Pricing Funeral (Burial) Insurance in a Microinsurance World with Emphasis on Africa

Abstract

Funeral (burial) insurance is one of the most common microinsurance policies sold in Africa. Funeral insurance is important because it pays for the cost of funeral arrangements, thus forming an important part of protection for low-income workers. We develop a model of a microinsurance market in an African context where funeral insurance policies are sold to low-income households through a burial society, which contracts with a risk-neutral profit-maximizing monopolistic insurer to supply funeral insurance policies. We assume the burial society is sufficiently large that it has bargaining power and can negotiate on behalf of its members. The burial society requires that the insurer offer separate policies that are affordable to both low-risk and high-risk individuals and waiting periods cannot exceed m months. Applicants for insurance are identical except for their mortality, which is known to the insurer except for a single parameter called the frailty parameter. As death benefits and premiums are low, the insurer cannot afford to use a costly but effective underwriting technology to provide accurate information on each applicant's future mortality. To mitigate the effect of adverse selection, the insurer uses a simple low-cost practical underwriting strategy that requires all applicants to complete a questionnaire on their personal/family medical history, certify their “good health,” and confirm that they are gainfully employed. The insurer also includes a waiting period before policyholders are eligible for death benefits. As there is no established actuarial theory that suggests how the optimum waiting period should be determined, the objective of this article is to establish a sound basis for determining both the premium and the waiting period for these policies. To this end we develop a discounted expected utility model of consumption by members of a burial society and use this model to determine the optimal premiums and waiting periods subject to solvency, lapse, and participation constraints.

Acknowledgments

This article is dedicated to the memory of Luis David Arcila's mother, Mrs. Nora Hoyos, who died in December 2010 while this research was being conducted. The authors thank Heng Sun and Peng Zhan for their assistance with some of the programming and calculations and an anonymous reviewer for his or her valuable suggestions that have improved this article. This research was supported by a Society of Actuaries CKER grant.

Notes

Though funeral insurance is valuable in less poor countries, it is less valuable in richer countries where more affordable and safer ways to prefund funerals exist such as personal savings, funeral bonds, and prepaid funeral arrangements (CPSA 2011).

Adverse selection does not necessarily arise from intentional deception or lying on an application; rather, it results from asymmetrical information between buyers and sellers (Akerlof 1970). In our case the applicant (the buyer) knows more about his or her mortality than the insurer (the seller).

In group microinsurance, for example, microinsurers may require a minimum percentage (say, 90%) of the group to actually purchase insurance.

Strange (2002) has argued that working-class concepts of “decent” burial were grounded in wider notions of respectful rather than respectable interment: The fulfillment of burial and mourning rites testified to the dignity and identity of the dead while facilitating expressions of loss, respect, and condolence.

McNeil (1998) notes that African funeral rituals vary, but they usually involve bringing the body into the home for at least one night, washing it, a public viewing, a graveside service, a big meal for the mourners, and a week-long period of mourning in which friends and relatives sleep in the widow's room and around the house. Besides the funeral itself, the family is expected to pay for the food and firewood, and some mourners even demand a bus fare home.

Jureidini and White (2000) provide an excellent review of the history of medical exams for life insurance purposes.

By complete insurance we mean that the insured's end-of-period wealth after insurance is the same regardless of the state of nature; that is, whether an accident occurs or not. In other words, losses are insured for their full amount. Otherwise, we have partial insurance.

This definition is based on and extends Sher's (1963) definition of funeral (burial) insurance.

This is assumption is made in order to simplify the mathematical development of the article. A more realistic assumption and its consequences is discussed in Section 7.

As Roth (2000) points out, in South Africa it is common for township residents to receive gifts from friends and neighbors when they have a funeral. However, the monetary gifts were not very significant when compared to the gifts of labor (such as assistance with food preparation) and other gifts including loans of cutlery, crockery, and furniture. In this article we will consider remittances to be monetary gifts, gifts of labor (such as assistance with food preparation), and other gifts, including loans of cutlery, crockery, and furniture from friends and relatives (at home or abroad) and neighbors, and we will assign a monetary value to all such gifts.

As each “individual” of interest is a “head-of-household,” the terms “individual” and “head-of-household” are used interchangeably.

For more on mortality heterogeneity, see, e.g., Wang and Brown (1998) or Tuljapurkar and Edwards (2011) and references therein.

Typically, the notation [x] refers to a “select” life age x with mortality following some select-ultimate mortality table. However, in this article [x] refers to a member of the x-cohort at time 0 and the standard mortality refers to aggregate or group mortality.

The high-risk versus low-risk model is commonly used in adverse selection problems since the pioneering works of Rothchild and Stiglitz (1976) and Stiglitz (1977).

Karlan (2010) notes that savings give people the ability to turn irregular cash flows into lump sums for larger purchases, emergencies, and investments. However, as the poor often do not have access to savings accounts, Banerjee and Duflo (2007) point out that a main challenge for the poor who try to save is to find safety and a reasonable return as stashing cash inside a pillow or elsewhere at home is neither safe nor well-protected from inflation. Rutherford (2000) describes many strategies the poor use to surmount this challenge such as forming savings clubs and rotating savings and credit associations that allow people to lend their savings to each other on a rotating basis.

For a discussion of other theories of smoothing consumption over time, see, for example, Deaton (1992) or Ljungqvist and Sargent (2004) and references therein.

Equation (10), without the inflation factor, is along the lines of Cropper and Sussman (1988, eq. [1]).

One of the insurer's main problems is solving U_I (G, w, z, x)=U_N (z, x), as a function of G. Assuming continuous risk-averse utility functions, it is clear that an insured household's discounted expected utility will decrease as G increases while the insurance benefits remain unchanged. Thus U_I is a continuous decreasing function in G with complicated derivatives. So it may be best to use methods such as the secant method or Brent's method (Brent 1973). By comparing equations (10) and (20) with G=0 and w⩾0, it is easy to see that U_I (0, w, z, x)⩾U_N (z, x), (i.e., “free insurance” is better than “no insurance”), while U_I (G, w, z, x)<U_N (z, x) as G→∞, (i.e., there exists some very expensive insurance that is worse than “no insurance”). Thus, equation U_I (G, w, z, x)=U_N (z, x) has a unique solution, G(w), for each w.