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Evaluating the quality of remote sensing products for agricultural index insurance

['Benson K. Kenduiywo', 'Department Of Environmental Science', 'Policy', 'University Of California', 'Davis', 'Ca', 'United States Of America', 'Department Of Geomatic Engineering', 'Geospatial Information Systems', 'Jomo Kenyatta University Of Agriculture']

Date: 2021-10

This section introduces basic insurance and economic concepts that can be used to evaluate the insurance benefit offered by remote sensing-based insurance contracts based on [6]. The section uses a series of graphical devices in order to explain the concepts and shows how we derived our new measure.

Each observation is a random draw from an underlying risk distribution for a household with about 1 tropical livestock unit expressed in US dollars ($); in this simplified example we only consider mortality, not herd growth through birth. See text for a description of the equations.

Using the data described in the methods section below, we estimate the probability distribution of livestock mortality and use it to calculate the probability that our stylized pastoralist household will enter next season with the different amounts of livestock wealth shown in Fig 1 . The histogram projected on the figure reflect the different probabilities the household faces. For example, the right most bar represents a near zero mortality rate, which occurs just under 5% of the time. In this case, the household would enter the next season with its $1000 in productive assets intact. At the other extreme, the leftmost bar represents a 65% mortality rate. While such extreme events happen less than 1% of the time, when they do occur the household would enter the next season with only $350 of livestock wealth as shown in the figure. More generally, Fig 1 displays the probability that the stylized household will enter the next season with the different amounts of livestock wealth shown on the horizontal axis. As can be seen, the risk of loss is substantial, suggesting that insurance could play an important role for pastoralists households.

Pastoralism is risky in arid and semi-arid climates like in our study region. Periodic droughts lead to livestock starvation and death, sometimes destroying more than half of a family’s livestock over the course of a few months. These large decreases in productive wealth generate large drops in income and increases in human suffering as families struggle to eat. For our illustrative household (denoted with the subscript i) that begins with 1 TLU of livestock wealth or capital, and is not insured (denoted with the superscript N) the amount of wealth they have the next season ( ) depends, in the absence of insurance, on the livestock mortality they experience (M i ). We can thus write next season’s livestock held by the household i as: (1) where k i1 is the household’s initial livestock, the mortality rate M i ranges between 0 (no mortality) and 1 (all animals die). Because we assume that k i1 = 1 for our example, the expression can be further simplified as shown in Eq ( 1 ). Note that to keep simplify our discussion we are ignoring the natural reproductive growth of the herd.

To illustrate the framework, we first consider a perfect livestock asset insurance contract for pastoralist households in our study region in Northern Kenya. These households economically depend almost exclusively on their livestock, which constitute more than 90% of total household wealth and generate 80% of household income. Livestock wealth is typically measured in Tropical Livestock Units (TLU), a measure that allows counts of different livestock species to be combined into a single number. In this convention, one TLU is equivalent to 1 cow, 0.7 camels, or 10 goats or sheep. While all but the most indigent households possess multiple TLU, to keep things simple here we analyze the case of a household that has only 1 TLU. In monetary terms, we will assume that 1 TLU is worth $US 1000, an amount roughly in line with animal prices during non-drought periods.

Insurance provides economic benefit to households by putting a floor or safety net by transferring resources from good times (when premiums are paid and no payouts received), when resources are relatively plentiful, to bad times, when resources are scarce and especially valuable to the household. The value of perfect insurance is intuitive from the wealth floor in Fig 1 . The household sacrifices wealth in the good years in order to avoid the draconian consequences of severe losses in bad years. We define the difference between the household’s wealth with and without insurance as Δ.

In our case, given the estimated probability distribution, AFP P = $20 and we assume it is marked up by a further 25%. When the family purchases insurance it reduces its wealth by $25. The payment of this premium is visible in Fig 1 as the reduction in wealth that the household would face in years of low mortality when no indemnity payments are received. When the stylized household purchases insurance, its livestock wealth next season will be: (5)

To pay for this protection, every season the household pays a premium, p. The starting point for calculation of an insurance premium is the “actuarially fair price” (AFP), which is defined as the long-run average or expected indemnity payout per-TLU insured. In our example, we estimate that the mortality rate can take on 25 different values as shown in Fig 1 . We denote each of these values as M ij and we denote π j as the probability that mortality rate j occurs. Using this notation, we can write the AFP for our perfect insurance contract as: (3) Unless subsidized, insurance is always sold at some mark-up rate, m, that covers the operating costs of the insurance provider. The final market price faced by the household can be expressed as follows: (4)

A well-functioning contract indemnifies households for losses and thereby puts a floor under household asset holdings. Fig 1 illustrates how a “perfect” insurance contract would work. By a perfect insurance contract, we mean a contract that observes and issues payouts based on the actual livestock mortality, M i , that a household experiences (later on we consider more realistic contracts where M i has to be estimated because it is prohibitively expensive to measure mortality for each and every household). We assume that this contract carries a 23% deductible, but then issues payments to compensate the household for any losses in excess of 23% according to the following perfect insurance payoff or indemnity function: (2) where the superscript P stands for perfect insurance contract and t is the deductible (sometimes called the payout trigger) which we set to 23% in our example. k i1 is the number of livestock units insured, which we assume to be 1 TLU in our example. Under this payoff function, the insurance pays nothing if the mortality rate is less than the trigger t, that is, t<23% in our example, and otherwise pays enough to restore the value of all livestock back to 77% of its initial level. As can be seen in Fig 1 , insurance puts a floor under the household asset holdings such that they will never fall below $744 in the subsequent season.

Valuing the quality of an insurance contract.

While the value of insurance to the household is fairly intuitive in the case of the perfect insurance contract illustrated in Fig 1, we will later consider imperfect contracts, which sometimes fail to pay when the household has losses (false negatives), and sometimes pays when households do not (false positives). In these cases, it is more difficult to discern when a contract is good enough to be worth buying [1], and even harder to judge which of several alternative contracts offers better economic support for households. In other words, we need a cogent standard to measure the quality of protection offered by an insurance contract, and to evaluate and select the remote sensing-based indices that it uses.

A number of studies in the economics literature have employed different metrics which speak to the quality of index insurance contracts. These can be grouped into (i) Measures that examine the impact of the insurance on some feature of the probability distribution for wealth; and, (ii) Measures that are based on an explicit normative or welfare metric designed to capture the economic well-being of the insured household. Studies in the first category include [10], who study the hedging effectiveness of insurance, and a number of studies that look at the risk-reducing potential of insurance [11–14]. In a similar spirit, [15] study the catastrophic performance ratio (defined as expected payouts, normalized by the sum insured, in catastrophic, left tail states of the world). Somewhat similarly, [9] study the impact of insurance on lower partial moments of the probability distribution. While these approaches all offer valuable insights, they focus on changes in the left tail portion of the probability distribution. While what happens in the left tail is very important for the value of insurance, index contracts that incorrectly issue payouts in the right tail (false positives) are also damaging to the welfare value of insurance. This because as we discuss more below, it always costs more than $1 to get $1 of a payout. Paying more than a dollar to get a dollar makes sense if a dollar is worth more than a dollar, as it is in bad states of the world. In good states of the world, however, a dollar is worth only a dollar and paying, say, $1.25 to get a $1 does not improve economic welfare.

The second type of quality measures—those based on explicit measures of individual well-being—consider the full distribution of outcomes and avoid the incompleteness of lower tail measures. As measures of well-being, they also open the door to a natural quality measure: a good insurance contract should increase the individual’s level of expected well-being compared to her well-being without insurance ([6] call this a minimum quality standard). While there are several well-developed economic approaches to measuring individual well-being in the face of risk, we will first explain the use of “expected utility” normative framework to measure quality and then discuss its strengths relative to alternative approaches.

The economist’s standard utility function expresses economic well-being, or utility, U as a function of available purchasing power or wealth. It is conventionally assumed that people always prefer to have more wealth or purchasing power. That is, additional wealth makes a household economically better off, and the additional value to the household of additional wealth (or “marginal utility”) is strictly positive (mathematically, that ). We label this marginal value of additional wealth as the shadow value of wealth or money as it represents the additional level of economic well-being a household can achieve with one more dollar of wealth. We denote this shadow value of purchasing power or money as λ(k), noting the value of additional wealth depends on how much wealth the household has.

The conventional assumption in economics is that as we become richer, the additional value of another dollar of wealth stays positive, but becomes smaller (i.e., marginal utility diminishes, or mathematically that ). In other words, λ(k) becomes smaller as we get richer. Flipped around, this same shadow value sensibly increases as wealth decreases and the family becomes more desperate (a dollar is worth more in times of economic stress than in better times). It is the curvature of the utility function which ultimately creates the asymmetric loss function that underlies the value of insurance measures defined below.

In the analysis that follows, we assume that the utility function follows a specific functional form (known as a constant relative risk aversion utility function) that conforms to the assumptions just discussed: (6) where the parameter ρ shapes the degree of sensitivity to risk and bad outcomes. Fig 2 shows how λ changes as wealth changes for the case where ρ = 2. As ρ increases, λ increases more steeply as wealth diminishes. Throughout the analysis here, we assume that ρ = 2, an assumption supported by experiments that measure ρ for small scale farmers in developing countries (see [16]).

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TIFF original image Download: Fig 2. Elements of insurance benefit: The shadow value of money (λ), insurance benefit (Δ), and an empirical distribution (histogram) of the probability of the livestock assets remaining after accounting for mortality expressed in USD. To combine these measures on one figure, the y-axis was scaled to values shown on the figure. https://doi.org/10.1371/journal.pone.0258215.g002

While the utility function given in Eq (6) gives us the economic well-being associated with a given outcome, when considering risk, we need to consider a multiplicity of possible outcomes that might occur. When considering the economic value of a risky prospect (e.g., well-being in a world without insurance), the conventional economic analysis employed by [6] suggests valuing that prospect by the average or expected utility that might occur given the underlying random variable (livestock mortality in our case). Average or expected utility is the sum of all the possible utility levels that might occur, each weighted by their probability of occurrence. Formally, expected utility (or expected economic well-being) for our stylized household with 1 TLU, but no insurance, EUN, is calculated as follows: (7) while expected utility for the case of perfect insurance can be written as: (8) Using these expressions, we can define the insurance benefit to the household under perfect insurance (IBP) as the difference between their expected well-being with and without perfect insurance: (9)

Analogue expressions to (8) and (9) can be defined for any (imperfect) index insurance contract J. A minimum quality standard for that alternative imperfect insurance contract would be that IBJ>0, i.e., insurance does not harm and at least improves expected well-being of the insured household [6]. Although IBJ>0 is clearly a minimum standard (ideally the insurance benefits should not only be positive, but also be sufficiently large to be meaningful), it is a non-trivial one to fulfill as illustrated by the failure of the rainfall-based index insurance contract discussed in [1].

While expected utility theory has deep roots within the discipline of economics, its descriptive accuracy has been called into question by a range of experimental studies that show that some people do not make choices that conform with those that would maximize their expected utility. In efforts to accommodate systematic behavioral deviations from the predictions of expected utility theory [17] and [18] assembled alternative theories of how individuals make decisions in the face of risk, known as cumulative prospect theory (CPT) and rank dependent utility (RDU). These two alternative frameworks share the perspective that people make decisions using a probability weight that may differ systematically from objective probabilities. In addition, CPT assumes that individuals value losses differently than gains, creating a kink in the smooth utility function shown in Eq (6), where the kink occurs at what they call the reference point that distinguishes losses from gains. While such a reference point can be easily established in a behavioral experiment, its definition is less obvious in real world circumstances, making it more difficult to use CPT as a general tool for evaluating the quality of index insurance [19]. While CPT and RDU approaches have both been used to evaluate the value of index insurance to an individual (see [20, 21]), it is not obvious that a welfare metric based on misperception of probabilities is appropriate to judge and design insurance quality. Put differently, while these alternative approaches may be descriptively more accurate than expected utility theory in predicting insurance uptake, it is not apparent that they form the better basis for normative judgements [17, 19] especially if the source of objectively wrong probability weights is simple misperception.

While these issues of normative bases for value judgements are philosophically complex, in the remainder of this paper we will maintain our focus on expected utilty based quality metrics as those metrics seem most defensible for the purposes at hand. In addition, while all this discussion may seem obtuse to non-economists, the expected utility based quality metric in Eq (9) above has the additional virtue that it can be decomposed into three parts that most economist would agree should underwrite a conceptually sound measure of insurance quality:

Δ(M j ), the difference in final wealth with and without insurance under each possible mortality rate, where Δ(M j ) = kP(M j ) = kN(M j ) = IP(M j )−pP (Fig 1), λ(θ j ), the shadow value of money when the difference occurs; and, π j , the probability that a given mortality rate and difference in wealth occurs.

Specifically, using a first order Taylor Theorem approximation of Eq (9), the quality metric for the perfect insurance contract, IBP, can be approximated as: (10) In words, the quality metric, IBP, is just the average (probability-weighed) difference in wealth with and without insurance times the shadow value of money when the difference occurs. Fig 2 graphs the elements of this expression. Note that Δ(M j ) will most frequently be negative because in most seasons an insured household is expected to pay a premium but not to receive a payout, and this will drive IB downwards. However, when insurance works well, the shadow value of money (λ(M j )) will be low when Δ is negative, but λ will be high when Δ is positive. As shown by the graph, the value of insurance can only be positive if insurance correctly pays off when things are bad and λ is high. As we will see below with actual examples of imperfect insurance, the economic value of insurance is reduced substantially if the insurance fails to pay off during high mortality shocks when λ is high. Indeed IB can be negative if the contract is of a poor enough quality, meaning the household would be better off not buying the insurance.

While the basic quality metric, IBJ, can be used to judge whether any particular contract J is good enough to provide economic benefit to the insured household, we propose a relative quality measure that will allow choosing between different alternative remote sensing indices (based on sensor(s) employed, statistical model used to predict losses based on sensor readings, etc.). Before forming that measure, we first transform the underlying expected utility measures, EUN and the EUJ into more interpretable “certainty equivalent income” metrics. In words, the certainty equivalent of the pastoralist’s expected utility without insurance, denoted CEN is the amount of income that if received for certain every year, would give the same pastoralist expected utility level EUN. The certainty equivalent income for the no insurance case is calculated as Analogue expressions can be used to calculate the certainty equivalent income for the case when the pastoralist has insurance. Using our data, CEN = $848 for a moderately risk averse person who has a coefficient of relative risk version of 2. This amount is below the expected value of wealth ($863), indicating that the person would take a guaranteed wealth below their expected wealth to avoid the severe suffering that attends years in which the livestock mortality rate is high.

After calculating the same object for the perfect insurance case, we can define a transformed analogue to Eq (9) which we write as: (11)

Using these transformed measures, we now propose a Relative Insurance Benefit (RIB) measure that allows us to benchmark any feasible insurance contract J against the insurance benefit offered by the perfect, failure proof, insurance contract: (12) where RIBJ is the relative insurance benefit of contract J. The measure is only meaningful in the case where perfect insurance offers benefits to insured households (IBP>0). Note that RIBJ≤1, with a value of 1 indicating the contract is just as good as perfect insurance and a value less than zero indicating that the contract J does not even pass the minimum quality standard test and offers no value to the household. An alternative contract design K, which has RIBK>RIBJ, would be strictly preferred to contract J based on this insurance quality metric. More generally, the remote sensing challenge is to find an index that pushes RIB as close to one as possible. In the analysis to follow, we will show that while related to predictive skill measures that are often used to evaluate alternative contract designs, the conceptually-based RIB measure captures relevant features of contract design that predictive skill measures do not.

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