# Modeling ILS Market Dynamics

The market for insurance-linked securities (ILS) has grown considerably over the past few years. As with any other financial market, we expect assets to be essentially fungible — in other words, we would expect two assets with the same risk exposure, duration, and structure to have the same price. Otherwise, an arbitrage opportunity would arise, an eagle-eyed investor would spot it and take advantage of it until the prices equalized.

The ILS market is considerably more heterogeneous than many other asset classes in the capital markets. In terms of diversity, they are more akin to the corporate bond market than to the commodities or futures markets. Consequently, the price of any given insurance-linked security is idiosyncratic and a function of its own characteristics.

However, this diversity, coupled with the efficiency of the market and the underlying fungibility of assets, allows us to tease out the dynamics of the ILS marketplace. We have a data set of several hundred insurance-linked securities. We know several key characteristics of each security, as well as the coupon payment for each security. In this post, I’ll apply statistical modeling techniques to estimate a function that links a security’s characteristics to its price in the market.

# Exploratory Data Analysis

We can start by plotting the coupon payment from each security against its issue date. The chart below shows expected loss (as a percentage of the total investment) on the y-axis and issue date on the x-axis. Each dot is a different security. The color of the dot represents the peril the security is exposed to, and the size of the dot represents the size of the security.

We can see that average coupons declined significantly over the period from 2012 to 2014. We can also plot the coupon against the expected loss for the security (expressed as a percentage of total amount invested). This plot has the same structure as the above plot, with only a different variable on the x-axis.

Immediately, we can see several things worth noting. First, most of the points in this plot lie on a line from the lower left to the upper right of the plot. This makes sense, as it shows that coupons increase as expected losses increase. The picture is somewhat muddied, however. There are about a dozen or so outliers at the bottom of the plot with coupons of 0 or 50 basis points. These extreme anomalies are not likely to be truly reflective of market dynamics — there may be other, unusual provisions in these contracts that distorts their overall value, or it may be simply a matter of dirty data.

Likewise, there are several securities with expected loss probabilities of 0.1% or less, including a few with expected loss probabilities of 0%. As with the coupon outliers, there is good reason to discard these outliers from our data set. We also see that almost all assets are on the order of tens to hundreds of millions of dollars, but there are a few extreme outliers: three assets with sizes of $3–4MM, and two assets worth $10.125B and $35B, respectively. These outliers would have an undue influence on our analysis if we were to retain them, so we will filter them out as well.

Finally, we can see that even within the main cluster of dots in our plot, the distribution is uneven. Many dots are in the lower-left corner of the cluster, while the upper-right corner is relatively sparse. This is a sign that the distribution of loss probability and coupon are what’s known in statistical jargon as “right-tailed”. We can correct for this by making the scales of the x- and y-axes of this plot logarithmic, rather than linear. The plot below shows the same data as above, but with the outliers removed and the scales adjusted. This plot represents 335 securities issued between 2010 and 2018 with a total notional value of $55.1B.

Now, the picture we see of the market is much clearer, and the strong correlation between probability of loss and coupon has been bought into focus. However, we can also see that we can’t assume a strictly linear relationship. Note that the trend is steeper for expected losses greater than 1%. This is entirely sensible. As a thought experiment, if there were an ILS with an expected loss of 0%, investors would still expect a coupon equivalent to the risk-free rate of return — after all, this hypothetical ILS would literally be a risk-free asset! Therefore, we are not at all surprised to see that there is some minimum coupon that investors expect in return for tying up their capital and gaining exposure to tail risk, regardless of how remote that risk may be. Furthermore, we expect this minimum coupon to be close to the risk-free rate of return.

# Model Description

This process of plotting and “eyeballing” the data is useful, but it has its limitations. First and foremost, it’s difficult to account for multiple effects simultaneously in this fashion. For example, it seems that some perils may earn higher coupons, but it’s difficult to say how much correcting for this would influence our estimate of the relationship between expected loss and coupon. Second, although eyeballing data is useful for identifying interesting relationships and formulating hypotheses about the structure of the data, it is less useful for making quantitative, rigorous estimations of those relationships. Fortunately, the discipline of statistics offers solutions to both of these problems.

I’ll share the results of a model I fit to the data, and walk through what the modeling results can tell us about the ILS market. The specific model I used is called a generalized additive model, or GAM. For those of you who are unfamiliar with GAMs, there are two important features to understand about them.

First, GAMs assume that the effects of predictors are strictly additive, or that the effect of each predictor in the model is independent of every other predictor. For example, this means that the model assumes the effect of peril on coupon is the same, regardless of the date of issue, the expected loss, or any other factor. I will freely admit that this is a simplifying assumption, and it’s unlikely to be exactly true. However, it’s a decent approximation to reality, our dataset is too small to reliably estimate complex interactions between predictors, and this assumption makes interpretation of the model much easier.

Second, GAMs have more flexibility than traditional linear regression when it comes to representing the effects of predictors. Linear models assumes that the relationship between each predictor and the response is a straight line (thus the name linear). By contrast, GAMs allow for some effects to be modeled with wiggly curves, and the model tries to find the least wiggly curve that fits the data best. For more technical detail, see Chapter 5 of The Elements of Statistical Learning.

Finally, before diving into the results of the model, I’ll note that statistical best practices include fitting several alternative model specifications to see how sensitive the results are to specification choices, to ensure that the model diagnostics do not point to any major issues, and so forth. For ease of exposition, I’m only showing the results from one model, but trust that I’ve fit quite a few models to this data, and the conclusions I’m sharing here are robust across a wide variety of alternative specifications.

# Model Results

The model I fit predicts coupon as a function of expected loss, issue date, peril, region, contract duration, and contract size. I’ll share the modeled effects for each of these predictors in turn.

First, let’s look at the modeled effect of expected loss on the coupon. The plot below shows the modeled effect. As before, the dots represent individual contracts in the underlying data, while the black line shows the fitted curve of the model effect. Technically speaking, the curve as shown is the modeled effect of expected loss on coupon, regardless of the values of the other variables, so the ratio between predicted coupons of, say, a 0.5% and 5% expected loss will hold, no matter the peril, region, and duration. However, to better ground the curve, the curve is shown for a typical contract of 3 years duration that was issued on December 15th, 2014 for $150MM of North American earthquake peril. The shaded area around the black line shows the 95% confidence interval for the fitted curve.

We can see that, as we had hypothesized earlier, the relationship between expected loss and coupon weakens as the probability of loss decreases. In the heart of the distribution, at a 3% expected loss, we expect a 0.61% increase in coupon for every 1% relative increase in probability of loss. (This means an increase from 3% to 3.03%, not from 3% to 4%.) We also see that the uncertainty for the curve is greater for extremely low and extremely high loss probabilities. This is a typical side effect of our estimation procedure, and it reflects the difficulty of extrapolating to extreme values where data is thinner. We will see similar effects in the other plots we examine.

Now, let’s look at the effect of duration on coupon. This plot is structured in the same way as the previous plot, except that the x-axis represents the contract duration, and the assumed expected loss of the contract used to generate the model effect curve is 2%.

We can see that shorter contracts are generally associated with higher coupons. After 4 years or so, the effect of duration on coupon bottoms out. The line of best fit shows a slight increase for longer contracts, but note that the confidence interval widens as well, so we can’t say for sure whether there is a true inflection point at around 4 years’ duration. Now, let’s turn to issue date.

We can see that in earlier years, coupons were significantly higher — at some points, nearly twice as high as typical recent coupons for equivalent securities! Yields decreased sharply in 2013 and 2014, and have been relatively stable since then. Now, let’s look at contract size.

Here, we see that if there is any relationship between contract size and coupon, that relationship is very weak. Note that we can draw a straight line through the graph such that the line is entirely contained within the 95% confidence interval! This means a sensible and conservative estimate of the effect of size on coupon is zero.

Now, let’s look at the effect of region. Region is a categorical variable, unlike the continuous variables we have considered up until now. The x-axis is now grouped by region, and we have replaced the fitted curve and shaded region with fitted black dots and vertical lines. The interpretation is much the same as before — the black dots represent the black point estimates of modeled effects of perils, and the black vertical lines represent the uncertainty around these estimates.

We can see that typical coupons for North America are higher than those for Europe and Japan. Finally, let’s look at the effect of peril.

We can see that typical coupons for earthquake securities are slightly higher than for other perils. At first glance, this result may seem surprising. Earthquake coupons are generally higher than windstorm coupons, but this is offset by the fact that earthquake expected losses are generally higher than windstorm expected losses. We could try to balance these effects by computing the ratio of coupon over expected loss and comparing this ratio by peril. The plot below does just that: the x-axis is issue date and the y-axis is coupon basis points per percent of expected loss.

Taken in isolation, this plot seems to imply that coupons for windstorm risks should be higher than for equivalent earthquakes risks. **However, this view is misleading!** We can see why if we plot the coupon / expected loss ratio as a function of expected loss.

We can see that the coupon / expected loss ratio is high for securities with very low expected losses. This is a direct consequence of the pattern we noticed earlier, where there appears to be some minimum coupon investors require in exchange for assuming any tail risk. We also see that the left-hand side of the graph is almost entirely windstorm securities. Therefore, the apparent premium in coupon / expected loss ratio for windstorm securities can be explained by distributional differences in expected loss by peril. In fact, once we adjust for these distributional effects, it turns that that earthquake perils earn higher coupons for the same risk!

Instead of taking the ratio of coupon over expected loss, we can directly adjust for the effect of expected loss on coupons. The plot below shows the loss-adjusted coupon ratio for each security, which we have calculated by dividing the coupon for each security by the coupon predicted for that security from a model that only uses expected loss as a predictor. For example, a loss-adjusted coupon ratio of +20% means that a security’s coupon is 20% higher than would be expected based on its expected loss alone.

Now, we can more clearly see that historically, earthquake coupons have been higher than windstorm coupons for equivalent risks. However, we can also see that the pricing difference between these two perils has slowly been eroding over time, and may have disappeared entirely within the past year.

# Conclusion

I hope you’ve enjoyed this exploration of dynamics in the ILS market. We can see that clear dynamics underlying the pricing of insurance-linked securities, and these make sense given prior knowledge about capital markets. The model I’ve shared is able to explain just over 90% of the variation in ILS coupons, using only six pieces of information about a security!

Ledger Investing is a leader in understanding the ILS market. For more about Ledger Investing, please visit our website at ledgerinvesting.com. Additionally, Ledger Capital Markets is now accepting registration requests from Insurers and Investors for its Investment Platform. You can contact us directly here to receive your invitation.