Transcript Combined Analysis of Paid and Incurred Losses

Posthuma Partners
Combined analysis of
paid and incurred losses
B. Posthuma
Washington, September 2008
E.A. Cator
Pp
Modern accounting (IFRS), capital management
and global regulation rules put more stringent
demands on loss reserving
Existing loss reserving methods are struggling
to provide an adequate, yet sufficiently flexible,
solution
Our combined analysis provides an excellent
tool for modern risk and capital management
Pp
Principles of modern Financial Economics
are:
1. expected present value of future cash
flows, and
2. their variances
Markowitz (1952), Sharpe (1964) and
Modigliani, Miller (1958) already applied
economic theory to business administration
Pp
International Financial Reporting Standards
(IFRS) are fully based on two basic
principles.
1. Fair Value =
expected present value plus margin for risk
or
market value
2. Actuarial prudence and solvency analyses
have to be in agreement with financial
economic theory
Pp
This requires:
1. stochastic loss reserving on a continuous
time basis, including discounting
2. adequate assessment of percentile ranges
3. flexibility in aggregating various datasets for
branches
Other prerequisites:
4. projections of expired and risk in force
5. adjustments in time to incurred loss
properly modeled
Pp
The datasets used often consist of two loss
triangles, for paid and reported incurred,
together with a measure for exposure
Many methods and models have been
developed for analyzing a single loss triangle
We are able to model the loss triangles for paid
and reported incurred simultaneously, and
show that this leads to a more accurate
analysis of the loss reserve
Pp
Loss period l
Development period k
(1)
lk
incremental paid losses
(2)
lk
incremental incurred losses
Y
Y
Pp
Start by supposing that all losses are
independent and normally distributed
Now note:
as all claims are settled eventually,
cumulative paid and incurred losses for
a given loss period must be equal
Pp
Therefore we condition the incremental
(1)
(2)
losses Ylk and Ylk on the event that
SY
k
(1)
lk
=
SY
(2)
lk
k
This conditioning preserves normality
Also, conditioning can be used to predict
future losses given the observed losses
Pp
Advantages of using the normal distribution:
– projections for expired insurance risk as well
as risk in force are readily available
– the formats of data such as time units for loss
period, development period or other
aggregations are easily handled
– aggregation of neighboring cells strengthens
the assumption of normality
– flexible projections for discounted values
naturally exists
– negative incrementals in incurred are not
uncommon
Pp
Now we need a parametric model for
(1)
(2)
means and variances of Ylk and Ylk
Pp
Means and variances:
EY = mlPk
(i)
(i)
lk
,
i = 1,2
Pp
where
m = Wl eXb
and
Wl is the exposure.
(i)
lk
var(Y )
(i )
~
= s i m lP k , i = 1,2
2
Development curves sum to 1 :
S P = S P =S
(1)
k
k
(2)
k
k
k
(1)
~
P =
k
Pp
SP =1
~ (2)
k
k
A four parameter family is constructed for
development curves
{f(x;b,g,m,s):b,g,s > 0, m  0}
with useful properties:
– integrate to 1
– can be negative if
m >1
– first and second primitive explicit
– direct control over boundary behavior
Pp
Pp
Ample flexibility, including negative values
A simulation experiment was conducted:
1. using a real dataset of partially observed paid
and incurred loss tables = loss triangles
2. estimating the parameters (b,g,m,s)
using maximum likelihood
3. generating 6000 pairs of complete tables
from our model with the estimated
parameters
4. predicting the reserve R on the basis of the
observed part for each of the 6000 samples
The combined analysis is compared to using
the single paid triangle only
Pp
single
Pp
combined
Variance of the combined analysis is 3x smaller!
Conclusion:
Our combined analysis of paid and
incurred proves to be a flexible and
accurate tool for loss reserving,
providing results for simulated and real
data that are superior to existing
methods
Pp