“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 Earned Exposures Change

X 100,000X+1 100,000 0.0%X+2 100,000 0.0%X+3 90,900 -9.1%X+4 78,500 -13.6%X+5 63,475 -19.1%X+6 48,575 -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 Paid Frequency ChangeX+2 0.200X+3 0.210 5.0%X+4 0.220 5.0%X+5 0.231 5.0%X+6 0.243 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 Pure Premium ChangeX+2 $38.00X+3 $42.84 12.7%X+4 $48.82 13.9%X+5 $55.28 13.2%X+6 $62.64 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 SeverityYear 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 Severity ChangeX+2 $190.00X+3 $204.00 7.4%X+4 $221.46 8.6%X+5 $238.81 7.8%X+6 $257.75 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 inCalendar 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

AdjustedYear Paid Frequency ChangeX+2 0.200X+3 0.200 0.0%X+4 0.200 0.0%X+5 0.200 0.0%X+6 0.200 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

AdjustedYear Pure Premium ChangeX+2 $38.00X+3 $39.90 5.0%X+4 $41.90 5.0%X+5 $43.99 5.0%X+6 $46.19 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

AdjustedYear Severity ChangeX+2 $190.00X+3 $199.50 5.0%X+4 $209.48 5.0%X+5 $219.95 5.0%X+6 $230.95 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 TrendBodily Injury Coverage

6 pt.actual curve of

Date data best fit9/ 01 3.97 4.10712/ 01 4.61 4.5753/ 02 5.23 5.0966/ 02 5.79 5.6779/ 02 6.44 6.32512/ 02 6.78 7.046

REGRESSION 6 pt.

Avg Ann Trend = 54.02%

Adjusted Paid Freq TrendBodily Injury Coverage

6 pt.actual curve of

Date data best fit9/ 01 3.23 3.47112/ 01 3.54 3.5133/ 02 3.80 3.5566/ 02 3.80 3.6009/ 02 3.72 3.64312/ 02 3.41 3.688

REGRESSION 6 pt.

Avg Ann Trend = 4.96%

Calendar Year Paid Sev TrendBodily Injury Coverage

6 pt.actual curve of

Date data best fit9/ 01 10,691 10,96712/ 01 11,788 11,4353/ 02 11,707 11,9236/ 02 12,680 12,4319/ 02 13,228 12,96212/ 02 13,155 13,515

REGRESSION 6 pt.

Avg Ann Trend = 18.19%

Adjusted Paid Sev TrendBodily Injury Coverage

6 pt.actual curve of

Date data best fit9/ 01 10,228 10,59712/ 01 11,194 10,7823/ 02 10,800 10,9716/ 02 11,436 11,1639/ 02 11,654 11,35812/ 02 11,144 11,557

REGRESSION 6 pt.

Avg Ann Trend = 7.18%

CY Paid Pure Premium TrendBodily Injury Coverage

6 pt.actual curve of

Date data best fit9/ 01 424 45012/ 01 544 5233/ 02 612 6086/ 02 734 7069/ 02 852 82012/ 02 892 952

REGRESSION 6 pt.

Avg Ann Trend = 82.04%

Adjusted Paid Pure Premium Trend

Bodily Injury Coverage

6 pt.actual curve of

Date data best fit9/ 01 330 36812/ 01 397 3793/ 02 410 3906/ 02 434 4029/ 02 434 41412/ 02 380 426

REGRESSION 6 pt.

Avg Ann Trend = 12.51%