Loss Modeling:
Introducing Simulation using
a Simplified Real World Problem
Domingo Castelo Joaquin
Illinois State University
dcjoaqu@ilstu.edu
Klugman, Stuart et al.(2004), Loss Models: From Data to Decisions, 2nd Ed., New York: Wiley, 618-621.
Joaquin, Domingo Castelo (2007), Loss Modeling using Spreadsheet-based Simulation, Risk Management and
Insurance Review, Vol. 10, No. 2, 283-297.
http://www.blackwell-synergy.com/loi/RMIR
Loss Payments
Target: PV of Payments for Losses covered by a
one-year policy
Simulating the PV of Loss Payments
covered by a one-year Policy
Cj = time of the jth loss, Cj-Cj-1 ~ i.i.d. Exponential(0.2), Co=0
Xj = amount of loss associated with loss event j
Xj ~ i.i.d. Pareto(1000,3)
Inter-arrival Times: Exponential
Loss Severity: Pareto
Claim Processing Time: Weibull
Lj = time from occurrence to payment for the jth loss
Lj ~ Weibull(1.5,LN(Xj+1)/6)
Tj= date of payment for the jth loss, Tj = Cj+Lj
Claim Processing Time: Weibull
Tj= date of payment for the jth loss, Tj = Cj+Lj
Target: PV of Payments for Losses covered by a
one-year policy
RΔT = required log return over ΔT years, RΔT ~ N(0.06 ΔT,0.02√ΔT)
Normal(0.067216, 0.021168)
X <= 0.0324
5.0%
40
X <= 0.1020
95.0%
35
30
25
20
15
10
5
0
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
V(T*n) = EXP(-RΔT*1) x EXP(-RΔT*2) x…x EXP(-RΔT*n)
V(T*n) = EXP(-Σi=1,..,n RΔT*i)
0.14
Frequency and Severity of Loss
Claim Settlement
Discount Rates and Present Values
PV of Specific Payments
Output Cells
Simulation Settings
Summary Output
A Note on Error Counts
If there are no
payments,
then ΔT is
undefined and
the standard
deviation for
the log return
also will be
undefined. A
discount rate
calculation will
not be carried
out and an
“error” will be
registered.
Beyond the error counts…
•
Do the numbers seem too small or too large or
have the wrong sign? For example, loss figures should be non-negative.
•
Do the numbers make sense relative to other
numbers? For example, retained loss should not exceed aggregate loss
•
Does a statistic look right by itself or in relation
to other statistics? For example, the average number of losses per
year should be close the variance if the inter-arrival times are exponentially
distributed.
•
Problems? Review the underlying cell formulas
for typos.
Tornado Diagram for
PV of Aggregate Loss Payments
•
Loss severity
•
Loss frequency
•
Claim processing time
•
Discount rates
•
The one-year policy period usually
would have elapsed after six loss
events
•
Most of the payments are made
in a short time
•
There is not much opportunity for
the power of compounding to
assert itself
References
Klugman, Stuart et al.(2004), Loss Models: From Data to
Decisions, 2nd Ed., New York: Wiley, 618-621.
Joaquin, Domingo Castelo (2007), Loss Modeling using
Spreadsheet-based Simulation, Risk Management and
Insurance Review, Vol. 10, No. 2, 283-297.
http://www.blackwell-synergy.com/loi/RMIR
dcjoaqu@ilstu.edu