Transcript document

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]
Assignment #1
Lineland Life Insurance Company
• 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.
Assignment #1
Lineland Life Insurance Company
Case 1 - Claim occurs at time T
The return is a continuous annuity of I
 R T
1 e
PV with claim  I
R
E PV withclaim 
 R 
t 1 e

0 I
R
z
F
G
H
q
 d
t
I
1  e  R t
q
1
R
t R
I
J
K
Assignment #1
Lineland Life Insurance Company
Case 2 - Claim does not occur
Return = PV[Annuity] + PV of Capital
 R t
1 e
PV without claim  I
R
e
 R t
F
1 e
E PV without claim  a
1  qf G

H 
 R t
I
R
e
 R t
I
J
K
Assignment #1
Lineland Life Insurance Company
• Receives P immediately.
• Receives annuity until claim occurs or
the term ends.
1 = P + E[PV with Claim] + E[PV without Claim]
Assignment #1
Lineland Life Insurance Company
I
R
q
6%
10%
0.100
t
P
P-q
1
2
3
4
5
6
7
0.131
0.160
0.185
0.208
0.229
0.248
0.264
0.031
0.060
0.085
0.108
0.129
0.148
0.164
P increases when capital must be held longer.
Background - Capital Requirements
Define Terms
X = Random Insurer Loss
F( x) = Pr{ X  x}
f ( x) = F  ( x)
af
LEV L 
z
a1 F(L)f
L
x

f
(
x
)
dx

L
0
 = Standard Deviation of X
C = Required Insurer Capital
Background - Capital Requirements
Three 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]
Background - Capital Requirements
Three Formulas
#2 Expected Policyholder Deficit (EPD)
a
f
LEV C  E[ x]
1

E[ X]
 is determined by judgment of insurer
management.
Sensitive to amount of insolvency
Background - Capital Requirements
Three Formulas
#3 Standard Deviation Formula
C  T
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
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
a f
F( x)   ; x / 
a
f
 GammaDist x, , , TRUE
Expected Value
a f
af
 1
EX 
 

Excel
Formula
Gamma Distribution Mathematics
Limited Expected Value (LEV) Function
a f a
af
   1
LEV L 
    1;L /   L  1     1;L / 

f b a
fg
a
f
 xa
1  GammaDist( x, , , TRUE)f
   exp GammLn(  1)  GammaLn()  GammaDist( x,   1, , TRUE)
Variance
E X2
a f
af
2 2

2 1
 
 2 Var X E X 2 E X
a f
2
2 
Excel
Formula
Assignment #2
Lineland Property Insurance Company
Probability of Ruin
• E[X] = 10,000
• F(12,472) = 0.99
Capital = 2,472 @ 1.0% Level
• Cost of capital = 247
Assignment #2
Lineland Property Insurance Company
Expected Policyholder Deficit
• E[X] = 10,000
• LEV[12,091] = 9,990
LEV[12,091]
EPD  1 
 0.0010
E[ X]
Capital = 2,091 @ 0.10% Level
• Cost of Capital = 209
Assignment #2
Lineland Property Insurance Company
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 Environment
e.g. interest rates
How long
Capital is held
Volatility of
Net Worth
Parameter Uncertainty
for 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  1  3b,  2  1,  3  1  3b
k p k
p
k
p
Pr   1  Pr    3  1 / 6 and Pr    2  2 / 3
E[] = 1 and Var[] = b
Assignment # 2´
Capital Requirements with
Parameter Uncertainty
b  0.02
1  100  1  75.51
 2  100   2  100.00
 3  100   2  124.49
a
af
f
a
f a
 , x / 1 2   , x / 2  , x / 3
FU x 


6
3
6
f
Assignment # 2´
Capital Requirements with
Parameter Uncertainty
Probability of Ruin
• E[X] = 10,000
• FU(14,443) = 0.99
Capital = 14,443 @ 1.0% Level
• Cost of capital = 444
Assignment # 2´
Capital Requirements with
Parameter Uncertainty
Probability of Ruin
Capital
Capital
Threshold w/o PU
with PU
1.0%
2,472
4,443
Expected Policyholder Deficit
Capital
Capital
Threshold w/o PU
with PU
0.10%
2,091
4,129
Standard Deviation
Capital
Capital
Threshold w/o PU
with PU
2.33
2,330
4,049
Assignment #3
Lineland Property Insurance Company
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
Assignment #3
Lineland Property Insurance Company
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
Probability of Ruin @ 1.0%
C
b
C-R
C
0.00 2,460.59 2,472.26 11.67
0.02 4,409.12 4,443.25 34.13
Expected Policyholder Deficit @0.1%
C
b
C-R
C
0.00 2,083.58 2,091.11 7.53
0.02 4,100.04 4,129.19 29.15
Standard Deviation @ 2.33
C
b
C-R
C
0.00 2,318.32 2,330.00 11.68
0.02 4,015.75 4,049.11 33.66
With Parameter
Uncertainty
Total Capital
 Double
Marginal Capital
 Triple +
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
Lag 0
Lag 1
Lag 2
Summary Statistics by Settlement Lag
E[Count] Std[Count] E[Severity] Std[Severity]
1,200
244
40,349
160,219
600
123
59,798
194,452
300
63
79,248
221,804
Assignment #4
Flatland Casualty Insurance Company
Outstanding Aggregate Loss Statistics
Lags 0-2
Lags 1-2
Lag 2
Aggregate Loss Statistics for OS Losses
E[Loss]
99th Pct
EPD = 0.1% Std Dev
108,071,943 158,505,938 155,520,667 19,835,337
59,653,299 91,387,990 90,579,282 12,265,291
23,774,319 40,533,916 41,250,295 6,283,149
The aggregate loss model included parameter uncertainty
affecting all claim count distributions simultaneously.
(g =.02 - analogous to b =.02 above.)
Assignment #4
Flatland Casualty Insurance Company
Capital is released over time as losses are paid.
Required Capital for OS Losses
Pr{Ruin}@1.0% [email protected]% Std Dev x 2.33
Lags 0-2
50,433,995
47,448,724
46,216,335
Lags 1-2
31,734,691
30,925,983
28,578,127
Lag 2
16,759,597
17,475,976
14,639,737
Assignment #4
Flatland Casualty Insurance Company
What is the cost of providing the capital?
i = Interest rate on invested capital
r = Rate of return needed to attract capital.
C0 = Capital needed at beginning of year 0.
Re lease t C t 1 (1 i) C t
The cost of capital, R, satisfies:
3 Re lease
t
C0 R 
t
r
t 1 1 
a f
Assignment #4
Given i = 6% and r = 10%
What is the cost of providing the capital?
Required Capital for OS Losses
Pr{Ruin}@1.0% [email protected]% Std Dev x 2.33
Lags 0-2
50,433,995
47,448,724
46,216,335
Lags 1-2
31,734,691
30,925,983
28,578,127
Lag 2
16,759,597
17,475,976
14,639,737
Time t
1
2
3
Expected Return at Time t
21,725,344
19,369,665
20,411,187
16,879,175
15,305,566
15,653,078
17,765,172
18,524,534
15,518,122
Cost of
Capital
3,386,713
3,272,953
3,065,288
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.
Event
1
2
3
4
5
6
7
8
9
10
Index
Value
100.0
89.04
87.56
83.48
83.20
82.15
80.95
80.55
79.19
77.48
Event
Probability
0.00000121
0.00000121
0.00000181
0.00000702
0.00000702
0.00000466
0.00000791
0.00005060
0.00000702
0.00000181
Max Event
Contract
Direct
Reinsurance Event Loss
Probability
Value
Insurer Loss
Recovery
Given Max
0.00000121 1,125,200,000 1,212,550,269 16,000,000
71,350,269
0.00000121 1,021,700,000 1,509,161,589 16,000,000
471,461,589
0.00000181 1,021,700,000 1,303,694,653 16,000,000
265,994,653
0.00000702 939,300,000
761,956,629 16,000,000 (193,343,371)
0.00000702 939,300,000
734,137,782 16,000,000 (221,162,218)
0.00000466 939,300,000
735,660,852 16,000,000 (219,639,148)
0.00000791 939,300,000 1,004,861,128 16,000,000
49,561,128
0.00005060 939,300,000 1,071,076,934 16,000,000
115,776,934
0.00000702 856,900,000
688,269,904 16,000,000 (184,630,096)
0.00000181 856,900,000 1,652,933,116 16,000,000
780,033,116
+ about 9000 more
• Compare variability before and after
• Is the risk reduction worth the cost?
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.
References
Missing 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 #5
Analyze 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 Search
Contract
Range
5-20
25-40
45-55
60-70
75-85
90-100
Number of Index Contracts
Insurer #1 Insurer #2 Insurer #3
47,400
93,100
0
74,400
118,100
6,300
59,500
67,900
0
47,600
28,600
0
81,400
545,100
0
37,200
634,800
0
Reinsurance
Retention 73,000,000 457,000,000 54,000,000
Limit
13,000,000 36,000,000 105,000,000
Financing With Reinsurance
and Catastrophe Options
Expected Net Loss
Cost of Capital
Cost of Reinsurance
Cost of Catastrophe Options
Cost of Financing Insurance
Insurer #1
Insurer #2 Insurer #3
16,315,629 62,086,995 1,464,410
53,470,927 143,662,761 12,914,922
2,088,287
1,848,530 1,726,342
22,252,015 42,409,101
249,427
94,126,858 250,007,387 16,355,100
Financing Without Reinsurance
and Catastrophe Options
Expected Net Loss
Cost of Capital
Cost of Reinsurance
Cost of Catastrophe Options
Cost of Financing Insurance
Insurer #1
Insurer #2 Insurer #3
34,839,348 95,417,229 2,385,629
68,768,384 166,962,499 15,356,683
0
0
0
0
0
0
103,607,732 262,379,728 17,742,312
Differences in Costs
Without Reins & Options
With Reins & Options
Difference
Pct Difference
Insurer #1
Insurer #2 Insurer #3
103,607,732 262,379,728 17,742,312
94,126,858 250,007,387 16,355,100
9,480,874 12,372,341 1,387,212
9.2%
4.7%
7.8%
Assignment #6
Spaceland 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 #6
Spaceland 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 #6
Information 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.
Relationship Between the
Capital Cost Allocation Methods
and the Efficient Frontier Methods
• 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
Relationship Between the
Capital Cost Allocation Methods
and the Efficient Frontier Methods
If two are equivalent, why did I switch?
• Easier to explain
• Easier to extend
– To different measures of risk
– To different capital holding times