A Day’s Work for New Dimensions an International Consulting Firm

A Day’s Work for New Dimensions

an International Consulting Firm

Glenn Meyers

Insurance Services Office, Inc.

CAS/ARIA Financial Risk

Management Seminar

DFA - Dynamic Financial Analysis

• Coined by the CAS in 1994.

• Best defined in terms of the problems it seeks to solve.– How much capital does an insurer need?– For how much time is the capital needed?– What decisions does an insurer make to

provide the greatest return on its capital?• Underwriting • Asset management (Include hedges)

Outline of Talk

• Multi-dimensional aspects of insurer capital management

• Provide simple (perhaps artificial) examples focusing on particular dimensions.– Short and long tailed lines– Catastrophe options and reinsurance

• Describe (but not solve) a multi-dimensional insurer problem in capital management.

• Compare approach with efficient frontier methods.

Assignment #1 Lineland Life Insurance Company

• Writes one life insurance policy

• Face value $1

• t is the term of the policy

• Mortality assumptions– Probability of death in [0,t] = q– Uniform distribution of deaths within [0,t]


• Investors provide $1 of capital.

• Capital is invested at rate I compounded continuously.

• In return for exposing the capital to loss they demand a return of R compounded continuously.

R > I

• Find minimum premium, P, it must get.


PV withclaime

I

T

R

R

1

E PV withclaime q

td

qe

t

IR

t

I

R

t

R

R

R

F

HGIKJ

z

1

11

0

Case 1 - Claim occurs at time T

The return is a continuous annuity of I


Case 2 - Claim does not occur

Return = PV[Annuity] + PV of Capital

PV withoutclaime

eI

t

R

tR

R

1

E PV withoutclaim qe

eI

t

R

tR

R

FHG

IKJ

1

1a f


• Receives P immediately.

• Receives annuity until claim occurs or the term ends.

1 = P + E[PV with Claim] + E[PV without Claim]


q6% 10% 0.100

t P P-q1 0.131 0.0312 0.160 0.0603 0.185 0.0854 0.208 0.1085 0.229 0.1296 0.248 0.1487 0.264 0.164

I R

P increases when capital must be held longer.

Background - Capital RequirementsDefine Terms

X = Random Insurer Loss

=

=

= Standard Deviation of X

C = Required Insurer Capital

F x X x

f x F x

LEV L x f x dx L F LL

( ) Pr{ }

( ) ( )

( ) ( )

zaf a f10

Background - Capital RequirementsThree Formulas

#1 Probabililty of Ruin

F C E X( [ ]) 1

is determined by judgment of insurer management.

Insurer management always knows what the rating agencies - NAIC, Best, S&P think they should have.

Value at Risk -- VaR = C+E[X]


#2 Expected Policyholder Deficit (EPD)

1

LEV C E x

E X

[ ]

[ ]a f

is determined by judgment of insurer management.

Sensitive to amount of insolvency


#3 Standard Deviation Formula

T is determined by judgment of insurer management.

Normal approximation to ruin formula, but you can use this formula as is.

Easiest to work with

C T

Assignment #2 Lineland Property Insurance Company

• Losses have a Gamma(100,100) distribution.

• Claims settle quickly– Time value of money is not an issue.

• Investors expect 10% ROE.

• Find the Cost of Capital.

Gamma Distribution Mathematics

Cumulative Distribution Function

Expected Value

F x x

GammaDist x TRUE

( ) ; /

, , ,

a fa f

E X

1a faf

ExcelFormula

Gamma Distribution Mathematics

Limited Expected Value (LEV) Function

Variance

LEV L L L L

11 1 1

a faf a f a fb g

; / ; /

exp ( ) ( ) ( , , , )

( , , , )

GammLn GammaLn GammaDist x TRUE

x GammaDist x TRUE

1 1

1

a fa f

E X

Var X E X E X

22

2

2 2 2 2

21

a faf a f

ExcelFormula


Probability of Ruin

• E[X] = 10,000

• F(12,472) = 0.99

Capital = 2,472 @ 1.0% Level

• Cost of capital = 247


Expected Policyholder Deficit

• E[X] = 10,000

• LEV[12,091] = 9,990

Capital = 2,091 @ 0.10% Level• Cost of Capital = 209

EPDLEV

E X 1

12 0910 0010

[ , ][ ]

.


Standard Deviation• E[X] = 10,000• Std[X] = 1000• Select T = 2.33

Capital = 2,330 • Cost of Capital = 233

Cost of Capital Depends Upon:

Economic Environmente.g. interest rates

Volatility of Net Worth

How long Capital is held

Parameter Uncertaintyfor Gamma(,)

• Let be a random variable– E[] = 1– Var[] = b

• Select at random

• Conditional distribution given Gamma(,)

Parameter Uncertainty for Gamma(,)

A simple, but nontrivial example

1 2 31 3 1 1 3 b b, ,

Pr Pr / Pr / 1 3 21 6 2 3k p k p k pand

E[] = 1 and Var[] = b

Assignment # 2´Capital Requirements with

Parameter Uncertainty

b

0 02

100 75 51

100 100 00

100 124 49

1 1

2 2

3 2

.

.

.

.

F xx x x

Uaf a f a f a f

, / , / , /1 2 3

6

2

3 6


Parameter Uncertainty

Probability of Ruin

• E[X] = 10,000

• FU(14,443) = 0.99

Capital = 14,443 @ 1.0% Level

• Cost of capital = 444


Parameter UncertaintyProbability of Ruin

ThresholdCapitalw/o PU

Capitalwith PU

1.0% 2,472 4,443

Expected Policyholder Deficit


Capitalwith PU

0.10% 2,091 4,129

Standard Deviation


Capitalwith PU

2.33 2,330 4,049


Considers Renewing a Policy

• The renewal business has a Gamma(100,1) loss distribution.

• Lineland has a Gamma(100,99) loss distribution without the renewal.

Property of the Gamma Distribution

• Lineland has a Gamma(100,100) loss distribution with the renewal.

This Property Assumes Independence


Considers Renewing a Policy

• What is the marginal capital needed for the renewal business?

• Calculate capital needed without the business.

• Calculate capital needed with the business.

• Marginal capital is the difference.

Assignment #3´Find Marginal Capital

Assuming Parameter Uncertainty

• The random variable affects all business (including renewal) simultaneously.

• The renewal’s parameter changes at the same time as the for the remaining business.

• The renewal’s losses are correlated with the rest of the losses.

In case you are interested -- = 0.195

Assignment #3 and #3´Results

Total Capital Double

Marginal Capital Triple +

With Parameter Uncertainty

Probability of Ruin @ 1.0%b C-R C C

0.00 2,460.59 2,472.26 11.670.02 4,409.12 4,443.25 34.13

Expected Policyholder Deficit @0.1%b C-R C C

0.00 2,083.58 2,091.11 7.530.02 4,100.04 4,129.19 29.15

Standard Deviation @ 2.33b C-R C C

0.00 2,318.32 2,330.00 11.680.02 4,015.75 4,049.11 33.66

How do you use the marginal cost of capital?

• Allocate the total cost of capital in proportion to the marginal cost of capital.– No consensus among actuaries yet.

• Add the allocated cost of capital to the expected loss and expense to see if you can make money at the “going market premium.”

• Can be done at individual insured level, or the line of business level.

Assignment #4 Flatland Casualty Insurance Company

• Claim count distribution is negative binomial - by settlement lag.

• Claim severity distribution is mixed exponential - by settlement lag.

Name E[Count] Std[Count] E[Severity] Std[Severity] Lag 0 1,200 244 40,349 160,219Lag 1 600 123 59,798 194,452Lag 2 300 63 79,248 221,804

Summary Statistics by Settlement Lag


Outstanding Aggregate Loss Statistics

E[Loss] 99th Pct EPD = 0.1% Std DevLags 0-2 108,071,943 158,505,938 155,520,667 19,835,337Lags 1-2 59,653,299 91,387,990 90,579,282 12,265,291

Lag 2 23,774,319 40,533,916 41,250,295 6,283,149

Aggregate Loss Statistics for OS Losses

The aggregate loss model included parameter uncertainty affecting all claim count distributions simultaneously.

(g =.02 - analogous to b =.02 above.)


Capital is released over time as losses are paid.

Pr{Ruin}@1.0% [email protected]% Std Dev x 2.33Lags 0-2 50,433,995 47,448,724 46,216,335Lags 1-2 31,734,691 30,925,983 28,578,127

Lag 2 16,759,597 17,475,976 14,639,737

Required Capital for OS Losses


What is the cost of providing the capital?

r = Rate of return needed to attract capital.i = Interest rate on invested capital

C0 = Capital needed at beginning of year 0.

Re ( )lease C i Ct t t 1 1

The cost of capital, R, satisfies:

C Rlease

rt

tt

01

3

1

Re

a f

Assignment #4Given i = 6% and r = 10%

What is the cost of providing the capital?

Pr{Ruin}@1.0% [email protected]% Std Dev x 2.33Lags 0-2 50,433,995 47,448,724 46,216,335Lags 1-2 31,734,691 30,925,983 28,578,127

Lag 2 16,759,597 17,475,976 14,639,737

Time t1 21,725,344 19,369,665 20,411,1872 16,879,175 15,305,566 15,653,0783 17,765,172 18,524,534 15,518,122

Cost of Capital

3,386,713 3,272,953 3,065,288

Required Capital for OS Losses

Expected Return at Time t

Asset Management Reinsurance and Catastrophe Options

• “Value will be determined not by the ability of an [insurance] enterprise to accumulate capital and sit on it.

• Rather it will be determined by a company’s franchise with its customers and its ability to originate risk.

• In this scenario the capital markets become the more efficient warehouse of [insurance] risk.”

Asset Management Reinsurance and Catastrophe Options

• Reduce the cost of financing insurance– Expected insurer costs– Cost of Capital– Cost of Capital Substitutes

• Reinsurance• Contracts on a catastrophe index

• Find the right mix of capital and capital substitutes

Quantifying the Cost of Capital

• We use the “easy” formula

Cost of Capital = K T Where:

= Standard deviation of total loss

T = Factor reflecting risk aversion

K = Rate of return needed to attract capital

Quantifying Basis Risk

Ran RMS cat model through insurers and index.

• Compare variability before and after• Is the risk reduction worth the cost?

Index Event Max Event Contract Direct Reinsurance Event LossEvent Value Probability Probability Value Insurer Loss Recovery Given Max

1 100.0 0.00000121 0.00000121 1,125,200,000 1,212,550,269 16,000,000 71,350,2692 89.04 0.00000121 0.00000121 1,021,700,000 1,509,161,589 16,000,000 471,461,5893 87.56 0.00000181 0.00000181 1,021,700,000 1,303,694,653 16,000,000 265,994,6534 83.48 0.00000702 0.00000702 939,300,000 761,956,629 16,000,000 (193,343,371)5 83.20 0.00000702 0.00000702 939,300,000 734,137,782 16,000,000 (221,162,218)6 82.15 0.00000466 0.00000466 939,300,000 735,660,852 16,000,000 (219,639,148)7 80.95 0.00000791 0.00000791 939,300,000 1,004,861,128 16,000,000 49,561,1288 80.55 0.00005060 0.00005060 939,300,000 1,071,076,934 16,000,000 115,776,9349 79.19 0.00000702 0.00000702 856,900,000 688,269,904 16,000,000 (184,630,096)

10 77.48 0.00000181 0.00000181 856,900,000 1,652,933,116 16,000,000 780,033,116

+ about 9000 more

Minimize Sum of Cost Elements

• Insurer Capital

Cost of Capital = K T (Net Losses)

• Reinsurance

Transaction Cost + Expected Cost

• Cat index contracts

Transaction Cost + Expected Cost

Use cat model results to back out transaction costs.

ReferencesMissing transaction costs are in the first paper.

• “The Cost of Financing Catastrophe Insurance” by Glenn Meyers and John Kollar - 1998 DFA Call Paper Program

• Catastrophe Risk Securitization: Insurer and Investor Perspectives” by Glenn Meyers and John Kollar - 1999 CAS Spring Meeting Call Paper Program

Assignment #5Analyze Three Insurers

• Insurer #1 - A medium national insurer

Highly correlated with the index

• Insurer #2 - A large national insurer

Moderately correlated with the index

• Insurer #3 - A small regional insurer

Slightly correlated with the index

Search for Best Strategy to Minimize Cost of Financing

Insurance

• Search for the combination of index and reinsurance purchases that minimizes total cost of providing insurance.

Questions• How many index contracts at each

strike price?

• What layer of reinsurance?

Results of SearchContractRange Insurer #1 Insurer #2 Insurer #3

5-20 47,400 93,100 025-40 74,400 118,100 6,30045-55 59,500 67,900 060-70 47,600 28,600 075-85 81,400 545,100 0

90-100 37,200 634,800 0

Retention 73,000,000 457,000,000 54,000,000Limit 13,000,000 36,000,000 105,000,000

Number of Index Contracts

Reinsurance

Financing With Reinsurance and Catastrophe Options

Insurer #1 Insurer #2 Insurer #3Expected Net Loss 16,315,629 62,086,995 1,464,410Cost of Capital 53,470,927 143,662,761 12,914,922Cost of Reinsurance 2,088,287 1,848,530 1,726,342Cost of Catastrophe Options 22,252,015 42,409,101 249,427Cost of Financing Insurance 94,126,858 250,007,387 16,355,100

Financing Without Reinsurance and Catastrophe Options

Insurer #1 Insurer #2 Insurer #3Expected Net Loss 34,839,348 95,417,229 2,385,629Cost of Capital 68,768,384 166,962,499 15,356,683Cost of Reinsurance 0 0 0Cost of Catastrophe Options 0 0 0Cost of Financing Insurance 103,607,732 262,379,728 17,742,312

Differences in Costs

Insurer #1 Insurer #2 Insurer #3Without Reins & Options 103,607,732 262,379,728 17,742,312With Reins & Options 94,126,858 250,007,387 16,355,100Difference 9,480,874 12,372,341 1,387,212Pct Difference 9.2% 4.7% 7.8%

Assignment #6Spaceland Property and Casualty

• Short tailed property exposure– Include catastrophe exposure

• Long tailed casualty exposure– Include unsettled claims from prior years

• Capital Management Questions– Catastrophe options/reinsurance?– Casualty reinsurance?

Assignment #6Spaceland Property and Casualty

Underwriting Management Decisions

• Allocate the cost of capital to the lines of insurance - in proportion to the marginal cost of capital.

• Allocate the cost of reinsurance and/or catastrophe options to the lines of insurance - in proportion to the marginal costs.

Assignment #6Information and Technology

Requirements

• An Aggregate Loss Model

• Size of loss distributions by settlement lag

• Correlation structure between lines of insurance

• A catastrophe model

• Exposure underlying catastrophe index

References• “Underwriting Risk” by Glenn Meyers

– 1999 CARe Call Paper Program

• “Estimating Between Line Correlations Generated by Parameter Uncertainty” by Glenn Meyers– 1999 DFA Call Paper Program

• These papers should be eventually available at CAS web site.

• Currently available on my personal web site

http://www.crimcalc.com/glenn.htm

Relationship Between this Capital Cost Allocation Method

and the Efficient Frontier Methods

• They are equivalent – (loosely speaking)

• I say “loosely speaking” because: – There is a lot of loose speaking about the

meaning of “risk.”– There is a lot of loose speaking about the

meaning of “allocated cost of capital.”

Relationship Between the Capital Cost Allocation Methods and the Efficient Frontier Methods

• The intuition

• Allocated cost of capital depends upon marginal risk.

• Making decisions that yield a higher return on marginal capital moves you closer to the efficient frontier.


• Some History from PCAS– Kreps: Risk loads from marginal capital

requirements, 1990– Meyers: Risk loads from efficient frontiers

(mimic CAPM), 1991– Heckman: Kreps and Meyers are

equivalent, 1993 (CAS Forum)– Meyers: Cat risk loads from marginal

capital requirements, 1997


If two are equivalent, why did I switch?

• Easier to explain

• Easier to extend– To different measures of risk– To different capital holding times

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A Day’s Work for New Dimensions an International Consulting Firm

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