The Effect of Changing Exposure Levels on Calendar Year Loss Trends by Chris Styrsky, FCAS, MAAA

1
The Effect of Changing Exposure Levels on
Calendar Year Loss Trendsby Chris Styrsky,
FCAS, MAAA 

• MAF Seminar
• March 22, 2005

2
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.

3
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

4
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)

5
Calendar Year Loss Trends

• Example Assumptions (cont.)
• The claim payment pattern does not change over
  time
• During calendar year X2, 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

6
Calendar Year Loss Trends

• Example Assumptions (cont.)

7
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

8
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

9
Calendar Year Paid Frequency Trend
10
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

11
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

12
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.
• Lets take a look at CY paid pure premiums.

13
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

14
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

15
Calendar Year Paid Pure Premium Trend
16
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
• Lets take a look at CY paid severity.

17
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

18
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

19
Calendar Year Paid Severity Trend
20
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

21
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.

22
Calendar Year Paid Severity Distribution by
Settlement Period
23
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?

24
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

25
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

26
Adjusted Paid Frequency Trend
27
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.

28
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

29
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

30
Adjusted Paid Pure Premium Trend
31
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

32
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

33
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

34
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.

35
Adjusted Paid Severity Trend
36
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)

37
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 years paid claim
• Recommend matching enough years where a
  significant portion of claims/losses have been
  paid (in PPA 8 years should be sufficient)

38
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

39
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

40
Lets take a look at some real examples
41
Calendar Year Paid Freq Trend
 
 
42
Adjusted Paid Freq Trend
 
 
43
Calendar Year Paid Sev Trend
 
 
44
Adjusted Paid Sev Trend
 
 
45
CY Paid Pure Premium Trend
 
 
46
Adjusted Paid Pure Premium Trend