1. Underwriting Cycles and Underwriter Sentiment
M. Martin Boyer,
Julia Eisenman,
J-François Outreville
HEC Montréal
Munich Behavioral Insurance Workshop

2. What do you see? It looks like two double-amputees dancing
It looks like the coast of Italy after an atom bomb attack, only mirrored
It looks like a pair of one-legged cannibals fighting over a victim
I love pudding
It looks like an RLFP DNA test result, with the phenotypes split
It looks like Satan's head, the white part in the middle (can't you see it?)
It looks like smudges, or maybe an inkblot

3. Underwriting results - US market

4. Is there a business cycle in insurance?

5. Historical Perspective
Premiums in year t are not independent of past loss experience and Witt (1978, 1981) noticed a cyclical pattern in loss ratio over time which could be due to the fact that insurers would set rates by using regression results derived from past losses.
Brockett and Witt (1982) were the first to show that an autoregressive process arises as a first order Taylor approximation to the loss ratio independently of the number of policy owners and companies

6. Historical debate
Naïve rate-making process: Outreville (1981), Venezian (1985), Berger (1988)
Price competition: Wilson (1981), Stewart (1984), Harrington & Danzon (1994)
Information lags: Cummins & Outreville (1987), Lamm-Tennant &Weiss (1997), Chen et al. (1999)
Interest rates: Doherty & Kang (1988), Smith (1989), Doherty & Garven (1992), Haley (1993)
Capital constraints: Winter (1989), Gron (1990, 1994) Cummins & Danzon (1997)
Shocks (losses, economic environment): Webb (1992), Grace & Hotchkiss (1995), Harrington & Niehaus (2000), Meier (2006)
Reinsurance: Meier & Outreville (2006, 2010)
Our rationale : “Biased behaviour” of underwriters.
This is, to our knowledge, a new approach.

7. Theory of Cycles What needs explaining?
Shocks are triggering mechanisms and are not inherently “causes” of cycles so that a rational expectations market could exist with shocks but no cycles
Boyer et al. (2012) find that any evidence of underwriting cycles in the property and casualty insurance market could simply be spurious because the naive prior is that cycles exist.

8. Theory of Cycles What needs explaining?
Shocks are triggering mechanisms and are not inherently “causes” of cycles so that a rational expectations market could exist with shocks but no cycles
Boyer et al. (2012) find that any evidence of underwriting cycles in the property and casualty insurance market could simply be spurious because the naive prior is that cycles exist.

9. Theory of Cycles What needs explaining?
Shocks are triggering mechanisms and are not inherently “causes” of cycles so that a rational expectations market could exist with shocks but no cycles
Boyer et al. (2012) find that any evidence of underwriting cycles in the property and casualty insurance market could simply be spurious because the naive prior is that cycles exist.
Doing a meta-analysis of papers focused on cycles using different countries / lines / measures of profitability, Boyer et al. (2012) find that cycles exist 60% of the time…

10. Do cycles exist? Meta-analysis of cycles: An example
10 out of 13 in Cummins and Outreville (1987)
10 out of 25 in Chen et al. (1999)
49 out of 80 in Lamn-Tennant and Weiss (1997)
11 out of 16 in Meier (2006)
Et glou et glou et glou…
For a total of 94 instances where an underwriting cycle appears to be present out of 155, or 61% (for an average cycle period of 7 years). What was our Bayesian prior again?
It appears that researchers started the analysis with strong priors in favor of the existence of a cycle.

11. Theory of Cycles
The real mystery of the cycle is why we observe alternating periods of hard and soft markets
Disequilibrium between supply and demand
Capital should quickly flow into the market, alleviating periods when prices seem “too high” and coverage is rationed
Capacity constraints: due to information asymmetries and imperfections, capital does not flow freely into and out of the insurance market
Risky debt: shocks drive insurers away from optimal capital structure, leading to supply shifts until.
External shocks
Interest rates, Imperfect information (delays and lags), Catastrophic losses, Reinsurance, General economic environment
Our rationale: “Biased behaviour” of underwriters.

12. What kind of bias?
The real world is defined as a random walk (+s, -s, ½).
Premium equal to expected future loss; what is that expectation (reality: Et(Lt+1)=Lt)?
Underwriters believe losses follow a reversion process or a momentum process.
Hidden Markov switching process in three matrices.

13. What kind of bias?
Barberis et al. (1998) demonstrates that investors tend to see trends in random walk sequences.
Provide an explanation of stock movements, based on two observed behaviors, over- and under-reaction
Behaviors associated with two phenomena (see also Bloomfield and Hales, 2002):
Conservatism
Representativeness

14. The paper’s contribution
Emission Matrix (R)
Emission Matrix (M)
Transition Matrix
We use underwriting data from the P&C insurance industry in the United States to calibrate the parameters in the transition and the two emission matrices. We thus have 4 parameters (p1, p2, πL, πH) to fit to the loss ratio data.
To calculate these parameters, we used Demster et al. (1997) Estimation-Maximization algorithm; EMMall toolbox in Matlab.

15. Basic Information
Annual loss ratios from 1967 to 2004 (AMBests’ P&C in the U.S): 38 years.
We find an industry average loss ratio for the time series of 0,7726, with a standard deviation of 0,0591. The loss ratios range from 0,6660 to 0,8870.
Basic statistics of the loss ratios
Mean
Standard deviation
Minimum
Maximum
Skewness
Kurtosis

16. Implementation of EM-algorithm
Random Initial state probabilities.
Uniformly distributed Initial transition (T) matrix probabilities, with lines summing to 1.
Uniformly distributed Initial emission (M&R) matrices probabilities, with lines summing to 1 (with appropriate constraints on both the upper and lower bounds).
The convergence threshold is set at 10−8.
The maximum number of iterations for the EM algorithm is set at 10,000.

17. Results from first step: EM algorithm with 10,000 simulations
Transition T
Period t+1
Mear-Reversion
Momentum
Period t
0.8710
0.1290
0.1212
0.8788
Emission R
positive shock
negative shock
0.0487
0.9513
Emission M
1.0000
0.0000

18. Cycle results from second step with 100,000 simulations
Initial belief: Model R
Initial belief: Model M
Initial belief: half&half
Cut data
Uncut data
# Simulations
100,000
# Cycles
53,175
53,377
52,683
52,897
53,554
53,796
Mean of period
6.6886
6.9272
6.5997
6.8727
6.6125
6.8945
Std.Dev.
3.1526
5.8261
3.1990
6.9434
3.1290
6.2634
Minimum
2.3654
2.2476
2.0214
Maximum
Skewness
4.1890
4.2056
4.1403
Kurtosis

19. Results from constrained-M first step: EM algorithm with 10,000 simulations
Transition T
Period t+1
Mear-Reversion
Momentum
Period t
0.8710
0.1290
0.1212
0.8788
Emission R
positive shock
negative shock
0.0512
0.9488
Emission M
0.9500
0.0500

20. Cycle results (constrained-M) from second step with 100,000 simulations
Initial belief: Model R
Initial belief: Model M
Initial belief: half&half
Cut data
Uncut data
# Simulations
100,000
# Cycles
60,574
60,852
60,759
61,016
60,533
60,827
Mean of period
7.1823
7.4802
7.1204
7.4465
7.1155
7.5628
Std.Dev.
3.2772
6.9005
3.2341
3.2485
Minimum
2.8950
3.0109
2.8636
Maximum
Skewness
3.9504
3.9880
4.0472
Kurtosis

21. Cycle results from second step versus reality
Initial belief: half&half
Initial belief: Model R
Reality: Meta-analysis
Free
Constrained
Classics (3)
All (5)
# Simulations
100,000
141
177
% Cycles
53.5%
60.5%
53.2%
60.6%
61%
66%
Mean of period
6.6125
7.1155
6.6886
7.1823
7,2186
7,6848
Std.Dev.
3.1290
3.2485
3.1526
3.2772
2,8711
3,2265
Minimum
2.0214
2.8636
2.3654
2.8950
4,0880
Maximum
21,9740
25,3600
Skewness
4.1403
4.0472
4.1890
3.9504
2,6146
2,8915
Kurtosis
9,5400
11,3456

22. Conclusion
A priori, cycles do not appear to exist (or at least they seem to result from a random walk).
So why do we “see a pattern” if there is no real cyclicality in losses: underwriters have biases.
We present an application to underwriting cycles of hidden Markov matrix approaches
Calibration of hidden Markov model on U.S. P&C insurance market
Simulation of cycles using the fitted parameters:
Probability of a cycle existing between 52% and 61%
Cycle length (std) between 6.5 (3.1) and 7.2 (3.3) years.