Lincolnshire+Room+(6 floor)+ ·...
Transcript of 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
Wtd Std Res vs Cal. Yr
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.
Limita>ons of Mack Method when calendar trends are 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
Limita>ons of Mack Method when calendar trends are present
Wtd Std Res vs Cal. Yr
97 98 99 00 01 02 03 04 05 06
-1
-0.5
0
0.5
1
1.5
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Acc. Yr
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
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
Wtd Std Res vs Acc. Yr
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).
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Acc. Yr
86 88 90 92 94 96 98
-3-2.5-2
-1.5-1
-0.50
0.51
1.5
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Acc. Yr
86 88 90 92 94 96 98
-3-2.5-2
-1.5-1
-0.50
0.51
1.5
Wtd Std Res vs Cal. Yr
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
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
0.5
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
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
0.450 .3 12 7
2 .5 8 2 6
Layers Lim1M, Lim2M and 1Mxs1M Lim2M=Lim1M+1Mxs1M
Very high process correla+ons (LeU to right 1M, 1Mxs1M, 2M)
43
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Wtd Std Res vs Cal. Yr
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
Layers Lim1M, Lim2M and 1Mxs1M Lim2M=Lim1M+1Mxs1M
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
MLE Variance vs Dev. Yr
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
MLE Variance vs Dev. Yr
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
Wtd Std Res vs Acc. Yr
97 98 99 00 01 02 03 04 05 062 2 3 2
-1.5
-1
-0.5
0
0.5
1
1.5
Wtd Std Res vs Acc. Yr
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
Risk Capital by LOB when Total Risk Capital is T-VaR at 95 %
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 %
HOFO PPA CALWC:PL(I) CMP
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
HOFO PPA CALWC:PL(I) CMP
MMOccMMCm
SL:PL(I)OLOcc
OLCm ReA ReB ReCPLOcc
PLCm AGR0102030405060708090100
Combined risk charge < underwri>ng risk charge + reserve risk charge, for any T-‐VaR . Why?
59
Risk Capital by LOB when Total Risk Capital is T-VaR at 95 %
Total Risk Capital 11,436,281; 1 Unit = $1,000
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
Risk Capital by LOB when Total Risk Capital is T-VaR at 95 %
Total Risk Capital 8,587,123; 1 Unit = $1,000
HOFO PPACAL
WC:PL(I) CMP
MMOcc
MMCm
SL:PL(I)
OLOcc
OLCm ReA ReB ReCPLOcc
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
Risk Capital by LOB when Total Risk Capital is T-VaR at 95 %
Total Risk Capital 5,080,962; 1 Unit = $1,000
HOFO PPACAL
WC:PL(I) CMP
MMOcc
MMCm
SL:PL(I)
OLOcc
OLCm ReA ReB ReCPLOcc
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