When catastrophe strikes, it is not unusual for the insurance payout to differ from the policyholder’s expectation. The possibility of such a discrepancy is referred to as “basis risk”. The term, however, can be ill-defined and easily misunderstood.

Therein lies the problem, without definition it is easy for the basis risk associated with a structure to remain unidentified and unquantified. If left unspoken, basis risk can lead to problems down the line, when events do occur. So, as a starting point, we can most simply define basis risk as the “difference between expectation and outcome”.

Parametric insurance is most commonly associated with basis risk, though when defined like this, it becomes clear that basis risk exists within all insurance structures.

For indemnity insurance, basis risk could stem from the possibility that a contract fails to pay because of a legal miswording; for a modeled loss trigger, the difference between modeled loss and measured loss after an event; and for pure parametric insurance, the difference between the index loss calculated from a wind speed measurement and the total actual loss.

The primary drivers of basis risk vary between structures. To quantify basis risk, it is first necessary to identify the primary sources of uncertainty with respect to each structure. Once identified, the basis risk can then be quantified and communicated. Once quantified and understood, the structure can then be tailored to modify the expectation as appropriate.

Let’s examine some typical methods for quantification of basis risk in two types of parametric structure. Pure parametric, in which a measured parameter is used to proxy the loss, and a simple modeled loss structure, where a modeled footprint is used with exposure, vulnerability, and financial models to create a modeled loss for the event.

**Pure Parametric**

A range of methods have been developed to assess basis risk in pure parametric structures, these can also be applied in modeled loss and indemnity cover.

To develop this idea, we will use a theoretical example of a parametric wind trigger in which there is a linear payout based on the wind speed measured at a single location.

Following the previous definition of basis risk, the expectation with this structure is that wind speed during an event correlates with total loss. Conveniently, it is possible to use catastrophe models to uncover this correlation.

Each point in Figure 1 represents a stochastic event, with its associated modeled loss and modeled wind speed at the measurement location.

Figure 1: Hazard against modeled loss

The form follows that of the vulnerability of the underlying exposure. The function that describes this relationship is contained within the index formula, which is used in combination with an exposure weight to transform the wind speed to an index loss. The results of this transformation of hazard to index loss are displayed in a basis risk plot (Figure 2):

FIigure 2: Basis risk plot

The correlation between the two variables is one measure of the overall basis risk. However, basis risk is a two-sided coin, it has positive and negative components, termed as overpayment and shortfall.

Overpayment describes the situation in which the payout from a structure is greater than the loss experienced during an event, whereas shortfall describes the more important situation for the risk holder, in which the payout is less than the loss.

It is more insightful to measure basis risk with respect to a target layer. Overpayment and shortfall can then be measured as a percentage of the total layer width. Figure 3 displays regions of shortfall and overpayment with respect to a target indemnity layer of 200 XS 200, for the same events as displayed in Figures 1 and 2.

Figure 3: Shortfall and overpayment with respect to a target indemnity layer

The degree of shortfall or overpayment can then be calculated. The following formulae are commonly used to calculate the basis risk for parametric structures with a linear payout.

We can calculate these basis risk metrics for all stochastic events, and produce conditional probability plots which describe the expected degree of overpayment or shortfall. Shortfall is conditional on an event exceeding an indemnity attachment, and conversely overpayment is conditional on an event attaching on the index side.

Shortfall and overpayment can be further quantified and visualized using conditional probability plots. Figure 4 displays the probability shortfall or overpayment of exceeding a range of thresholds. As can be implied from Figure 3, and quantified in Figure 4, the example structure shows a moderate bias towards shortfall.

Figure 4: Conditional probability plot

Using these methods, it is possible to understand and communicate the basis risk associated with the structure. This helps to refine the trigger mechanisms to a point where all stakeholders are comfortable with the levels of associated basis risk.

It is important to note that while hazard and loss uncertainty can and should also be factored in, we should remember that basis risk metrics calculated within a model cannot account for all the uncertainty that exists. Indeed, basis risk can be most well understood if measured using independent models and methods.

**Minimizing Parametric Basis Risk**

Once quantified, there are ways to close the gap between the expectation and outcome. The simplest of these is to change the attachment level.

If the expectation from a risk holder is that a payout should be received after any event that makes the news, then the index attachment level can be lowered, and the trigger can be biased towards overpayment. This transfer of basis risk from shortfall to overpayment comes at the cost of a higher premium.

A less costly option may be to introduce a phased attachment, where there is a small binary payout at a lower attachment threshold, with the remaining principal more closely tied to an indemnity layer. This stepped payout mechanism may help to manage the reputational risk associated with a binary parametric trigger structure.

A more technical approach to reducing basis risk requires a more detailed understanding of the sources of uncertainty. Within the confines of the model, the overall basis risk in pure parametric structures primarily derives from:

- The displacement of the exposure from the measurement station(s)
- The ability of the index formula to capture the vulnerabilities of the exposure

One way to reduce the basis risk is to increase the number of measurement stations, and assign exposure to the station which best proxies the hazard at the exposure. Another is to use more index formulae to better capture the range of vulnerabilities.

**Modeled Loss**

The obvious extension is that a simple modeled loss based trigger solves both of these problems, where we effectively know the hazard at all locations, and can use the true vulnerabilities of the exposure. Surely this reduces the basis risk to zero?

Not necessarily. The primary sources of uncertainty in a modeled loss trigger are often distinct from those in pure parametric. The nature of the uncertainty is also more structure specific, making it difficult to generalize the drivers of basis risk in “modeled loss”. For example, in different forms of modeled loss triggers, the process used to generate the modeled hazard footprint, or the development of loss curves from limited historical data might drive basis risk in a modeled loss trigger.

The example of modeled loss draws out an important feature of basis risk; some sources of uncertainty can be quantified easily using models, and some cannot. A model is well placed to quantify the correlation between hazard and loss in a pure parametric structure, but less well equipped to convey the uncertainty that exists within itself, which can be relatively more significant in a modeled loss structure.

In light of this, when assessing basis risk in modeled loss triggers we need step outside the confines of the model at hand, and assess the structure independently using external models and the real world. A daunting task, with project resources and the historical record often restricting the insight that can be gained, though one which can greatly enhance the success of the structure.

**On What Basis**

Importantly, and somewhat contrary to common perception, both modeled and un-modeled sources of basis risk exist in all structures. The balance between the two shifts depending on the dominant driver of uncertainty, be it within pure parametric, modeled loss, or indemnity. In all cases, identification, quantification, and communication are the keys to understanding basis risk.

The benefits of a clear understanding of the basis risk are well worth the effort in attaining it. Structural decisions can be made with greater confidence, potential options can be measured against one another, and the expectation that a structure always pays out when there is a loss can be set appropriately.

Ultimately, the potential for surprises is reduced with greater understanding, and the risk transfer is more likely to function as expected. Basis risk can never be entirely eradicated, though with the right analytical approaches it becomes much more manageable.