Nanyang Business School Forum on Risk Management and Insurance

Tags: index insurance, agricultural insurance, mortality securities, catastrophe insurance
More from: Jinggong Zhang, Ken Seng Tan, Chengguo Weng

Editor’s Note: Posted by Jinggong Zhang, Assistant Professor, Nanyang Business School, Nanyang Technological University; Professor, Department of Statistics and Actuarial Science, University of Waterloo; Chengguo Weng, Associate Professor, Department of Statistics and Actuarial Science, University of Waterloo


The purpose of this paper is to provide an in-depth analysis on a class of insurance known as the index-indemnifying insurance, or simply the index insurance. As opposed to the traditional loss-indemnifying insurance for which its payout (indemnity) is a function of the actual loss incurred by the policyholder, the payout of an index insurance depends exclusively on a pre-determined index or some appropriately chosen indicators.

Prominent applications of index insurance can be found in insurance coverage provided to agricultural producers. In fact in recent years there is a surge of interest in piloting index insurance for agricultural households in developing economics. In these applications, an index may be an average county crop yield, the number of heating days, the amount of rainfall received by a particular area during the growing season, or based on remote sensing satellite vegetation. Other than hedging agricultural and livestock risks, index-based securities that are issued in the capital market have been effective in securitizing the catastrophic risks.

The popularity of index insurance stems from a number of reasons. The first and foremost reason lies in its potential of reducing or even eliminating completely the moral hazard and adverse selections since the indemnity payments are based on an index that is transparent, well defined, and cannot be manipulated by either the insured or the insurer. The second reason is its low operational (such as the cost associated with the underwriting, administration, loss assessment). Because the indemnity payments are completely determined by an index, there is no need to assess the losses actually incurred by the agricultural producers. The loss assessment can be expensive and prohibitive, especially in rural areas where accessibility can be problematic. The number of small agricultural households further aggravates the cost if insurer needs to assess loss for all households. As a result, the claim settlement can also be processed more efficiently and more timely whenever there is a claim from an index insurance.

Despite all the aforementioned advantages, the challenge with the index insurance is the basis risk, which arises due to the discrepancy between the indemnity payments dictated by the index and the actual losses incurred by the insured. The imperfect correlation between the adopted index and the loss random variable casts doubt on the effectiveness of index insurance in hedging agricultural production risk and as such leads to low demand in some pilot index insurance programs.

The presence of basis risk implies that the index must be chosen meticulously. A logical line of inquiry is the determination of an index that optimally minimizes the basis risk. This basically relates to the optimal design of index insurance. The optimal design of loss-indemnifying contract is a widely studied problem in the actuarial literature. It is, however, important to point out some subtle differences between the formulation of optimal loss-indemnifying contract and optimal index-indemnifying contract. More specifically, the indemnity function in a loss-indemnifying insurance contract needs to be non-decreasing, bounded from above by the actual loss, and has a non-zero deductible, in order to avoid moral hazard. In contrast, the indemnity function of an index insurance can have very flexible structure. The indemnity is not necessarily increasing in the underlying indices. The indemnity payment can even exceed the loss incurred by the insured.

In this article, we adopt a utility maximization framework for the design of index insurance (Raviv, 1979) and define the optimal index insurance as the one that maximizes the insureds’ expected utility. The variance minimization problem can be viewed as a special case in our general utility maximization framework when a quadratic utility function is adopted.

We contribute to the literature in the following aspects. First, we provide a rigorous mathematical examination on the existence and uniqueness of the optimal index insurance arrangement. Second, explicit form of the optimal index insurance is derived for utility functions commonly adopted in insurance economics including quadratic and exponential utility functions. For a general strictly concave utility function, the optimal solution is characterized by an implicit ordinary differential equation (ODE), for which the solution can be easily obtained numerically, for example, by the Runge-Kutta method (Burden and Faires, 2001). Third, an empirical agricultural index insurance is conducted and it shows that the index based contract from our results significantly outperforms those existing index contracts from the literature. Choosing the average temperature as the underlying index, we find that the optimal indemnity function generally follows a “first decreasing and then increasing” pattern and its specific shape relies on the premium level charged by the insurance contract, the maximum indemnity paid and the form of utility function. For quadratic utility function, the design is equivalent to minimizing the variance of insured’s resulting position, and our numerical results show that the effectiveness in terms of variance deduction does not continue to improve with the premium level after the premium exceeds certain threshold. This observation provides important and useful insights for government agency in making agricultural insurance premium subsidiaries. Further, our results also show that the proposed optimal contract generally outperforms the linear-type insurance contracts, and that the multi-index contracts can further reduce basis risk, when compared to the single-index ones.

The complete paper is available at:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3241297.

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