allowing for the intensity to be
adjusted by LGD. Namely, in the simple
constant intensity model, with intensity , one
could adjust the intensity to * =  LGD.
• In a discrete time setting, the one-period
interest rate (r +  LGD)/(1 –  LGD)
The case of OPS Argentina
• In 2005 Argentina offered a swap program of
bonds defaulted in 2001 for two structures of
bonds
• Discount bonds: maturity 12/2033, coupons
7.82% (capitalized until 12/2013). Exchange
ratio 33.70%
• Par bonds: maturity 09/2038, lower coupons
step up (2.26%, 3.38%, 4.74%) amortizing
schedule 20 equal payments starting in the
last 10 years. Exchange ratio: 100%
28/03/2010
28/01/2010
28/11/2009
28/09/2009
28/07/2009
28/05/2009
28/03/2009
28/01/2009
28/11/2008
28/09/2008
28/07/2008
28/05/2008
28/03/2008
28/01/2008
28/11/2007
28/09/2007
28/07/2007
28/05/2007
28/03/2007
28/01/2007
28/11/2006
28/09/2006
28/07/2006
28/05/2006
28/03/2006
28/01/2006
28/11/2005
The prices of the OPS bonds
120
100
80
60
Discount
Par
40
20
0
Argentina CDS
 CDS n 
Q(n)  Q(n  1) 1 
LGD 

CDS n   CDS n  1 n 1

v0, i Qi  1

v0, n LGD
i 1
Maturity
1
2
3
4
5
6
7
8
9
10
Swap Rate Discount
1,12
0,989
1,40
0,973
1,72
0,950
2,03
0,922
2,31
0,891
2,56
0,857
2,76
0,823
2,93
0,789
3,07
0,756
3,189
0,723
CDS
714,41
823,16
836,18
841,90
845,32
829,51
818,17
813,84
810,43
807,79
v(0,i)Q(i-1)
0,989
0,8567589
0,6892958
0,5705105
0,470926
0,3874916
0,3306688
0,2815426
0,236256
0,1981686
Q(n)
0,881
0,726
0,619
0,529
0,452
0,402
0,357
0,313
0,274
0,24
DP(n)
11,91%
17,62%
14,73%
14,55%
14,52%
11,13%
11,18%
12,39%
12,35%
12,35%
CDP(n) Intensity
11,907% 12,68%
27,432% 19,39%
38,124% 15,94%
47,128% 15,72%
54,806% 15,69%
59,836% 11,80%
64,326% 11,86%
68,747% 13,23%
72,606% 13,18%
75,990% 13,19%
Recovery: PAR
0,1
0,09
0,08
0,07
0,06
Face
Treasury
0,05
0,04
0,03
0,02
0,01
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
0
Recovery: DISCOUNT
0,07
0,06
0,05
0,04
Face
Treasury
0,03
0,02
0,01
57
55
53
51
49
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
0
OPS bonds
•
•
•
•
•
•
•
•
Type: Discount
Market price: 63.974
Fair Value: 151.12
EL (Face): 67.46
EL (Treasury): 87.22
Value (Face): 83.66
Value (Trs): 63.91
Recovery: 21.92%
•
•
•
•
•
•
•
•
Type: Par
Market price: 31.25
Fair Value: 91.22
EL (Face): 64.50
EL (Treasury): 59.22
Value (Face): 26.73
Value (Trs): 32.00
Recovery: 21.92
Portfolios of exposures
• Assume we have a portfolio of exposures (for simplicity with the
same LGD). We can distinguish between a very large number of
exposures and a limited number of them. In a retail setting we
are obviously interested in the former case, even though to set
up the model we can focus on the latter one (around 50-100).
• We want define the probability of loss on the portfolio. We define
Q(k) the probability of observing k defaults (Q(0) being survival
probability of the portfolio). Expected loss is
n
EL  LGD kQk 
k 1
“First-to-default” derivatives
• Consider a credit derivative providing
“protection” the first time that an element in
the basket of obligations defaults. Assume the
protection is extended up to time T.
• The value of the derivative is
FTD = LGD v(t,T)(1 – Q(0))
• Q(0) is the survival probability of all the
names in the basket:
Q(0) Q(1 > T, 2 > T…)
“First-x-to-default” derivatives
• As an extension, consider a derivative
providing protection on the first x
defaults of the obligations in the basket.
• The value of the derivative will be
x
n
k 1
k  x 1
FTDx   LGD kQk   xLGD  Qk 
Probability Q(x) specification
• Evaluating basket credit derivatives and the
credit risk of a portfolio requires to specify the
joint distribution of defaults Q(x)
• This distribution depends on two elements
– Default probability of each obligation in the basket.
– The correlation (dependence) structure of
defaults of the obligations in the portfolio.
Models for Q(x)
• Hypotheses about individual exposures are
– Homogeneous pool of exposures (same default
probability)
– Heterogeneous pool of exposures (different
default probabilities)
• Hypotheses about dependence structure are
–
–
–
–
Indipendent defaults
Multivariate reduced form models (Marshall Olkin)
Copula functions
Factor copulas (conditionally indipendent defaults)
Indipendent defaults
• If we assume the defaults to be independent the most
obvious choices for the joint probability are
– Binomial distribution
– Poisson distribution
 n x
n x


Qx    q 1  q 
 x



exp   i T  t   i T  t 
 i 1
 i 1

Q x  
x!
n
n
x
Portfolio intensity
• Poisson model is particularly useful because it allows
immediate extension of reduced form models to the
multivariate setting.
• The indipendence hypothesis implies that
Q(0) = Q(1 > T, 2 > T…) = Q(1 > T) Q(2 > T)…
and, in intensity based models
Q(1 > T) Q( 2 > T)…= exp[– (1 + 2 +…)(T – t)]
• We then obtain a porfolio intensity as the sum of
individual intensities
 = 1 + 2 +…
The value of a first-to-default
• Remember that the value of the a first-todefault swap is given by
FTD = LGD v(t,T)(1 – Q(0))
• In the independent default case we get then
LGDv(t,T)(1 – exp[– (T – t)])
=
LGDv(t,T)(1 – exp[– (1 + 2 +…)(T – t)])
• The problem is to extend this model to the
case in which defaults are dependent.
Marshall Olkin distribution
• Marshall Olkin distribution is the natural extension of
Poisson distribution to the multivariate case.
• Assume a case with two exposures. According to
Marshall Olkin distribution we have
Q(1 > T, 2 > T) = exp[– (1 + 2 + 12)(T – t)]
where 12 is the intensity of an event “killing” both the
obligations.
• The correlation between the survival times is
12 = 12 /(1 + 2 + 12)
Portfolio intensity
• The idea behind Marshall Olkin distribution is that
several shocks affect subset of the obligations in the
portfolio.
• The problem is that one may conceive a very high
number of such shock, making calibration of the
model unfeasible.
• Generally the specification is proposed is with a
unique common factor, and the intensity is
n
   i  123....n
i 1
Intensity risk and contagion
• In Marshall Olkin distribution each exposure in the
portfolio is subject to idiosyncratic shocks and shocks
that are common to other exposures.
• The shock that triggers default of all exposures is the
ideal representation of a systemic event.
• Notice that since in the Marshall-Olkin model all
events (idiosyncratic and common) are independent,
the models does not allow for contagion
• In a possible extension, contagion can be introduced
with dependence between idiosyncratic and common
shocks,