Premium principle
An insurer that seeks to maximise its expected profits subject to a survival constraint would price premiums according to
$$\begin{aligned} p_{c}=L_{c}+y\left( Z_{f'}-Z_{f}\right) , \end{aligned}$$
(1)
where \(p_{c}\) is the premium on contract c, \(L_{c}\) is the expected loss, y is the cost of capital and \(Z_{i}\) represents the insurer’s capital reserves held against portfolio \(i\in \left\{ f,f^{\prime }\right\}\).Footnote 8 When the insurer takes on contract c, the insurer’s portfolio is \(f^{\prime }=f+c\). Thus expected profit maximisation implies the insurer prices the contract at its expected loss. However, the need to satisfy a survival constraint implies an additional load on the premium, equal to the cost of the additional capital required to ensure the portfolio-wide survival constraint is still met.
Expected losses and capital loads
Assuming a single probability distribution is appropriate to characterise the insurer’s losses on its portfolio, the expected loss \(L_{c}\) is precisely estimated, as are the capital reserves, which can be set so that they are just sufficient to cover the loss \((-)x\) at a pre-specified probability \(\theta\) (e.g. 1/200 years or 0.005).Footnote 9 For portfolio f the capital reserves are
$$\begin{aligned} Z_{f}=\min \left\{ x:P_{f}(-x)\le \theta \right\} , \end{aligned}$$
(2)
where \(P_{f}(x)\equiv P_{f}\left( z:z<x\right)\), i.e. it is shorthand for the probability that portfolio f pays out any amount less than x, or equivalently that the loss is more than x.
The key question is what to do when multiple conflicting estimates of \(P_{f}(x)\) are available and there is no basis for assuming one estimate is precisely correct. This might often be the case when insuring catastrophe risks such as hurricanes. The insurer may have at its disposal a set of estimates from the various catastrophe models available.
One approach is to blend the various estimates into a single probability distribution and proceed exactly as above. There are several ways to blend models, but the principal methods are frequency and severity blending.Footnote 10 Their workings will be described below.Footnote 11
Dietz and Walker (2019) propose an alternative approach. Where \(\pi\) is an estimate of the probability distribution of losses—call it a ‘model’ for short—the expected loss \(L_{c}\) is simply the expectation of the expected losses estimated by each \(\pi\).Footnote 12 On the other hand, in setting capital reserves the insurer may not be neutral towards the existence of ambiguity and this implies not simply taking expectations. Thus, applying recent developments in economic theory, Dietz and Walker (2019) propose that capital reserves be set according to
$$\begin{aligned} Z_{f}=\min \left\{ x:\alpha \cdot \left[ \underset{\pi \in \varPi }{\mathrm {max}}P_{f}^{\pi }(-x)\right] +(1-\alpha )\cdot \left[ \underset{\pi \in \varPi }{\mathrm {min}}P_{f}^{\pi }(-x)\right] \le \theta \right\} , \end{aligned}$$
(3)
where \(\varPi \in {\mathbb {Z}}^{+}\) is the set of all models. The insurer computes \(Z_{f}\) by taking a weighted average of the highest and lowest estimates of the loss \((-)x\) at probability \(\theta\). The weight factor \(\alpha\) captures the insurer’s attitude to ambiguity.Footnote 13 Notice that ambiguity aversion affects the capital reserves, but not the expected loss. This just follows from the assumption that the insurer’s objective is to maximise expected profit subject to a survival constraint.
Dietz and Walker (2019) go on to show that, if one portfolio is more ambiguous than anotherFootnote 14 in a specific sense, then an insurer holds more capital if and only if \(\alpha > 0.5\). In other words, an insurer with \(\alpha > (<)\,0.5\) is ambiguity-averse (-seeking) and charges higher (lower) premiums for contracts that increase the ambiguity of the portfolio, all else being equal. This comes about because an insurer with \(\alpha >0.5\) places more weight on the highest loss estimate, the worst case. In the limit of \(\alpha =1\), the insurer sets its capital reserves based exclusively on the worst case, which is analogous to (unweighted) maxmin decision rules that have been proposed as a means of making rational decisions under ambiguity/ignorance (e.g. Gilboa and Schmeidler 1989; Hansen and Sargent 2008). The specific case Dietz and Walker analyse is a set of models \(\varPi\) that is centrally symmetric. One portfolio is more ambiguous than another, all else being equal, when it has the same loss distribution at the centre, but where the loss distribution depends more strongly on the true model \(\pi\). If \(\varPi\) is not centrally symmetric, then it is not true in general that \(\alpha >0.5\) corresponds to ambiguity aversion.
Model blending
As mentioned above, instead of Dietz and Walker’s (2019) approach, the insurer could blend the set of models into a single probability distribution and set its capital reserves just using (2). This approach does not allow for ambiguity aversion (seeking). Frequency blending works by taking the weighted average of the probabilities estimated by each of the set of models \(\varPi\) of a given loss:
$$\begin{aligned} P_{f}(-x)=\sum _{\pi }\gamma _{\pi }P_{f}^{\pi }(-x), \end{aligned}$$
(4)
where \(\gamma _{\pi }\) is the weight assigned to model \(\pi\) (see Fig. 1 for a schematic representation). Clearly if each model weight in the set is equal to \(1/\varPi\), then (4) is equivalent to computing the arithmetic mean of the probabilities at a given loss, which is a common and natural starting point.
Severity blending involves taking the weighted average of the losses estimated by the models at a given probability:
$$\begin{aligned} P_{f}(-x)=P_{f}\left[ \sum _{\pi }\gamma _{\pi }\text { inv}P_{f}^{\pi }(-x^{\pi })\left| P_{f}^{j}=P_{f}^{k},\forall \pi =j,k\right. \right] . \end{aligned}$$
(5)
See Fig. 1. If each model weight is equal to \(1/\varPi\), then (5) is equivalent to computing the arithmetic mean of the losses at a given probability.
Although severity blending might appear to be the inverse of frequency blending, it is not. It is well known that the two techniques tend to produce different estimates of the composite loss distribution, even in relatively trivial examples where there are two models and \(\gamma _{1}=\gamma _{2}=0.5\) (see Calder et al. 2012, pp. 18–21). Although severity blending involves applying the inverse of the loss distribution function, \(\text {inv}P_{f}^{\pi }(-x)\), the weighted average loss at a given probability need not lead to the same result as the weighted average probability at a given loss. Only in two cases will frequency and severity blending yield exactly the same estimate of \(P_{f}(-x)\). One is the trivial case where the models agree exactly on the probability of a given loss, \(P_{f}^{j}=P_{f}^{k},\forall \pi =j,k\). The other is when the slopes of the loss distribution functions are equal, that is, for all pairs of models j and k, when \(P_{f}^{j\prime }(-x^{j})=P_{f}^{k\prime }(-x^{k})\) over the interval \(\left[ -x^{j},-x^{k}\right]\). See Appendix 1.
It has been argued that frequency blending is superior to severity blending. While severity blending is easy to perform, doing so breaks the link with the underlying event set, which makes it difficult to make comparisons, such as the accumulation of risk from a given peril across the insurer’s whole portfolio, or the comparison between losses gross and net of excess (Calder et al. 2012; Cook 2011). Nevertheless both techniques are common and so we will evaluate both.
It is possible to envisage a situation in which the application of the \(\alpha\)-maxmin reserving rule in (3) is mathematically equivalent to using frequency blending to mix models so that reserving rule (2) can be applied (or to using severity blending under the limited circumstances described above). This would be the case if, for example, there were two models and the ambiguity parameter \(\alpha\) happened to coincide with the model weights \(\left\{ \gamma _{\pi }\right\} _{\pi =1}^{2}\). For instance, the insurer might set \(\alpha =0.5\) and consider each of the models to be equally likely to be the true model \((\gamma _{1}=\gamma _{2}=0.5\)). However, it is important to stress that \(\alpha\) is a behavioural/preference parameter that is intended to capture the insurer’s attitude to ambiguity, whereas the model weights \(\gamma _{\pi }\) reflect the insurer’s beliefs about how likely each model is to be the true model. They are conceptually quite different.