Best Estimate Reserves and the Claims Development
Results in Consecutive Calendar Years
Annina Saluz and Alois Gisler
ETH Zurich
May 22, 2013
ASTIN Colloquium
The Hague, Netherlands
Overview
1
Introduction
2
Martingale Argument
3
Examples
One Jump Model
Normal-Normal Model
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
2 / 27
Introduction
Importance of claims reserves
Most important item on the liability side of the balance sheet
Wrong reserves ⇒ wrong premiums
Reserve risk = main insurance risk in solvency
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
3 / 27
Introduction
Importance of claims reserves
Most important item on the liability side of the balance sheet
Wrong reserves ⇒ wrong premiums
Reserve risk = main insurance risk in solvency
Perception of management
Want to be sure that the actuarial reserves are ‘best estimates’
First check: look at the CDR
I
I
if it is negative over several years: actuaries are blamed to be responsible for
the losses in the annual profit and loss statement
if it is positive over several years: actuaries are blamed that prices in the past
were not competitive and that the company has missed profitable opportunities
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
3 / 27
Introduction
Importance of claims reserves
Most important item on the liability side of the balance sheet
Wrong reserves ⇒ wrong premiums
Reserve risk = main insurance risk in solvency
Perception of management
Want to be sure that the actuarial reserves are ‘best estimates’
First check: look at the CDR
I
I
if it is negative over several years: actuaries are blamed to be responsible for
the losses in the annual profit and loss statement
if it is positive over several years: actuaries are blamed that prices in the past
were not competitive and that the company has missed profitable opportunities
Question:
Are claims development results with the same sign over several
consecutive calendar years a contradiction to best estimate reserves?
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
3 / 27
Notation
Incremental claims Xi,j
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
4 / 27
Notation
Incremental claims Xi,j
Cumulative claims Ci,j , Ui = Ci,J ultimate claim
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
4 / 27
Notation
Incremental claims Xi,j
Cumulative claims Ci,j , Ui = Ci,J ultimate claim
Information at time I: DI = {Xi,j ; i + j ≤ I} and external information II .
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
4 / 27
Notation
Incremental claims Xi,j
Cumulative claims Ci,j , Ui = Ci,J ultimate claim
Information at time I: DI = {Xi,j ; i + j ≤ I} and external information II .
Outstanding liabilities at time I:
(I)
Ri
J
X
=
Xi,j = Ui − Ci,I−i+1 ,
j=I−i+1
Annina Saluz and Alois Gisler
R(I) =
I
X
(I)
Ri .
i=I−J+1
Best Estimates and the CDR
May, 2013
4 / 27
Notation
Incremental claims Xi,j
Cumulative claims Ci,j , Ui = Ci,J ultimate claim
Information at time I: DI = {Xi,j ; i + j ≤ I} and external information II .
Outstanding liabilities at time I:
(I)
Ri
J
X
=
Xi,j = Ui − Ci,I−i+1 ,
j=I−i+1
I
X
R(I) =
(I)
Ri .
i=I−J+1
Claims Reserves:
(I)
R̂i
(I)
= Ûi − Ci,I−i+1 = prediction of Ri ,
R̂(I) =
I
X
(I)
R̂i .
i=I−J+1
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
4 / 27
Claims Development Result (CDR)
(I+1)
After one year: New predictions R̂i
CDR(I+1) =
I
X
(I+1)
, Ûi
(I)
Ûi
and R̂(I+1) .
(I+1)
− Ûi
i=I−J+1
=
I
X
(I)
R̂i
(I+1)
− Xi,I−i+1 − R̂i
.
i=I−J+1
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
5 / 27
Best Estimate Reserves
Neither optimistic nor pessimistic, neither on the prudent nor on the
aggressive side.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
6 / 27
Best Estimate Reserves
Neither optimistic nor pessimistic, neither on the prudent nor on the
aggressive side.
For best estimate reserves: Expected value of the CDR is zero.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
6 / 27
Best Estimate Reserves
Neither optimistic nor pessimistic, neither on the prudent nor on the
aggressive side.
For best estimate reserves: Expected value of the CDR is zero.
(I)
In mathematical terms: R̂i
Annina Saluz and Alois Gisler
(I)
= E[Ri |II , DI ].
Best Estimates and the CDR
May, 2013
6 / 27
Martingale Argument
Best estimate predictions of the ultimate claims:
(I)
Ûi
Annina Saluz and Alois Gisler
= E[Ui |II , DI ].
Best Estimates and the CDR
May, 2013
7 / 27
Martingale Argument
Best estimate predictions of the ultimate claims:
(I)
= E[Ui |II , DI ].
n
o
(I)
The sequence Ûi : I = i, i + 1, . . . is a martingale.
o
n
(I)
(I−1)
(I)
⇒ CDRi = Ûi
− Ûi : I = i, i + 1, . . .
i
h
(I)
= 0.
are uncorrelated with E CDRi
Ûi
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
7 / 27
Martingale Argument
Best estimate predictions of the ultimate claims:
(I)
= E[Ui |II , DI ].
n
o
(I)
The sequence Ûi : I = i, i + 1, . . . is a martingale.
o
n
(I)
(I−1)
(I)
⇒ CDRi = Ûi
− Ûi : I = i, i + 1, . . .
i
h
(I)
= 0.
are uncorrelated with E CDRi
Ûi
Often drawn conclusion: CDR’s with the same sign (losses or gains) over
several consecutive calendar years are a contradiction to best estimate
reserves.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
7 / 27
Example from Practice
Motor Liability in Switzerland after the mid nineties.
Most companies: negative CDR over several consecutive years
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
8 / 27
Example from Practice
Motor Liability in Switzerland after the mid nineties.
Most companies: negative CDR over several consecutive years
Situation:
I
I
I
I
I
emergence of a new phenomenon: ‘whiplash cases’
creeping change of legislation leading to higher indemnities for such claims
reserving actuary at the beginning of this period:
increase of incurred claims in newest diagonal (but not in paid triangle);
not clear whether this was systematic or random;
not full weight given to observations in newest diagonal;
only with time ‘new situation’ is fully reflected by the reserves;
consequence of this behaviour: negative CDR’s over several years.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
8 / 27
Example from Practice
Motor Liability in Switzerland after the mid nineties.
Most companies: negative CDR over several consecutive years
Situation:
I
I
I
I
I
emergence of a new phenomenon: ‘whiplash cases’
creeping change of legislation leading to higher indemnities for such claims
reserving actuary at the beginning of this period:
increase of incurred claims in newest diagonal (but not in paid triangle);
not clear whether this was systematic or random;
not full weight given to observations in newest diagonal;
only with time ‘new situation’ is fully reflected by the reserves;
consequence of this behaviour: negative CDR’s over several years.
Questions:
I
I
Did all these actuaries violate the principle of best estimates?
Or did we oversee something in the martingale argument?
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
8 / 27
Example from Practice
Motor Liability in Switzerland after the mid nineties.
Most companies: negative CDR over several consecutive years
Situation:
I
I
I
I
I
emergence of a new phenomenon: ‘whiplash cases’
creeping change of legislation leading to higher indemnities for such claims
reserving actuary at the beginning of this period:
increase of incurred claims in newest diagonal (but not in paid triangle);
not clear whether this was systematic or random;
not full weight given to observations in newest diagonal;
only with time ‘new situation’ is fully reflected by the reserves;
consequence of this behaviour: negative CDR’s over several years.
Questions:
I
I
Did all these actuaries violate the principle of best estimates?
Or did we oversee something in the martingale argument?
Answer:
I
I
I
Yes we have overseen something in the martingale argument
Nothing wrong in the mathematics
But we have to be careful in the interpretation what the martingale property
exactly means.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
8 / 27
Interpretation of the Martingale Property
Claims development depends on ‘hidden’ characteristics (e.g. economic
environment, legislation)
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
9 / 27
Interpretation of the Martingale Property
Claims development depends on ‘hidden’ characteristics (e.g. economic
environment, legislation)
These ‘hidden’ characteristics are best modelled in a Bayesian way as random
variables.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
9 / 27
Interpretation of the Martingale Property
Claims development depends on ‘hidden’ characteristics (e.g. economic
environment, legislation)
These ‘hidden’ characteristics are best modelled in a Bayesian way as random
variables.
Martingale argument holds in the average over these state space
characteristics.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
9 / 27
Interpretation of the Martingale Property
Claims development depends on ‘hidden’ characteristics (e.g. economic
environment, legislation)
These ‘hidden’ characteristics are best modelled in a Bayesian way as random
variables.
Martingale argument holds in the average over these state space
characteristics.
What we observe in a claims development triangle are conditional
observations (given one specific trajectory of the state space process).
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
9 / 27
Interpretation of the Martingale Property
Claims development depends on ‘hidden’ characteristics (e.g. economic
environment, legislation)
These ‘hidden’ characteristics are best modelled in a Bayesian way as random
variables.
Martingale argument holds in the average over these state space
characteristics.
What we observe in a claims development triangle are conditional
observations (given one specific trajectory of the state space process).
These conditional observations are no longer increments of a martingale.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
9 / 27
Interpretation of the Martingale Property
Claims development depends on ‘hidden’ characteristics (e.g. economic
environment, legislation)
These ‘hidden’ characteristics are best modelled in a Bayesian way as random
variables.
Martingale argument holds in the average over these state space
characteristics.
What we observe in a claims development triangle are conditional
observations (given one specific trajectory of the state space process).
These conditional observations are no longer increments of a martingale.
Consequence: CDR’s with same sign over several consecutive calendar years
are not (necessarily) a contradiction to best estimate reserves.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
9 / 27
Basic Assumptions
Bayesian framework:
Each calendar year t characterised by a calendar year effect Ψt
Ψt modelled as hidden random variable
Ψ = (Ψ0 , Ψ1 , . . .)
E[Xi,j |Ψ] = µi γj Ψi+j
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
10 / 27
One Jump Model
Situation considered:
Calendar years t ≤ s0 − 1:
I
I
I
‘stable’ situation
no diagonal effects visible in the triangle
no indication of a change in the future
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
11 / 27
One Jump Model
Situation considered:
Calendar years t ≤ s0 − 1:
I
I
I
‘stable’ situation
no diagonal effects visible in the triangle
no indication of a change in the future
Calendar year s0
I
I
I
external information: change of legislation
impact:
increase of expected value of future claim payments by some not exactly
known factor Ψs0
no indication of other changes in the future
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
11 / 27
One Jump Model
Situation considered:
Calendar years t ≤ s0 − 1:
I
I
I
‘stable’ situation
no diagonal effects visible in the triangle
no indication of a change in the future
Calendar year s0
I
I
I
external information: change of legislation
impact:
increase of expected value of future claim payments by some not exactly
known factor Ψs0
no indication of other changes in the future
Calendar years t ≥ s0 + 1
I
stable situation, no further changes in claims environment
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
11 / 27
One Jump Model
Model Assumptions:
i) Given Ψ, Xi,j are independent and normally distributed with
E[Xi,j |Ψ] = µi γj Ψi+j ,
Var(Xi,j |Ψ) = µi γj σ 2 ,
PJ
where j=0 γj = 1.
ii) For t ≤ s0 − 1:
Conditional on {It , Dt }: Ψl = 1 for l ≤ t and
E[Ψt+l |It , Dt ] = E[Ψt |It , Dt ] = 1 for l ≥ 1.
iii) For t ≥ s0 :
Conditional on {It , Dt }
Ψl = 1 for l ≤ s0 − 1;
Ψl = Ψs0 for l = s0 , . . . , t;
E [ Ψt+l | It , Dt ] = E [ Ψt | It , Dt ] = E [ Ψs0 | It , Dt ] for l ≥ 1,
where Ψs0 | {It , Ds0 −1 } ∼ N (m, τ 2 ).
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
12 / 27
Best Estimates
Best estimate reserves for t ≥ s0 :


J
J
X
X
E[Ψs0 |It , Dt ]
R̂(t) =
γj µt+k−j 
k=1
j=k
(t)
Let Ψs0 = E[Ψs0 |It , Dt ].
Exact Credibility case with
0 +l)
= E[Ψs0 |Is0 +l , Ds0 +l ] = αs0 +l D̄s0 +l + (1 − αs0 +l )m,
Ψ(s
s0
qs(s00 +l) = Var(Ψs0 |Is0 +l , Ds0 +l ) = (1 − αs0 +l )τ 2 ,
where
Ps0 +l PJ
D̄s0 +l =
t=s0
j=0 Xt−j,j
Ps0 +l PJ
t=s0
j=0 µt−j γj
Annina Saluz and Alois Gisler
Ps0 +l PJ
t=s
and αs0 +l = Ps0 +l P0 J
Best Estimates and the CDR
t=s0
j=0
j=0
µt−j γj
µt−j γj +
May, 2013
σ2
τ2
.
13 / 27
Recursive estimation of Ψs0
i) For t = s0
PJ
0)
Ψ(s
s0
j=0
= α̃s0
Xs0 −j,j
ws0
+ (1 − α̃s0 )m,
qs(s00 ) = (1 − α̃s0 )τ 2 ,
where
α̃s0
ws0
=
2
ws0 + στ 2
and ws0 =
J
X
µs0 −j γj .
j=0
ii) For t > s0
PJ
Ψs(t)
0
= α̃t
j=0
Xt−j,j
wt
,
+ (1 − α̃t )Ψ(t−1)
s0
qs(t)
= (1 − α̃t )qs(t−1)
,
0
0
where
α̃t =
wt
σ2
wt + (t−1)
and wt =
qs0
Annina Saluz and Alois Gisler
Best Estimates and the CDR
J
X
µt−j γj .
j=0
May, 2013
14 / 27
Claims Development Result
Assume µi ≡ µ:
CDR(t+1) = Ψ(t+1)
µ(1 − β0 ) −
s0
J
X
(t+1)
Xt+1−j,j + (Ψ(t)
)µ
s0 − Ψs0
j=1
|
{z
}
J
X
|
{z
(t+1)
)µ
Xt+1−j,j + (Ψ(t)
s0 − Ψs0
{z
difference forecast and observations
Annina Saluz and Alois Gisler
}
change in update
j=1
|
(1 − βk−1 )
k=1
difference updated forecast and observations
= Ψ(t)
s0 µ(1 − β0 ) −
J
X
J
X
(1 − βk−1 ),
k=2
}
|
Best Estimates and the CDR
{z
}
change future
May, 2013
15 / 27
Sign of the CDR
Observations: Conditional data, given a specific realisation of Ψs0 .
Expected Value of the CDR for t + 1 ≥ s0 + 1
E[CDR(t+1) |It+1 , Ψs0 ]
= (m − Ψs0 )µ (1 − β0 )(1 − αt ) + (αt+1 − αt )
J
X
!
(1 − βk−1 )
k=2
If βj ≤ 1 for all j:
I
I
I
I
I
Sign of E[CDR(t+1) |It+1 , Ψs0 ] is the same for all t + 1 ≥ s0 + 1.
Absolute value is the bigger, the bigger |m − Ψs0 | is.
Absolute value is the bigger, the longer the development is.
Absolute value is decreasing in t since αt is increasing in t.
For t → ∞ the CDR is conditionally unbiased since αt → 1 (t → ∞).
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
16 / 27
Simulation Example
Parameter choices: µ = 100 000, m = 1.02, σ = 0.04, τ = 0.02.
j
γj
0
0.2
1
0.23
2
0.12
3
0.11
4
0.09
5
0.07
6
0.058
7
0.05
8
0.043
9
0.029
Table: Development pattern γj .
Estimation of Ψs0 :
Ψ(t)
s0 = αt
t X
J
X
1
Xl−j,j + (1 − αt )m,
µ(t − s0 + 1)
j=0
l=s0
where
αt =
t − s0 + 1
(t − s0 + 1) +
σ2
τ2
.
Assume jump occurs at time s0 = 10, Ψ10 = 1.035.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
17 / 27
Simulated Data
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
0
2308
1731
1757
1868
1970
1915
1766
2027
2041
2236
2008
2197
2098
1918
1741
2203
2122
2238
2128
2091
2143
2004
1
2151
2612
2036
2190
2342
1959
2293
2039
2278
2325
2507
2399
2362
2448
2693
2294
2523
2176
1960
2121
2833
2
1083
1189
1014
1098
1091
1066
1028
1476
986
1008
1289
1449
1366
1301
1417
1280
1581
1232
1297
1264
3
1067
1050
1106
746
1098
1241
1070
1344
1209
1369
1081
1139
883
1133
1106
1283
1201
1165
1235
4
701
915
926
680
1021
1011
841
904
1014
1002
1046
1008
855
935
935
1225
845
813
5
641
807
634
654
626
763
844
618
685
687
718
764
851
699
649
687
765
6
502
854
651
773
533
764
711
490
594
563
551
667
529
593
470
637
7
657
470
373
501
468
661
516
524
591
540
680
614
723
692
631
8
520
573
446
354
329
415
431
465
441
419
381
430
256
457
9
324
321
239
227
370
284
230
145
286
395
334
302
431
Table: Incremental claims
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
18 / 27
Predictions of Ψs0 over time
(t)
Figure: Predictions Ψs0 of Ψs0 = 1.035.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
19 / 27
CDR’s
Figure: CDR and E[CDRt |Ψs0 , It ].
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
20 / 27
CDR’s
Figure: CDR and split in ‘difference forecast and observations’ plus ‘change in future’.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
21 / 27
Results
We observe
negative CDR’s in 6 consecutive years
expected value of the CDR is negative
9 times a loss in the 10 years from calendar year 10 to 19.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
22 / 27
AR(1) Process
Idea: Based on external information It assume
AR structure for future diagonal effects
d
(t)
{Ψt+l : l ≥ 0} = {Ψ̃t+l = ρ(t) Ψ̃t+l−1 + (1 − ρ(t) )m(t) + ∆t+l , l ≥ 0}
I
(t)
iid
(t)
∆t+l |It ∼ N 0, δ 2
I
Given It , ∆t+l , l ≥ 1, independent from Ψ̃t
I
Given It , ∆t independent from Ψ̃t−1
Ψt |It normally distributed.
I
(t)
(t)
Updating of AR parameters for future diagonal effects is possible.
Past diagonal effects not updated due to new external information:
d
Ψt−l |It = Ψt−l |It−l
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
23 / 27
More General Model
(t)
(t)
(t)
(t)
(t)
Idea: Ψt = at Ψt−1 + bt + ∆t , where ∆t |It ∼ N (0, δt2 ) independent
of Dt−1 and Ψt−1 .
I
I
(t)
(t)
Parameters at , bt based on updated a priori information It and on past
(t)
data Dt−1 , and δt2 based on It .
(t)
(t)
(t)
Parameters at , bt and δt2 will not change due to new external information
It+l .
(t)
(t)
(t)
(t)
(t)
2
Future: Ψt+l = at+l Ψt+l−1 + bt+l + ∆t+l , where the ∆t+l |It ∼ N (0, δt+l
)
are mutually independent and independent of Dt and Ψt .
I
I
(t)
(t)
(t)
2
Parameters at+l , bt+l and δt+l
reflect a priori assumptions of process in later
years based on It and Dt .
(t)
(t)
2(t)
Parameters at+l , bt+l and δt+l
will be replaced in later years due to new
information.
Updating the external ‘a priori’ information It is possible.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
24 / 27
Normal-Normal Model
i) Conditionally, given Ψ and It , Xi,j independent and normally distributed
with
E [ Xi,j | Ψ, It ] = E [ Xi,j | Ψi+j ] = µi γj Ψi+j ,
Var ( Xi,j | Ψ, It ) = µi ηj2 σ 2 ,
PJ
PJ
where j=0 γj = j=0 ηj2 = 1.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
25 / 27
Normal-Normal Model
i) Conditionally, given Ψ and It , Xi,j independent and normally distributed
with
E [ Xi,j | Ψ, It ] = E [ Xi,j | Ψi+j ] = µi γj Ψi+j ,
Var ( Xi,j | Ψ, It ) = µi ηj2 σ 2 ,
PJ
PJ
where j=0 γj = j=0 ηj2 = 1.
ii) Conditionally, given It and Dt−1 , Ψt normally distributed with
(t,t−1)
Ψt
(t,t−1)
qt
(−1)
(t)
(t−1)
(t)
= E[Ψt |It , Dt−1 ] = at Ψt−1 + bt
2
(t)
(t)
(t−1)
= Var(Ψt |It , Dt−1 ) = at
qt−1 + δt2 ,
(−1)
for t ≥ 0, and Ψ−1 = 1 and q−1 = τ 2 .
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
25 / 27
Normal-Normal Model
i) Conditionally, given Ψ and It , Xi,j independent and normally distributed
with
E [ Xi,j | Ψ, It ] = E [ Xi,j | Ψi+j ] = µi γj Ψi+j ,
Var ( Xi,j | Ψ, It ) = µi ηj2 σ 2 ,
PJ
PJ
where j=0 γj = j=0 ηj2 = 1.
ii) Conditionally, given It and Dt−1 , Ψt normally distributed with
(t,t−1)
Ψt
(t,t−1)
qt
(−1)
(t)
(t−1)
(t)
= E[Ψt |It , Dt−1 ] = at Ψt−1 + bt
2
(t)
(t)
(t−1)
= Var(Ψt |It , Dt−1 ) = at
qt−1 + δt2 ,
(−1)
for t ≥ 0, and Ψ−1 = 1 and q−1 = τ 2 .
iii) Conditionally, given It and Dt , {Ψt+l : l ≥ 1} are normally distributed with
(t)
(t)
(t)
(t)
Ψt+l := E[Ψt+l |It , Dt ] = at+l Ψt+l−1 + bt+l
2
(t)
(t)
(t)
2(t)
qt+l := Var ( Ψt+l | It , Dt ) = at+l qt+l−1 + δt+l
.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
25 / 27
Normal-Normal Model
i) Conditionally, given Ψ and It , Xi,j independent and normally distributed
with
E [ Xi,j | Ψ, It ] = E [ Xi,j | Ψi+j ] = µi γj Ψi+j ,
Var ( Xi,j | Ψ, It ) = µi ηj2 σ 2 ,
PJ
PJ
where j=0 γj = j=0 ηj2 = 1.
ii) Conditionally, given It and Dt−1 , Ψt normally distributed with
(t,t−1)
Ψt
(t,t−1)
qt
(−1)
(t)
(t−1)
(t)
= E[Ψt |It , Dt−1 ] = at Ψt−1 + bt
2
(t)
(t)
(t−1)
= Var(Ψt |It , Dt−1 ) = at
qt−1 + δt2 ,
(−1)
for t ≥ 0, and Ψ−1 = 1 and q−1 = τ 2 .
iii) Conditionally, given It and Dt , {Ψt+l : l ≥ 1} are normally distributed with
(t)
(t)
(t)
(t)
Ψt+l := E[Ψt+l |It , Dt ] = at+l Ψt+l−1 + bt+l
2
(t)
(t)
(t)
2(t)
qt+l := Var ( Ψt+l | It , Dt ) = at+l qt+l−1 + δt+l
.
d
iv) For 0 ≤ l ≤ t, Ψt−l |(It , Dt−l ) = Ψt−l |(It−l , Dt−l ).
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
25 / 27
Recursive Calculation of the Ψt ’s, Normal-Normal Model
For t ≥ 0:
a) Update in the newest diagonal:
(t)
(t,t−1)
Ψt = αt X̄t + (1 − αt ) Ψt
,
where
αt =
X̄t =
wt
,
wt + κt
with wt =
J∧t
X
µt−j
j=0
γj2
,
ηj2
κt =
σ2
(t,t−1)
qt
,
J∧t
1 X γj
Xt−j,j ,
wt j=0 ηj2
and
(t)
(t,t−1)
qt = (1 − αt )qt
.
b) Forecast of future diagonal effects: For k ≥ 1
(t)
Ψt+k
Annina Saluz and Alois Gisler
(t)
(t)
(t)
= at+k Ψt+k−1 + bt+k .
Best Estimates and the CDR
May, 2013
26 / 27
Summary
Models that allow to
I
I
describe diagonal effects,
update external a priori information.
Conclusion:
I
I
Same sign of CDR over several consecutive calendar years is not a
contradiction to best estimates.
Potential applications for pricing.
Annina Saluz and Alois Gisler
Best Estimates and the CDR
May, 2013
27 / 27