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Property ACTSC and Casualty
Loss Reserving
1
Examples From Previous
Calculate the ULAE reserve using both the traditional approach and the Kittel Refinement and the
following information:
CY 2010 Paid Losses
Total Reserves at 12/31/2009
Total Reserves at 12/31/2010
IBNR at 12/31/2009
IBNR at 12/31/2010
CY 2010 Paid ULAE
CY 2010 Incurred ULAE
900
9000
10000
3600
4000
90
210
2
Examples From Previous
Using the Wendy Johnson technique for determining ULAE reserves, determine the weighted
number of open claims for each future calendar year. Use the information below:
AY
2007
2008
2009
End of Year
1
2
3
4
Cumulative
Reported Claims
at 12/31/2009:
45
41
40
% Reported
% Closed
40%
30%
85%
60%
100%
85%
100%
3
Loss Development Using
Credibility
4
Least Squares Development
> Want to determine future development
points using equations of the form
= +
> Use historical data to solve for and
> =
2 2
and =
> If is the loss at time then is the loss at
time + 12
5
Example
Use the given information to calculate the ultimate losses for AY 2008 using least-squares.
Assume the tail factor from 48 months to ultimate is 1.0375.
AY
2006
2007
2008
2009
2010
12
100
110
120
130
140
24
120
130
140
150
36
130
150
150
48
140
160
6
60
145
Example
= 140
= 155.5
= 21,875
2 = 19,700
21,875 140 155.5
=
= 1.05
19,700 19,600
= 155.5 1.05 140 = 8.5
2008 = 8.5 + 150 1.05 = 166
7
Least Squares Development
> Special Cases
and are totally uncorrelated:
= 0, = (Expected loss technique)
Loss development factors are all equal
= 0, = (Loss development technique)
Bornhuetter-Ferguson has similar equation
= 1, = +
> Recall that B-F is a credibility weighting of
expected loss and loss development
(remember Hugh Whites question)
8
Least Squares Development
> Least squares is similar to B-F but has
more flexibility by allowing the multiplier to
known data to fluctuate (automatically
adjusts weights in credibility formula)
> Test hypothesis that least squares
produces a better estimate of ultimate than
traditional approaches using known
distributions and Bayesian analysis
9
Bayesian Development
> Define random variable Y for number of
claims which can be 0 or 1 with equal
probability
> Define random variable X as the # reported
by year end with 50% probability of claim
being reported
> We want to determine:
= = , expected total claims
= = = , Reserve
10
Bayesian Development
> Remember Bayes Theorem!
> =
( )
> In the example:
> = 0 = 0 =
> = 1 = 0 =
1
2
( )( 1)
1
2
1
1
3
11
11
+( ) ( )
22
=
2
3
Bayesian Development
> 0 = 0
2
3
+1
1
3
=
1
3
> = 1 1
2
1
1
2
> = + , = , =
3
3
3
3
> Since both and are both not 0 the solution is different
then what we would be achieved by the standard simple
approaches, suggesting the least squares method would
produce a better estimate
> i.e. The least squares method could produce the exact
solution given enough historical data
12
Bayesian Development
> Whats the problem with Bayesian
Development?
We dont have the distributions!
The real world is far more complex, then the
illustrative example
That leads to finding an approximation to the
Bayesian estimate
The best linear approximation is given by minimizing
2
Bayesian Credibility!
13
Bayesian Credibility
> The best linear approximation to Q is given by:
=
,
+ ()
> This can be used to answer Hugh Whites
Question (if actual losses are higher than
expected what do you do?):
If Cov(X,Y)<Var(X), decrease the reserve
If Cov(X,Y)=Var(X), leave the reserve the same
If Cov(X,Y)>Var(X), increase the reserve
Easier to understand with an example
14
Hugh White Example
,
=
+ ()
> Suppose you have a line of business with expected
ultimate losses of $900 (E[Y]) and expected reported
losses at time 1 of $500 (E[X])
> If actual losses at time 1 are $600 we can demonstrate
the result on the previous slide
> The expected reserve at time 1 is 900 500 = 400
> But 600 = 600 500
R 600 = 600 600
,
15
+ 900 and the reserve is
Hugh White Example
> For Cov(X,Y)=Var(X), R(600)=400
> For Cov(X,Y)<Var(X), R(600)<400
> For Cov(X,Y)>Var(X), R(600)>400
16
Bayesian Credibility
> ,
2
, () 2 2
2
> , ()
> Substituting into L(X) gives the least squares estimate
> It can be shown that the least squares estimate has less
bias then the loss development technique
> Least squares can also be used with Berquist-Sherman
type adjustments
17
Bayesian Credibility
> L(X) can be written in the form of a credibility weighting to
give additional insight into the relative use of the loss
development and expected loss techniques
> Define:
EVPV =
VHM =
> If there is a real number 0, = =
, :
> =
> =
()
+
+
+ 1 ,
=
+
18
Bayesian Credibility
> E(Y) is clearly the expected loss method
> x/d is the loss development technique
It can be shown that =
and =
19
Final Exam
> Saturday, December 10th 7pm
> Will cover everything from lectures, slides, Friedland and
Brosius
Note: We didnt get to the Mack paper or the Clark paper so they
will not be included on the exam
> E-mail me with any questions
20