“The Effect of Changing
Exposure Levels on Calendar
Year Loss Trends”
by Chris Styrsky, FCAS, MAAA
MAF Seminar
March 22, 2005
Why are loss trends important?
Loss trends are used to project the historical
data to the future experience period so
accurate loss costs will be reflected in the
rates charged.
How should data be organized for
loss trends?
• Accident Year/Policy Year
Benefit
– Best matching of risk with exposure
Drawback
– Most recent years requires loss development
• Calendar Year
Benefit
– Ease of use
Drawback
– Mismatching risk with exposure
Calendar Year Loss Trends
Example Assumptions:
• All policies are written on January 1st and are 12 month
policies
• The ultimate claim frequency for every risk in existence is
0.20
• 50% of the ultimate claims are paid within 12 months of the
date the policy was written, 30% between 12 and 24 months,
and 20% between 24 and 36 months (no claims paid past 36
months)
Calendar Year Loss Trends
Example Assumptions (cont.):
• The claim payment pattern does not change over time
• During calendar year X+2, claims that were paid within 12
months of the date the policy was written were settled for
$100, $200 for claims between 12 to 24 months, and $400 for
claims between 24 to 36 months
• Annual inflation is 5% for all claims
Calendar Year Loss Trends
Example Assumptions (cont.):
Calendar Year
X
X+1
X+2
X+3
X+4
X+5
X+6
Earned Exposures
100,000
100,000
100,000
90,900
78,500
63,475
48,575
Change
0.0%
0.0%
-9.1%
-13.6%
-19.1%
-23.5%
Calendar Year Paid Frequency
CYX Paid Frequency = (C0,12,X + C12,24,X + …) / EX
Where:
• CYX = Calendar year X
• CT,T + 12,X = # of claims paid during CYX that were
paid between T and T + 12 months after the claim
occurred
• EX = Earned Exposures from calendar year X
Calendar Year Paid Frequency
Year X + 2 = (100,000 * 0.2 * 0.5 + 100,000 * 0.2 *
0.3 + 100,000 * 0.2 * 0.2) / 100,000
= 0.2
Year X + 6 = (48,575 * 0.2 * 0.5 + 63,475 * 0.2 *
0.3 + 78,500 * 0.2 * 0.2) / 48,575
= 0.243
Calendar Year Paid Frequency
Trend
Calendar Year
X+2
X+3
X+4
X+5
X+6
Paid Frequency
0.200
0.210
0.220
0.231
0.243
Change
5.0%
5.0%
5.0%
5.0%
Why was there a trend???
There was a mismatch between the claims and
exposures!
For example:
Calendar Year X + 6 paid claims come from
Accident Years X + 4, X + 5, and X + 6 but
are matched to Calendar Year X + 6 earned
exposures
Will there always be an impact to
paid frequency trends?
There are two factors that need to occur to see
a distortion:
• Changing exposure levels
• Significant amount of time between
accident date and settlement date
CY Paid Pure Premium Trend
Since CY paid frequency trend is 5% and
inflation is 5% we would expect the CY
paid pure premium to about 10%.
Let’s take a look at CY paid pure
premiums….
Calendar Year Paid Pure
Premium
CYX Paid Pure Premium= (L0,12,X + L12,24,X +
…) / EX
Where:
• LT,T + 12,X = losses paid during CYX that were
paid between T and T + 12 months after the
claim occurred
Calendar Year Paid Pure
Premium
Year X + 2 = (100,000 * 0.2 * 0.5 *100 + 100,000 *
0.2 * 0.3 * 200 + 100,000 * 0.2 * 0.2 * 400) /
100,000
= $38.00
Year X + 6 = (48,575 * 0.2 * 0.5 * 100 * 1.05 4 +
63,475 * 0.2 * 0.3 * 200 * 1.05 4 + 78,500 * 0.2 *
0.2 * 400 * 1.05 4 ) / 48,575
= $62.42
Calendar Year Paid Pure
Premium Trend
Calendar Year
X+2
X+3
X+4
X+5
X+6
Pure Premium
$38.00
$42.84
$48.82
$55.28
$62.64
Change
12.7%
13.9%
13.2%
13.3%
CY Paid Severity Trend
In this example we know that inflation is 5%,
so we want a measure that will produce a
5% severity trend
Let’s take a look at CY paid severity….
Calendar Year Paid Severity
CYX Paid Severity= (S0,12,X * C0,12,X + S12,24,X *
C12, 24,X + …) / (C0,12,X + C12,24,X + …)
Where:
• ST,T + 12,X = losses paid during CYX that were
paid between T and T + 12 months after the
claim occurred
Calendar Year Paid Severity
Year X + 2 = (100,000 * 0.2 * 0.5 *100 + 100,000 *
0.2 * 0.3 * 200 + 100,000 * 0.2 * 0.2 * 400) /
(100,000 * 0.2 * 0.5 + 100,000 * 0.2 * 0.3 +
100,000 * 0.2 * 0.2)
= $190.00
Year X + 6 = (48,575 * 0.2 * 0.5 * 100 * 1.05 4 +
63,475 * 0.2 * 0.3 * 200 * 1.05 4 + 78,500 * 0.2 *
0.2 * 400 * 1.05 4 ) / (48,575 * 0.2 * 0.5 + 63,475
* 0.2 * 0.3 + 78,500 * 0.2 * 0.2)
= $257.75
Calendar Year Paid Severity
Trend
Calendar Year
X+2
X+3
X+4
X+5
X+6
Severity
$190.00
$204.00
$221.46
$238.81
$257.75
Change
7.4%
8.6%
7.8%
7.9%
Calendar Year Paid Severity
Calendar Year Paid Severity represents a
weighted average of the severities from the
different settlement periods where the
weights are the percentage of total paid
claims from that specific settlement period
What Happened???
This example assumes uniform inflation of 5%
annually, but the paid severity varies depending on
how long it takes to settle the claim.
With the declining exposures, the percentage paid
claims from the early settlement times decreases
with respects to total paid claims.
Calendar Year Paid Severity
Distribution by Settlement Period
% of Total Paid Claims Settled in
Calendar Year 0-12 mths 12-24 mths 24-36 mths
X+2
50.0%
30.0%
20.0%
X+3
47.6%
31.4%
21.0%
X+4
45.4%
31.5%
23.1%
X+5
43.2%
32.1%
24.7%
X+6
41.1%
32.3%
26.6%
What can you do to measure the
correct paid frequency?
Calendar Year Paid Frequency was distorted
by the mismatch between paid claims and
exposures, why not match the paid claims to
the exposures that produced them?
Adjusted Paid Frequency
Adjusted Paid Frequency (APF) = C0,12,X / EX
+ C12,24,X / EX-1 + C24,36,X / EX-2 + …
This formula can be thought of as adding the
incremental frequencies
Adjusted Paid Frequency
Year X + 2 = 100,000 * 0.2 * 0.5 / 100,000 +
100,000 * 0.2 * 0.3 /100,000 + 100,000 * 0.2 * 0.2
/ 100,000
= 0.2
Year X + 6 = 48,575 * 0.2 * 0.5 /48,575 + 63,475 *
0.2 * 0.3 / 63,475 + 78,500 * 0.2 * 0.2 / 78,500
= 0.2
Adjusted Paid Frequency Trend
Year
X+2
X+3
X+4
X+5
X+6
Adjusted
Paid Frequency
0.200
0.200
0.200
0.200
0.200
Change
0.0%
0.0%
0.0%
0.0%
What about paid pure premium?
Calendar Year Paid Pure Premium is also
distorted by the mismatch between paid
claims and exposures, so a similar
adjustment would seem warranted.
Adjusted Paid Pure Premium
Adjusted Paid Pure Premium (APPP) = L0,12,X /
EX + L12,24,X / EX-1 + L24,36,X / EX-2 + …
This formula can be thought of as adding the
incremental pure premiums
Adjusted Paid Pure Premium
Year X + 2 = 100,000 * 0.2 * 0.5 * 100 / 100,000 +
100,000 * 0.2 * 0.3 * 200 /100,000 + 100,000 *
0.2 * 0.2 * 400 / 100,000
= $38.00
Year X + 6 = 48,575 * 0.2 * 0.5 * 100 * 1.05
4/48,575 + 63,475 * 0.2 * 0.3 *200 * 1.05 4/
63,475 + 78,500 * 0.2 * 0.2 * 400 * 1.05 4/ 78,500
= $46.19
Adjusted Paid Pure Premium
Trend
Year
X+2
X+3
X+4
X+5
X+6
Adjusted
Pure Premium
$38.00
$39.90
$41.90
$43.99
$46.19
Change
5.0%
5.0%
5.0%
5.0%
What about paid severity?
Since we have formulas for adjusted paid
frequency and adjusted paid pure premium,
the formula for paid severity can be backed
into using the relationship of:
Frequency * Severity = Pure Premium
Adjusted Paid Severity
Adjusted Paid Severity (APS)= (L0,12,X / EX + L12,24,X /
EX-1 + L24,36,X / EX-2 + … )/(APF)
= (L0,12,X / EX)/APF + (L12,24,X / EX-1)/APF + …
= ((L0,12,X / C0,12,X ) * (C0,12,X / EX))/APF + ((L12,24,X /
C12,24,X ) * (C12,24,X / EX-1))/APF + …
= (S0,12,X * (C0,12,X / EX))/APF + (S12,24,X * (C12,24,X /
EX-1))/APF + …
Adjusted Paid Severity
Adjusted Paid Severity represents a weighted
average of the severities from the different
settlement periods where the weights are the
percentage of total paid frequency from
that specific settlement period
Adjusted Paid Severity
You have a formula to derive adjusted paid
severity, but you can use the same
relationship used to derive that formula and
just divide the Adjusted Paid Pure Premium
by the Adjusted Paid Frequency.
Adjusted Paid Severity Trend
Year
X+2
X+3
X+4
X+5
X+6
Adjusted
Severity
$190.00
$199.50
$209.48
$219.95
$230.95
Change
5.0%
5.0%
5.0%
5.0%
Benefits of using Adjusted Loss
Trends
• Adjusted loss trends remove the implicit
assumption with CY loss trends that exposure
levels are constant
• If exposure levels are constant, CY loss trends are
equal to adjusted loss trends
• No development needed (issue w/ AY)
• No issues with seasonality of reporting patterns,
plus adjustment is made for severity issues (issue
w/ reported frequency)
Pitfalls or Issues to Watch for if
using this method
#1
• How many years to match claims/losses with
exposures?
– Claims can be paid many years after the accident
occurred
– Not practical to match every accident year within a
calendar year’s paid claim
– Recommend matching enough years where a
“significant” portion of claims/losses have been paid (in
PPA 8 years should be sufficient)
Pitfalls or Issues to Watch for if
using this method (cont.)
#2
• What to do with the claims/losses from the years
not match?
– Recommend creating an “all others” accident year
category where all of the paid claims/losses are
summed
– These “all others” paid claims/losses should then be
matched to the calendar year exposures from the most
recent year that falls in the “all others” group since this
should be most representative of the exposure level of
the claims/losses
Pitfalls or Issues to Watch for if
using this method (cont.)
#3
• Some older CY earned exposures could be very
small if company is relatively new, potentially
causing unusual results
– Ex. There might be 1 paid claim matched to 2 earned
exposures causing frequencies to look extremely high
• Could remove incremental frequency that is distorted
• Could match back to years w/ at least X exposures
– Actuarial judgment should be used as to what the
appropriate action should be
Let’s take a look at some real
examples…
Calendar Year Paid Freq Trend
Bodily Injury Coverage
Date
9/01
12/01
3/02
6/02
9/02
12/02
actual
data
3.97
4.61
5.23
5.79
6.44
6.78
6 pt.
curve of
best fit
4.107
4.575
5.096
5.677
6.325
7.046
REGRESSION
6 pt.
Avg Ann Trend = 54.02%
Adjusted Paid Freq Trend
Bodily Injury Coverage
Date
9/01
12/01
3/02
6/02
9/02
12/02
actual
data
3.23
3.54
3.80
3.80
3.72
3.41
6 pt.
curve of
best fit
3.471
3.513
3.556
3.600
3.643
3.688
REGRESSION
6 pt.
Avg Ann Trend =
4.96%
Calendar Year Paid Sev Trend
Bodily Injury Coverage
Date
9/01
12/01
3/02
6/02
9/02
12/02
actual
data
10,691
11,788
11,707
12,680
13,228
13,155
6 pt.
curve of
best fit
10,967
11,435
11,923
12,431
12,962
13,515
REGRESSION
6 pt.
Avg Ann Trend =
18.19%
Adjusted Paid Sev Trend
Bodily Injury Coverage
Date
9/01
12/01
3/02
6/02
9/02
12/02
actual
data
10,228
11,194
10,800
11,436
11,654
11,144
6 pt.
curve of
best fit
10,597
10,782
10,971
11,163
11,358
11,557
REGRESSION
6 pt.
Avg Ann Trend =
7.18%
CY Paid Pure Premium Trend
Bodily Injury Coverage
Date
9/01
12/01
3/02
6/02
9/02
12/02
actual
data
424
544
612
734
852
892
6 pt.
curve of
best fit
450
523
608
706
820
952
REGRESSION
6 pt.
Avg Ann Trend =
82.04%
Adjusted Paid Pure Premium
Trend
Bodily Injury Coverage
Date
9/01
12/01
3/02
6/02
9/02
12/02
actual
data
330
397
410
434
434
380
6 pt.
curve of
best fit
368
379
390
402
414
426
REGRESSION
6 pt.
Avg Ann Trend =
12.51%