Jacques J., Raimondo M. Semi-markov risk models for finance, insurance and reliability
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Reliability and credit risk models 359
Effec.
AAA AA A BBB BB B CCC D N.R. Total
165
AAA
90.3 6.1 0 0.61 0 0 0 0 3.03 100
560
AA
0.18 90 5.71 0.18 0 0 0 0 4.29 100
1095
A
0.09 1.5 87.22 5.11 0.18 0 0 0 5.94 100
896
BBB
0 0 2.79 84.93 4.46 0.67 0.22 0.34 6.59 100
619
BB
0.32 0.2 0.16 5.33 75.44 5.98 2.75 0.65 9.21 100
649
B
0 0 0.15 0.62 6.16 76.3 5.09 4.47 7.24 100
30
CCC
0 0 3.33 0 0 20 33.3 36.67 6.67 100
N.R.
0 0 0 0 0 0 0 0 100 100
4014
Table 3.3: example with rating withdrawn
Here, we see for example that companies in state AA will not be in default the
next year but that 5.71 % of them will degrade to simple A and 18 % to a BBB
and 0.18 will upgrade to an AAA.
Under the assumption of a homogeneous Markov chain, we obtain the following
results:
(i) the probability that an AA company defaults after two years:
P
(2)
(D/AA)=0.0018⋅ 0.0034=0.0006%,
which is still very low.
(ii) the probability that a BBB company defaults in one of the next two years :
This probability is given by:
(/;2)(/)(/)(/)
( / )( / ) ( / )( / ) ( / )( / )
0.34%+(84.93% 0.34%)
+(4.46% 0.65%)+(0.67% 4.47%)+(0.22% 36.67%)
=0.77%.
PD BBB PD BBB PBBB BBBPD BBB
PBB BBBPD BB PB BBBPD B PCCC BBBPD CCC
=
+
+++
=⋅
⋅⋅⋅
(iii) the probability for a company BBB to default between year 1 and year 2:
Using the standard definition of conditional probability ( see Chapter 1) we get
P(D at 2/non-def. at 1) = P(D at 2 & non-def. at 1)/ P(non-def. at 1)
=(0.77%-0.34%)/(1-0.34%)
=0.43%.
Let us point out that these illustrative results are true under the homogeneous
Markov chain model and moreover give similar results for all the companies of
the panel in the same credit rating.
In fact, in real life applications, credit rating agencies also study each company
on its own account so that specific information is also determining for giving the
final grade.
360 Chapter 8
3.3.4 Rating And Spreads On Zero Bonds
Let us first recall that a zero-coupon bond is a contract paying a known fixed
amount called the principal, at some given future date, called the maturity date.
So if the principal is one monetary unit and T the maturity date, the value of this
zero-coupon at time 0 is given by:
(0, )
T
B
Te
δ
−
= (3.26)
if
δ
is the considered constant instantaneous intensity of interest rate.
Of course, the investor in zero-coupons must take into account the risk of default
of the issuer. To do so, we consider that, in a risk neutral framework, the investor
has no preference between the two following investments:
(i) to receive almost certainly at time 1 the amount e
δ
as counterpart of the
investment at time 0 of one monetary unit,
(ii) to receive at time 1 the amount
()
(0)
s
es
δ
+
> with probability (1
−
p) or 0 with
probability p, as counterpart of the investment at time 0 of one monetary unit, p
being the default probability of the issuer.
The positive quantity s is called the spread with respect to the non-risky
instantaneous interest rate
δ
as counterpart of this risky investment in zero-
coupon bonds.
From the indifference given above, we obtain the following relation:
()
(1 )
s
epe
δδ
+
=−
(3.27)
or
1(1 ),
s
p
e=−
(3.28)
ln(1 );
s
p
=
−− (3.29)
s
p
sp p
≈
≅+
,
.
1
2
2
(3.30)
Let us now consider a more positive and realistic situation in which the investor
can get an amount
,(0 1)
α
α
<<
if the issuer defaults at maturity or before.
In this case, the expectation equivalence principle relation (3.27) becomes:
(1 ) ,
s
epepe
δ
δδ
α
+
=− + (3.31)
or
1(1 ) .
s
p
ep
α
=− +
(3.32)
It follows that in this case the value of the spread satisfies the equation
1
1
s
p
e
p
α
−
=
−
(3.33)
and so the spread value is
Reliability and credit risk models 361
1
ln .
1
p
s
p
α
−
=
−
(3.34)
As above, using the Mac Laurin formula respectively of order 1 and 2, we obtain
the two following approximations for the spread:
2
(1 ),
1
1
(1 ) (1 ) .
121
p
s
p
pp
s
pp
α
αα
≈−
−
⎛⎞
≈−− −
⎜⎟
−−
⎝⎠
(3.35)
4 CREDIT RISK AS A RELIABILITY MODEL
4.1 The Semi-Markov Reliability Credit Risk Model
As we already know, the credit risk problem can be seen as a reliability problem
in which the rating process, carried out by the rating agency, gives a reliability
degree of a firm bond and moreover, the default state can be seen as a down state
and an absorbing state.
From relations (2.15) and (2.19) it results that in this case the concept of
reliability and availability coincide.
We know that rating agencies like Standard & Poor’s, Moody’s or Fitch give
each examined firm a rating. In the preceding subsections, we used the S&P
simplified model giving eight kinds of ratings:
AAA, AA, A, BBB, BB, B, CCC, D,
where the states are in decreasing order depending on the “reliability” of their
debts, and the state D means default (for the precise definition of each state see
Crouhy et al (2001)).
In order to apply reliability models in a credit risk environment it is possible to
consider, following S&P classification, the first seven states as “good” states and
the D state as the only bad state and apply our semi-Markov reliability models to
the credit risk problem.
The state D will be an absorbing state, because once the state is reached, in the
sense that the firm is not in position to pay its debts and so therefore defaults, it is
not possible to exit from the state.
Furthermore in this case we are interested only in the R(t) function; the A(t) and
M(t) functions are meaningless in this environment.
()
i
R
t
gives the probability that the system was always working up to the time t
given that the system was in the working state i at time 0.
362 Chapter 8
In this case the reliability model is substantially simplified and to get all the
results that are relevant in the credit risk case, it suffices to solve the semi-
Markov evolution equation only once to get the following probabilities:
1) ( )
ij
t
φ
and ( , )
ij
s
t
φ
representing respectively the probabilities to be in the state j
after a time t starting in the state i at time 0 in the homogeneous case and starting
at time s in the state i in the non-homogeneous case. The semi-Markov
environment takes into account the different probabilities of state changes during
the permanence of the system in the same state (duration problem);
2) () ()
iij
jU
R
tt
φ
∈
=
∑
and ( , ) ( , )
iij
jU
R
st st
φ
∈
=
∑
, representing respectively the
probabilities that the system never goes into the default state in a time t in the
homogeneous case and from time s to time t in the non-homogeneous one;
3)
1()
i
H
t−
and
1(,)
i
H
st−
, representing the probabilities that in a time interval
t, in the homogeneous case, and from time s to time t, in the non-homogeneous
case, no one new rating evaluation was done for the firm;
4) ( )
ij
t
ϕ
and ( , )
ij
s
t
ϕ
representing the probabilities to get the rank j at the next
rating if the previous state was i and not one rating evaluation was made up to the
time t in the homogeneous case and from time s to time t in the non-
homogeneous one. In this way, for example, if the transition to the default state is
possible and if the system doesn’t move for a time t from the state i, we know the
probability that in the next transition the system will go to the default state.
They are defined by the following relations:
()
() ,
1()
ij ij
ij
i
p
Qt
t
Ht
ϕ
−
=
−
(4.1)
() (,)
(,) .
1(,)
ij ij
ij
i
p
sQst
st
Hst
ϕ
−
=
−
(4.2)
4.2. A Homogeneous Case Example
Now we give an example using the transition matrix given in Jarrow et al (1997).
This example will be chosen in the homogeneous case.
This matrix was constructed starting from the one year transition matrix given in
Standard & Poor’s Credit Review (1993).
We report the matrix for the sake of completeness.
AAA AA A BBB BB B CCC D
AAA
0.891 0.0963 0.0078 0.0019 0.003 0 0 0
AA
0.0086 0.901 0.0747 0.0099 0.0029 0.0029 0 0
A
0.0009 0.0291 0.8896 0.0649 0.0101 0.0045 0 0.0009
BBB
0.0006 0.0043 0.0656 0.8428 0.0644 0.016 0.0018 0.0045
BB
0.0004 0.0022 0.0079 0.0719 0.7765 0.1043 0.0127 0.0241
B
0 0.0019 0.0031 0.0066 0.0517 0.8247 0.0435 0.0685
CCC
0 0 0.0116 0.0116 0.0203 0.0754 0.6492 0.2319
D
0 0 0 0 0 0 0 1
Table 4.1: 1 year transition matrix
Reliability and credit risk models 363
The matrix value for d.f. are not known and so we construct these d.f. by means
of random number generators.
We report the results at time 5 and at time 10 of the matrix ( )
ij
t
φ
respectively in
Table 4.2.1 and Table 4.2.2.
For example the element 0.04326 that is in row
AA and in column A represents
the probability that a firm that at time 0 has a rating
AA will have rating A at
time 5.
AAA AA A BBB BB B CCC D
AAA
0.93129 0.06044 0.00504 0.00148 0.00164 0.00009 0.00000 0.00001
AA
0.00464 0.94420 0.04326 0.00519 0.00100 0.00165 0.00002 0.00005
A
0.00051 0.01505 0.94403 0.02950 0.00697 0.00330 0.00004 0.00060
BBB
0.00030 0.00295 0.03704 0.90384 0.04110 0.00976 0.00105 0.00397
BB
0.00023 0.00148 0.00572 0.04727 0.85624 0.05887 0.00908 0.02111
B
0.00000 0.00096 0.00195 0.00351 0.03377 0.89002 0.02404 0.04575
CCC
0.00000 0.00004 0.00474 0.00535 0.01258 0.03479 0.85292 0.08958
D
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000
Table 4.2.1: probabilities (5)
ij
φ
AAA AA A BBB BB B CCC D
AAA
0.83968 0.13696 0.01488 0.00375 0.00415 0.00047 0.00003 0.00008
AA
0.01084 0.86440 0.10055 0.01526 0.00433 0.00421 0.00012 0.00030
A
0.00141 0.03991 0.84668 0.08517 0.01638 0.00807 0.00032 0.00206
BBB
0.00086 0.00749 0.08702 0.78071 0.08579 0.02549 0.00327 0.00937
BB
0.00056 0.00344 0.01366 0.09229 0.69959 0.13097 0.01814 0.04135
B
0.00003 0.00279 0.00512 0.01162 0.06732 0.75319 0.05575 0.10419
CCC
0.00001 0.00029 0.01329 0.01436 0.02313 0.07803 0.61935 0.25154
D
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000
Table 4.2.2: probabilities (10)
ij
φ
Table 4.3 gives the reliability results
()
i
R
t
, probabilities to have no default in a
time t (row index) starting in the state i (column) at time 0.
AAA AA A BBB BB B CCC D
1 1.00000 1.00000 0.99987 0.99933 0.99846 0.99642 0.99294 0.0
2 1.00000 1.00000 0.99975 0.99884 0.99461 0.98808 0.98146 0.0
3 1.00000 0.99999 0.99969 0.99789 0.98908 0.97527 0.96374 0.0
4 0.99999 0.99997 0.99961 0.99715 0.98624 0.97029 0.94233 0.0
5 0.99999 0.99995 0.99940 0.99603 0.97889 0.95425 0.91042 0.0
6 0.99998 0.99992 0.99917 0.99505 0.97436 0.94749 0.89800 0.0
7 0.99997 0.99989 0.99888 0.99418 0.97144 0.93795 0.84898 0.0
8 0.99995 0.99984 0.99856 0.99334 0.96771 0.92535 0.79778 0.0
9 0.99994 0.99978 0.99825 0.99210 0.96446 0.90689 0.77184 0.0
10 0.99992 0.99970 0.99794 0.99063 0.95865 0.89581 0.74846 0.0
Table 4.3: probabilities of not having a default
364 Chapter 8
Table 4.4 gives probabilities 1 ( )
i
H
t
−
to remain always in the starting state
without transitions.
AAA AA A BBB BB B CCC D
1 0.98490 0.89746 0.86572 0.92634 0.86317 0.86674 0.94774 1.0
2 0.82635 0.82919 0.74506 0.77373 0.78411 0.75612 0.92103 1.0
3 0.74275 0.75242 0.68724 0.65713 0.64732 0.66181 0.87814 1.0
4 0.57977 0.73210 0.55915 0.59711 0.60133 0.58323 0.79454 1.0
5 0.47763 0.51794 0.47518 0.45947 0.47929 0.43098 0.65872 1.0
6 0.37730 0.41739 0.35444 0.36779 0.42974 0.30765 0.56190 1.0
7 0.30913 0.32773 0.26773 0.29968 0.30514 0.24540 0.41717 1.0
8 0.23808 0.25246 0.22929 0.17914 0.27208 0.15297 0.25461 1.0
9 0.11174 0.21338 0.12389 0.14214 0.13721 0.11744 0.15293 1.0
10 0.08543 0.02793 0.06785 0.05651 0.04622 0.07177 0.05478 1.0
Table 4.4: probabilities to remain in the starting state
Lastly,
Tables 4.5.1 and 4.5.2 give probabilities ( )
ij
t
ϕ
at 5 years and 10 years.
For example 0.07128 represents the probability that a firm that was at time 0 in
state
AA and remained in this state up to time 5 will then have the next transition
in state
A.
AAA AA A BBB BB B CCC D
AAA
0.89611 0.09045 0.00858 0.00157 0.00329 0.00000 0.00000 0.00000
AA
0.00861 0.90209 0.07128 0.01090 0.00417 0.00296 0.00000 0.00000
A
0.00102 0.03494 0.86517 0.08440 0.00935 0.00405 0.00000 0.00107
BBB
0.00078 0.00436 0.06867 0.83756 0.06478 0.01815 0.00234 0.00336
BB
0.00041 0.00223 0.00690 0.06408 0.78794 0.11412 0.01027 0.01405
B
0.00000 0.00264 0.00357 0.00866 0.05087 0.81139 0.05240 0.07046
CCC
0.00000 0.00000 0.01073 0.01009 0.01212 0.06289 0.68286 0.22131
D
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000
Table 4.5.1: probabilities (5)
ij
ϕ
AAA AA A BBB BB B CCC D
AAA
0.98240 0.00781 0.00808 0.00063 0.00108 0.00000 0.00000 0.00000
AA
0.03042 0.83757 0.08618 0.02700 0.01034 0.00849 0.00000 0.00000
A
0.00107 0.04218 0.85463 0.09107 0.00919 0.00071 0.00000 0.00115
BBB
0.00072 0.00028 0.04913 0.87719 0.04199 0.02433 0.00251 0.00386
BB
0.00033 0.00285 0.00795 0.11255 0.64711 0.17648 0.01645 0.03627
B
0.00000 0.00211 0.00409 0.00684 0.06930 0.79561 0.03869 0.08336
CCC
0.00000 0.00000 0.00456 0.00604 0.03402 0.10958 0.50718 0.33864
D
0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000
Table 4.5.2: probabilities (10)
ij
ϕ
Reliability and credit risk models 365
4.3 A Non-Homogeneous Case Example
Now we give a non-homogeneous example using as basis the transition matrices
given in Table 15 of Standard & Poor’s (2001).
In these matrices the state No Rating was present. Starting from the data reported
in the S&P publication, the non-homogeneous transition matrix was constructed.
Each element ( )
ij
p
s of the embedded non-homogeneous Markov chain should be
constructed directly from the data.
Constructing the MC, all possible transitions from state i to state j starting from
the year s should be taken into account. But we do not have the raw data and so
we use the one year transition matrices given in the Standard & Poor’s
publication.
The publication reports 20 years of history (one year transition matrices from
1981 up 2000). The example covers from time 0, corresponding to the year 1981,
to time 19, corresponding to the year 2000.
Table 4.6 reports three years of the non homogeneous embedded MC.
TRANSITION MATRICES
MATRIX AT TIME 0
AAA AA A BBB BB B CCC D
AAA
0.92450 0.07550 0 0 0 0 0 0
AA
0.01990 0.91045 0.06965 0 0 0 0 0
A
0 0.04760 0.88406 0.06624 0.00210 0 0 0
BBB
0 0 0.04870 0.90260 0.04870 0 0 0
BB
0 0 0.00924 0.04631 0.62960 0.31023 0.00462 0
B
0 0 0.01240 0 0.04940 0.91351 0.02470 0
CCC
0 0 0 0 0 0.09090 0.90910 0
D
0 0 0 0 0 0 0 1.00000
MATRIX AT TIME10
AAA AA A BBB BB B CCC D
AAA
0.97070 0.02930 0 0 0 0 0 0
AA
0.00485 0.88460 0.11055 0 0 0 0 0
A
0 0.02129 0.88672 0.07783 0.01240 0.00176 0 0
BBB
0 0 0.04247 0.89077 0.05151 0.00915 0 0.00610
BB
0 0 0.00397 0.06742 0.74600 0.10714 0.03575 0.03972
B
0 0.00975 0.00321 0.00654 0.03912 0.78181 0.05862 0.10095
CCC
0.02269 0 0 0 0.02269 0.04549 0.56819 0.34093
D
0 0 0 0 0 0 0 1.00000
MATRIX AT TIME19
AAA AA A BBB BB B CCC D
AAA
0.96115 0.02776 0.01108 0 0 0 0 0
AA
0.00986 0.87568 0.11113 0.00332 0 0 0 0
A
0 0.02493 0.90134 0.06779 0.00428 0.00083 0 0.00083
BBB
0 0.00178 0.02259 0.92584 0.03703 0.00638 0.00272 0.00366
BB
0 0 0.00362 0.04074 0.87290 0.05871 0.01202 0.01202
366 Chapter 8
B
0 0 0.00338 0.00338 0.03738 0.83230 0.04533 0.07824
CCC
0 0 0 0 0.01296 0.06489 0.59745 0.32470
D
0 0 0 0 0 0 0 1.00000
Table 4.6: Embedded NHMC
To apply the model it is necessary to construct also the d.f. of the waiting times
in each state i, given that the state successively occupied is known. As we do not
have these data either, we construct them by means of random number
generators.
Probabilities
(,)
i
H
st
to remain in the state from s to t without any transition are
reported in
Table 4.7. For example the element 0.55706 represents the
probability that the rating
AA has no other rating evaluation from the time 0 up
to the time 9.
Probability no movement
TIMES
AAA AA A BBB BB B CCC
0 1 0.90856 0.96379 0.98697 0.93658 0.93951 0.92026 0.94220
0 2 0.84173 0.92113 0.94230 0.88791 0.90524 0.86158 0.90722
0 3 0.78727 0.88494 0.88535 0.84758 0.85096 0.77361 0.89632
0 8 0.58795 0.59798 0.69578 0.65079 0.66009 0.62475 0.69739
0 9 0.51816 0.55706 0.68355 0.57614 0.58619 0.60454 0.64461
0 10 0.51338 0.50773 0.59941 0.52915 0.50759 0.59869 0.54138
0 17 0.13144 0.10865 0.14796 0.16612 0.16478 0.15852 0.13142
0 18 0.09384 0.09523 0.09996 0.09903 0.09947 0.10168 0.10572
0 19 0.02224 0.03585 0.03916 0.04847 0.07476 0.07823 0.06299
1 2 0.91830 0.94635 0.93876 0.96978 0.99286 0.90339 0.94856
1 11 0.42990 0.46490 0.48222 0.55546 0.44814 0.49398 0.45959
1 19 0.00536 0.07610 0.06425 0.02936 0.08739 0.06138 0.05097
2 3 0.94570 0.96836 0.90135 0.93726 0.98392 0.98624 0.97455
2 11 0.53603 0.44457 0.41834 0.42495 0.55929 0.52844 0.47755
2 19 0.09757 0.05980 0.07554 0.07681 0.06381 0.07286 0.05074
5 6 0.96537 0.92527 0.93849 0.97609 0.85933 0.97116 0.94845
5 13 0.55308 0.42610 0.43405 0.45723 0.38602 0.47904 0.47031
5 19 0.06883 0.02407 0.01707 0.01610 0.03967 0.03351 0.02644
7 8 0.94296 0.90031 0.90565 0.86546 0.86446 0.90422 0.88623
7 14 0.35994 0.44772 0.40121 0.33883 0.37718 0.42686 0.53458
7 19 0.05654 0.04441 0.02632 0.05958 0.01941 0.07475 0.07226
10 11 0.88047 0.94281 0.82722 0.88183 0.79898 0.88194 0.86912
10 15 0.40490 0.53878 0.39667 0.45283 0.32490 0.50327 0.41918
10 19 0.06716 0.00235 0.09389 0.06345 0.02003 0.08191 0.03364
13 14 0.96214 0.73636 0.90271 0.69666 0.65322 0.86464 0.88569
13 17 0.52826 0.27128 0.46206 0.30303 0.17601 0.45055 0.48379
13 19 0.00934 0.08777 0.03932 0.06692 0.08494 0.02828 0.02014
17 18 0.59856 0.51260 0.73049 0.30187 0.27078 0.73709 0.32717
17 19 0.04785 0.06337 0.00726 0.06841 0.07776 0.01435 0.05934
Table 4.7: probabilities to remain in the starting state without transitions
Reliability and credit risk models 367
Tables 4.8.1
and 4.8.2 give the probabilities ( , )
ij
s
t
ϕ
that the next transition from
the state i will be to the state j given that there is no transition from the time s to
the time t.
(,)
ij
s
t
ϕ
Prob. Next State Without Transitions from s to t
TIME 0-1
AAA AA A BBB BB B CCC D
AAA
0.92523 0.07477 0 0 0 0 0 0
AA
0.02061 0.90773 0.07166 0 0 0 0 0
A
0 0.04379 0.88959 0.06456 0.00206 0 0 0
BBB
0 0 0.05006 0.90070 0.04924 0 0 0
BB
0 0 0.00926 0.04740 0.63802 0.30051 0.00481 0
B
0 0 0.01327 0 0.05037 0.91098 0.02537 0
CCC
0 0 0 0 0 0.09064 0.90936 0
D
0 0 0 0 0 0 0 1.00000
TIME 0-10
AAA AA A BBB BB B CCC D
AAA
0.92792 0.07208 0 0 0 0 0 0
AA
0.01978 0.90729 0.07293 0 0 0 0 0
A
0 0.04515 0.89622 0.05644 0.00219 0 0 0
BBB
0 0 0.04520 0.91235 0.04245 0 0 0
BB
0 0 0.00933 0.04906 0.63862 0.29823 0.00476 0
B
0 0 0.01053 0 0.04094 0.92444 0.02410 0
CCC
0 0 0 0 0 0.07973 0.92027 0
D
0 0 0 0 0 0 0 1.00000
TIME 0-19
AAA AA A BBB BB B CCC D
AAA
0.73240 0.26760 0 0 0 0 0 0
AA
0.03252 0.83111 0.13637 0 0 0 0 0
A
0 0.11417 0.85701 0.02554 0.00328 0 0 0
BBB
0 0 0.08342 0.83961 0.07698 0 0 0
BB
0 0 0.00180 0.03527 0.77843 0.18296 0.00154 0
B
0 0 0.01377 0 0.06237 0.91291 0.01095 0
CCC
0 0 0 0 0 0.03438 0.96562 0
D
0 0 0 0 0 0 0 1.00000
Table 4.8.1: probabilities (0, )
ij
t
ϕ
For example the element 0.07293 gives the probability that the next transition
from the rating
AA will be to the rating A, given that from the time 0 up to the
time 10 there will be no real or virtual transitions; by virtual transition we denote
the fact that the next transition is in the same state.
(,)
ij
s
t
ϕ
Prob. Next State Without Transitions from s to t
TIME 15-16
AAA AA A BBB BB B CCC D
AAA
0.94075 0.05343 0.00582 0 0 0 0 0
368 Chapter 8
AA
0.00410 0.93332 0.06258 0 0 0 0 0
A
0 0.02803 0.95461 0.01667 0.00070 0 0 0
BBB
0.00156 0 0.07437 0.90203 0.02028 0.00175 0 0
BB
0 0 0.00782 0.06731 0.86642 0.04816 0.00506 0.00522
B
0 0 0.00266 0.00548 0.09608 0.84881 0.01684 0.03012
CCC
0 0 0 0 0.05549 0.11808 0.77923 0.04720
D
0 0 0 0 0 0 0 1.00000
TIME 15-17
AAA AA A BBB BB B CCC D
AAA
0.95837 0.03828 0.00335 0 0 0 0 0
AA
0.00763 0.87026 0.12211 0 0 0 0 0
A
0 0.02601 0.95488 0.01829 0.00082 0 0 0
BBB
0.00217 0 0.07649 0.89866 0.02092 0.00176 0 0
BB
0 0 0.00836 0.10192 0.84363 0.03558 0.00396 0.00655
B
0 0 0.00237 0.00557 0.08279 0.87605 0.01026 0.02296
CCC
0 0 0 0 0.06733 0.14355 0.74361 0.04552
D
0 0 0 0 0 0 0 1.00000
TIME 15-18
AAA AA A BBB BB B CCC D
AAA
0.94688 0.05050 0.00262 0 0 0 0 0
AA
0.00377 0.92617 0.07006 0 0 0 0 0
A
0 0.02797 0.93524 0.03576 0.00104 0 0 0
BBB
0.00215 0 0.04336 0.93813 0.01437 0.00199 0 0
BB
0 0 0.00497 0.08134 0.83046 0.06627 0.00643 0.01053
B
0 0 0.00188 0.00649 0.08795 0.84445 0.01090 0.04833
CCC
0 0 0 0 0.07399 0.07638 0.83271 0.01691
D
0 0 0 0 0 0 0 1.00000
TIME 15-19
AAA AA A BBB BB B CCC D
AAA
0.71135 0.20969 0.07895 0 0 0 0 0
AA
0.00329 0.95760 0.03911 0 0 0 0 0
A
0 0.02745 0.96394 0.00822 0.00038 0 0 0
BBB
0.00687 0 0.27043 0.58332 0.13605 0.00334 0 0
BB
0 0 0.01711 0.08730 0.83929 0.04680 0.00753 0.00197
B
0 0 0.00330 0.00732 0.05632 0.90751 0.02157 0.00397
CCC
0 0 0 0 0.16897 0.21855 0.51403 0.09845
D
0 0 0 0 0 0 0 1.00000
Table 4.8.2: probabilities (15, )
ij
t
ϕ
Tables 4.9.1 and 4.9.2 report the probabilities ( , )
ij
s
t
φ
.
(,)
ij
s
t
φ
EVOLUTION EQUATION MATRICES
TIME 0-1
AAA AA A BBB BB B CCC D
AAA
0.99243 0.00757 0 0 0 0 0 0
AA
0.00004 0.99938 0.00059 0 0 0 0 0
A
0 0.00438 0.99303 0.00252 0.00007 0 0 0
BBB
0 0 0.00182 0.99560 0.00258 0 0 0