Why Pooling Works

CAJPA Spring 2017

Mujtaba DatooActuarial Practice Leader, Public EntitiesAon Global Risk Consulting

1

Discussion Points

Mathematical preliminaries Why insurance works Pooling examples Loss distributions Simulation Extreme events: low frequency, high severity To pool or not to pool…

Mathematical and StatisticalPreliminaries

3

Trivia…Statistics Defined

Latin origin: statisticum collegium meaning “council of state”

Italian connection: statista meaning “statesman”

German context: statistik meant analysis of data about the state

English term: “political arithmetic”

After 19th Century, “statistics” was introduced to refer to any collection and classification of data

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Trivia…Mathematics Defined

Pythagorean origin “That which is learned”

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Definitions

Risk: Deviation or variability around an expected outcome– Uncertainty in timing and amount of payment

Liquidity: Ability to pay in cash short term liabilities, usually in one year Solvency: Ability to pay all liabilities in the long run

6

Law of Large Numbers

Sample mean converges to population mean as sample size increases Variance of means

–

– As n gets larger, variance is minimized– Coefficient of Variation (CV)

• Ratio of the standard deviation to the mean• Normalizes the scale• Compare entities (not distorted by volume or size)

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Law of Large NumbersCV decreases with increasing sample size

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Central Limit Theorem

For data from any distribution, if we repeatedly take n independent random samples– As “n” becomes large

• The distribution of the sample means will approach a normal distribution.

9

Common Measures of Risk

Measure dispersion:– Variance, standard deviation, average absolute deviation, Value at

Risk (VaR), Tail Value at Risk (TVaR), Coefficient of Variation (CV) Other measures:

– Mean (aka average, expected), median, mode

10

Key Measure: Coefficient of Variation

Coefficient of Variation (CV) =

The smaller the CV, the smaller the variability around the average

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Random Number Generation

Used for simulation To “mimic” a distribution where standard functions cannot be

“‘accurately” fit Can generate random numbers in Excel

– RAND() – generates numbers between 0 and 1– RANDBEWTEEN(low, high) – generates whole numbers between

selected low and high number

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Generating 1,000 Random Numbers Between 0 and 11 0.6152 0.7903 0.8794 0.3315 0.1636 0.9527 0.0678 0.0609 0.934

10 0.12511 0.64412 0.99213 0.66614 0.71215 0.70516 0.93417 0.232 Bin Freq Cum % Inc %18 0.684 0.0 0.0 0 0.0%19 0.730 0.1 0.1 99 9.9% 9.9%20 0.544 0.2 0.2 107 20.6% 10.7%21 0.664 0.3 0.3 93 29.9% 9.3%22 0.080 0.4 0.4 81 38.0% 8.1%23 0.901 0.5 0.5 112 49.2% 11.2%24 0.176 0.6 0.6 98 59.0% 9.8%25 0.484 0.7 0.7 107 69.7% 10.7%26 0.613 0.8 0.8 96 79.3% 9.6%27 0.271 0.9 0.9 104 89.7% 10.4%28 0.404 1.0 1.0 103 100.0% 10.3%29 0.00530 0.29831 0.59632 0.12233 0.29334 0.88235 0.21636 0.91437 0.11938 0.28739 0.19940 0.28641 0.30142 0.032

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Histogram

Freq Cum %

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Law of Large NumbersCoin toss example

Coin toss Probability (heads) = 0.5 As more coins tossed,

Pr(H) 0.5 (p=1/2) e.g. random 10 tosses

produces Pr(H) = 0.7,however as number of tosses increase Pr(H) will converge to 0.5

Trial#

Result H = 1T = o

Sum of Heads

Pr(H)(4)/(1)

(1) (2) (3) (4) (5)

1 H 1 1 1.00

2 T 0 1 0.50

3 T 0 1 0.33

4 H 1 2 0.50

5 H 1 3 0.60

6 H 1 4 0.67

7 H 1 5 0.71

8 T 0 5 0.63

9 H 1 6 0.67

10 H 1 7 0.70

….

1000 H 1 506 0.51

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Simulated Coin Toss

Trial # Rand() ResultH=1, T=0

Sum of H Pr(H)

1 0.68 T 0 0 0%2 0.34 H 1 1 50%3 0.95 T 0 1 33%4 0.58 T 0 1 25%5 0.66 T 0 1 20%6 0.18 H 1 2 33%7 0.95 T 0 2 29%8 0.96 T 0 2 25%9 0.59 T 0 2 22%

10 0.68 T 0 2 20%11 0.56 T 0 2 18%12 0.07 H 1 3 25%13 0.82 T 0 3 23%14 0.24 H 1 4 29%15 0.39 H 1 5 33%16 0.86 T 0 5 31%17 0.90 T 0 5 29%18 0.62 T 0 5 28%19 0.15 H 1 6 32%20 0.39 H 1 7 35%

0%10%20%30%40%50%60%70%80%90%

100%

1 101 201 301 401 501 601 701 801 901Trial #

Pr(H)

Why Insurance Works

16

Why Insurance Works

Example from Society of Actuary paper “Risk and Insurance” The example is about the loss amount of a car owner for a certain year

Loss Amount ProbabilityNumber of Policyholders

1 10 1,000 1,000,000

$0 80% $0 $0 $0500 10% 50 500 50,000

5,000 8% 400 4,000 400,00015,000 2% 300 3,000 300,000

Average Loss per Policyholder

$750 $750 $750

Average Total Losses (μ)

750 7,500 750,000

Variance (σ ) ≈ 6 million ≈ 60 million ≈ 6 billionStandard

Deviation(σ)2,442 7,722 77,217

Coefficient of Variation (CV)

3.26 1.03 0.10 ≈ 0

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Why Insurance Works

People seek security, insurance spreads the risk and can provide the security

To measure the potential variability of the loss (Risk), we use the standard deviation.

The existence of the insurance industry does not decrease the frequency or severity of loss.

Policyholders are willing to pay a gross premium for an insurance contract, which exceeds the expected value of their loss, in order to substitute the fixed low variance premium payment for an unmanageable amount of risk inherent in not insuring

Pooling Example

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Scatter Diagram – Below $100,000

Single Entity Pool

$0

$25,000

$50,000

$75,000

$100,000

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Scatter Diagram – Above $100,000

Single Entity Pool

$100,000

$200,000

$300,000

$400,000

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$600,000

$700,000

$800,000

$900,000

$1,000,000

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$100,000

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Scatter Diagram – Combined

Below $100,000 Above $100,000

$0

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$100,000

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Single Entity Pool

$100,000

$200,000

$300,000

$400,000

$500,000

$600,000

$700,000

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$900,000

$1,000,000

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Single Entity Pool

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Loss Rates: Pool vs Single Entities

$0.00

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$8.0020

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2016

PoolABCDEFGH

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Risk MeasuresPool has lower CV than individual entities

Std. Dev. Average CV

POOL 0.3 $2.03 0.17

A 0.3 $0.97 0.35

B 0.5 $2.47 0.21

C 0.9 $1.50 0.58

D 1.3 $1.62 0.80

E 1.6 $2.19 0.71

F 2.1 $3.72 0.57

G 2.6 $1.73 1.51

H 0.8 $2.11 0.40

0.17

0.35

0.21

0.58

0.80 0.71

0.57

1.51

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Poo

l A B C D E F G H

CV

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Funding: Pool vs Single Entity

Single Entity is relatively volatile, therefore– Need more capital to absorb greater variability– Potential overfunding or underfunding in any given year is more

likely

Building a Workers CompensationProbability Distribution

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WC Empirical Distribution

payroll $100,000,000 frqunce per $1M 1expected number of claims 100

avg annual salary $50,000 number of workers 2,000 Prob of claim per worker 5%prob of NO claims = 95%

Med only % of claims 70%TD % of claims 25%PD% of claims 5%

severity average range - can fit to a loss ditributionMed Only $1,000 50 to 1,500 TD $20,000 10,000 to 30,000 PD $200,000 100,000 to 300,000

Probabilties random number rangeNO claim 95.00% 0.9500 0.0001 to 0.9500Med only 3.50% 0.9850 0.9500 to 0.9850TD 1.25% 0.9975 0.9850 to 0.9975PD 0.25% 1.0000 0.9975 to 1.0000

Loss Distribution Curve Fitting

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Fit a Curve

Empirical histogram, (from scatter diagram) Fit data to “standard curves”

– Estimate parameters from data– e.g. Normal (bell-shaped symmetrical curve)– e.g. skewed distributions: Weibull, lognormal, Burr, Gamma, Pareto

Simulation– Ensure it “mirrors” the (empirical) data

Allows interpolating or extrapolating beyond observed experience– e.g. if no large claims emerged, but can extrapolate from the fitted

curve

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Distribution of Incurred Losses

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Claim Data Adjustments

Check for reasonability Detect outliers Develop – bring claims to ultimate value Trend – bring to a common date, like CPI

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Data Morphing

Historical data may not reflect future changes– e.g. medical technological improvement such as nano-surgery

• Impacts future medical costs– Tort threshold changes, e.g. North Dakota– Judicial impact – e.g. Florida Castellanos, et al.

Increase in life expectancy, e.g. impact on PD claims Economic disturbances, structural transformation (technology impact)

Make judgmental adjustments

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Stratified Losses – HistogramUsed to fit a curve

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Fit the “Best” Curve: Lognormal, Weibull, Gamma

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‒ Lognormal‒ Weibull‒ Gamma

Excess Pool Loss Distribution Example

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Mill

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Distribution of Losses (Developed, Trended)Losses> $25,000 – about 1,300 over 10 years

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Methodology, Stratify ClaimsAbove $100,000

Project losses above $100,000:– Use last 10 years of data– Sparse data– Volatile, less predictable– Use modeling:

• Pareto distribution

37

Methodology: Claims Above $100,000Frequency x Severity

Frequency– Number of claims per

exposure unit– Adjust historical data to a

common year, say 2016/17– Select from 10-year history

Severity– Average cost per claim– Adjust historical data to a

common year, say 2016/17– Fit a Pareto statistical

distribution to the underlying data

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10-year history of claimsDeveloped and trended

$0

$1

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$3

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Over 10 Years, 600 claims over $100K250 claims over $250K100 claims over $500K40 claims over $1M15 claims over $2M6 claims over $3M5 claims over $4M3 claims over $5M

39

Pareto Statistical Distribution

Used in actuarial literature to estimate large claims Calculate within each layer, say $1,000,000 to $2,000,000:

– % of claims– Average cost per claim

Has “fat tail” – allows for very large claims to be considered

40

Stratify Claims Over $100,000Fit to a “theoretical” curve

ClaimsAbove

Threshold

Numberof

Claims

Ratio to $100,000Claims

ParetoFit to

$100,000$100 K 600 100% 100%

$250 K 250 42% 38%

$500 K 100 17% 19%

$1 M 40 7% 9%

$2 M 15 3% 4%

$3 M 6 1% 3%

$4 M 5 0.8% 2.1%

$5 M 3 0.5% 1.7%

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FrequencyClaims Above $100,000

Calculate number of claims per exposure unit

Review for reasonability Select frequency, say for 2016/17

= 0.0375 claims for every $1 million of payroll

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Projected 2016/17 Claims Over $100,000From Pareto Fit

Layer Payroll Frequency

% ofClaimsin Layer

ProjectedNumber of

Claims

$1 - $2 M $1.5 B 0.0375 9% 5.0

$2 - $3 M $1.5 B 0.0375 4% 2.5

$3 - $4 M $1.5 B 0.0375 3% 1.6

$4 - $5 M $1.5 B 0.0375 2% 1.2

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Projected 2016/17 Severity Over $100,000From Pareto Fit

Layer StraightAverage

ParetoFit

$1 - $2 M $1,500,000 $1,683,067

$2 - $3 M 2,500,000 2,804,004

$3 - $4 M 3,500,000 3,857,807

$4 - $5 M 4,500,000 4,888,368

• Pareto fit reflects skewed nature of large claims

• Of the 600 claims above $100,000, about 9% will exceed $1 million

44

Projected Losses for 2016/17From Pareto Fit

LayerProjectedNumber of

ClaimsSeverity Projected

Losses

$1 - $2 M 5.0 $683,067 $3.4 M

$2 - $3 M 2.5 804,004 2.0 M

$3 - $4 M 1.6 857,807 1.4 M

$4 - $5 M 1.2 888,368 1.1 M

45

Size of Loss Distribution

Layer

Total Reported Claims

Percent of Total

Cumulative Percent of

Total

Total Trended Developed

LossesPercent of Total

Cumulative Percent of

Total$25,000 to $500,000 1,159 89.8% 89.8% $139,694,990 49.1% 49.1%

$500,000 to $1M 91 7.1% 96.9% 62,628,246 22.0% 71.1%

$1M to 2M 27 2.1% 99.0% 33,370,164 11.7% 82.9%

$2M to 3M 7 0.5% 99.5% 16,252,543 5.7% 88.6%

$3M to 4M 1 0.1% 99.6% 3,639,981 1.3% 89.9%

$4M to 5M 2 0.2% 99.8% 9,029,413 3.2% 93.0%

$5M to 10M 3 0.2% 100.0% 19,795,046 7.0% 100.0%

Over $10M 0 0.0% 100.0% 0 0.0% 100.0%

Total 1,290 100% $284,410,384 100%

Simulation Demonstration

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Why Simulate?

Simulated results generate a distribution of aggregate losses (frequency times severity)

From this aggregate distribution, can derive:– Average (expected value)– Percentiles

• Rank results in increasing order• Divide (or mark off) various percentiles, e.g. 70th, 90th

• Calculate VaR, TVaR Adjust for various limits (SIR), aggregate losses, deductibles, etc.

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Monte Carlo Simulation

For each claim, select sizeSelect number of claims

Use claim stratification information to simulate claims process

Number of claims (frequency)

Average claim size (severity)

$1,000 $2,000 $3,000

$4,000 $5,000

$2,000

$6,000

$5,000

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Monte Carlo SimulationNumber of Claims

0%

5%

10%

15%

20%

1 2 3 4 5 6

% o

f Obs

erva

tions

Number of Claims

Uniform Distribution,i.e. equally likely to occur

Number of Claims

Poisson Distribution,Mean = 100

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Monte Carlo SimulationClaim Size

0%

5%

10%

15%

20%

25%

30%

35%

1,000 2,000 3,000 4,000 5,000 6,000

% o

f Cla

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Claim Size

Lego Distribution

100 500 2,500 10,000 50,000 100,000250,000Claim Size

Lognormal Distribution

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Monte Carlo SimulationExample – Simulated Claims

TrialNumber

Numberof Claims

Claim Amount for Claim #Aggregate

Losses1 2 3 4 5 61

2

3

4

5

…

9,999

10,000

Average

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Monte Carlo SimulationExample – Simulated Claims

TrialNumber

Numberof Claims

Claim Amount for Claim #Aggregate

Losses1 2 3 4 5 61 2 $2,000 $4,000 $6,000

2 4 5,000 3,000 4,000 3,000 15,000

3 2 4,000 3,000 7,000

4 5 1,000 6,000 4,000 1,000 3,000 15,000

5 4 6,000 4,000 4,000 2,000 16,000

…

9,999 3 4,000 4,000 4,000 12,000

10,000 5 2,000 3,000 3,000 4,000 3,000 15,000

Average 3.5 $12,000

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Monte Carlo SimulationClaim Size Comparison

0%

5%

10%

15%

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30%

35%

1,000 2,000 3,000 4,000 5,000 6,000

% o

f Cla

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Claim Size

Lego Distribution

1,000 2,000 3,000 4,000 5,000 6,000Claim Size

Simulation Distribution

54

Monte Carlo SimulationExample – Ranked Simulated Claims

Trial Number

Aggregate Losses

1 $5,348,825

2 4,884,701

3 5,925,967

4 3,551,488

5 4,166,714

6 4,647,452

7 2,522,083

…

9,999 3,825,756

10,000 5,353,735

Average $5,571,671

RankAggregate

Losses

1 $2,282,852

2 2,393,690

3 2,522,083

…

5,000 5,310,869

…

7,000 6,117,601

…

9,000 7,615,029

10,000 27,533,442

Average $5,571,671

PercentileConfidence Level Factor

70% 1.10

90% 1.37

$6.1M / $5.6M =

$7.6M / $5.6M =

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Monte Carlo SimulationExample – Confidence Level

0%

10%

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3.3 3.5 3.8 4.1 4.5 4.8 5.1 5.6 5.8 6.1 6.7 7.6 8.5 11.4

% o

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Aggregate Losses ($Million)

70% confidence

level

Expected level

99.5%1 in 200 year

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Model RiskOne in a 100-year event

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1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

169,396 games played215 no-hitters19 perfect games

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Model RiskParameters shift with time

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1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

Frequency first 80 years 0.00012%

5 timesFrequency next 20 years 0.00066%

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Risk Margin

An amount that recognizes uncertainty VaR (say at 90%) minus E(X) = risk margin Other measures to set risk margin

– Financial ratios– Risk-based capital– Solvency II, ORSA

Several considerations in setting risk margin– Solvency– Catastrophic events– Ratings– Etc.

Extreme EventsLow Frequency, High Severity

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Extreme Events

The black swan event Cannot quantify readily extreme events in tails – need for complex

statistical approaches (copulas) and simulations– Be wary of its limitations– Isolated or correlated extreme events is where “upper” bound of

surplus can be targeted Understand well and use judgment to evaluate

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Extreme EventPictorially

Assets = Reserves + Surplus

Median Expected Claim Costs50th Percentile

Expected Policyholder Deficit of 1% to 2%

40%30%20%10%

Black swan lives here!

50% 90%

Pro

babi

lity

of

Cla

im C

osts

RBC

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VaR and TVaR

Value-at-Risk (VaR)– Threshold value that losses to a certain confidence level, say 95%

of cases would not be exceeded– Solvency II calibrates 99.5% over a one-year horizon

Tail Value-at-Risk (TVaR)– Takes the average of all the values in the tail above VaR threshold

for a specific time period– Average loss amount of “extreme” events– Actual experience may not exist– Mostly estimated by simulation

63

VaR and TVaRSimulation Example

Trial Number

Aggregate Losses

1 $5,348,825

2 4,884,701

3 5,925,967

4 3,551,488

5 4,166,714

6 4,647,452

7 2,522,083

…

9,999 3,825,756

10,000 5,353,735

Average $5,571,671

RankAggregate

Losses

1 $2,282,852

2 2,393,690

3 2,522,083

…

5,000 5,310,869

…

9,500 8,464,392

9,501 8,473,869

…

10,000 27,533,442

Average $5,571,671

VaR at 95% = $8,464,392

Average = $9,794,388= TVaR at 95%

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Asset Liability Matching

Duration of investments– Investment policy– Restrictions, conservative (usually bonds)

Payout pattern Large claim

– WC– Liability

Liquidate bond (asset)– Pricing loss (risk)

65

The Formula That Killed Wall Street

By Felix Salmom WIRED MAGAZINE 02.23.2009

To Pool or Not to Pool...That is the Question

67

Minimizing TCOR

Expected losses + expenses (overhead, reins, claims admin) + risk margin

Most (public entities) seek stability for budgeting – Stability is key– Smooth out large random, volatile claims– If not stable, need source of funding

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Types of Risk

Process risk:– Associated with projection of future contingencies that are

inherently variable Parameter risk:

– Associated with selection of parameters of the model (e.g., selecting inapplicable LDF)

Model risk: – Misidentifying a process model (e.g., Poisson for frequency)

Surplus provides protection against variation

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Risk CategoriesPublic entities: mostly underwriting risk

Investment risk Credit risk

– Reinsurance recoverable– Other credit risks

Underwriting risk– Premium (pricing) risk– Loss reserve risk

Operating risk Catastrophe risk

– Floods– Earthquake

premium written 32%

credit 10%

loss reserves

27%

investm't31%

70

Balance Sheet

02468

101214161820

Assets Liabilities

Bonds

Stocks

Cash

Receivables

RealEstate

Case Reserves

IBNR

ULAEOther

Surplus

$Mill

ions

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Variability

Pooling in itself does NOT reduce frequency or severity Reduces variability

– Yields stability Higher SIRs imply

– greater volatility– more liquidity needed– More surplus needed

72

Pooling: Long Term Stability

Pooling – Produces more stable long term averages

Individual entity– Volatile– Not conducive to budgeting

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Pooling

Advantages

Stability of rates Economies of scale

– Services– Loss control– Litigation management

Leveraging expertise Purchasing power Equity belongs to members Adds homogeneous risks to pooling,

increases volume (credibility) of data

Concerns (from individual entity’s perspective) Assessable Joint & Several liability “Sharing” – allocation equity Diversification

– Geographical– Homogeneity of risks

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Ken Hearnsberger, thanks

Master practitioner Suggested the subject matter

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Questions?

Mujtaba Datoo, ACAS, MAAA, FCAActuarial Practice LeaderAon Global Risk Consulting(949) 608-6332mujtaba.datoo@aon.com

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Thank You!

Pyx Chamber,

Westminster Abbey

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The name “Pyx” refers to small boxes, containing the official samples of gold and silver coinage which were also kept here. New coins were annually tested against these samples in a public “Trial of the Pyx,” held in the Palace of Westminster.

1859 engraving, showing the Pyx Chamber still containing cupboards for state documents.

Thank You!