MOTORCAR INSURANCE I

MOTORCAR INSURANCE I I Acc. II Acc. III Acc. Sex Year Month Day 19970602 0 0 M 1966 4 11 19820101 19840801 0 M 1926 3 25 19820801 19840712 0 F 1952 2 19 19781222 19810507 0 M 1952 3 23 19821110 19870614 19951222 M 1957 11 9

MOTORCAR INSURANCE II year frequencies cum. freq. mean numb. 1 0.003519 0.003519 0.003519 2 0.011986 0.015505 0.01551738 5 0.022133 0.089832 0.09107341 17 0.000204 0.141676 0.16261899 18 0.000219 0.141895 0.16340732 36 0.0 0.14315 0.16698588 37 1.46E-05 0.143165 0.16701685

TAXI CAB PROBLEM 0.5 0.4 0.1 P 0.3 0.6 0.1. 0.2 0.1 0.7

A Social Insurance Example.90.10 0 0 0 0.95.05 0 0 P 0 0.90.05.05 0 0 0.90.10 0 0.05.05.90

BONUS MALUS ITALIAN RULES Arriving state in function of claims Starting state 0 claims 1 claims 2 claims 3 claims 4 or more 1 1 3 6 9 12 2 1 4 7 10 13 3 2 5 8 11 14 4 3 6 9 12 15 5 4 7 10 13 16 6 5 8 11 14 17 7 6 9 12 15 18 8 7 10 13 16 18 9 8 11 14 17 18 10 9 12 15 18 18 11 10 13 16 18 18 12 11 14 17 18 18 13 12 15 18 18 18 14 13 16 18 18 18 15 14 17 18 18 18 16 15 18 18 18 18 17 16 18 18 18 18 18 17 18 18 18 18

BONUS MALUS MARKOV TRANSITION MATRIX I states 1 2 3 4 5 6 1 0,941655 0 0,056264 0 0 0,001973 2 0,935097 0 0 0,062379 0 0 3 0 0,941646 0 0 0,056611 0 4 0 0 0,948892 0 0 0,049364 5 0 0 0 0,945231 0 0 6 0 0 0 0 0,949204 0 7 0 0 0 0 0 0,934685 8 0 0 0 0 0 0 9 0 0 0 0 0 0 10 0 0 0 0 0 0 11 0 0 0 0 0 0 12 0 0 0 0 0 0 13 0 0 0 0 0 0 14 0 0 0 0 0 0 15 0 0 0 0 0 0 16 0 0 0 0 0 0 17 0 0 0 0 0 0 18 0 0 0 0 0 0

BONUS MALUS MARKOV TRANSITION MATRIX II states 7 8 9 10 11 12 1 0 0 0,000081 0 0 0,000027 2 0,002427 0 0 0,000097 0 0 3 0 0,001574 0 0 0,000169 0 4 0 0 0,001744 0 0 0 5 0,052354 0 0 0,002314 0 0 6 0 0,04908 0 0 0,00157 0 7 0 0 0,061856 0 0 0,00339 8 0,92227 0 0 0,073137 0 0 9 0 0,914103 0 0 0,082621 0 10 0 0 0,923854 0 0 0,071989 11 0 0 0 0,92933 0 0 12 0 0 0 0 0,930156 0 13 0 0 0 0 0 0,937854 14 0 0 0 0 0 0 15 0 0 0 0 0 0 16 0 0 0 0 0 0 17 0 0 0 0 0 0 18 0 0 0 0 0 0

BONUS MALUS MARKOV TRANSITION MATRIX III states 13 14 15 16 17 18 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 4 0 0 0 0 0 0 5 0,000067 0 0 0,000034 0 0 6 0 0,000146 0 0 0 0 7 0 0 0,000069 0 0 0 8 0,004246 0 0 0,00026 0 0,000087 9 0 0,003185 0 0 0 0,000091 10 0 0 0,003827 0 0 0,00033 11 0,066723 0 0 0,003696 0 0,000251 12 0 0,066697 0 0 0,002994 0,000153 13 0 0 0,059651 0 0 0,002495 14 0,920681 0 0 0,074704 0 0,004615 15 0 0,885204 0 0 0,107143 0,007653 16 0 0 0,777568 0 0 0,222432 17 0 0 0 0,876733 0 0,123267 18 0 0 0 0 0,888614 0,111386

Motor Car Adiacency Matrix 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 2 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 3 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 4 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 5 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 6 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 7 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 8 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 9 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 10 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 11 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 12 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 13 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1 14 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

Reachability Matrix 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 14 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 15 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 18 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

THE PROBLEM disabling professional diseases study of the illness evolution in the time The states occupied by the process S 1 S 2 S 3 S 4 S 5 = from 0 up to 10% disablement = from 10% up to 30% disablement = from 30% up to 50% disablement = from 50% up to 70% disablement = from 70% up to 100% disablement

INPUTS P ij n n ij i ij G () t 1 e t ij nij ni number of transitions from Si number of observed elements in Si ij estimated mean sojourn time in state Si given the state S successively visited j

OUTPUTS S i m j1 ij () t S j S j disability degree mean value inside the j-th state Φ() t computed at epochs 1, 2,..., 10,..., 20,...,30

EMBEDDED M.C..00000 1.00000.00000.00000.00000.00000.82629.16012.00906.00453.00273.01639.85247.11475.01366.00000.01869.05607.68225.24299.00000.00000.03279.01639.95082

CONDITIONAL MEAN DISABLEMENT DEGREE 10 years forecasting. Starting state 1 2 3 4 5 Mean degree 33.70 34.12 52.94 74.99 98.09 20 years forecasting Starting state 1 2 3 4 5 Mean degree 34.38 34.40 53.22 75.20 98.06 30 years forecasting Starting state 1 2 3 4 5 Mean degree 34.43 34.44 53.25 75.21 98.06

TOTAL MEAN DISABILITY DEGREE years 10 20 30 Mean degree 82.96 83.04 83.05

PREFERENCE RELATION ( St, ), St, Dated sum ( S, t ) ( S, t ) 1 1 2 2 ( S, t ) ( S, t ) 1 1 2 2 ( S, t ) ( S, t ) 1 1 2 2

FUNDAMENTAL FINANCIAL RELATIONS ( St, ) ( St, ) 1 2 t t if S 0 1 2 t t if S 0 1 2 ( S, t) ( S, t) 1 2 iff 1 2 S S

FIXED & VARIABLE RATES S t S t 1 2 ( 1, 1) ( 2, 2) iff (1 ) t (1 ) t i i S S 1 2 1 2 ( S1, t1) ( S2, t2) iff (1 ) t (1 ) t it it S S 1 2 1 2

NON HOMOGENEOUS & HOMOGENEOUS APPROACH ( S1, t1) R ( S2, t2) iff ( S1, t1) ( S2, t2) static approach S ( t, t, S ) 2 1 2 1 discretizing the sums I S, S,..., S, S S S 1 2 n 1 2 ( S, t ) ( S, t ) ( S, t h) ( S, t h) h S 1 1 2 2 1 1 2 2 ( S, t) 2 1 m

STOCHASTIC STATIC APPROACH S 1 at beginning of the operation S [ S', S"] S as a random variable p(s) r.v. density at the end S " S S p( S) ds S ' S " _ 2 2 ( S) ( S S) p( S) ds S '

MODEL DESCRIPTION,,P P probability measure S S t t t1 t n ( ( ),,..., ) process p( S, S, t) P S( t) S S(0) S S1, S2 [ S', S"] 1 2 2 1 I S1 S2 S m Tn,,,. (, 0) time of the n-th sum transition n X n P p ij (X-T) continuous time homogeneous semi-markov process

FINANCIAL MEANING Qij () t probability that the value is S given that the value was S j i Gij () t the probability that i S in a time length t will become j HSMP Z represents the value at time t of the dated sum () t 1 S () t Q () s ( t s) ds ij ij i ik kj ke 0 1 S ( t) ij t ke 0. i Q () s ( t s) ds ik kj t. S

DISCRETE TIME MODEL discrete time, discrete state space t b () t P[ X j, T T t X i], ij n1 n1 n n () t (1 S ()) t b ( ) ( t ). ij ij i ih hj h1 s m t

FUTURES MODEL Y(t) market value of the future contract at time t Yt (); t 0,1,, T Ymin, Ymax Ymax Ymin m 1 I Y, Y, Y 2,, Y ( m2), Y min min min min max S1, S2,, Sm

DATA DESCRIPTION 1 7408 records (Fib30) March 27 th 1998 September 17 th 1998 10394 unit operations made by 36 different traders obtained contract prices belong to the range [27.955, 39.490] T # 0-1 5917 1-2 1495 2-3 707 3-4 549 4-5 445 5-6 331 6-7 209 7-8 153 8-9 86 9-10 6 10-11 6 11-12 6 12-13 6 13-14 10 14-15 6 15-16 6 16-17 6

DATA DESCRIPTION 2 m=122 (number of states of SMP). T=9 (number of periods for the transient analysis of SMP) The transition matrix P of the embedded MC in SMP square lower-triangular block matrix T G of order 10

VALUE AT EXPIRY DATE starting state price at time 10 sigma square p.v. at time 0 VaR 28.300 29.822 877 29.795 28.245 28.400 29.974 906 29.947 28.363 28.500 30.014 899 29.987 28.412 28.600 30.002 929 29.975 28.386 28.700 30.056 879 30.028 28.526 28.800 29.894 874 29.867 28.492 38.700 37.965 517 37.930 37.312 38.800 38.460 442 38.425 37.784 39.100 38.079 555 38.045 37.317 39.200 38.756 587 38.720 37.807 39.400 38.115 567 38.080 36.930

VALUES AFTER FIVE DAYS starting state price at time 5 sigma square p.v. at time 0 VaR 28.300 29.193 445 29.179 28.536 28.400 29.317 727 29.303 28.401 28.500 29.452 411 29.438 28.706 28.600 29.624 631 29.610 28.571 28.700 29.863 548 29.848 29.001 28.800 29.686 412 29.671 29.303 38.700 38.615 288 38.597 37.930 38.800 38.813 286 38.794 38.215 39.100 38.539 307 38.520 37.850 39.200 38.818 471 38.799 38.025 39.400 38.178 325 38.159 37.829

INTRADAY VALUES starting state price at time 1 sigma square p.v. at time 0 VaR 28.300 28.732 122 28.730 28.504 28.400 28.563 222 28.562 28.315 28.500 28.644 150 28.642 28.410 28.600 28.926 340 28.925 28.514 28.700 28.892 200 28.890 28.610 28.800 28.800 0 28.798 28.705 38.700 38.719 39 38.717 38.606 38.800 38.877 158 38.875 38.706 39.100 39.100 0 39.098 39.005 39.200 39.227 45 39.225 39.107 39.400 39.400 0 39.398 39.305

HSMP INTEREST RATE r ri r', r" r I r1, r2,, r m implied stochastic interest rates G () t increasing waiting time d. f. of interest rate change ij ij () t (1 S ( t)) ij i m t h1 0 b ih ( ) ( t ) hj

INTEREST RATE EVALUATION i ( ) r.v. interest rate related to the period ( 1, ) (0, t) m i ij j j1 E( ( )) ( ) r m m 2 2 ( i( )) ij( ) rj ij( ) rj j1 j1 2

DISCOUNT FACTOR MEAN 1 ( ) 1 ( ) uniperiodical discount factor i A( h) ( ) i h 1 i m i E( ( )) ( )(1 r ) i ij j j1 h ( h) E( A( h)) E( ( )) i i i 1 1

DISCOUNT FACTOR VARIANCE m m 2 2 1 ( i( )) Pij( )(1 rj) ij( )(1 rj) j1 j1 2 2 h h A ( h) S M q q i C D 1 q1

SYMBOL MEANING S 2 C ( ), 1,, q i r Cq r1 M D q h r1 Cq C(h, 2 ( ),,, D if 2 2 i ( ) ( i( )) i( ) E( i( )) Cq Dq 1,, h i r 1 h q 1 if h h

AN EXAMPLE maximum value: state 15 =.10 intermediate values: state 14 =.095 state 13 =.09 state 12 =.085 state 11 =.08 state 10 =.075 state 9 =.07 state 8 =.065 state 7 =.06 state 6 =.055 state 5 =.05 state 4 =.045 state 3 =.04 state 2 =.035 minimum value: state 1 =.03.

INSURANCE TRAJECTORY

SMP TRAJECTORY

GENERAL INSURANCE m STATE MODEL

INTEREST RATE EVOLUTION EQUATION () t (1 S ()) t ( t ) b ( ) ij ij i j i 11 n t E r1, r2,, r n I 1, 2,, m

HEALTH INSURANCE m STATE MODEL

REAL ESTATE EVALUATION STATEVALUE 1 100,000 2 120,000 3 140,000 4 160,000 5 180,000 6 200,000 INPUTS T G, T P

VALUES AT YEAR 14 S1 S2 S3 S4 S5 S6 0 156402 157604 164197 172921 177117 185100 1 149909 161888 164860 172098 180438 187121 2 151793 157836 162764 172795 181729 188499 3 150595 155939 165437 173683 181367 190427 4 150131 158171 164283 174851 182043 188605 5 146518 157644 163887 173290 183554 193242 6 149232 157543 165342 172358 183543 190664 7 146416 153576 162808 176039 185553 193935 8 145166 155003 164899 174062 183478 192231 9 145952 151365 163839 176422 183176 195389 10 144678 151559 164546 177217 186779 194998 11 142459 153907 165238 174879 190582 195729 12 143166 151110 161906 177975 189038 195932 13 140880 151989 163663 177240 192641 200737

E.P.V. at YEAR 0 S1 S2 S3 S4 S5 S6 1 116639 134054 152240 171276 190227 185100 2 113472 128419 145569 163317 179490 195985 3 110185 123325 138061 155522 169699 184172 4 106454 118387 131368 147823 160093 173712 5 102987 113099 125078 140207 151717 163960 6 99184 108832 119324 133115 142748 153941 7 96095 104507 113934 126250 134674 144855 8 92647 100470 108659 120027 127705 136888 9 89905 96513 103531 113915 120692 128956 10 87968 92518 98636 108063 113812 121057 11 85956 88670 94322 102242 107225 113624 12 83273 85510 90008 97086 101126 106562 13 80733 82193 86659 91960 95444 99586 14 78994 79601 82931 87337 89456 93488

ASSET VALUES maximum value: state 11 = 20000 intermediate values: state 10 = 19000 state 9 = 18000 state 8 = 17000 state 7 = 16000 state 6 = 15000 state 5 = 14000 state 4 = 13000 state 3 = 12000 state 2 = 11000 minimum value state 1 = 10000.

TIME 0 TRANSITION PROBABILITIES i y 0.115081 0.205079 0.148765 0.090851 0.146841 0.0515925 0.0967928 0.024714 0.0690959 0.0188871 0.0323013 0.152056 0.0923734 0.136775 0.137232 0.0666064 0.119935 0.102492 0.0807004 0.046316 0.0541403 0.0113729 0.0750198 0.155426 0.141359 0.106367 0.108168 0.113165 0.100611 0.0476689 0.0602072 0.0403808 0.0516274 0.108723 0.108299 0.133361 0.136165 0.0982937 0.126759 0.102252 0.0983709 0.0534774 0.0193613 0.014937 0.054895 0.0695269 0.139799 0.100845 0.104654 0.147106 0.0929341 0.0872038 0.0769513 0.0685584 0.0575263 0.0732438 0.0678411 0.0872094 0.11715 0.134706 0.108331 0.0866107 0.119068 0.0581916 0.0747954 0.0728533 0.061881 0.0799785 0.0708427 0.0898525 0.120231 0.117384 0.08538 0.127598 0.0959383 0.0654861 0.0854276 k0.0226702 0.0856967 0.0662277 0.0747256 0.113652 0.073347 0.128928 0.151991 0.12141 0.0773896 0.0839625 { 0.00447285 0.0439141 0.03405 0.0625003 0.0982934 0.139534 0.101136 0.15492 0.12453 0.0959437 0.140706 0.0160766 0.0328816 0.0595976 0.0716554 0.104363 0.0710514 0.0716975 0.138139 0.136131 0.13948 0.158928 0.0221572 0.0129829 0.0366481 0.0234278 0.0639569 0.128072 0.151977 0.118464 0.112809 0.145908 0.183598

WAITING TIME D.F. 1.2 2.5 0.0686693 0-1, 0.048234 1-2 0.0869034 0-2, 0.0517992 1-3 0.125819 0-3, 0.103144 1-4 0.173014 0-4, 0.109809 1-5 0.177101 0-5, 0.172370 1-6 0.239573 0-6, 0.239397 1-7 0.271002 0-7, 0.286419 1-8 0.325844 0-8, 0.351679 1-9 0.394935 0-9, 0.400787 1-10 0.445495 0-10, 0.470971 1-11 0.513563 0-11, 0.504282 1-12 0.558356 0-12, 0.519115 1-13 0.602921 0-13, 0.541922 1-14 0.620377 0-14, 0.545760 1-15 0.645842 0-15, 0.562321 1-16 0.665479 0-16, 0.610480 1-17 0.715788 0-17, 0.645973 1-18 0.731454 0-18, 0.653449 1-19 0.781820 0-19, 0.726265 1-20 0.820131 0-20, 0.733316 1-21 0.882271 0-21, 0.812839 1-22 0.898686 0-22, 0.888162 1-23 0.917190 0-23, 0.926003 1-24. 0.972981 0-24.

E. ASSET VALUE at 24 i y 13887.7 14220.7 14475.7 14526.3 14799.4 15120.1 15141.9 15547.4 15686.9 15942.5 15922.7 13887.7 14324.2 14399.5 14602.1 14810.9 14954.9 15245.4 15525.7 15607.3 15923.4 16032.9 13899.6 14166.5 14187.1 14753.1 14650. 14897.9 15190.6 15497.3 15460.7 15918.3 16287. 13699. 13949.4 14330.8 14414.8 14781.4 14915.7 15216.2 15561.8 15830.1 16133.4 16266.9 13655.8 14156.6 14345.6 14389.5 14705.7 15010.2 15266.1 15535.9 15822.2 15917.8 16343.5 13759.5 13742.4 14151.1 14442.2 14700.1 14962. 15276.6 15583.7 15842.7 16230. 16267.3 13467.1 13913.6 14093.2 14471.5 14627.4 14915.8 15314.2 15708. 15802.7 16454.4 16435.9 13492.4 13761.9 14064.6 14300.6 14674.3 15218.5 15256.2 15666.3 15928.9 16335.6 16701.8 13190.1 13776. 14152.8 14150.5 14758.5 15117.3 15374.2 15762.5 16096.4 16468.7 16640.9 13324.2 13649.4 13971.8 14263.9 14552.3 15101.4 15415.6 15845.7 16264.3 16422.3 16693.6 13477.6 13534.6 13816.2 14244.7 14614.8 14962.8 15468.3 15756.1 16079.8 16452. 16736.8 13206.5 13407.6 13865.1 14205.7 14656.1 15125.9 15475. 15786.8 16148.3 16434.1 17000.5 13133.6 13399.2 13773.3 14189.2 14554.6 14988.4 15308.9 15672.9 16295.1 16439.8 17007.2 12956.1 13365.5 13917.3 14178.7 14507.9 14808.7 15527.7 15819. 16226.5 16483.1 16861.6 12982. 13287.1 13610.1 14322.6 14574.2 15084.8 15484.2 15835.4 16419.5 16765.2 17064.3 12981.9 13613. 13828.3 14168.5 14541.2 14910.9 15420. 15831.8 16384. 16624.7 17011.2 12800.2 13077.3 13725.6 14041.6 14507.8 15056.1 15446.6 15848. 16351.2 16912.2 17443.7 12791.4 13316. 13748.1 14064.7 14481.2 15149.4 15473.1 16107.1 16293.7 16886.9 17182.7 12878.3 13115.3 13524.2 14192.8 14480.2 15119.7 15555.4 15847.3 16314. 16975.9 17500.6 12907.3 13180.7 13655.6 13765.5 14461.8 15196.2 15636.1 16158.2 16347.7 16900.8 17626.8 k12658.1 13024. 13646.9 14010. 14572. 14971.9 15622.9 16060.3 16630.6 17045.5 17409.7 { 12588.6 12892.9 13572.1 14039. 14475.6 14899.3 15511.5 15951.9 16671.5 17181.2 17378. 12525. 12834.2 13298.2 13819.9 14278.1 15160.3 15643.3 16085.3 16794.7 17271.1 17783.3 12272.9 12707.5 12789.5 13600.2 14370.4 14978.1 15573.9 16272.5 16721.3 17398.3 17984.5

E. ASSET P.V. at 24 i y 12596.6 12898.6 13129.9 13175.8 13423.5 13714.4 13734.2 14101.9 14228.5 14460.3 14442.4 12647.9 13045.4 13114. 13298.5 13488.7 13619.8 13884.4 14139.6 14214. 14501.9 14601.5 12710.2 12954.3 12973.2 13490.7 13396.4 13623.2 13890.8 14171.3 14137.8 14556.2 14893.4 12577.9 12807.8 13158. 13235.1 13571.7 13695. 13971. 14288.3 14534.6 14813. 14935.6 12589.3 13050.9 13225.2 13265.7 13557.2 13838. 14073.9 14322.6 14586.5 14674.7 15067.1 12736.6 12720.8 13099.1 13368.6 13607.3 13849.7 14140.9 14425.1 14664.9 15023.4 15058. 12516.7 12931.7 13098.6 13450.2 13595.1 13863.1 14233.5 14599.5 14687.5 15293.2 15276. 12591.3 12842.8 13125.3 13345.6 13694.2 14202.2 14237.3 14620.1 14865.1 15244.6 15586.4 12359.4 12908.4 13261.4 13259.3 13829. 14165.2 14405.9 14769.7 15082.6 15431.5 15592.8 12535.9 12841.8 13145.1 13419.9 13691.3 14208. 14503.5 14908.2 15302. 15450.7 15705.9 12731.8 12785.7 13051.7 13456.5 13806.1 14134.9 14612.4 14884.3 15190.1 15541.7 15810.7 12526.6 12717.3 13151.3 13474.3 13901.5 14347.1 14678.3 14974. 15316.9 15588. 16125.3 12508.2 12761.1 13117.4 13513.5 13861.5 14274.7 14579.9 14926.6 15519.1 15657. 16197.3 12389.4 12780.9 13308.6 13558.5 13873.3 14161. 14848.6 15127.1 15516.7 15762.2 16124.1 12464.8 12757.7 13067.9 13751.9 13993.5 14483.8 14867.3 15204.4 15765.3 16097.3 16384.4 12515.4 13123.8 13331.5 13659.4 14018.7 14375.2 14866. 15262.9 15795.3 16027.3 16400. 12390.5 12658.8 13286.3 13592.2 14043.5 14574.2 14952.3 15340.8 15827.9 16371. 16885.4 12432.5 12942.4 13362.3 13670. 14074.8 14724.3 15038.9 15655.2 15836.5 16413. 16700.6 12567.9 12799.3 13198.3 13850.8 14131.2 14755.3 15180.5 15465.3 15920.8 16566.8 17078.9 12647.5 12915.4 13380.8 13488.5 14170.7 14890.4 15321.4 15833. 16018.7 16560.6 17272.1 k12453.9 12813.9 13426.8 13784. 14336.9 14730.4 15370.9 15801.3 16362.4 16770.5 17128.9 { 12436. 12736.6 13407.5 13868.8 14300.2 14718.7 15323.4 15758.5 16469.4 16972.9 17167.3 12423.5 12730.2 13190.5 13708. 14162.5 15037.5 15516.6 15955.1 16658.7 17131.2 17639.3 12223.1 12655.9 12737.6 13545. 14312.1 14917.4 15510.7 16206.5 16653.4 17327.7 17911.5

ASSET VALUES S.D. at 24 i y 2945.32 2931.76 2936.07 2807.93 2935.02 2830.54 2842.49 2882.96 2958.36 2942.82 3027.92 3031.45 3011.7 2919.36 2820.12 2795.97 2835.06 2859.42 2851.3 2861.96 2986.28 3018.44 3112.06 2987.87 2861.31 2825.86 2810.62 2858.81 2790.04 2909.96 2973.15 2994.99 3015.95 3100.93 2959.05 2930.68 2831.92 2752.46 2711.21 2841.09 2749.8 2832.25 2946.07 3082.82 3031.87 2986.54 2892.22 2782.2 2759.18 2807.95 2827.19 2819.08 2934.5 3012.88 3048.81 3005.04 2975.57 2862.41 2735.23 2731.4 2769.86 2743.7 2745.6 2897.99 2963.98 3077.71 3101.01 2973.38 2862.45 2743.56 2634.69 2723.93 2643.49 2711.8 2897.48 2842.36 3068.15 3111.46 2939.75 2804.91 2712.97 2658.96 2684.61 2721.33 2713.28 2782.1 2897.5 3040.09 3006.94 2948.85 2863.26 2659.4 2717.89 2629.58 2668.67 2674.53 2801.46 2877.16 3065.44 3123.31 2940.39 2782.94 2719.73 2613.03 2578.5 2623.37 2636.7 2685.77 2917.07 3036.7 3161.66 2931.69 2740.55 2595.16 2605.48 2684.86 2567.88 2636.11 2857.67 2939.25 3083.58 3149.16 2823.56 2768.35 2612.21 2560.61 2561.78 2564.93 2731.56 2807.51 3008.36 2972.4 3072.35 2860.77 2686.85 2609.39 2578.19 2555.73 2527.35 2659.68 2709.5 2936.22 3010.17 2982.52 2876.59 2805.31 2606.24 2509.9 2524.34 2584.31 2621.65 2748.5 2917.66 3061.48 3013.89 2774.64 2581.4 2680.07 2538.6 2540.74 2562.43 2653.44 2662.93 2827.24 3025.65 3070.54 2924.04 2757.09 2533.51 2474.38 2538.4 2489.4 2527.65 2680.6 2884.75 3088.82 2930.4 2722.91 2597.36 2522.51 2500.84 2360.46 2561.71 2552.84 2695.6 2806.26 2898.16 3011.96 2811.06 2757.68 2545.89 2427.26 2423.98 2390.71 2526.51 2782.87 2662.06 3022.11 3113.65 2751.27 2613.62 2601.35 2350.67 2441.07 2444.38 2526.58 2717.57 2804.11 2849.96 3086.23 2837.51 2704.53 2413.08 2503.16 2458.42 2340.27 2493.49 2655.75 2810.74 2819.82 k2958.3 2800.17 2653.11 2582.17 2369.21 2470.96 2417.71 2589.53 2605.31 2641.04 2994.69 { 2966.92 2622.44 2476.45 2402.34 2393.14 2408.47 2274.77 2471.71 2486.05 2657.82 2954.01 2894.5 2543.25 2507.12 2229.63 2301.37 2405.16 2376.9 2440.64 2398.54 2541.86 2837.91 2806.12 2461.85 2019.97 2290.37 2157.13 2295.64 2298.47 2248.78 2306.57 2349.61 2678.51

E. ASSET V. from 1 to 24 S. 0 i y 10112.9 11108.1 12064.3 13033.6 14008.5 14984.2 15982.5 17005.5 17915.8 18860. 19902. 10271.6 11215.1 12138.9 13089.1 14008.8 14997.1 15979.9 16950.4 17866.5 18768.8 19744.6 10369.5 11337. 12244.9 13150. 14001.7 15007.9 15901.4 16896.9 17793.9 18711.1 19535.7 10429.1 11478.2 12301.7 13148.3 14025.1 14985.4 15899.1 16836.6 17720. 18594.7 19409.5 10587.8 11618. 12409.6 13188.1 14066.1 14993.8 15835.3 16791.5 17608.7 18500.4 19230.7 10708.2 11756.1 12461.7 13241.5 14098.4 14988.1 15771.7 16758. 17558.3 18411.3 19057.2 10815.4 11863.4 12525.4 13275. 14115.1 15020.3 15766.7 16702.4 17453.3 18293.9 18911.8 10903.6 11998.1 12615.9 13346.9 14175.9 15002.2 15746.7 16665.6 17371.9 18192.3 18751.9 11034. 12103.8 12692.9 13422.4 14185.9 15033.8 15693.9 16614.2 17277.9 18081.3 18581. 11169.7 12209.1 12794.1 13456.3 14215.9 15009.6 15644.4 16552.4 17197.4 17986.1 18402.5 11322.8 12317.6 12835. 13492.8 14233.8 15054.3 15605.7 16553.7 17108.1 17867.7 18261.9 11445.3 12409.8 12928.9 13544.9 14244.2 15053.9 15584.4 16525.8 17013.3 17804.1 18099.9 11599.8 12517.9 12998.6 13585.1 14298.1 15063.9 15558.9 16456. 16914. 17668.9 17953.5 11794.8 12649.5 13100.3 13659.3 14348. 15080.4 15538.8 16386.8 16840.4 17558.7 17810.9 11935.5 12764.3 13227.9 13728.7 14376.7 15086. 15510.5 16325.6 16771.7 17426.5 17657.4 12076. 12892.1 13367.5 13795.8 14418.1 15065.4 15447.3 16261.5 16707.2 17322.3 17479.1 12218.6 12987.4 13504.4 13841.4 14428.3 15041.8 15415.6 16174.2 16600.6 17186.9 17294.4 12417.7 13149.4 13626.4 13922.6 14475.2 15040.5 15372.3 16128.9 16489.3 17054.1 17125.6 12602.9 13328.8 13754.2 14021.8 14548.9 15042. 15348.8 16034.7 16389.1 16909.4 16936.5 12805.8 13474.4 13882.2 14092.3 14572.1 15038.2 15301.8 15945.5 16251.6 16753.5 16768.5 k12995.2 13631. 14013.8 14170.8 14613.2 15056.5 15269.6 15867.8 16099.5 16580. 16568.5 { 13209.7 13796.9 14142.6 14271.6 14684.3 15061.4 15204. 15762.4 15972.7 16387.9 16379.1 13495.3 13989.1 14309.5 14388.1 14777.6 15097.1 15206.7 15679. 15834.8 16212.1 16188.3 13887.7 14220.7 14475.7 14526.3 14799.4 15120.1 15141.9 15547.4 15686.9 15942.5 15922.7

E. ASSET P.V. from 1 to 24 S. 0 i y 10071.9 11063. 12015.3 12980.7 13951.6 14923.4 15917.6 16936.5 17843.1 18783.5 19821.3 10188.4 11124.3 12040.6 12983.1 13895.4 14875.6 15850.5 16813.1 17721.8 18616.8 19584.7 10243.8 11199.6 12096.4 12990.6 13832. 14825.9 15708.6 16692. 17578.2 18484.3 19298.8 10260.9 11293.1 12103.3 12936.2 13798.9 14743.7 15642.6 16565. 17434.2 18294.8 19096.4 10374.7 11384.2 12159.9 12922.7 13783.1 14692.1 15516.6 16453.6 17254.3 18128.1 18843.7 10450.1 11472.8 12161.4 12922.4 13758.6 14626.9 15391.6 16354.2 17135.1 17967.6 18597.9 10511.9 11530.5 12173.9 12902.5 13719. 14598.8 15324.3 16233.8 16963.6 17780.6 18381.1 10554.6 11614.1 12212.2 12919.8 13722.2 14522. 15242.7 16132.2 16815.9 17610.1 18151.8 10637.5 11668.9 12236.8 12940.1 13676.1 14493.6 15130. 16017.2 16657.1 17431.7 17913.4 10724.7 11722.7 12284.4 12920.1 13649.5 14411.5 15021.1 15893. 16512.2 17269.5 17669.2 10827.6 11778.8 12273.6 12902.7 13611.2 14395.8 14923.1 15829.7 16359.8 17086.2 17463.1 10900.3 11818.8 12313.2 12899.9 13565.9 14337. 14842.2 15738.8 16203.2 16956.2 17238. 11002.6 11873.5 12329.4 12885.7 13562. 14288.3 14757.9 15608.8 16043.2 16759.2 17029.1 11142.2 11949.6 12375.4 12903.5 13554.1 14245.9 14679. 15480. 15908.6 16587.2 16825.4 11229.3 12009.1 12445.3 12916.4 13526.1 14193.4 14592.8 15359.7 15779.4 16395.4 16612.7 11315.4 12080.2 12525.6 12926.9 13510. 14116.6 14474.4 15237.3 15655. 16231.3 16378.2 11402.6 12120. 12602.5 12917. 13464.7 14037.2 14386.1 15094. 15491.9 16039.1 16139.4 11541.3 12221.4 12664.8 12940. 13453.6 13979.1 14287.4 14990.6 15325.6 15850.5 15917.1 11665.9 12337.9 12731.7 12979.3 13467.3 13923.7 14207.7 14842.7 15170.7 15652.3 15677.4 11805.7 12422.1 12798. 12991.7 13434.1 13863.7 14106.7 14700.2 14982.4 15445.1 15458.9 k11931.7 12515.4 12866.9 13011.1 13417.3 13824.3 14019.9 14569.2 14781.9 15223.1 15212.5 { 12079.4 12616.4 12932.5 13050.5 13427.9 13772.7 13903.1 14413.7 14606. 14985.7 14977.6 12290.5 12740.2 13032.1 13103.6 13458.3 13749.3 13849.1 14279.3 14421.1 14764.7 14743. 12596.6 12898.6 13129.9 13175.8 13423.5 13714.4 13734.2 14101.9 14228.5 14460.3 14442.4

AN EXAMPLE maximum value: state 15 =.10 intermediate values: state 14 =.095 state 13 =.09 state 12 =.085 state 11 =.08 state 10 =.075 state 9 =.07 state 8 =.065 state 7 =.06 state 6 =.055 state 5 =.05 state 4 =.045 state 3 =.04 state 2 =.035 minimum value: state 1 =.03.

HSMP EXAMPLE I 1993 S&P TRANSITION MATRIX 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

HSMP EXAMPLE II PROBABILITIES ij (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

HSMP EXAMPLE III PROBABILITIES ij (10) 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

HSMP EXAMPLE IV NO DEFAULT PROBABILITIES 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

HSMP EXAMPLE V PROBABILITIES TO REMAIN IN THE STARTING STATE 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

HSMP EXAMPLE VI Distribution Functions DEFAULT PROBABILITIES 1 2 3 4 5 AAA 0,00000 0,00000 0,00000 0,00001 0,00001 AA 0,00000 0,00000 0,00001 0,00003 0,00005 A 0,00013 0,00025 0,00031 0,00039 0,00060 BBB 0,00067 0,00116 0,00211 0,00285 0,00397 BB 0,00154 0,00539 0,01092 0,01376 0,02111 B 0,00358 0,01192 0,02473 0,02971 0,04575 CCC 0,00706 0,01854 0,03626 0,05767 0,08958 D 1,00000 1,00000 1,00000 1,00000 1,00000

HSMP EXAMPLE VII Distribution Function DEFAULT PROBABILITIES 6 7 8 9 10 AAA 0,00002 0,00003 0,00005 0,00006 0,00008 AA 0,00008 0,00011 0,00016 0,00022 0,00030 A 0,00083 0,00112 0,00144 0,00175 0,00206 BBB 0,00495 0,00582 0,00666 0,00790 0,00937 BB 0,02564 0,02856 0,03229 0,03554 0,04135 B 0,05251 0,06205 0,07465 0,09311 0,10419 CCC 0,10200 0,15102 0,20222 0,22816 0,25154 D 1,00000 1,00000 1,00000 1,00000 1,00000

NON HOMOGENEOUS NUMERICAL EXAMPLE I TRANSITION MATRIX MATRIX AT TIME 0 I PART AAA AA AA- A A- BBB BBB- AAA 0.9245 0 0.0755 0 0 0 0 AA 0.019899 0.910452 0 0 0.069648 0 0 AA- 0.017699 0 0.892452 0 0.089848 0 0 A 0 0.0476 0 0.884057 0 0 0.066239 A- 0 0.0414 0 0 0.847057 0 0.106239 BBB 0 0 0 0.0487 0 0.9026 0 BBB- 0 0 0 0.0357 0 0 0.8726 BB 0 0 0 0.009243 0 0.046313 0 BB- 0 0 0 0.008143 0 0.037313 0 B 0 0 0 0.012399 0 0 0 B- 0 0 0 0.010399 0 0 0 CCC 0 0 0 0 0 0 0 CCC- 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0

NON HOMOGENEOUS NUMERICAL EXAMPLE II TRANSITION MATRIX MATRIX AT TIME 0 - II PART BB BB- B B- CCC CCC- D AAA 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 AA- 0 0 0 0 0 0 0 A 0 0.002104 0 0 0 0 0 A- 0 0.005304 0 0 0 0 0 BBB 0 0.0487 0 0 0 0 0 BBB- 0 0.0917 0 0 0 0 0 BB 0.629596 0 0 0.310227 0 0.004621 0 BB- 0 0.599596 0 0.334227 0 0.020721 0 B 0.049395 0 0.913509 0 0 0.024698 0 B- 0.044395 0 0 0.886509 0 0.058698 0 CCC 0 0 0.0909 0 0.9091 0 0 CCC- 0 0 0.0839 0 0 0.9161 0 D 0 0 0 0 0 0 1

NON HOMOGENEOUS NUMERICAL EXAMPLE III TRANSITION MATRIX MATRIX AT TIME 10 I PART AAA AA AA- A A- BBB BBB- AAA 0.899545 0 0.095701 0 0.004754 0 0 AA 0 0.918598 0 0 0.081402 0 0 AA- 0 0 0.87046 0 0.12954 0 0 A 0.00172 0.005059 0 0.919263 0 0 0.070619 A- 0.001502 0.004106 0 0 0.872126 0 0.100619 BBB 0 0.008356 0 0.052747 0 0.858366 0 BBB- 0 0.007356 0 0.043275 0 0 0.815366 BB 0 0 0 0 0 0.080374 0 BB- 0 0 0 0 0 0.072374 0 B 0 0.003848 0 0 0 0.003848 0 B- 0 0.003102 0 0 0 0.003483 0 CCC 0 0 0 0 0 0.018525 0 CCC- 0 0 0 0 0 0.014452 0 D 0 0 0 0 0 0 0

NON HOMOGENEOUS NUMERICAL EXAMPLE IV TRANSITION MATRIX MATRIX AT TIME 10 II PART BB BB- B B- CCC CCC- D AAA 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 AA- 0 0 0 0 0 0 0 A 0 0.003339 0 0 0 0 0 A- 0 0.021647 0 0 0 0 0 BBB 0 0.061103 0 0.008356 0 0.005536 0.005536 BBB- 0 0.101103 0 0.010356 0 0.006536 0.016008 BB 0.799118 0 0 0.075855 0 0.017861 0.026791 BB- 0 0.754912 0 0.104586 0 0.027861 0.040267 B 0.061352 0 0.747004 0 0 0.034524 0.149423 B- 0.052135 0 0 0.7002 0 0.124524 0.116555 CCC 0.03705 0 0.074099 0 0.518468 0 0.351858 CCC- 0.034205 0 0.06741 0 0 0.483847 0.400086 D 0 0 0 0 0 0 1

NON HOMOGENEOUS NUMERICAL EXAMPLE V PROBABILITIES NO MOVEMENT I times AAA AA AA- A A- BBB BBB- 0 1 0.872685 0.801149 0.864196 0.883794 0.858587 0.952123 0.925977 0 2 0.773958 0.738359 0.851322 0.812449 0.809074 0.922824 0.77301 0 3 0.700077 0.718044 0.683984 0.660399 0.795356 0.843209 0.721275 0 4 0.579162 0.607474 0.602195 0.560755 0.752769 0.679192 0.714172 0 5 0.495302 0.430286 0.444241 0.501583 0.68336 0.646963 0.656272 0 6 0.375061 0.370131 0.324484 0.465195 0.632127 0.597326 0.479142 0 7 0.301492 0.325259 0.260044 0.337689 0.512233 0.413345 0.313155 0 8 0.206683 0.271936 0.176109 0.223134 0.359028 0.229397 0.293567 0 9 0.118951 0.206603 0.139362 0.115929 0.171761 0.114175 0.120906 0 10 0.048605 0.030271 0.057611 0.081737 0.037575 0.084027 0.076363 6 7 0.814112 0.78414 0.903473 0.795045 0.764451 0.730398 0.817988 6 8 0.515732 0.455292 0.790893 0.638901 0.570437 0.441683 0.573782 6 9 0.313335 0.307999 0.187242 0.245239 0.266835 0.299954 0.279234 6 10 0.041924 0.026452 0.085297 0.058358 0.037503 0.01082 0.04734

NON HOMOGENEOUS NUMERICAL EXAMPLE VI PROBABILITIES NO MOVEMENT II times BB BB- B B- CCC CCC- 0 1 0.989431 0.894801 0.835928 0.910767 0.884764 0.981664 0 2 0.953923 0.849355 0.687272 0.735475 0.720782 0.860022 0 3 0.864952 0.72999 0.613613 0.707733 0.684136 0.69544 0 4 0.703253 0.640092 0.574038 0.678904 0.584032 0.549688 0 5 0.57982 0.591852 0.53899 0.663917 0.50686 0.534244 0 6 0.469104 0.47757 0.402361 0.492298 0.487587 0.459526 0 7 0.365078 0.351639 0.302954 0.458244 0.34713 0.290185 0 8 0.245702 0.213933 0.242703 0.291599 0.191329 0.225957 0 9 0.135979 0.152695 0.188321 0.112469 0.142328 0.105861 0 10 0.066222 0.044292 0.010004 0.088224 0.042743 0.091513 6 7 0.642073 0.598058 0.682122 0.77757 0.689264 0.6604 6 8 0.542752 0.508902 0.639685 0.459139 0.382102 0.471916 6 9 0.266475 0.173278 0.261105 0.18707 0.154006 0.316151 6 10 0.05534 0.078582 0.048732 0.072883 0.083662 0.020525

NON HOMOGENEOUS NUMERICAL EXAMPLE VII PROBABILITIES TO GO IN THE RANK J AT NEXT TRANSITION - I AAA AA AA- A A- BBB BBB- AAA 0.917993 0 0.082007 0 0 0 0 AA 0.020148 0.899131 0 0 0.080721 0 0 AA- 0.015326 0 0.879766 0 0.104908 0 0 A 0 0.060946 0 0.846243 0 0 0.089902 A- 0 0.034944 0 0 0.863655 0 0.096407 BBB 0 0 0 0.046109 0 0.901409 0 BBB- 0 0 0 0.031356 0 0 0.888297 BB 0 0 0 0.010585 0 0.029786 0 BB- 0 0 0 0.008 0 0.027965 0 B 0 0 0 0.011653 0 0 0 B- 0 0 0 0.010812 0 0 0 CCC 0 0 0 0 0 0 0 CCC- 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0

NON HOMOGENEOUS NUMERICAL EXAMPLE VIII PROBABILITIES TO GO IN THE RANK J AT NEXT TRANSITION - II BB BB- B B- CCC CCC- D AAA 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 AA- 0 0 0 0 0 0 0 A 0 0.002909 0 0 0 0 0 A- 0 0.004994 0 0 0 0 0 BBB 0 0.052482 0 0 0 0 0 BBB- 0 0.080347 0 0 0 0 0 BB 0.710572 0 0 0.244362 0 0.004695 0 BB- 0 0.618223 0 0.325257 0 0.020555 0 B 0.049741 0 0.912045 0 0 0.026561 0 B- 0.029961 0 0 0.907159 0 0.052068 0 CCC 0 0 0.084153 0 0.915847 0 0 CCC- 0 0 0.110386 0 0 0.889614 0 D 0 0 0 0 0 0 1

NON HOMOGENEOUS NUMERICAL EXAMPLE IX ij (0,4) PROBABILITIES - I AAA AA AA- A A- BBB BBB- AAA 0.914297 0 0.085703 0 0 0 0 AA 0.021064 0.901159 0 0 0.077777 0 0 AA- 0.016519 0 0.889573 0 0.093908 0 0 A 0 0.053557 0 0.867697 0 0 0.076311 A- 0 0.045204 0 0 0.833427 0 0.115872 BBB 0 0 0 0.044675 0 0.907406 0 BBB- 0 0 0 0.037216 0 0 0.872348 BB 0 0 0 0.008973 0 0.033339 0 BB- 0 0 0 0.007345 0 0.031525 0 B 0 0 0 0.01488 0 0 0 B- 0 0 0 0.011445 0 0 0 CCC 0 0 0 0 0 0 0 CCC- 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0

NON HOMOGENEOUS NUMERICAL EXAMPLE X ij (0,4) PROBABILITIES - II BB BB- B B- CCC CCC- D AAA 0 0 0 0 0 0 0 AA 0 0 0 0 0 0 0 AA- 0 0 0 0 0 0 0 A 0 0.002435 0 0 0 0 0 A- 0 0.005496 0 0 0 0 0 BBB 0 0.04792 0 0 0 0 0 BBB- 0 0.090436 0 0 0 0 0 BB 0.64796 0 0 0.305254 0 0.004473 0 BB- 0 0.630491 0 0.309835 0 0.020803 0 B 0.058717 0 0.897847 0 0 0.028556 0 B- 0.045216 0 0 0.881392 0 0.061947 0 CCC 0 0 0.088142 0 0.911858 0 0 CCC- 0 0 0.082797 0 0 0.917203 0 D 0 0 0 0 0 0 1

NON HOMOGENEOUS NUMERICAL EXAMPLE XI RELIABILITY - I times AAA AA AA- A A- BBB BBB- 0 1 1 1 1 1 1 1 1 0 2 1 0.999997 0.999971 0.999966 0.999557 0.999984 0.999847 0 3 1 0.999993 0.99994 0.999939 0.999112 0.999907 0.999631 0 4 1 0.999987 0.999898 0.999827 0.998523 0.999745 0.999342 0 5 1 0.999984 0.999887 0.999792 0.998349 0.999586 0.998944 0 6 0.999996 0.999937 0.999796 0.999599 0.997803 0.99918 0.998202 0 7 0.999993 0.999885 0.999685 0.999444 0.997007 0.998784 0.997432 0 8 0.999983 0.999799 0.999534 0.999312 0.996421 0.998155 0.996363 0 9 0.999959 0.999619 0.999264 0.998968 0.995586 0.9969 0.994263 0 10 0.999806 0.998802 0.997949 0.995926 0.99085 0.987962 0.974278 6 7 1 1 1 1 1 1 1 6 8 1 1 1 0.999985 0.999982 0.99987 0.999895 6 9 0.999997 0.999986 0.999953 0.999792 0.999562 0.999071 0.998771 6 10 0.999946 0.999731 0.999026 0.997798 0.995813 0.990239 0.979062

NON HOMOGENEOUS NUMERICAL EXAMPLE XI RELIABILITY - II times BB BB- B B- CCC CCC- D 0 1 1 1 1 1 1 1 0 0 2 0.999962 0.9995 0.999227 0.999309 0.995818 0.999765 0 0 3 0.999888 0.998705 0.998268 0.998373 0.991999 0.998878 0 0 4 0.999468 0.997207 0.996049 0.996217 0.987275 0.990783 0 0 5 0.997387 0.994572 0.992605 0.992034 0.978779 0.974065 0 0 6 0.994542 0.990933 0.988516 0.987031 0.96813 0.961347 0 0 7 0.991588 0.986831 0.979859 0.981979 0.955272 0.94721 0 0 8 0.986241 0.979546 0.972571 0.973973 0.935277 0.910461 0 0 9 0.975078 0.965145 0.957014 0.948575 0.819837 0.861098 0 0 10 0.938266 0.92036 0.902538 0.878429 0.680215 0.672408 0 6 7 0.999035 0.996466 0.98767 0.998919 0.958468 0.971783 0 6 8 0.997658 0.98804 0.977098 0.994086 0.913587 0.866085 0 6 9 0.991951 0.974409 0.959634 0.939398 0.780581 0.813993 0 6 10 0.967335 0.9316 0.891448 0.868098 0.609677 0.657879 0