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More information will be available at the St. James Room 4 th floor 7 – 11 pm.PowerPoint Presentation

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### Unstable Trends, Low Process Variability Invariance propertyData (ABC)

### Link Ratios can give answers trend kicks in at earlierthat are much too high (LR high) - Case Study 5.

### Comparing PL vs CRE we assume that trend only

### Do link ratio methods have any predictive power for this years, and latedata?

### Paids vs CREs the best we can do in ELRF.

More information will be available at the St. James Room 4th floor 7 – 11 pm.

ICRFS-ELRF will be available for FREE!

Much of current discussion included in the software

MPTF

PTF

LRT

SEE NEXTSLIDE

BF/ELR

Unique Benefits afforded by Paradigm Shift

“Best Estimates for Reserves” is now included in 2005 CAS Syllabus of Examinations. Reviewhttp://casact.org/pubs/actrev/may01/latest.htm

Models – Coin Versus Roulette Wheel

MPTF discussed in Session 3 (3:15 Arlington) and 6 (10:30am White Hill)

Summary- Examples

- Many myths grounded in a flawed paradigm
Ranges? Confidence Intervals? Myths

Loss Reserve Upgrades. Myths

- Link Ratios cannot capture trends and volatility
- Link Ratios can give very false indications
- Must model Paids and CREs separately
Cannot determine volatility in paids from incurreds!

Some real life examples taken from “Best Estimates..”

Unique Benefits afforded by Paradigm Shift

Modeling

MPTF

PTF

PAD

MULTIPLE LINES/ SEGMENTS/LAYERS

REINSURANCE

V@R

RELATIONSHIPS/CORRELATIONS

ADVERSE DEVELOPMENTCOVER

CAPITAL ALLOCATION

CREDIBILITY MODELLING

EXCESS OF LOSS

The Pleasure of Finding Things Out !

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ProbabilisticModelling

e.g. trends in the development year direction

If we graph the data for an accident year against development year, we can see two trends.

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Probabilistic Modelling

Could put a line through the points, using a ruler.

Or could do something formally, using regression.

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Variance =

Introduction to Probabilistic Modelling

Models Include More Than The Trends

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(y – ŷ)

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- The model is not just the trends in the mean, but the distribution about the mean

(Data = Trends + Random Fluctuations)

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Introduction to Probabilistic ModellingSimulating the Same “Features” in the Data

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- Simulate “new” observations based in the trends and standard errors

- Simulated data should be indistinguishable from the real data

x1,…,xn

- Random Sample from fitted distribution:

y1,…,yn

What does it mean to say a model gives a good fit? e.g. lognormal fit to claim size distribution

Does not mean we think the model generated the data

fitted lognormal

- Fitted Distribution

y’s look like x’s: —

Model has probabilistic mechanisms that can reproduce the data

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Real

Data

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Based on Ratios

Simulated triangles cannot be distinguished from real data – similar trends, trend changes in same periods, same amount of random variation about trends

Models project past volatility into the future

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X = Cum. @ j-1

Y = Cum. @ j

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ELRF(Extended Link Ratio Family)x is cumu. at dev. j-1 and y is cum. at dev. j- Link Ratios are a comparison of columns

- We can graph the ratios of Y to X

y/x

y/x

Intercept (Murphy (1994))

Since y already includes x: y = x + p

Incremental Cumulative

at j at j -1

Is b -1 significant ? Venter (1996)

Cumulative

Incremental

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j-1 j

} p

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Use link-ratios for projection

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Abandon Ratios - No predictive power

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Is assumption E(p|x) = a + (b-1) xtenable?

Note: If corr(x, p) = 0, then corr((b-1)x, p) = 0

If x, p uncorrelated, no ratio has predictive power

Ratio selection by actuarial judgement

can’t overcome zero correlation.

Condition 3:

Incremental

Review 3 conditions:

Condition 1: Zero trend

Condition 2: Constant trend, positive or negative

Condition 3: Non-constant trend

FORECASTING AND STATISTICAL MODELS

FUTURE PAST

(i) RECOGNIZE POTENTIAL ERRORS

(ii) ON STRAIGHT STRETCHES NAVIGATE

QUITE WELL

Future

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Probabilistic Modelling

Trends occur in three directions:

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Development year

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Calendar year

t = w+d

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Accident year

d

M3IR5 Data0 1 2 3 4 5 6 7 8 9 10 11 12 13

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072 7427

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080 9072

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534 11080

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530 13534

100000 81873 67032 54881 44933 36788 30119 24660 20190 16530

100000 81873 67032 54881 44933 36788 30119 24660 20190

100000 81873 67032 54881 44933 36788 30119 24660

100000 81873 67032 54881 44933 36788 30119

100000 81873 67032 54881 44933 36788

100000 81873 67032 54881 44933

100000 81873 67032 54881

100000 81873 67032

100000 81873

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alpha = 11.513

-0.2

PAID LOSS = EXP(alpha - 0.2d)

WHEN CAN ACCIDENT YEARS BE REGARDED AS DEVELOPMENT YEARS?GLEN BARNETT, BEN ZEHNWIRTH AND EUGENE DUBOSSARSKYAbstractThe chain ladder (volume-weighted average development factor) is perhaps the most widely used of the link ratio (age-to-age development factor) techniques, being popular among actuaries in many countries. The chain ladder technique has a number of interesting properties. We present one such property, which indicates that the chain ladder doesn’t distinguish between accident years and development years. While we have not seen a proof of this property in English language journals, it appears in Dannenberg, Kaas and Usman [1]. The result is also discussed in Kaas et al [2]. We give a simple proof that the chain ladder possesses this property and discuss its other implications for the chain ladder technique. It becomes clear that the chain ladder does not capture the structure of real triangles.

The Chain Ladder( Volume Weighted Average )

Transpose Invariance property

Use Volume Weighted Average to project incremental data: Take incremental array, cumulate across, find ratios, project, and difference back to incremental data.

Now: transpose incremental*, do Volume Weighted Average , transpose back same forecasts!

(equivalently, perform chain ladder ‘down’

not ‘across’: cumulate down, take ratios

down, project down, difference back)

The Chain Ladder( Volume Weighted Average )

The Chain ladder (Volume weighted aveage) - Transpose Invariance property

Chain ladder does not distinguish between accident and development directions. But they are notalike:

raw data

adjusted for trend in other direction

The Chain Ladder Invariance property( Volume Weighted Average )

Additionally, chain ladder (and ratio methods in general) ignore abundant information in nearby data.

* If you left out a point,

how would you guess

what it was?

- observations at same

delay very informative.

The Chain Ladder Invariance property( Volume Weighted Average )

Additionally, chain ladder (and ratio methods in general) ignore information in nearby data.

* If you left out a point,

how would you guess

what it was?

- observations at same

delay very informative.

- nearby delays also

informative (smooth trends)

(could leave out whole development)

Chain ladder ignores both

Major Calendar Year shifts satisfying Condition 3

The plots indicate a shift from calendar periods 84-85-86. However, we cannot adjust for accident period trends to diagnostically view what is left over along the calendar periodsas we can with PTF models.This example is in “Best Estimates”

Do U assign zero weight to all years save last two or three? However, we cannot adjust for

The link-ratio type models cannot capture changes along the calendar periods (diagonals).Determine the optimal model and note that several of the ratios are set to 1. The residualsof the optimal model are displayed below.

Model Display calendar periods

Note that as you move down the accident years the 16%+_ trend kicks in at earlierdevelopment periods. If variancewas not so small, we would not be able to see this on thegraphs of the data themselves – the trend change would be ‘obscured’ by randomfluctuation.

1977 Run-off

1978 Run-off

1979 Run-off

Overview

This case study illustrates how the residuals in ELRF can be very powerful indemonstrating that methods based on link ratios can sometimes give answers that aremuch too high.

The ELRF module also allows us to assess the predictive power of link ratio methodscompared to trends in the incremental data. For the data studied, trends in the incremental data have much more predictive power. Moreover, link ratio methods do not capture many ofthe features of the data.

Bring up the Weighted Residual Plot using the trend kicks in at earlierbutton.

Residuals represent the data minus what has been fitted to the data. Observe that theresiduals vs calendar years (Wtd Std Res vs Cal Year) trend downwards (negative trend).

This means that the trends fitted to the data are much higher than the actual trends in thedata.Accordingly any forecast produced by this method will assume trends that are muchhigher than the trends in the data. Therefore, the forecast will be much too high.

Below are the forecasts based on volume weighted average ratio and the arithmeticaverage ratio.

Table 5.1 - Summary of ForecastsThe Best Model in ELRF rarely uses link ratios and treats development periods as separate problems- Show them!

A good model for this data has the following trends and volatility about trends

With the calendar year volatility about trendstrend of 8.71% ± 0.97%, we obtain a distribution with a mean $593,506,000,and a st. deviation of $42,191,000. Scenario 1If we revertto the trend of 18.3% ± 2.6% experienced from 1981-1984, we obtain a reservedistribution with the mean of$751,912,000 with a standard deviation of $79,509,000. Scenario 2.Returning to the calendar year trend changes, it is important to try and identify whatcaused those changes. We cannot just assume that the most recent trend (8.71% ±0.97%) will continue for the next 17 years. Modelling other data types such as CaseReserve Estimates (CRE) and Number of Claims Closed (NCC) can assist in formulatingassumptions about future calendar year trends. See below.This would make it easier to decide on a future trend scenario along the calendar years.

The estimated trend between 1974-1978 is 47.5% ± 3.2%. If we assume that trend onlyfor the next calendar year (1991-1992) and revert to a trend of 18.3% ± 2.6% thereafter,we obtain a reserve distribution with a mean of $1,007,496,000 and a standard deviationof $112,234,000. Scenario 3

Conclusions we assume that trend only

There is some evidence that even the mean of $593,506,000 based on scenario 1 is too high. Accordingly, scenarios 2 and 3 appear tobe even more unlikely given the features in these three triangles in the past.

Overview

This case study illustrates how comparing a model for the Paid Losses with a model forthe Case Reserve Estimates (CRE) gives additional critical information that cannot beextracted from the Incurred Losses triangle. For most portfolios, we find that CREs lagPaid Losses in respect of calendar year trends.

Bring up the Weighted Residual Plot using the we assume that trend onlybutton.

The residual display is quite informative. Early accident years, and lateaccident yearsand calendar years are under-fitted.The link ratio regression model does not describe (nor capture) thestructure in the data.Most importantly, we cannot extract any information about the CREs and Paid Losses,and their relationship. This is done in Section 7.3 below using the PTF modellingframework.

All the trends are not captured even with this model that is the best we can do in ELRF.More importantly, we cannot extract any information about the trends in the data and thevolatility about the trends.

In which year was this company purchased?

- Loss Reserving Myths the best we can do in ELRF.
- Reserve Upgrades
- Ranges and Confidence Intervals

More information will be available at the St. James Room 4 the best we can do in ELRF.th floor 7 – 11 pm.

ICRFS-ELRF will be available for FREE!

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