Basis risk and the demand for catastrophic rainfall insurance

Abstract

Rainfall is an important source of covariate shock in developing countries. Insurance against a rainfall index has, therefore, held much promise as a formal insurance product to protect the livelihoods of poor farmers. But how good is rainfall as a measure of covariate shocks? The imperfect association between them has been flagged as a reason for low demand for index insurance. Using district crop yields and rainfall data for India, we find that tail dependence characterizes the association between aggregate crop yields and rainfall. Could this property be exploited to design catastrophic loss insurance programs that have low basis risk? Using simulations of the estimated copulas, we show that the value of index-based insurance relative to actuarial cost is higher for insurance against extreme losses (of the index) than for insurance against all losses. We conclude that index insurance could find greater acceptability if it only insured extreme losses.

1. Introduction

Rainfall is an important source of covariate or aggregate shocks in developing countries (Walker and Ryan 1990; Giné, Townsend and Vickery 2008; Mobarak and Rosenzweig 2012; Cole et al. 2013; De Janvry, Ramirez Ritchie and Sadoulet 2016; Jensen, Mude and Barrett 2018). Insurance against a rainfall index has, therefore, held much promise as a formal insurance product to protect the livelihoods of poor farmers. Although there exist various informal mechanisms in rural communities that allow farmers to pool their idiosyncratic risks, such insurance even when effective, does not protect against covariate weather risks such as rainfall.

The design of rainfall insurance must, however, contend with the problem of basis risk (Giné, Townsend and Vickery 2008; Elabed and Carter 2015; Clarke 2016; Hill, Robles and Ceballos 2016) because of which producers might not receive insurance payouts when they need it (Morsink, Clarke and Mapfumo 2016). The severity of basis risk depends on answering the question: How good is rainfall as a measure of covariate shocks? In this paper, we examine the basis risk in contracts that insure against catastrophic or extreme losses in the rainfall index. Such contracts could also be considered as drought insurance. The suggestion has been made that ‘‘...basis risk should be lowest when weather index insurance is specifically designed to protect against catastrophic loss events....” (Collier, Skees and Barnett 2009). While it may be intuitive that weather indices could possibly capture spatially correlated loss events such as drought or extreme temperatures (rather than moderate risks), a detailed examination is necessary to establish its salience.

In Bokusheva (2018), the idea that dependence between farm yield and a weather index might be stronger and more stable for extreme observations (corresponding to a catastrophe) is explicitly investigated by modeling the joint distribution of weather index and farm yield as a copula. The weather index realizations below the first, second, and third deciles of its distribution are used to define threshold levels of the weather index that would signify extreme drought.

Like Bokusheva (2018), this paper also explores the hypothesis that basis risk is lower in catastrophic index insurance. We focus on the possibility that lower tail dependence might characterize the association between yields and rainfall. If so, associations between yield and index losses are stronger for large deviations than for small deviations. That would, in turn, imply lower basis risk in catastrophic index insurance. While this paper shares the objectives of Bokusheva (2018), our approach is different. Firstly, we use standard maximum likelihood methods to estimate copulas instead of Bayesian methods employed in Bokusheva (2018). Secondly, the focus of Bokusheva’s work is to quantify the reduction in downside risk using various statistical measures. We do that too but within a standard expected utility framework using the metric of catastrophic performance ratio (Morsink, Clarke and Mapfumo 2016) that allows us to enlarge the inquiry and to ask how tail dependence impacts the demand for crop insurance. Investigating this question is the key contribution of this paper. We find that the demand for catastrophe insurance is much greater than what would be predicted by models that ignore tail dependence.

We do this by estimating tail dependence in the joint distribution of weather (i.e., rainfall) and yields for a large district-level data set for all India and for nine major crops. Using maximum likelihood methods, the paper estimates a number of copulas from the parametric families of Elliptical copulas and the Archimedean copulas. Within a conceptual model of an insurance purchaser, the best-fitting copulas are simulated allowing us to measure the reliability of index insurance (following Clarke and Dercon 2016; Morsink, Clarke and Mapfumo 2016) and to estimate the optimal insurance cover for a variety of insurance contracts that vary according to the index threshold value that triggers payout. These results are compared with those obtained from a copula without tail dependence (the Gaussian copula).

Our findings confirm lower tail dependence in the joint distribution of rainfall and district-level crop yields. As a result, catastrophic rainfall insurance has lower basis risk than would be predicted by models that assume linear association between yields and rainfall. Consequently, we find that the demand for catastrophic insurance is greater than what would be predicted by models that ignore tail dependence. In addition, we find that the demand for catastrophic insurance is greater than insurance that also covers moderate losses.

The paper is organized as follows. The next section reviews the relevant literature and describes the contribution of this paper. Section 3 illustrates why tail dependence between yields and weather may follow from tail dependence between rainfall at different locations. We, therefore, expect the yield-weather tail dependence to hold in countries other than India as well. Section 4 describes the data and presents estimates for the copula function of average yields and rainfall. The estimated copulas exhibit tail dependence. The implications for rainfall insurance are investigated in Section 5. The paper concludes with Section 6.

2. Relation to literature

Despite the promise of index insurance, the record is mixed. In particular, the uptake of index insurance is poor, especially when it is not subsidized (Binswanger-Mkhize 2012; Jensen and Barrett 2017; Jensen, Barrett and Mude 2016). The literature has highlighted many reasons for the low uptake. These include a lack of familiarity about formal insurance, the lack of trust in the insurance provider, and the difficulties of communication resulting in poor understanding of the insurance product. Poor farmers also face liquidity constraints and insurance demand is highly sensitive to price (Cole et al. 2013; Cole, Stein and Tobacman 2014; Giné, Townsend and Vickery 2008; Matsuda and Kurosaki 2019).

An important strand of this literature highlights the fundamental constraint of basis risk that occurs because of imperfect correlation between the index and farmer losses. Research has shown that basis risk reduces the demand for insurance (Giné, Townsend and Vickery 2008; Elabed et al. 2013; Clarke and Dercon 2016; Hill, Robles and Ceballos 2016). The importance of acknowledging basis risk is stressed in a recent study that states “Discerning the magnitude and distribution of basis risk should be of utmost importance for organizations promoting index insurance products, lest they inadvertently peddle lottery tickets under an insurance label” (Jensen, Barrett and Mude 2016).

The basis risk in rainfall index insurance stems from two sources—first, the rainfall index may not accurately represent covariate shocks (the design risk) and second, covariate shocks may itself account for only a small part of the variation in individual farmer yield (Jensen, Barrett and Mude 2016). By design, index insurance can only protect against covariate shocks to local average area yield (Carter, Cheng and Sarris 2016; Ramaswami and Roe 2004; Miranda 1991; Carter et al. 2014). Hence, the importance of the question: To what extent does rainfall vary with local area average yield?¹

From an analysis of 270 weather insurance contracts in a state of India, Clarke et al. (2012) quantified the design risk in weather insurance. They estimated that there was a one-in-three chance of not receiving insurance payout in the event of a total production loss (of area average yield). In a follow-up analysis, Clarke (2016) argued that, if these contracts were priced commercially (i.e., unsubsidized), the basis risk in them was so great as to reduce optimal demand to zero. In a similar vein, Morsink, Clarke and Mapfumo (2016) proposed two measures of the reliability of index insurance. The first metric is the probability of not receiving an insurance payout in the event of a catastrophic loss. The second measure is the ratio of expected payout to premium in the event of a catastrophic loss.

This paper builds on these contributions to similarly quantify the design risk in rainfall insurance. However, more fundamentally, departing from the literature, the paper also examines the sensitivity of design basis risk to the trigger. A catastrophe loss contract insures only severe losses and, is therefore, triggered by a larger loss in the index compared with a contract that also compensates small losses. This way the paper is able to examine the hypothesis that basis risk is lower in catastrophic insurance.

Such a possibility arises when the distribution of rainfall and area yields exhibit lower tail dependence. Previous work has found lower tail dependence with regard to the spatial association of weather indices including rainfall (Kuhn et al. 2007; Liu and Miranda 2010; Aghakouchak, Ciach and Habib 2010; Okhrin, Odening and Xu 2013). If yields depend on weather, then their spatial association would inherit the property of tail dependence. This has been confirmed in the literature (Goodwin 2014; Goodwin and Hungerford 2015; Du et al. 2017). Given these associations, tail dependence between rainfall indices and average yield would be highly likely. We demonstrate this possibility in the next section.

We note that tail dependence between average yields and rainfall was investigated in Bokusheva (2011, 2018). In this paper, we too use copula-based methods to examine tail dependence between yields and rainfall. Our goal is use the estimates of tail dependence to derive its implications for basis risk as well as for the demand for catastrophic insurance.

3. Tail dependence

As noted earlier, the literature has found tail dependence in the spatial association of rainfall in various regions. In this section, we construct an illustrative example to argue that this may imply tail dependence between average yield and a rainfall index as well. The example motivates our paper.

To see this in a simple manner, suppose that the area yield is defined over two locations l = {a, b}. Tail dependence in the joint distribution of rainfall at these locations would imply that the lower quantile dependence probability is bounded away from zero, i.e.,

for arbitrarily small t and where R_l and F_l are, respectively, rainfall and its cumulative probability at location l. Assume yields depend on rainfall in the following manner: |$Y_l =\overline{Y}_l R_l$|⁠. Let G_l(.) denote the cumulative distribution of yield Y_l. It follows that |$G_l (Y_l) = G_l (\overline{Y}R_l) = F_l (R_l)$| for l = {a, b}. Thus, it is clear that yields at both locations would also be tail dependent, i.e.,

Equation (1) has another implication. Let f(R_a, R_b) be a linear transformation of R_a and R_b where f is a rainfall index. If R_a and R_b are tail dependent, it follows that the index f(R_a, R_b) is also tail dependent with R_a and also R_b. This is because the events that lead to large rainfall shortfalls in the two locations will also cause the rainfall index to be in the lower extremity of its distribution. Similarly, if we let g(Y_a, Y_b) be the average farm yield, then it is clear that average farm yield is also tail dependent with Y_a and with Y_b. To write this, let f(R_a, R_b) = γR_a + (1 − γ)R_b and g(Y_a, Y_b) = αY_a + (1 − α)Y_b. Also let H be the cumulative distribution function for f and K be the cumulative distribution function for g. Then, for l = {a, b}, we have

Similarly,

But |$G(Y_l )=G(\overline{Y} R_l ) = F(R_l )$|⁠. Hence, Equation (4) becomes

This implies average yields are lower tail dependent with rainfall in location a and with rainfall in location b. However, rainfall at either location is tail dependent with the rainfall index. Hence, it follows that

Thus, tail dependence in the spatial distribution of weather may lead to tail dependence in the spatial distribution of yields.

We complement the above reasoning with a simulation exercise. We first generate two random variables R_a and R_b from Clayton copula. These exhibit lower tail dependece by design. Assume that these are rainfalls at two locations. We then simulate yields at two locations as a linear stochastic function of rainfalls at the two locations. Finally, we generate f(R_a, R_b) and f(Y_a, Y_b) as simple averages of rainfall and yields at two locations. Table 1 presents the lower quantile dependence probabilities from our simulation exercise. As can be observed from the estimates in the table, all lower quantile dependence probabilities are positive. The simulation reconfirms that tail dependence in the joint distribution of rainfalls flows through to tail dependence in average yield and average rainfall.

4. The joint distribution of average area yields and average area rainfall

4.1 Data

District yields are collected from the district database of the International Crops Research Institute for the Semi-Arid Tropics that is compiled from various official sources.² To maintain consistency and comparability of time series across districts, the data refer to the district boundaries as they were in 1966.

India receives 85% of the annual rainfall during the monsoon months of June–September. Crops grown during this period depend on rainfall as it represents a major source of uncertainty. These are the kharif season crops (June–October).³ A rainfall insurance contract is meaningful, therefore, for crops grown during this period. In the data set, the kharif crops comprise Maize, Cotton, Sorghum, Finger millet, Pigeon pea, Soybean, Pearl millet, Groundnut, and Rice. Our analysis spans these nine crops across 311 districts in 19 states from the year 1966–67 to 2011–12. Crop yields typically exhibit significant upward trends overtime due to technological changes. Yield deviations are estimated by fitting a linear trend to log yields of each crop for each district.

The high resolution gridded rainfall data from the Indian Meteorological Department is used to construct district level kharif season rainfall as cumulative rainfall for the months from June to October. The cumulative seasonal rainfall is transformed as standard deviations from their long-term district average.

4.2 Joint distribution estimation using copulas

Table 2 presents the coefficients of linear and rank correlation between log yield and rainfall deviations. The table also reports the bootstrapped standard errors for the correlation coefficients. As expected, both linear and rank correlations show a statistically significant positive association between yield and rainfall deviations, despite some difference in their magnitude.

We use copula functions to capture the asymmetric dependence between yield and rainfall deviations by fitting copulas to rank-based empirical marginal distributions of yield and rainfall deviations. The copula function provides a flexible way to bind the univariate marginal distributions of random variables to form a multivariate distribution and can accommodate different marginal distributions of the variables (Nelsen 2006; Trivedi and Zimmer 2007). A two-dimensional copula can be defined as a function |$C(u,v):[0,1]^2\longrightarrow [0,1]$| such that

Where θ represents the strength of dependence and G(.) and F(.) are the marginal distribution functions of Y and X, respectively. The joint probability density function can be expressed as

Sklar (1959) showed that for a continuous multivariate distribution, the copula representation, as described in Equations (7) and (8) with the probability density function in Equation (9), holds for a unique copula C. The copula representation holds for any marginal distribution. This construction allows us to estimate separately the marginal distributions and the joint dependence of the random variables. There are several parametric families of copula available in the literature. The frequently used ones are the elliptical copulas and the Archimedean copulas. Note that the nature of dependence among the random variables will depend on the copula function chosen for estimation. The statistical properties of the copulas that we use in this paper are given in Appendix Table A1.

We use two-step maximum likelihood procedure to estimate the copula function wherein the marginals are estimated in the first step and the dependence in the second step by substituting the estimated marginal distributions in the selected copula function (Trivedi and Zimmer 2007). A non-parametric estimator is used to estimate the univariate marginal distribution for crop yield deviations and rainfall deviations. This makes the model semiparametric. Estimation of copula using a non-parametric distribution does not affect the asymptotic distribution of the estimated copula dependence parameter (Chen and Fan 2006).

A simple maximum likelihood estimator can be used to choose the best-fitting copula and estimate the dependence parameter (Patton 2013). Selection of the copula model can be made based on the Akaike Information Criterion (AIC) or Bayesian Information Criterion. If all the copulas have equal number of parameters, then the choice of a model based on these criteria is equivalent to choosing a copula with the highest log likelihood (Trivedi and Zimmer 2007). The log likelihood function of the copula can be written as

where |$U_{Y_{i}}$| and |$U_{X_{i}}$|⁠, are the cumulative probabilities from the marginal distribution of yield and rainfall deviations.⁴ The copula parameter can be estimated by maximizing the likelihood function using numerical methods. This procedure gives the “Inference Functions for Margins” estimator as θ is conditional on the model that is used to transform the raw data (Trivedi and Zimmer 2007; Patton 2013). All copula models and tail dependence statistics are estimated using Patton’s (2013) procedure and Matrix Laboratory (MATLAB) codes.

Recall that we have a panel of observations on district yield and rainfall for 45 years across 311 districts. For the pooled data, Table 3 presents the log likelihood values from the eight copula models fitted to the data. For all crops, the Clayton copula is the best model to describe the dependence between yield and rainfall deviations. This is not surprising as Clayton copula exhibits only lower tail dependence and no upper tail dependence. Copula models that allow only upper tail dependence perform the worst (Gumbel, rotated Clayton). Table 4 presents the parameters of the Clayton copula with bootstrapped standard errors and lower tail dependence based on the fitted copula parameter.

The estimated Clayton copula density for different crops is presented in Fig. 1. As expected, all crops show significantly higher density at the lower tail. This further confirms that the association between yield and rainfall deviations is stronger at the lower tail. This means that when the rainfall index is abnormally low, so is the average yield. Therefore, the basis risk is low for extreme shortfall in rainfall.

In a second exercise, we depart from the pooled analysis. We fit all the selected eight copula models to each district that has at least 40 data observations. Based on the log likelihood values and the AIC criterion, we choose the one that best describes the dependence. Table 5 summarizes the results. For example, in the case of rice, Clayton copula gives best fit for 40 percent of the 274 rice growing districts. Student’s t copula is the next best. Across all crops, about 70 percent of the best-fitting cases are accounted by either the Clayton copula or the Student’s t copula. These findings clearly indicate nonlinearity in association between weather and yield risk and have implications for the demand for insurance and thus, its design.

Copula estimation is based on the assumption that the association between yield and rainfall deviations is monotonic. To see if our findings are sensitive to this assumption, we quantify the probability mass in the lower and upper tail of the joint distribution using conditional quantile dependence probabilities. The conditional quantile dependence probabilities are estimated independently for the lower and upper extremes of the variables and, hence, do not impose any monotonic functional form on the data. The estimates from the upper and lower conditional quantile dependence probabilities are presented in Appendix  A and reinforce the point that there is a stronger association between yield and rainfall deviations in the lower tail of the joint distribution (Appendix Fig. A1 and A2).

5. Implications for rainfall insurance

5.1 The catastrophic performance ratio

Our findings show that the joint density of yield and rainfall exhibit lower tail dependence, i.e., a stronger association between yield and rainfall when rainfall is abnormally low. This implies that the basis risk varies across the joint distribution of yield and index. It opens up the possibility of designing insurance such that it covers the losses with the least basis risk. Here, we analyze the implications of these findings for the demand and design of index insurance.

We assume that the payout from one unit of rainfall-based insurance contract is given by

where X is the rainfall index with distribution function F(X) and |$\hat{X}$| is the rainfall threshold set by the insurance selling agency. If the threshold is lower, the trigger in the insurance payouts is higher. The contract triggers payouts only if actual rainfall falls below |$\hat{X}$|⁠. The implicit assumption in offering such a contract is that farmers’ yield and the rainfall index are correlated, such that in periods of low rainfall, crop yields will also be lower. The actuarially fair price P of such a contract is just the expectation of I.

The net profits per unit land of a farmer purchasing a rainfall insurance contract can be written as

where Y is the yield and α is the number of insurance units purchased and m is the markup over actuarially fair insurance. As before, Y is distributed as G(.). Starting from no insurance, the increment to expected utility because of insurance is given by

where u(.) is the utility function of the farmer with |$u^{{\prime }}(.)\gt 0$| and |$u^{{\prime \prime }} (.)\lt 0$| and η is expected utility expressed as a function of α. Hence, we have

where H(X∣Y) is the density of rainfall conditional on yield. This can be rewritten as

The term inside the square brackets is nothing but E(I ∣ Y) − mP. Hence, we have

From the above, it can be seen that the insurance demand is 0 if E(I∣Y) ≤ mP, for all values of Y. This result is a restatement of a theorem in Clarke (2016). Following Morsink, Clarke and Mapfumo (2016), we define the catastrophic performance ratio as the following:

For every level of output Y, the catastrophic performance ratio measures the average amount a farmer gets back as claims received per dollar of commercial premium.⁵ If the basis risk is large, the catastrophic performance ratio will be small even at low output levels, i.e., the insurance payouts are small even when losses are large. In such a case, it may be optimal not to buy any index insurance. Clarke shows that if κ(Y) ≤ 1 over the entire yield distribution, then α = 0 for all risk-averse individuals. In our model, this result follows from Equation (17).

Clarke et al. (2012) used the payout structure of 270 weather-based crop insurance products sold to Indian farmers in one state in 1 year and combined it with historical data to simulate payouts over the period 1999–2007. Their work found the ratio κ(Y) to be almost flat over the entire yield distribution. The ratio is below 1 for most values of Y and is barely above 1 for very low levels of Y. It follows then that the basis risk in these contracts is so large that it would be optimal not to purchase them.

Morsink, Clarke and Mapfumo (2016) proposed that the catastrophic performance ratio be used as a measure of reliability of weather insurance contracts. Concerned about basis risk during catastrophic losses, they argued for evaluating Equation (18) at low yield levels. In particular, it should not be too low. They further suggested that the ratio could be used to “improve the quality of products, protect consumers, and reduce reputational risk”. The higher the ratio, the greater are the payouts when it really matters, i.e., when losses are large. This is a necessary condition for insurance demand to be positive. In their example drawn from Clarke et al. (2012), the catastrophic performance ratio was barely above 1 even when losses exceeded 70 percent.

In what follows, we use the catastrophic performance ratio as a metric to examine how tail dependence matters to basis risk. More specifically, we ask whether the catastrophic performance ratio varies between the Gaussian copula (with zero tail dependence) and copulas with lower tail dependence. We also ask how the catastrophic performance ratio varies with trigger levels.

A hypothetical rainfall insurance contract of the form in Equation (11) is considered. The payoffs are simulated using 10,000 draws of rainfall and yield from a Gaussian copula and from a copula exhibiting lower tail dependence. The correlation between the two variables is held constant across the two copulas. The comparison of the performance ratio across the two copulas is, then, revealing about the effect of tail dependence.

The exact procedure is as follows: For both these copulas, the marginal distribution of yield and rainfall is assumed to be normal with a mean of 2,000 and standard deviation of 300. In the last section, the best-fitting copula to the joint distribution of rice yields and rainfall was found to be the Clayton copula with a parameter of 0.42. The marginal distributions are combined in a Clayton copula with a parameter of 0.42 to generate 10,000 observations of yield and rainfall. These observations are used to compute the insurance payoffs. The linear correlation between rainfall and yield draws from the Clayton copula is combined with the assumed marginal distributions to generate another 10,000 observations from a Gaussian copula.

Thus, we have two empirical joint distributions such that they share the same marginal distributions and the same correlation between rainfall and yield. The only difference is that yield and rainfall index simulated from Clayton copula exhibit lower tail dependence, while the other does not.

Figure 2 a plots the non-parametrically estimated relationship between claims to commercial premium ratio and yield from the simulated data, i.e., the catastrophic performance ratio

where the insurance contract parameter |$\hat{X}$| is assumed to be 1 standard deviation below the mean rainfall and m is assumed to be 1.56 times the actuarially fair premium. At this premium level, the catastrophic performance ratio was below 1 for the rainfall insurance contracts considered by Clarke et al. (2012). This is not true, however, for the payouts from rainfall contracts in Fig. 2a. The ratio from the Gaussian copula and from the Clayton copula is above 1 for low yield levels. There is, however, a substantial divergence between the Gaussian copula distribution and the Clayton copula at these low yield levels. The catastrophic performance ratio is substantially higher for the Clayton copula though it is lower than a perfect insurance contract that triggers payout for all yield shortfalls below 1 standard deviation of the mean yield. Thus, by the measures proposed by Morsink, Clarke and Mapfumo (2016), accounting for tail dependence markedly reduces basis risk.

The denominator in the catastrophic performance ratio is the premium cost of covering losses. Given lower tail dependence, most of the benefits of insurance could be preserved by restricting coverage to severe or catastrophic losses while reducing the cost of insurance substantially. Hence, it would seem that the catastrophic performance ratio could be boosted by a higher trigger. Figure 2b plots the Clayton copula-based catastrophic performance ratio for different levels of the trigger. |$\hat{X}$| is chosen to be either the mean, or 0.5 standard deviation below the mean, or 1 standard deviation below the mean. It can be seen that as the trigger rises (i.e., |$\hat{X}$| falls) so does the catastrophic performance ratio. Catastrophic insurance carries the least basis risk when insurance covers only the yield losses beyond 1 standard deviation below the mean. The catastrophic performance ratio at a 50 percent loss (yield of 1,000) is above 6 (i.e., $6 of payout for a dollar of premium). By contrast, the catastrophic performance ratio halves when all losses below the mean are covered. The difference in the ratio is due to the lower premium costs in the first case.

5.2 Demand for insurance

In Fig. 2a and b, the catastrophic performance ratio is well above unity at lower levels of output. This suggests that insurance demand may well be positive. That depends on the evaluation of Equation (17) which, in turn, depends on the extent of risk aversion. Equation (17) can also be written as

Risk aversion and the expected shape of the regression E(I∣Y) guarantees the first term to be positive. When insurance is actuarially fair, the second term is 0, and it is optimal for farmers to buy some insurance. When m > 1, the answer would depend on risk aversion and the markup over the fair premium.

To investigate these issues, we consider data from two districts in India, Mahabubnagar and Anantapur, that have been heavily researched for the extent of local risk sharing (e.g., Townsend 1994). These districts are characterized by dependence on rainfed agriculture and vulnerability to droughts. Households in these districts have also been recently surveyed for their risk aversion using Binswanger-type lotteries (Binswanger 1980; Cole et al. 2013), and we use those estimates. Using the procedures described in the previous section, a best-fitting copula model is selected for rice yields and rainfall in each of the two districts. Table 6 displays the results. Unlike the previous exercise that generated Fig. 2a and b, we do not assume here the normality of marginal distributions of rainfall and yield. Instead, we consider various parametric form and choose the best-fitting functional form (see Figures A3 and A4 and Table 6).⁶ Rainfall appears to be log normal in both districts. Yields appear to follow a Weibull distribution in Anantapur and follow a gamma distribution in Mahabubnagar. Plots of estimated parametric distributions against the observations are presented in the Appendix.

These marginal distributions are combined in the appropriate copula (as in Table 6) to generate 10,000 observations of yield and rainfall. These observations are used to compute the insurance and payoffs. The linear correlation between these rainfall and yield draws is combined with the selected marginal distributions to generate another 10,000 observations from a Gaussian copula.

Fig. 3a and b show the catastrophic performance ratios for these districts. These pictures are very much like Fig. 2a and b. Once again, basis risk is much lower relative to a Gaussian copula. Further, basis risk falls with a larger trigger.

Next, we move to an evaluation of Equation (17). For a constant risk aversion utility function with parameter γ, Equation (17) becomes

Based on the work of Cole et al. (2013), the risk aversion parameter is assumed to be 0.57. The above equation can be used to compute the markup over the actuarially fair premium for which insurance demand is positive. From the results displayed in Fig. 4, it can be seen that the m that extinguishes insurance demand is higher for a tail-dependent copula as compared with a Gaussian copula. This is simply a reflection of the lower basis risk that comes with lower tail dependence. A second finding of Fig. 4 is that the maximum markup for which insurance demand is positive is higher when the trigger is larger. This again is a reflection of the earlier Fig. 3b that showed the basis risk is lowest in contracts with the smallest rainfall threshold.

For the constant relative risk aversion utility function, the optimal insurance units can be solved from

The payouts and the premium that were simulated to compute the catastrophe performance ratios can also be used to evaluate Equation (22). We continue to use γ = 0.57. Optimal insurance cover is computed with and without tail dependent yield and rainfall distribution and for insurance contracts that vary according to the index threshold value that triggers payout. The results are displayed in Fig. 5 where the computations assume m = 1. What is noteworthy about the results is that the optimal insurance cover is much larger with a tail dependent copula than with a Gaussian copula. This is consistent with the lower basis risk that accompanies a tail dependent copula.

These results will continue to hold for actuarially unfair premiums as log as the markup is close to 1. For sufficiently high m, however, demand will be driven to zero in all of these cases. From Fig. 4, it is clear that this would happen even with moderately high premiums (above 1.08). In this data set, weather insurance does not appear to be commercially viable. If index insurance is subsidized to lower m below 1.08, then we would once again get the same qualitative picture as in Fig. 5.

Finally, we must note a caveat to our results. We have implicitly assumed that farmers are fully insured against idiosyncratic risks and care only about aggregate risks. If risk sharing is partial, farmers bear some idiosyncratic risks as well and that would mean the basis risk is greater than what we have calculated.

6 Conclusions

Although cost effective and free from moral hazard and adverse selection, index-based crop insurance products have seen poor uptake because of imperfect association between the index and crop loss that reduces the value of insurance and, therefore, its demand.

In this paper, we found the association between crop yield and rainfall index, for nine major crops in India, to be characterized by the statistical property of “tail dependence”. As our analysis has explained, the important implication of our findings is that, for farmers, the utility of index-based insurance relative to actuarial cost is greater for insurance limited to catastrophic losses than for insurance against all losses. For this reason, tail dependence boosts the demand for catastrophic insurance. This may, therefore, be one route for rainfall index insurance to receive greater uptake and for it to be an effective, if limited, risk management strategy.

Crop insurance is frequently subsidized in many countries. Collier, Skees and Barnett (2009) proposed that the subsidy should be limited to catastrophic losses because otherwise the subsidy might impede producer adaptation to climate change. The results of this paper suggest an additional argument for this proposal. Since basis risk is least for catastrophic insurance, a subsidy on the extreme layer of risk is likely to maximize net social benefits.

Our model of insurance decisions assumed an expected utility maximizing agent. In future work, the implications of tail dependence should be investigated in other contexts as well. Extension to other behavioral theories, such as prospect theory, is a promising direction of future work since theories that emphasize downside risks and loss aversion may make catastrophic rainfall insurance even more salient (Chavas 2019). Another worthwhile extension would be to estimate lower tail dependence using time-series data of yields and rainfall at lower aggregation levels when such data are available. Some risks that are idiosyncratic at higher aggregation levels merge with aggregate risks at lower levels of aggregate insurance (Ramaswami and Roe 2004). As a result, smaller spatial units have a greater scope for index insurance. However, it may well be that tail dependence is still strong enough to justify catastrophic insurance cover over normal insurance.

We note that farmers may not purchase insurance for other reasons as well including poor understanding of the product, credit constraints, low trust of the insurance seller, and optimism about yields. If these are binding constraints, then a reduction in basis risk may not impact the demand for insurance.

Finally, we wish to point out that tail dependence is unlikely to be India specific since it flows from the nature of spatial associations of weather. Therefore, although our results are based on Indian data, the general lessons are available for other countries too.

Acknowledgments

We would like to thank the two anonymous reviewers and the editor for valuable comments and feedback during the review process.

Data availability

The data supporting this article is available in the public domain. Codes to replicate the main results of the paper would be available from the corresponding author upon reasonable request.

Footnotes

Appendix A. Local measures of association

We quantify the probability mass in the lower tail of the joint distribution by estimating the conditional quantile dependence probabilities for the lower (p^L) and higher (p^U) extremes of the transformed variables as:

where |$U_{Y_{i}}$| and |$U_{X_{i}}$|⁠, are the cumulative probabilities from the marginal distribution of yield and rainfall deviations and are estimated non-parametrically. Fig. A1 plots the estimated upper and lower tail quantile dependence plots. A corollary is that the joint distribution of crop yield and rainfall deviations exhibit asymmetric tail dependence. The difference between the upper and lower quantile dependence is statistically significant and is greater at lower quantiles (Fig. A2). These results clearly reveal that there is more mass in the lower tail of the joint distribution of yields and rainfall.

B. Copula models

References