On the Design of Regional Insurance Markets for East Asia

Abstract

This chapter identifies the central issues in designing a possible regional insurance scheme or mechanism for East Asia with a focus on a risk sharing mechanism for catastrophe risks households in the region. I apply theoretical observations that provide a consistent explanation for the apparent anomalies concerning the demand for catastrophe insurance within the subjective expected utility framework. The key observation is that the number of observations would be inevitably insufficient to warrant a robust probability estimate for a rare event. The inherent lack of a robust probability estimate leads to diverse probability beliefs. I evaluate the various insurance schemes in terms of social welfare. In doing so, I adopt a measure that is based on the ex post social welfare concept in the sense of Hammond (Economica 48, 235–250, 1981), as the standard Pareto optimality criterion is problematic in the presence of diverse beliefs, for it ignores the regrets or pleasure ex post caused by ‘incorrect’ beliefs. Although the ex post social welfare may have an expected utility form, I only focus on the ex post utility frontier rather than specifying a particular social probability. A desirable insurance scheme is the one that eliminates any personal catastrophe state.

Appendix: The Large Deviation Property

In what follows, we reproduce the exposition of Lemma 1.1.9 of Dembo and Zeitouni (1998) in Nakata, et al. (2010). Let random variable X _t denote the loss in period t, and let X ₁, X ₂, …, X _T be an i.i.d. sequence. Also, let $\mathcal{P}\left(\mathcal{A}\right)$ denote the space of all probability laws on $\mathcal{A}:=\left\{{a}_1,{a}_2,\dots,\ {a}_S\right\}$. Furthermore, for a finite sequence (of realisations) x ^T = (x ₁, x ₂, …, x _T ), we define the empirical measure of a _s as follows:

$$ {m}_T^{\mathbf{x}}\left({a}_s\right):=\frac{1}{T}{\displaystyle {\sum}_{t=1}^T{1}_{a_s}\left({x}_t\right)},\kern0.5em \forall s, $$

where ${1}_{a_s}\left(\cdot \right)$ is an indicator function such that

$$ {1}_{a_s}\left({x}_t\right)=\left\{\begin{array}{c}\hfill 1\kern0.5em \mathrm{if}\kern0.75em {x}_t={a}_s;\hfill \\ {}\hfill 0\kern0.5em \mathrm{otherwise}.\hfill \end{array}\right. $$

Then, we define type m ^x _T of x ^T as

$$ {m}_T^{\mathbf{x}}:=\left({m}_T^{\mathbf{x}}\left({a}_1\right),{m}_T^{\mathbf{x}}\left({a}_2\right), \dots, {m}_T^{\mathbf{x}}\left({a}_S\right)\right). $$

Let ℳ _T denote the set of all possible types of sequences of length T, i.e.

$$ {\mathrm{\mathcal{M}}}_T:=\left\{\nu :\nu ={m}_T^{\mathbf{x}}\ \mathrm{for}\ \mathrm{some}\ {\mathbf{x}}^T\right\}. $$

Also, the empirical measure m ^x _T associated with a sequence of random variables X ^T := (X ₁, X ₂, …, X _T ) is a random element of ℳ _T .

Let P _π denote the probability law associated with an infinite sequence of i.i.d. random variables X = (X ₁, X ₂, …) distributed following $\pi \in \mathcal{P}\left(\mathcal{A}\right)$. Also, the relative entropy of probability vector ν with respect to another probability vector π is $H\left(\nu \Big|\pi \right):={\displaystyle {\sum}_{s=1}^S{\nu}_s \ln \frac{\nu_s}{\pi_s}}$.

Proposition (Lemma 1.1.9; Dembo and Zeitouni 1998)

For any ν ∈ ℳ _T ,

$$ {\left(T+1\right)}^{-S}{e}^{- TH\left(\nu \Big|\pi \right)}\le {P}_{\pi}\left({m}_T^{\mathbf{x}}=\nu \right)\le {e}^{- TH\left(\nu \Big|\pi \right)} $$

(10.2)

The proposition states that the probability of observing type ν for a sequence of length T with respect to probability law π ^π has the lower and upper bounds as specified in Eq. (10.2).⁸ Clearly, both the lower and upper bounds are decreasing in H(ν|π). Note that this result (and the results in the literature of large deviations) is very useful, since it may well be rather difficult to compute the exact probability P _π (m ^x _T  = ν) in many cases. This difficulty arises from the fact that we need to consider all possible paths/sequences that belong to the specified type, which involves combinatorics. Moreover, from this result, we know that the relative entropy H(ν|π) characterises the probability P _π (m ^x _T  = ν), although the bounds may not be very tight in some cases.