lincolnshire+room+(6 ﬂoor)+ ·...

More informa+on will be available at

Lincolnshire Room (6th floor)

7:00pm-‐11:00pm

Monday September 14

(Also learn about the Bootstrap technique for tes+ng validity of a model or method)

1

Economic (& Cost of) Capital for the Combined Reserve and Underwri>ng Risk

across all Long Tail Lines of Business

2

Economic (& Cost of) Capital for the combined reserve and underwri>ng risk across all long tail lines of business

•  How is Economic and Cost of Capital to be calculated on a company-‐wide level, for reserve and underwri>ng risks taking into account diverse LOBs?

• What is the difference between variability and uncertainty?

• How can vola>lity for a LOB be described succinctly?

• What are the different types of calendar year trends and how are they manifested? • When should there be Total Reserve increases from year to year? By how much? • How do we maintain consistent es>mates of prior year ul>mates from year to year?

• How do es>mates of ul>mates change condi>onal on next calendar year’s paid losses? Equivalently, one year horizon.

•  Case Reserve Es>mates versus Paid Losses. Should they be modelled separately? CREs oSen lag paid losses in respect of trends.

•  What are the different types of correla>ons between LOBs and how are they manifested?

•  How do we know if two LOBs have common drivers and accordingly are highly correlated?

•  What is the impact of correla>on on risk capital alloca>on?

•  How much do different companies (in par>cular, Berkshire Hathaway, Swiss Re and The HarTord) have in common in the same LOB and between LOBs? Do they share vola>lity and correla>ons?

We use real companies’ data to answer these ques>ons.

Economic Capital and Cost of Capital computa>ons depend on VaR and or T-‐VaR calcula>ons

• VaR and T-‐VaR must based on an “accurate” distribu>on of aggregate reserves by LOB and correla>ons between them. Assump>ons should be transparent and auditable

• Need to allocate risk capital by LOB and calendar year. Cost of capital is based on calendar year payment stream distribu>ons. Therefore need to model paid losses.

• Ra>ng agencies apply an addi>ve risk charge for combined reserve and underwri>ng risk. Incorrect! An>thesis of basic principle of insurance.

• New business is mostly renewal business and affords addi>onal risk diversifica>on. This means that typically

Combined reserve and underwri>ng risk charge

< reserve risk charge + underwri>ng risk charge, for any T-‐VaR

Three types of correla+ons 1. Process correla+on (linear) 2. Parameter correla+on (linear) 3. Reserve (and Ul+mate) distribu+on correla+ons (not linear) •  Reserve distribu+on correla+on is usually considerably less

than process correla+on. •  Segments of the same LOB such as 1. net of reinsurance and

gross, 2. indemnity versus medical, 3. layers, for example, limited to 500K and limited to 1M, have common drivers and are highly correlated.

•  Different LOBs very oUen do not have common drivers. That is, the trend structure (especially along calendar years) is not the same and process correla+on is zero.

6

• The Mack method is a regression formula>on (weighted average trend through the origin) of volume weighted average link ra>os.

• It is important to check the weighted standardized residuals versus development period, accident period, calendar period and fi_ed values. • Tradi>onal actuarial methods, including the Mack method and bootstrapping*, oSen give grossly inaccurate assessment of risks. Yet they are regarded as “best prac>ce” and are used by Fitch, S&P and Moody’s. We illustrate this with real data Berkshire Hathaway PPA and Everest Re HO/FO.

• Tradi>onal actuarial methods do not have descriptors of features (vola>lity) in the data-‐ there is absence of a story! •  Inaccurate informa>on on risk is at best misguided and at worst very dangerous

We use the bootstrap technique to test

(validate) the model

Variability and Uncertainty are different concepts; not interchangeable

"Variability is a phenomenon in the physical world to be measured, analyzed and where appropriate explained. By contrast uncertainty is an aspect of knowledge."

Sir David Cox Our knowledge (measurement) of variability has associated uncertainty Insurance is the transfer of variability from one risk bearing en>ty to another

Variability and Uncertainty

Fair Coin 100 tosses fair coin (#H?)

Mean = 50

Std Dev = 5 CI [50,50]

In 95% of experiments with the coin the number of heads will be in interval [40,60].

“Balanced Roulette Wheel" No. 0,1, …, 100 Mean = 50 Std Dev = 29 CI [50,50]

In 95% of experiments with the wheel, observed number will be in interval [2, 97].

… 1 0 1 0 0

Where do you need more risk capital?

In each example there is no uncertainty in our knowledge about the variability. We know the whole probability distribution.

Introduce uncertainty into our knowledge - if coin or roulette wheel are mutilated then conclusions could be made only on the basis of observed data.

Example: Coin vs Roule_e Wheel

Risk diversifica>on by pooling risks CV(X+Y)< W1CV(X)+W2CV(Y) provided X & Y>0, and correla>on is not one, where W1 and W2 are weights propor>onal to the means CV is a measure of % variability Coefficient of Varia>on (CV)=SD/M Mean (M) = average of outcomes weighted by the probabili>es Standard Devia>on (SD)= (approx.) average distance of an outcome from the mean (M), weighted by the probabili>es

Basic Principles of Insurance

Basic Principles of Insurance

SD(Sum)<=Sum of SD’s-‐ basic principle of sta>s>cs It is easier to forecast the sum of many numbers than the sum of fewer

numbers SD(Sum) = Sum of SDs if correla>ons are all 1 SD(Sum)<<Sum of SDs if correla>on close to zero The difference between SD(Sum) and the Sum of SDs is related to the

correla>ons (degree of diversifica>on)

Calendar year “effects” (trends and shiUs) (i) “Infla+on” = social, economic or a legisla+ve shock (ii) Mortgage insurance-‐ defaults related to economic/financial variables. Are there leading indicators? (iii) Legisla+ve changes (shock) Cannot immunize (insulate) against social infla+on

When should there be Total Reserve increases from year to year? By how much?

Suppose true infla+on prior 1998 is 4%, post 1998 is 15%. True Reserves Assume (falsely) 15% kicks in in 1994. Case 1 Assume (falsely) paid losses con+nue to increase from 1998 only at 4%. Case 2.

Unrecognized infla+onary trends consume capital exponen+ally.

Total Reserve increases from year to year (with same exposure as previous year) • What does a calendar year trend (infla+on) of 15% imply in terms of loss reserves and underwri+ng each year? The following are simple arithme+c facts. § Each year the company needs to increase its total reserves by at least 15%. § The ul+mates for prior accident years will remain consistent with each increase in total reserves. Indeed it is only by forecas+ng along the 15% trend that they would remain consistent. §  Each year the company needs to increase its premium (price) by at least 15%. §  Ul+mates increase by at least 15% from one accident year to the next. §  These are not reserve upgrades. §  Mack and related methods give inconsistent ul+mates from year to year on upda+ng. See discussion on one year horizon

Wtd Std Res vs Cal. Yr

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

(LeU) Residuals aUer applying Mack method to the loss array for company ABC. Note the sharp trend aUer 1984. (Right) Probability Trend Family model picks up the change in trend structure in this direc+on. How can we improve Mack to deal with this?

Cal. Yr Trends

77 78 79 80 81 82 83 84 85 86 87

-1

-0.5

0

0.5

1

1.5

2

0 .0 6 5 2+-‐ 0 .0 0 3 5

0 .10 8 3+-‐ 0 .0 12 3

0 .16 9 1+-‐ 0 .0 0 7 4

Limita>ons of Mack Method when calendar trends are present


77 78 79 80 81 82 83 84 85 86 87

-1-0.8-0.6-0.4-0.20

0.20.40.60.81

A natural way to try to deal with the foregoing problem is to omit the early years and keep only the last three years of data for which the infla+on trend appears to be constant. Unfortunately this does not work as calendar trends are inherently invisible to link ra+o based methods. Mack does not contain any sensible descriptors of the vola+lity in the data.

(LeU) residuals aUer applying the Mack method with all but the last three years omihed from data. The sharp trend is s+ll present.


The best solu+on is to consider a modelling framework that possesses calendar year, development year and accident year (trend) parameters plus vola+lity about the trend structure.

This involves four pictures that provide a story about the business

Assump+ons about the future are explicit (transparent) and can be related to past experience

Require a modelling framework that can incorporate detailed business knowledge in the formula+on and explora+on of future scenarios including events that give rise to correla+ons



97 98 99 00 01 02 03 04 05 06

-1

-0.5

0

0.5

1

1.5


97 98 99 00 01 02 03 04 05 06

-1

-0.5

0

0.5

1

1.5

2

Mack BH PPA Mack ER HO/FO

Here trend in method << trend in data Here trend in method >> trend in data

Residuals represent the difference of two trends

Residuals = trend in data -‐ trend esBmated by the method

4,423,054T+_ 219,857T Too High! 37,178T +_10,708T Much Too low!

Mack method Berkshire Hathaway and Everest Re, Schedule P 2006


97 98 99 00 01 02 03 04 05 06

-1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.20.40.60.811.21.4

Mack method Berkshire Hathaway and Everest Re, Schedule P 2006 Measuring ‘average’ calendar year trend


97 98 99 00 01 02 03 04 05 06

-1.5

-1

-0.5

0

0.5

1

1.5

2

BH PPA ER HO/FO

PTF with iden>fied 7%+_ forward trend yields 3,939,208T+_130,818T versus Mack 4,423,054T+_ 219,857T 500M difference!

PTF with zero forward trend yields 52,311T+-‐14,656T versus Mack 37,178T+_10,708T

15M difference. S>ll much too low. If we con>nue the 40%+_ trend 03-‐06 to the en>re forecast period we obtain 132,363T+_54,266T

TG LR HIGH Mack can yield Reserve Mean much too high

ELRF-‐ Mack Trend es+mated by method >> Trend in data

Mack= 896,133T +_104,117T Arithme>c averages= 1,167,464T +-‐ 234,466T


74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

-1.5

-1-0.50

0.511.5

22.5

20

TG LR HIGH

Residuals= Trend in data-‐ Trend es>mated by method

Fihed (20%)

Observed (data) (10%)= trend in data

Residual = Data trend –Fitted Trend

+ve

-ve

21

TG LR HIGH Trend in data close to trend in method. Best ELRF model, link ra+os =1,

and s+ll not very good. Trend assump+on of method not known 468,439T+-‐ 47,346T

Very different answer! Mack= 896,133T +_104,117T


74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91

-1.5

-1

-0.5

0

0.5

1

1.5

2

22

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 .0 0 0 0+-‐0 .0 0 0 0

-‐0 .2 8 4 6+-‐0 .0 2 2 2

0 .0 6 8 8+-‐0 .0 2 4 9

0 .2 3 9 4+-‐0 .0 2 6 4

0 .0 6 8 8+-‐0 .0 2 4 9

COMPANY XYZ Modelling Incurred data does not yield distribu+on of calendar year payment streams and their correla+ons required for cost of capital

calcula+ons.

CREs usually lag paid losses in respect of trends CREs versus Paids

When was the company sold?

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98 99 00 01 02 03

-2.5-2-1.5-1-0.500.511.522.533.544.55

0 .2 9 5 8+-‐ 0 .0 15 9

0 .0 9 6 4+-‐ 0 .0 0 5 7

0 .2 9 5 8+-‐ 0 .0 15 9

A number of reinsurers lost a lot of money on the aggregate stop loss!

•  Three types of correla+ons Process correla+on Parameter (trend) correla+on Similar trend structure implying commonality in calendar year drivers

•  Cannot measure these correla+ons unless LOB trend structure and process variability (vola+lity) modeled accurately

•  Most important direc+on is the calendar year

•  Reserve distribu+on correla+on << Process correla+on

•  Highest Process correla+on have seen is 0.6!

•  Highest Reserve distribu+on correla+on 0.2!

•  Most long tail LOBs exhibit close to zero correla>on

•  Each company is different

Correla>ons between LOBs

Correla+on and Linearity Correla>on, linearity, normality, weighted least squares, and linear regression are closely related concepts. The idea of correla+on arises naturally for two random variables that have a joint distribu+on that is bivariate normal. For each individual variable, two parameters a mean and standard devia+on are sufficient to fully describe its probability distribu+on. For the joint distribu+on, a single addi+onal parameter is required the correla+on. If X and Y have a bivariate normal distribu+on, the rela+onship between them is linear: the mean of Y, given X, is a linear func+on of X ie:

( ) βXαY|XE +=

The slope β is determined by the correla+on ρ, and the standard devia+ons and :

The correla+on between Y and X is zero if and only if the slope β is zero. Also note that, when Y and X have a bivariate normal distribu+on, the condi+onal variance of Y, given X, is constant ie not a func+on of X:

,XY σρσ=β

( ) 2|XYY|XVar σ=

( ).),( YXσσρ YXCov=where

This is why, in the usual linear regression model Y = α + βX + ε the variance of the "error" term ε does not depend on X. However, not all variables are linearly related. Suppose we have two random variables related by the equa+on

where T is normally distributed with mean zero and variance 1. What is the correla+on between S and T ?

2TS =

Linear correla+on is a measure of how close two random variables are to being linearly related. In fact, if we know that the linear correla+on is +1 or -‐1, then there must be a determinis+c linear rela+onship Y = α + βX between Y and X (and vice versa). If Y and X are linearly related, and f and g are func+ons, the rela+onship between f( Y ) and g( X ) is not necessarily linear, so we should not expect the linear correla+on between f( Y ) and g( X ) to be the same as between Y and X. (Answer to ques+on on previous slide is zero)

A common misconcep+on with correlated

lognormals Actuaries frequently need to find covariances or correla+ons between variables such as when finding the variance of a sum of forecasts (for example in P&C reserving, when combining territories or lines of business, or compu+ng the benefit from diversifica+on).

Correlated normal random variables are well understood. The usual mul+variate distribu+on used for analysis of related normals is the mul+variate normal, where correlated variables are linearly related. In this circumstance, the usual linear correla+on ( the Pearson correla1on ) makes sense.

However, when dealing with lognormal random variables (whose logs are normally distributed), if the underlying normal variables are linearly correlated, then the correla+on of lognormals changes as the variance parameters change, even though the correla+on of the underlying normal does not.

All three lognormals below are based on normal variables with correla+on 0.78, as shown leU, but with different standard devia+ons.

We cannot measure the correla+on on the log-‐scale and apply that correla+on directly to the dollar scale, because the correla+on is not the same on that scale. Addi+onally, if the rela+onship is linear on the log scale (the normal variables are mul+variate normal) the rela+onship is no longer linear on the original scale, so the correla+on is no longer linear correla+on. The rela+onship between the variables in general becomes a curve:

Time

5 10 15 20

-3-2

-10

12

3

Time

5 10 15 20

-3-2

-10

12

3

Time

5 10 15 20

-3-2

-10

12

3

Series corr. = 0

Series corr. = 0.5

Series corr. = -‐0.5

Series corr. = 0.8

Correla>on in >me-‐series

These two triangular loss arrays have process corr. = 0.9 aUer modelling their respec+ve trend structures. Cannot detect from data plot.

3D plot of data

We call the correla>on of the random component (aSer modeling the trend structure in the three

direc>ons) of two loss development arrays: process correla>on

Two LOBs: LOB1 and LOB3 Actually same LOB different territories

35

Cal. Yr Trends

94 95 96 97 98 99 00 01 02 03

-1

-0.5

0

0.5

1

1.5

0 .0 0 0 0+-‐0 .0 0 0 0

0 .119 3+-‐0 .0 3 0 4

LOB1

Cal. Yr Trends

94 95 96 97 98 99 00 01 02 03

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 .0 0 0 0+-‐0 .0 0 0 0

0 .0 8 14+-‐0 .0 2 9 4

LOB3

Both LOBs have a calendar year trend change in 2000 That should be regarded as a concern!

Two LOBs: LOB1 and LOB3 Actually same LOB different territories

36

LOB1 LOB3 Wtd Std Res vs Cal. Yr

94 95 96 97 98 99 00 01 02 03

-1.5

-1

-0.5

0

0.5

1

1.5

2


94 95 96 97 98 99 00 01 02 03

-2.5-2

-1.5-1

-0.50

0.51

1.52

Note 98-00 slight negative trend, 00-02 slight positive trend and 02-03 zero trend LOB1 and slight negative trend LOB3.

Process correlation=0.35

PTF Calendar trends and Accident levels, LOB A (leS) LOB B (right) – close resemblance in placement of trend changes is evidence of common drivers.

Cal. Yr Trends

86 88 90 92 94 96 98 00 02 04 06

-2-10123456789

1.2 6 8 8+-‐ 0 .10 5 5

0 .2 19 8+-‐ 0 .0 2 2 1

0 .0 0 0 0+-‐ 0 .0 0 0 0

0 .12 6 0+-‐ 0 .0 12 8

Acc. Yr Trends

86 88 90 92 94 96 98 00 02 04 06-2-10123456789

-‐ 0 .0 4 4 7+-‐ 0 .0 6 0 9

-‐ 0 .3 2 5 8+-‐ 0 .0 6 6 3

0 .3 7 0 5+-‐ 0 .0 5 5 0

-‐ 0 .4 8 7 6+-‐ 0 .114 4

Acc. Yr Trends

86 88 90 92 94 96 98 00 02 04 06

012345678910

0 .9 10 0+-‐ 0 .0 6 4 0

0 .2 15 1+-‐ 0 .0 4 8 4

0 .3 5 8 4+-‐ 0 .0 6 8 9

-‐ 0 .5 7 3 5+-‐ 0 .0 7 6 1

Cal. Yr Trends

86 88 90 92 94 96 98 00 02 04 06

-3-2-101234567

0 .7 9 0 8+-‐ 0 .10 13

0 .13 19+-‐ 0 .0 2 12

0 .0 0 0 0+-‐ 0 .0 0 0 0

0 .114 1+-‐ 0 .0 115

When do two LOBs (LOB A & LOB B) have common drivers? When they have “same” trend structure and process correla+on is high

Calendar year 97-06 parameter correlation is 0.74. Process correlation next slide is 0.84


86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06

-2.5-2

-1.5-1

-0.50

0.51

1.52


86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 03 04 05 06

-3-2.5-2

-1.5-1

-0.50

0.51

1.52

2.5

Wtd Std Res vs Acc. Yr

86 88 90 92 94 96 98 00 02 04 063 2 2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5


86 88 90 92 94 96 98 00 02 04 061 2 3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Residual plots by calendar year LOB A (leU) LOB B (right)

Blue line is trace (versus accident year) of (single) calendar year (2006)

Process Correla>on = 0.85


86 88 90 92 94 96 98 00 02 04 063 2 2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5


86 88 90 92 94 96 98 00 02 04 061 2 3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

When do two LOBs (LOB A & LOB B) have common drivers? • Two LOBs have “same” trend structure and high process correla+on • Visible in trace of calendar year 2006 versus accident years. • Note high process correla+on (of 0.85).


86 88 90 92 94 96 98 00 02 04 06

-4-3.5-3-2.5-2-1.5-1-0.500.511.522.5


86 88 90 92 94 96 98 00 02 04 06

-4.5-4-3.5-3-2.5-2-1.5-1-0.500.511.52

Residuals above are for data adjusted for development and accident period trends only.

Therefore correla>ons are trend + process.

The two LOBs A & B are in fact gross and net of reinsurance paid losses that do share common drivers!

Two segments of WC Each segment only adjusted for development year trends

41

Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9

-3-2.5-2

-1.5-1

-0.50

0.51

1.5


86 88 90 92 94 96 98

-3-2.5-2

-1.5-1

-0.50

0.51

1.5


91 92 93 94 95 96 97 98 99

-3-2.5-2

-1.5-1

-0.50

0.51

1.5

Wtd Std Res vs Fitted

5 6 7 8 9 10

-3-2.5-2

-1.5-1

-0.50

0.51

1.5

Wtd Std Res vs Dev. Yr

0 1 2 3 4 5 6 7 8 9

-3-2.5-2

-1.5-1

-0.50

0.51

1.5


86 88 90 92 94 96 98

-3-2.5-2

-1.5-1

-0.50

0.51

1.5


91 92 93 94 95 96 97 98 99

-3-2.5-2

-1.5-1

-0.50

0.51

1.5

Wtd Std Res vs Fitted

6 7 8 9 10

-3-2.5-2

-1.5

-1-0.50

0.5

11.5

Accident year parameter correlations equal 1 after modelling accident years- major implications also for future underwriting years, where correlation in distributions of

ultimates exceed 0.99

Layers Lim1M, Lim2M and 1Mxs1M Lim2M=Lim1M+1Mxs1M

The trend structure is the same for each layer (LeU to right 1M, 1Mxs1M, 2M)

42

Dev. Yr Trends

0 2 4 6 8 10 12

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

1.0 8 9 9+-‐ 0 .12 16

0 .0 0 0 0+-‐ 0 .0 0 0 0

-‐ 0 .3 8 9 0+-‐ 0 .0 7 8 0

Acc. Yr Trends

85 87 89 91 93 95 97

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

11

-‐ 0 .2 0 8 0+-‐ 0 .19 9 0

-‐ 0 .4 4 2 5+-‐ 0 .18 3 4

0 .4 6 2 8+-‐ 0 .2 2 8 0

-‐ 0 .0 0 3 0+-‐ 0 .18 8 7

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 .10 3 5+-‐ 0 .0 3 9 3

MLE Variance vs Dev. Yr

0 2 4 6 8 10 12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 0 .3 12 7

2 .5 8 2 6

Dev. Yr Trends

0 2 4 6 8 10 12

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

1.2 8 16+-‐ 0 .13 4 0

0 .0 0 0 0+-‐ 0 .0 0 0 0

-‐ 0 .3 0 7 7+-‐ 0 .0 8 6 0

Acc. Yr Trends

85 87 89 91 93 95 97

4

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

-‐ 0 .2 6 0 1+-‐ 0 .2 19 3

-‐ 0 .5 18 5+-‐ 0 .2 0 2 1

0 .3 9 5 2+-‐ 0 .2 5 13

-‐ 0 .16 0 1+-‐ 0 .2 0 7 9

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 .0 5 6 3+-‐ 0 .0 4 3 3


0 2 4 6 8 10 12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55 0 .3 12 7

2 .5 8 2 6

Dev. Yr Trends

0 2 4 6 8 10 12

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

1.14 8 4+-‐ 0 .12 3 9

0 .0 0 0 0+-‐ 0 .0 0 0 0

-‐ 0 .3 5 9 0+-‐ 0 .0 7 9 5

Acc. Yr Trends

85 87 89 91 93 95 97

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

11

11.5

-‐ 0 .2 4 6 0+-‐ 0 .2 0 2 9

-‐ 0 .4 9 11+-‐ 0 .18 6 9

0 .4 2 7 8+-‐ 0 .2 3 2 4

-‐ 0 .0 6 5 9+-‐ 0 .19 2 3

Cal. Yr Trends

89 90 91 92 93 94 95 96 97 98

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 .0 8 7 6+-‐ 0 .0 4 0 0


0 2 4 6 8 10 12

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.450 .3 12 7

2 .5 8 2 6


Very high process correla+ons (LeU to right 1M, 1Mxs1M, 2M)

43


89 90 91 92 93 94 95 96 97 98

-2.6-2.4-2.2-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.20.40.60.811.21.41.61.8


89 90 91 92 93 94 95 96 97 98

-2.2-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.20.40.60.811.21.41.61.822.2


89 90 91 92 93 94 95 96 97 98

-2.4-2.2-2-1.8-1.6-1.4-1.2-1-0.8-0.6-0.4-0.200.20.40.60.811.21.41.61.8


Tables of process correla+ons (linear) and calendar year parameter correla+ons (linear)

44

This type of equivalent trend structure and high parameter and process correla+ons has not been observed for two LOBs

Reserve distribu>on correla>ons between two dis>nct LOBs-‐ a very different story

45

•  Highest process correla+on observed between two different LOBs is about 0.6 (in our experience)

•  But Reserve distribu+on correla+on is typically lower.

•  Trend structures for two LOBs typically different •  Parameter correla+ons low or zero •  See Private Passenger Automobile (PPA) versus Commercial Auto Liability (CAL)for Berkshire Hathawy below, for example

Line A:

Res. $100m CV=5%

Line B:

Res. $50m

CV = 15%

Aggregate:

Res. $150m

CV = ?

Assume Risk Capital at 98th percen+le = 2 Standard Devia+ons

Risk Capital for Line A = $10m

Risk Capital for Line B = $15m

Aggregate Risk Capital (ARC) = $25m ?

Risk Capital Alloca>on

Line A:

Res. $100m CV=5%

Line B:

Res. $50m

CV = 15%

Aggregate:

Res. $150m

CV = ?

Assume Risk Capital at 98th percen+le = 2 Standard Devia+ons

Risk Capital for Line A = $10m

Risk Capital for Line B = $15m

Aggregate Risk Capital (ARC) = $25m ?

Risk Capital Alloca>on

The answer depends on the correla+on.

If Corr = +1.0, ARC = $25m

If Corr = 0.0 ARC = $18m

Benefit =

Sum of individual risk capital assessments – aggregate risk capital assessment from joint distribu1on of the two (correlated) lines.

Risk Capital Alloca>on: Diversifica>on benefit

Berkshire Hathaway Schedule P 2006 No LOBs have the “same” tend structure and most LOBs have

zero process correla+on. Consider Private Passenger Automobile and Commercial Auto Liability

49

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9

-7.5-7

-6.5-6

-5.5-5

-4.5-4

-3.5-3

-2.5-2

-1.5-1

-0.50

-‐ 0 .3 9 8 1+-‐ 0 .0 18 4

-‐ 1.0 6 4 1+-‐ 0 .0 18 5

-‐ 0 .8 3 3 7+-‐ 0 .0 10 9

-‐ 0 .9 4 9 3+-‐ 0 .0 2 2 5

Acc. Yr Trends

97 98 99 00 01 02 03 04 05 069.510

10.511

11.512

12.513

13.514

14.515

15.516

16.517

0 .0 9 4 3+-‐0 .0 2 3 1

-‐0 .17 0 7+-‐0 .0 2 4 1

Cal. Yr Trends

97 98 99 00 01 02 03 04 05 06

-3-2.5-2

-1.5-1

-0.50

0.51

1.52

2.53

3.54

0 .2 113+-‐ 0 .0 12 4

0 .118 6+-‐ 0 .0 10 0

0 .0 7 7 9+-‐ 0 .0 0 7 8


0 1 2 3 4 5 6 7 8 90

2e-3

4e-3

6e-3

8e-3

0.01

0.012

0.014

0.016

Dev. Yr Trends

0 1 2 3 4 5 6 7 8 9

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 .3 10 1+-‐ 0 .0 6 9 1

-‐ 0 .2 3 4 1+-‐ 0 .0 3 9 2

-‐ 0 .7 0 3 8+-‐ 0 .0 3 9 8

Acc. Yr Trends

97 98 99 00 01 02 03 04 05 06

7.5

8

8.5

9

9.5

10

10.5

11

11.5

0 .2 2 10+-‐ 0 .0 6 7 3

0 .14 8 6+-‐ 0 .0 6 10

-‐ 0 .16 5 7+-‐ 0 .0 8 4 6

Cal. Yr Trends

97 98 99 00 01 02 03 04 05 06

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 .17 3 1+-‐ 0 .0 4 8 1

0 .0 0 0 0+-‐ 0 .0 0 0 0


0 1 2 3 4 5 6 7 8 90

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

LOB PPA LOB CAL

Note LOBs have very different trend structure and process variance

Berkshire Hathaway Schedule P 2006

50


97 98 99 00 01 02 03 04 05 062 2 3 2

-1.5

-1

-0.5

0

0.5

1

1.5


97 98 99 00 01 02 03 04 05 063 3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

LOB PPA LOB CAL

Note zero process correlation. Blue lines represent trace of calendar year 2006

Forecast lognormal distribu+ons (and their correla+ons) for each future cell for each LOB based on

an explicit forecast scenario

51

Blue is observed, black is mean of lognormal and red is standard deviation of lognormal

Berkshire Hathaway Schedule P 2006 One Year Time Horizon- “Change” in Ultimates conditional

on next year’s paid losses? When do estimates of prior year ultimates stay consistent on updating (next valuation

period) ? No need for simulations! Why?

Related Question:

Which risk characteris+cs of the data do calendar year payment stream distribu+ons depend on? 1. Base development period trends, and 2. Calendar year trend assump+ons for the future

53

Distribu+on of Aggregate Reserves=Sum of all lognormals for each future cell for each LOB

54

HistogramKernelLognormalGamma

Density Comparisons (Acc Year: Total)

1 Unit = $1,000,000,00018 22 26 30 34 38

00.020.040.060.08

0.10.120.140.160.18

0.20.22

Note skewness (even) of aggregate distribution. Mean=23.9B, VaR at 95%=3.6B and T-VaR at 95%=5B

55

Mean Reserve as a % of Total Reserve, and Reserve CV of aggregate versus CV for each LOB. WC has largest CV but is not smallest LOB by Mean Reserve ReA has large CV and is not so large proportion of total business. Reserve distribution correlations essentially zero!

Reserve Mean by LOB as percentage of Total Reserve Mean

PPA 21.26%

WC:PL(I) 4.06%

MMOcc 3.88%

MMCm 3.30%

OLOcc 7.59%

OLCm 2.90%

ReA 15.56%

ReB 35.67%

Other 5.77%

CV (%) of Reserve Distribution by LOB

CV of aggregate reserve = 8.54%HOFO PPA CAL WC:PL(I) CMP MMOcc MMCm SL:PL(I) OLOcc OLCm ReA ReB ReC PLOcc PLCm

05

101520253035404550

CV of aggregate less than most CVs

Risk alloca+on based on a variance/covariance formula

56

Risk Capital Allocation (Totals)

WC:PL(I) 5.97%

ReA 55.86%

ReB 33.81%

Other 4.36%

Risk capital for ReA if this is the only line wri_en is not much less than risk capital for aggregate of all 15 LOBs!

Much credit for risk diversifica>on by wri>ng the

other LOBs that are essen>ally uncorrelated

57

Risk Capital by LOB when Total Risk Capital is T-VaR at 95 %

Total Risk Capital 5,210,028; 1 Unit = $1,000

HOFO PPA CALWC:PL(I) CMP

MMOccMMCm

SL:PL(I)OLOcc

OLCm ReA ReB ReCPLOcc

PLCm0

500,000

1,000,000

1,500,000

2,000,000

2,500,000


Total Risk Capital 4,369,548; 1 Unit = $1,000ReA:PL(I)

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

Only LOB ReA

Berkshire Hathaway Schedule P 2006

58

Risk Capital as a Percentage of Mean by LOB for T-VaR at 95 %


MMOccMMCm

SL:PL(I)OLOcc OLCm ReA ReB ReC

PLOccPLCm AGR

0

10

20

30

40

50

60

70

Mean and Risk Capital as a Percentage of Total by LOB for T-VaR at 95 %

Risk Capital Mean


MMOccMMCm

SL:PL(I)OLOcc


PLCm AGR0102030405060708090100

Combined risk charge < underwri>ng risk charge + reserve risk charge, for any T-‐VaR . Why?

59



HOFOPPA

CAL

WC:PL(I) CMP

MMOcc

MMCm

SL:PL(I)

OLOcc

OLCmReA ReB ReC

PLOccPLCm

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

4,500,000

5,000,000

5,500,000

6,000,000

6,500,000

7,000,000

7,500,000

8,000,000

8,500,000

9,000,000



HOFO PPACAL

WC:PL(I) CMP

MMOcc

MMCm

SL:PL(I)

OLOcc


PLCm

0

500,000

1,000,000

1,500,000

2,000,000

2,500,000

3,000,000

3,500,000

4,000,000

4,500,000

5,000,000

5,500,000

6,000,000

6,500,000

7,000,000

7,500,000

8,000,000



HOFO PPACAL

WC:PL(I) CMP

MMOcc

MMCm

SL:PL(I)

OLOcc


PLCm

0

100,000200,000

300,000400,000

500,000

600,000700,000

800,000900,000

1,000,0001,100,000

1,200,000

1,300,0001,400,000

1,500,0001,600,000

1,700,000

1,800,0001,900,000

2,000,0002,100,000

2,200,0002,300,000

2,400,000

2,500,0002,600,000

2,700,0002,800,000

Combined Underwriting Reserve

Percentile for Value at Risk

Perc

ent

1009590858075

300

250

200

150

100

50

0

Variable

Aggregate

ReA+WC CombReA+WC ReserveReA ReserveWC Reserve

Value at Risk as percentage of Mean ReserveThis graph compares risk capital, calculated as Value at Risk and expressed as a percentage of the Forecast Mean Reserve. Only WC and ReA, two of the high CV lines are shown.

Blue = WC and Green = ReA Red = WC+ReA

Black = WC+ReA including next underwri+ng year.

Orange = Aggregate of all BH lines.

Berkshire Hathaway – Diversifica>on Effects

Comparison of CVs across three companies

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CALCMP

HOFO

MM Cm

MM Occ

OL Cm

OL OccPL Cm

PL Occ PPA

ReAReB

ReC SLWC Intl

Line of Business

Coefficient of V

ariation for F

orecast M

ean Reserve

Comparative size of each LOB by reserve for three companies

0

10

20

30

40

50

60

CALCMP

HOFO

MM Cm

MM Occ

OL Cm

OL Occ

PL Cm

PL Occ PPA

ReAReB

ReC SLWC Intl

Line of Business

Share of Total (%

)

Berkshire Hathaway

Swiss Re

The Hartford

Berkshire Hathaway, Swiss Re and The HarTord, comparison by Line of Business

For OLOcc,Mean Hart> Mean Swiss Re > Mean BH & CV Hart>CV Swiss Re >CV BH

Higher Mean does not necessarily mean lower CV

For PPA , BH Mean>HF Mean, and CV BH> CV HF

WC for BH versus SR reserve distribu+ons by calendar year

62

Distributions of payment streams by calendar year are different These depend on the base development period trends and the assumed forecast scenario calendar year trends.

Risk capital alloca+on (%) by calendar year for WC, BH versus SR

63

Risk Capital Allocation Percentage (BH WC:PL(I) - Cal. Years)

07 08 09 10 11 12 13 14 15 16 17 18 19 20 210

1

2

3

4

5

6

7

8

9

Risk Capital Allocation Percentage (SR WC:PL(I) - Cal. Years)

07 08 09 10 11 12 13 14 15 16 17 18 19 20 210123456789101112

% allocation very different. It is based on a variance/covariance formula that is a function of three factors. 1. The future calendar year trend (mean and standard deviation thereof) assumption, 2. Development period trends and 3. process variance.

• All companies are different and no company is the same as the industry.

• Consistent es+mates of prior year ul+mates on upda+ng can only be maintained within a sound modelling framework that incorporates calendar year parameters and assump+ons about the future are explicit and auditable. The assump+ons can easily be monitored on upda+ng.

• Reserve distribu+on correla+on is usually considerably less than process correla+on.

• Segments of the same LOB such as 1. net of reinsurance and gross, 2. indemnity versus medical, 3. layers, for example, limited to 500K and limited to 1M; have common drivers and are highly correlated.

• Different LOBs very oUen do not have common drivers. That is, the trend structure (especially along calendar years) is not the same and process correla+on is zero. •  A sound measurement of vola+lity and correla+ons (from the data) is essen+al to calculate risk capital alloca+on by LOB and calendar year, irrespec+ve of capital risk measure.

Summary

Summary

•  Combined reserve and underwri+ng risk charge < reserve risk charge + underwri+ng risk

•  Mack and related methods can give answers that are wildly too low or too high and cannot capture the vola+lity in the data.

65

More informa+on will be available at

Lincolnshire Room (6th floor)

7:00pm-‐11:00pm

Monday September 14

(Also learn about the Bootstrap technique for tes+ng validity of a model or method)

66

lincolnshire+room+(6 ﬂoor)+ ·...

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