# A closer look at H.B. 3622

## Analysis

I have undertaken an analysis of H.B. 3622 that is going to be discussed in the House Insurance Committee at a hearing at 2 p.m. this Tuesday, April 30, 2013.  The one sentence summary is that, although it has some good features, H.B. 3622 is an economic disaster for the Texas coast and the rest of Texas because it does not create a high enough stack to protect against tropical cyclones. The probability of TWIA going bankrupt, even if it does not grow, over the next 20 years under this bill is about 22%. Here are the bullet points.

### Baseline scenario

I conducted 1000 simulations of H.B. 3622 over its plausible shelf life of 20 years using models based on data provided to TWIA by AIR and RMS. TWIA policyholders end up paying via operating funds, reinsurance premiums and contributions to the catastrophe reserve fund for about 66% of the amount of TWIA losses.  There is thus about 34% subsidization in H.B. 3622.  The remaining losses are paid for approximately as follows: 9% by coastal insureds for paying off 70% of the Class 2 bonds, 12% by insurers (and, derivatively, their insureds) by low attachment Class1 Funding assessments, paying off 30% of Class 2 bonds, and high attachment Class 3 Funding assessments, 3% by the State of Texas via premium tax credits given to insurers that partly offset assessments, and, a disturbing 11% absorbed without insurance by TWIA policyholders when TWIA lacks funds with which to pay claims due to an inadequate stack. The pie chart below illustrates this distribution. For some caveats on this computation, see the note below.

In 222 of those 1000 simulations, (22.2% of the time) TWIA became insolvent at some point during those 20 years. At first, I thought this had to be a mistake in my simulation. But, I did a back of the envelope computation that suggests it is an accurate result.  This high risk exists because, particularly over the next 5 years or so, the stack protecting TWIA policyholders is very low relative to potential losses.  Some depopulation of TWIA via, for example, lowering maximum policy limits or reducing moral hazard through higher deductibles and coinsurance would reduce this probability. My “envelope” containing the computation is set forth in the notes below.

### Low reinsurance scenario

Reductions in the purchase of reinsurance produce yet worse results.  The 20-year risk of insolvency is now 29%. And TWIA policyholders pay for even less of the risk they create.  The pie chart below shows the distribution.

Additional premiums paid in by TWIA policyholders could lower the risk of insolvency and increase their responsibility for losses. By increasing premiums 25%, the probability of insolvency is reduced to 21%, still far too high a number. The pie chart below shows, however, that TWIA policyholders now pay a larger proportion of losses suffered.

### Higher Maximum CRTF Payment Scenario

The rate of subsidization and the risk of insolvency would decrease significantly, if H.B. 3622 liberated the CRTF to do its job.  H.B. 3622 would be improved if the $1 billion ceiling in its section 6 (amending section 2210.072) placed on CRTF payments were replaced with$3 billion, as the maximum amount of CRTF funds that could be used to pay for losses. A conforming amendment should also be made to proposed section 2210. 4522. Such an amendment, although it would do little for the next 5 to 8 years, at least reduces the risk of insolvency in years down the road provided no major hurricane has previously hit the Texas coast.  Insolvency risk over the 20 year period would decline to 18% — still way too high but smaller.  And the distribution pie chart shows that now 73% of the losses are born through insurance by TWIA policyholders, though 9% is still unfunded.

The failure to index the parameters to H.B. 3622, such as the maximum amount of the catastrophe reserve fund that can be used to pay a claim or the maximum assessments against insurers means that the insolvency risk grows if, as coastal interests desire, the value of property on coast continues to grow.

## Conclusion

This bill, if were to be passed by a 2/3 majority, at least makes a dent in urgent crisis facing Texas for the 2013 hurricane season. It gets rid of the “bug” in current law that I have discussed in this blog recently. And it does away with the worst of post-event bonding as a funding mechanism. The bill, however, still suffers from several fundamental problems that threaten to destroy the Texas coast.  Unlike S.B. 18 that woud somewhat deconcentrate TWIA risk, it continues the concentration of correlated risk in a single entity. This placing of a lot of eggs in the single TWIA basket inevitably leads to extraordinarily high prices for reinsurance, which in turn prevents TWIA from building up adequate internal reserves in timely fashion. By insulating coastal Texas from market forces, the bill distorts development patterns and discourages risk mitigation. It perpetuates the economically unjustifiable large-scale subsidization from the poor in non-coastal Texas to the middle class and wealthy in coastal Texas. It continues to do so in an opaque manner by complexities such as insurer assessments and premium tax credits.  And it leaves the Texas coast and, derivatively, the rest of Texas extremely vulnerable over the reasonable lifespan of this bill to a devastating insolvency — a threat which itself is likely to retard coastal development.

## Assumptions and Qualifications

I assume the AIR and RMS models are reasonable.  There is some evidence to suggest that the reinsurance industry believes these models are optimistic about the risk of severe tropical cyclones in Texas.  If that is true, the insolvency problem highlighted here becomes yet more serious.

My original analysis contained some errors; I attempt to fix them here. Most relate to my prior lack of complete recognition that the bill does away with Class 1 post-event bonds, the alternative Class 2 post-event bonds, and with Class 3 post-event bonds and substitutes assessment mechanisms for them.

I assume that insurers pay for about 20% of that portion of assessments for which a premium tax credit is available.  This percentage is a crude estimate of the time value of money.

I assume that insurers incur no costs in having to stockpile money to pay assessments.  This is an assumption made for purposes of simplicity and is obviously false.  Taking risk costs into account would mean that insurers bear even more of the costs of a system such as H.B. 3622.

I use a model of reinsurance pricing consistent with that in the literature under which reinsurance prices are based on the sum of the expected claims costs and a fraction of the maximum exposure. I have attempted to calibrate the model, particularly with respect to the fraction used to multiply maximum exposure, by looking at the amount TWIA has paid for reinsurance in recent years.  I continue my concern that TWIA is paying too much for reinsurance and substitute mechanisms for catastrophic risk transfer ought to be explored.

A copy of the Mathematica notebook underlying the assertions in this blog post is available here. I have not had the time to annotate it fully, but am happy to explain it and run different simulations should any legislator desire.

## Three back of the envelope computations confirming a high probability that H.B. 3622 will leave TWIA insolvent over the next 20 years.

### Method 1

If you have a stack like this one for 2013 that is likely to be at best only $2.98 billion high ($180 million CRTF, $800 million Class 1 Funding and$1 billion Class 2 Bonds plus an optimistic $1 billion in low attaching reinsurance) and you have roughly a 1.9% probability of a tropical cyclone losses that exceeds that sum, over 20 years, the cumulative probability of having at least one loss in excess of the stack is 31%. (The survival function at 0 of a negative binomial distribution with 20 trials and a negative probability of 98.1% per trial). It’s only because the stack can grow by perhaps$100 million per year on average (due to increases in the CRTF) and the fact that there the probability in the simulation drops to a still frightening 21.3%.

### Method 2

I also performed a second simplified analysis in which one computed the height of the stack as a function of time under the optimistic assumption that TWIA suffered no major losses.  The height of the stack was set to increase as contributions to the CRTF increased.  I then computed the numeric probabilities for solvency each year.  I then multiplied these probabilities together.  By subtracting these values from 1, one obtains the probability at the end of each 20 year period that TWIA has become insolvent. I again see results between 15-25% depending on what assumptions are made.  These results are consistent with the findings made using the more elaborate methodology.

### Method 3

I generated 10,000 storms from the AIR/RMS derived distribution.  I then partitioned these storms into groups of 20 and found the largest storm.  I then plotted the “Exceedance Curve” or “Survival Function” of this empirical order distribution.  I show the results below.  As one can see the probability of the largest storm being more than $3 billion is about 20%. Even at$5 billion, the probability is above 15%.

Exceedance Curve for Largest Storm in 20 years

# It’s (close to) a Weibull — again!

You recall that in my last post, I went through an involved process of showing how one could generate storm losses for individuals over years.  That process, which underlies a project to examine the effect of legal change on the sustainability of a catastrophe insurer, involved the copulas of beta distributions and a parameter mixture distribution in which the underlying distribution was also a beta distribution. It was not for the faint of heart.

One purpose of this effort was to generate a histogram that looks like the one below that shows the distribution of scaled claim sizes for non-negligible claims. This histogram was obtained by taking one draw from the copula distribution for each of the $y$ years in the simulation and using it to constrain the distribution of losses suffered by each of the $n$ policyholders in each of those $y$ years.  Thus, although the underlying process created an $y \times n$ matrix, the histogram below is for a single “flattened” $y \times n$ vector of values.

Histogram of individual scaled non-negligible claim sizes

But, if we stare at that histogram for a while, we recognize the possibility that it might be approximated by a simple statistical distribution.  If that were the case, we could simply use the simple statistical distribution rather than the elaborate process for generating individual storm loss distributions. In other words, there might be a computational shortcut that could approximate the elaborate proces.  If that were the case, to get the experience of all $n$ policyholders — including those who did not have a claim at all — we could just upsample random variates drawn from our hypothesized simple distribution and add zeros; alternatively, we could create a mixture distribution in which most of the time one drew from a distribution that was always zero and, when there was a positive claim, one drew from this hypothesized simple distribution.

# It’s a Weibull

To understand the premiums charged by the Texas Windstorm Insurance Association and the current legal and financial issues being debated in Austin, you have to get your hands a little dirty with the actuarial science.  You need to have some ability to model the damages likely to be caused by a tropical cyclone on the Texas coast.  Now, to do this really well, it might be thought you need an awful lot of very fine data.  In fact, however, you can do a pretty good job of understanding TWIA’s perspective by just reverse engineering publicly available information.

What I want to show is that the perceived annual exposure to the Texas Windstorm Association can be really well modeled by something known in statistics as a Weibull Distribution. To be fancy, it’s a zero-censored three parameter Weibull Distribution:

CensoredDistribution[{0, ∞},
WeibullDistribution[0.418001, 1.26765*10^8, -4.81157*10^8]]

We can plot the results of this distribution against the predictions made by TWIA’ s two consultants: AIR and RMS. The x-axis of the graph are the annual losses to TWIA.  The y-axis of the graph is the probability that the losses will be less than or equal to the corresponding amount on the x-axis. As one can see, it is almost a perfect fit.  For statisticians, the “adjusted R Squared” value is 0.995.

How did I find this function? Part of it is some intuition and some expertise about loss functions.  But a lot of it comes from running a “non-linear regression” on data in the public domain.  Here’s a chart (an “exceedance table”) provided by reinsurance broker Guy Carpenter to TWIA.  It shows the estimates of two consultants, AIR and RMS, about the losses likely to be suffered by TWIA.  Basically, you can use statistics software (I used Mathematica) to run a non-linear regression on this data and assume the underlying model is a censored Weibull distribution of some sort.  And, in less than a second, out pop the parameters to the Weibull distribution that best fit the data. As shown above it fits the AIR and RMS data points really well.  Moreover, it calculates the “AAL” (the mean annual loss to TWIA) pretty well too.

In some forthcoming posts, I’ m going to show what the importance of this finding is, but suffice it to say, it explains a lot about the current controversy and suggests some matters to be examined with care.