1998 CAS DFA Seminar
Managing Risk in a Portfolio Context
Reinsurance and Reserves
Gary G. Venter
Sedgwick Re Insurance Strategy, Inc.
(INSTRAT)
Investment and Reinsurance Risk
Goal
Manage investment and reinsurance risk simultaneously
Test strategies by their impact on bottom line income probability distributions
Create risk/return efficient frontiers from strategies tested
Risk/Return Efficient Frontier @ probability = p
Any strategy above or to the
right of the efficient frontier
provides less benefit than points
on or below the efficient
frontier.
Inefficient options can be
quickly recognized.
By establishing an efficient
frontier of options, one can
discover and create new or
hybrid solutions that provide
greater benefit
Measuring Risk
Problems with standard deviation
Inconsistent meaning among distributions
Treats upside and downside risk the same
Look at key individual percentiles
Preference might be to increase mean return, reduce downside risk by giving up
chance of big gain
Risks to Income
Investment performance
Market results
Need to liquidate to pay losses
Underwriting result
Current year
Reserve development
Reinsurance
Modeling Requirements
Model the risk elements
Investment market, current u/w, develop-ment, reinsurance, cash flow, taxes,
GAAP, statutory
Get probability distribution of income for each strategy to find efficient frontier
of strategies at each probability level
Need to generate scenarios by probability, not just wide variety
Modeling Investment Risk
Yield curve
Diffusion model
Other assets
Regression models
Why a Diffusion Model?
Rates are moving continuously
Process generating movement of short-term rates also generates yield curve
Can guarantee arbitrage-free yield curve movements in model
Can calibrate to market - e.g., to bond options
The CIR Model
dr = a(b - r)dt + sr1/2dz.
r is short-term rate
b is reverting mean
a is speed of reversion
s is volatility measure
What is z?
z is standard Brownian motion
Starts at zero
After time t, z is normally distributed with mean 0 and variance t
Notation sdz means that after time t, variance is s2t
Thus for CIR sr1/2dz means that after time t, variance is s2rt
Use this to simulate short intervals t
Yield Curve under CIR
Y(T) = A(T) + rB(T) for term T
A(T) = -2(ab/s2T)lnC(T) – 2aby/s2
B(T) = [1 – C(T)]/yT
C(T) = (1 + xyeT/x – xy)-1
x = [(a - )2 + 2s2]-1/2 y = (a - x)/2
This has a free parameter  called the market price of risk
Yield is a linear function of r for fixed T
Testing - Yield Curve Model Should:
Closely approximate current yield curve
Produce patterns of changes in the short-term rate that match history
Over longer simulations, produce distribu-tions of yield curves that, for any
given short-term rate, match historical conditional distribution of yield curves
Historical Yield Curve Distribution
Measure shape by 1st and 2nd differences
E.g., 1 year rate minus 3 month rate
Look at historical shape measures as a function of r
Yield Spread 3-Month to 1-Year
1-Year to 3-Year Yield Spread
CIR Compared to Historical
Andersen & Lund Model Comparison
Andersen & Lund Comparison
A&L + Variable Market Price of Risk
Long spread matches historical
Modeling Loss Development
Need paid development for cash flow
Need reserve development for liabilities
One strategy: simulate paid diagonal and develop triangle to get new incurred
Simulating whole diagonal - as opposed to just simulating sum - requires model
of development process
Diagonal Development Issues
Is development proportional to emerged?
Is development proportional to ultimate?
Is it inflation sensitive?
What is variance proportional to?
Can you tell from data?
Classifying Development Models
Six yes-no questions
Can test the questions using triangles
Provides 64 classes of development models
May be several models in each class
Other class schemes possible
Six Questions
Development depends on emerged?
Purely multiplicative development?
Independent of diagonal effects?
Stable parameters - e.g. factors?
Normally distributed disturbances?
Constant variance of disturbances?
For each age - not proportional to anything
E.g., All Yes
new incrmnt = f*[prev emerged] + e
e normal in 0 and s2
MLE of f is chain ladder (i.e., link ratio) estimate
Alternatives
Development proportional to ultimate - BF
new incrmnt = f*ultimate + e
Additive not multiplicative - Cape Cod
new incrmnt = a + e
Diagonals inflation sensitive - separation
new incrmnt = f*[prev emerged]*(1+i) + e
More Alternatives
Factors change over time - smoothing
Last 3 diagonals, exponential smoothing . . .
Lognormal disturbances - take logs
Disturbance proportional - weighted MLE
Inverse weights to variance
Proportional to emerged to date: weighted least squares takes ratios of column
means
Still More
Combined models - e.g.:
new incrmnt = a + f*ultimate*(1+i) + e
Reduced parameters - e.g.:
new incrmnt = (1+i)*ultimate/(1+j)k + e
Which model does the data like?
Fit models to the triangle
Measure goodness of fit
Statistical significance of parameters
Sum of squared errors
Penalize for extra parameters
Divide sse by (obs - params)2
Testing a Triangle
0 to 1 is a constant
Lag 1 vs. Lag 0 Losses
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
1000
2000
3000
4000
5000
6000
Adjusted SSE’s for Incrementals
SSE
Model Prms Simulation Formula
157,902
CL
9
qw,d = fdcw,d + e
81,167
BF
18
qw,d = fdhw + e
75,409
CC
9
qw,d = fdh + e
52,360
BF-CC 9
qw,d = fdhw + e
h5 - h9 same, h3=(h2+h4)/2, f1=f2, f6=f7, f8=f9
44,701
BF-CC+ 7
qw,d = fdhwgw+d + e
4 unique age factors, 2 diagonal, 1 acc. yr.
Stability of Factors
2 nd t o 3 r d 5 - t er m m oving aver age
2 .4
2 .2
2
1 .8
1 .6
1 .4
1 .2
1
0 .8
Normal Residuals
Is Disturbance Proportional?
Errors as Function of Independent Var
Check for Linearity of Fit
3000
2000
1000
0
-1000
-2000
-3000
-4000
0
1000
2000
3000
4000
5000
6000
Reserve Model Choice Affects Risk
No Inflation on Reserves
Short
Medium
Mean
3422
3443
1%
3085
2600
10%
3220
3024
90%
3626
3786
99%
3821
4096
Long
Stocks+
3470
3540
2182
1762
2801
2287
4117
4661
4784
6120
With Post-Event Inflation
Short
Medium
Mean
3429
3438
1%
3021
2589
10%
3205
2972
90%
3635
3816
99%
3859
4289
Long
Stocks+
3538
3569
1899
1358
2848
2294
4242
4613
4879
6197
Reinsurance & Investment Scenarios
Higher Retention
Short
Medium
Mean
3422
3443
1%
3085
2600
10%
3220
3024
90%
3626
3786
99%
3821
4096
Long
Stocks+
3470
3540
2182
1762
2801
2287
4117
4661
4784
6120
Lower Retention
Short
Medium
Mean
3227
3255
1%
3271
2884
10%
3351
3156
90%
3500
3749
99%
3618
3951
Long
Stocks+
3345
3473
2197
1773
2865
2564
4202
4909
4630
5642
Conclusions
DFA can be used to jointly manage reinsurance and investment strategy
Requires probabilistic generation of asset and underwriting scenarios
Modeling issues
Getting reserve model right necessary for relevant analysis
Interest rates follow complex processes that require careful attention