1 Introduction
Inclusive insurance (or microinsurance) relates to the provision of insurance services to lowincome populations with limited access to mainstream insurance or alternative effective risk management strategies. Many individuals excluded from basic financial services and those microinsurance aims to proctect, live below the minimum level of income required to meet their basic needs. Currently fixed at $1.90 USD per day, 9.2% of the population were estimated to live below the international extreme poverty line in 2017
(Online:WorldBank2021). Increases in the number of new poor and those returning to poverty as a result of the COVID19 pandemic are expected to reverse the historically declining poverty trend (Book:WorldBank2020).Fundamental features of the microinsurance environment such as the nature of low income risks, limited financial literacy and experience, product accessibility and data availability, create barriers to penetration, particularly in relation to the affordability of products. For the proportion of the population living just above the poverty line, premium payments heighten the risk of poverty trapping and induce a balance between profit and loss as a result of insurance coverage, dependent on the entity’s level of capital. Here, poverty trapping refers to the inability of the poor to escape poverty without external help (Article:Kovacevic2011).
Highlighting vulnerability reduction and investment incentive effects of insurance, Article:Janzen2020 observe a marked reduction in longterm poverty and the social protection costs required to close the poverty gap following introduction of an asset insurance market. Calibrating their model to riskprone regions in Africa, their study suggests that those in the neighbourhood of the poverty line do not optimally purchase insurance (without subsidies), suppressing their consumption and mitigating the probability of trapping. Article:Kovacevic2011 propose negative consequences of insurance uptake for members of lowincome populations closest to the poverty line, applying ruintheoretic approaches to calculation of the trapping probability. Article:Liao2020 support these findings in their analysis of a multiequilibrium model with agricultural output risks on data from rural China. Voluntary insurance would enable individuals close to the poverty threshold to opt out of insurance purchase in favour of alternative risk management strategies, in order to mitigate this risk.
In line with the findings of Article:Singh2020 on the effectiveness of social protection mechanisms for poverty alleviation, Article:Jensen2017 observe a greater reduction in poverty through implementation of an integrated social protection programme in comparison to pure cash transfers. Government subsidised premiums are the most common form of aid in the context of insurance. Besides reducing the impact on household capital growth, lowering consumer premium payments has the potential to increase microinsurance takeup, with wealth and product price positively and negatively influencing microinsurance demand, respectively (Article:Eling2014).
Poverty traps are typically studied in the context of economics, with a large literature focus on why economic stagnation below the poverty line occurs in certain communities. While the poor could readily grow their way out of poverty by adopting profitable strategies such as productive asset accumulation, opportunistic exchange and implementation of costeffective production technologies, poverty traps are underlined by poverty reinforcing behaviours induced by the state of being poor (Article:Barrett2016). A detailed description of the mechanics of the poverty trap state is provided by Book:Matsuyama2010. In studying the probability of falling into such a trap, “trapping” describes the event in which a household falls underneath the poverty line and into the area of poverty.
In this paper, we adopt the ruintheoretic approach to calculating the trapping probability of households in lowincome populations presented by Article:Kovacevic2011, adapting the piecewise deterministic Markov process such that households are subject to large shocks of random size. In line with the poverty trap ideology, we assume that the area of poverty to be an absorbing state and so consider only the state of events above the poverty threshold. Obtaining explicit solutions for the trapping probability, we compare the influence of three structures of microinsurance on the ability of households to stay above the poverty line. Specifically, we consider a (i) proportional, (ii) subsidised proportional and (iii) subsidised proportional with barrier microinsurance scheme. Aligning with the essential place for governmental support in the provision of social protection which encompasses risk mitigation, we assess for the first time in this context, to the best of our knowledge, the impact of a (government) subsidised insurance scheme with barrier strategy. We optimise the barrier level in the context of the trapping probability and the governmental cost of social protection, identifying the proportion of the population for which such a product would be beneficial. Here, the cost of social protection is defined to account for the provision of government subsidies, in addition to the cost of lifting a household from poverty, should they fall underneath the threshold. The benefit of subsidy schemes for poverty reduction is measured through observation of this governmental cost, in addition to the trapping probability of the households under consideration.
The remainder of the paper will be structured as follows. In Section 2, we introduce the household capital model and its associated infinitesimal generator. The (trapping) time at which a household falls into the area of poverty is defined in Section 3, and subsequently the explicit trapping probability and the expected trapping time are derived for the basic uninsured model. Links between classical ruin theoretic models and the trapping model of this paper are stated in Sections 2 and 3. Microinsurance is introduced in Section 4, where we assume a proportion of household losses are covered by a microinsurance policy. The capital model is redefined and the trapping probability is derived. Sections 5 and 6 consider the case where households are proportionally insured through a government subsidised microinsurance scheme, with the impact of a subsidy barrier discussed in Section 6. Optimisation of the subsidy and barrier levels is presented in Sections 5 and 6, alongside the associated governmental cost of social protection. Concluding remarks are provided in Section 7.
2 The Capital Model
The fundamental dynamics of the model follow those of Article:Kovacevic2011, where the growth in accumulated capital of an individual household is given by
(2.1) 
where . The capital growth rate incorporates household rates of consumption, income generation and investment or savings, while represents the threshold below which a household lives in poverty. Reflecting the ability of a household to produce, accumulated capital is composed of land, property, physical and human capital, with health a form of capital in extreme cases where sufficient health services and food accessibility are not guaranteed (Article:Dasgupta1997). The notion of a household in this model setting may be extended for consideration of poverty trapping within economic units such as community groups, villages and tribes, in addition to the traditional household structure.
The dynamical process in (2.1) is constructed such that consumption is assumed to be an increasing function of wealth (for full details of the model construction see Article:Kovacevic2011). The poverty threshold represents the amount of capital required to forever attain a critical level of income, below which a household would not be able to sustain their basic needs, facing elementary problems relating to health and food security. Throughout the paper, we will refer to this threshold as the critical capital or the poverty line. Since (2.1) is positive for all levels of capital greater than the critical capital, points less than or equal to are stationary (capital remains constant if the critical level is not met). In this basic model, stationary points below the critical capital are not attractors of the system if the initial capital exceeds , in which case the capital process grows exponentially with rate .
Using capital as an indicator of financial stability over other commonly used measures such as income enables a more effective analysis of a household’s wealth and wellbeing. Households with relatively high income, considerable debt and few assets would be highly vulnerable if a loss of income was to occur, while lowincome households could live comfortably on assets acquired during more prosperous years for a longperiod of time (Book:Gartner2004).
In line with Article:Kovacevic2011, we expand the dynamics of (2.1) under the assumption households are susceptible to the occurrence of large capital losses, including severe illness, the death of a household member or breadwinner and catastrophic events such as floods and earthquakes. We assume occurrence of these events follows a Poisson process with intensity , where the capital process follows the dynamics of (2.1) between events. On the occurrence of a loss, the household’s capital at the event time reduces by a random amount . The sequence is independent of the Poisson process and i.i.d. with common distribution function . In contrast to Article:Kovacevic2011, we assume reduction by a given amount rather than a random proportion of the capital itself. This adaptation enables analysis of a tractable mathematical model without threatening the core objective of studying the probability that a household falls into the area of poverty.
A household reaches the area of poverty if it suffers a loss large enough that the remaining capital is attracted into the poverty trap. Since a household’s capital does not grow below the critical capital , households that fall into the area of poverty will never escape. Once below the critical capital, households are exposed to the risk of falling deeper into poverty, with a risk of negative capital due to the dynamics of the model. A reduction in a household’s capital below zero could represent a scenario where total debt exceeds total assets, resulting in negative capital net worth. The experience of a household below the critical capital is, however, out of the scope of this paper.
We will now formally define the stochastic capital process, where the process for the interevent household capital (2.2
) is derived through solution of the first order ordinary differential equation (
2.1). This model is an adaptation of the model proposed by Article:Kovacevic2011.Definition 1.
Let be the event time of a Poisson process with parameter , where Let
be a sequence of i.i.d. random variables with distribution function
, independent of the process . For , the stochastic growth process of the accumulated capital is defined as(2.2) 
At the jump times , the process is given by
(2.3) 
The stochastic process is a piecewisedeterminsitic Markov process (Article:Davis1984) and its infinitesimal generator is given by
(2.4) 
The capital model as defined in (2.2) and (2.3) is actually a wellstudied topic in ruin theory since the 1940s. Here, modelling is done from the point of view of an insurance company. Consider the insurer’s surplus process given by
(2.5) 
where is the insurer’s initial capital, is the constant premium rate, is the riskfree interest rate, is a Poisson process with parameter which counts the number of claims in the time interval , and is a sequence of i.i.d. claim sizes with distribution function . This model is also called the insurance risk model with deterministic investment, which was first proposed by Article:Segerdahl1942 and subsequently studied by Article:Harrison1977 and Article:Sundt1995. For a detailed literature review on this model prior to the turn of the century, readers can consult Article:Paulsen1998.
Observe that when , the insurance model (2.5) for positive surplus is equivalent to the capital model (2.2) and (2.3) above the poverty line . Subsequently, the capital growth rate in our model corresponds to the riskfree investment rate of the insurer’s surplus model. More connections between these two models will be made in the next section after the first hitting time is introduced.
3 The Trapping Time
Let
(3.1) 
denote the time at which a household with initial capital falls into the area of poverty (the trapping time), where is the infinitetime trapping probability. To study the distribution of the trapping time, we apply the expected discounted penalty function at ruin concept commonly used in actuarial science (Article:Gerber1998), such that with a force of interest and initial capital , we consider
(3.2) 
where is the deficit at the trapping time and is an arbitrary nonnegative penalty function. For more details on the so called GerberShiu risk theory, the interested reader may wish to consult Book:Kyprianou2013. Using standard arguments based on the infinitesimal generator, can be characterised as the solution of the IntegroDifferential Equation (IDE)
(3.3) 
where
(3.4) 
Due to the lack of memory property, we consider the case in which losses (
) are exponentially distributed with parameter
. Specifying the penalty function such that , becomes the Laplace transform of the trapping time, also interpreted as the expected present value of a unit payment due at the trapping time. Equation (3.3) can then be written such that(3.5) 
Applying the operator to both sides of (3.5), together with a number of algebraic manipulations, yields the second order homogeneous differential equation
(3.6) 
Letting , such that is associated with the change of variable , (3.6) reduces to Kummer’s Confluent Hypergeometric Equation (Book:Slater1960)
(3.7) 
for and , with regular singular point at and irregular singular point at (corresponding to and , respectively). A general solution of (3.7) is given by
(3.8) 
for arbitrary constants . Here,
(3.9) 
is Kummer’s Confluent Hypergeometric Function (Article:Kummer1837) and denotes the Pochhammer symbol (Book:Seaborn1991). In a similar manner,
(3.10) 
is Tricomi’s Confluent Hypergeometric Function (Article:Tricomi1947). This function is generally complexvalued when its argument is negative, i.e. when in the case of interest. We seek a realvalued solution of over the entire domain, therefore an alternative independent pair of solutions, here, and , to (3.7) are chosen for .
To determine the constants and , we use the boundary conditions at and at infinity. Applying equation (13.1.27) of Book:Abramowitz1964, also known as Kummer’s Transformation , we write (3.8) such that
(3.11) 
for . For , it is wellknown that
(3.12) 
and
(3.13) 
(see for example, equations (13.1.4) and (13.1.8) of Book:Abramowitz1964). Asymptotic behaviours of the first and second terms of (3.11) as are therefore given by
(3.14) 
and
(3.15) 
respectively. For , (3.14) is unbounded, while (3.15) tends to zero. The boundary condition , by definition of in (3.2), thus implies that . Letting in (3.5) and (3.8) yields
(3.16) 
Hence, and the Laplace transform of the trapping time is given by
(3.17) 
Remarks.

[label=()]

Figure 1(a) shows that the Laplace transform of the trapping time approaches the trapping probability as tends to zero, i.e.
(3.18) As , (3.17) yields
(3.19) Figure 1(b) displays the trapping probability for the stochastic capital process . We can further simplify the expression for the trapping probability using the upper incomplete gamma function . Applying the relation
(3.20) (see equation (6.5.3) of Book:Abramowitz1964) and the fact that for , we have
(3.21) 
As an application of the Laplace transform of the trapping time, one particular quantity of interest is the expected trapping time. This can be obtained by taking the derivative of , where
(3.22) As such, we differentiate Tricomi’s Confluent Hypergeometric Function with respect to its second parameter. Denote
(3.23) A closed form expression of the aforementioned derivative can be given in terms of series expansions, such that
(3.24) where corresponds to equation (6.3.1) of Book:Abramowitz1964, also known as the digamma function. Thus, using expression (3.24), we obtain the expected trapping time
(3.25) In line with intuition, the expected trapping time is an increasing function of both the capital growth rate and initial capital . However, since the capital process grows exponentially, large initial capital and capital growth rates significantly reduce the trapping probability and increase the expected trapping time to the point where it becomes nonfinite, making the indicator function in the expected discounted penalty function (3.2) tend to zero. A number of expected trapping times for varying values of are displayed in Figure 2.

The ruin probability for the insurance model (2.5) given by
(3.26) is found by Article:Sundt1995 to satisfy the IDE
(3.27) Note that when , (3.27) coincides with the special case of (3.3) when , , and . Thus, the household’s trapping time can be thought of as the insurer’s ruin time. Indeed, the ruin probability in the case of exponential claims when as shown in Section 6 of Article:Sundt1995 is exactly the same as the trapping probability (3.21) when .
4 Introducing Microinsurance
As in Article:Kovacevic2011, we assume that households have the option of enrolling in a microinsurance scheme that covers a certain proportion of the capital losses they encounter. The microinsurance policy has proportionality factor , where , such that percent of the damage is covered by the microinsurance provider. The premium rate paid by households, calculated according to the expected value principle is given by
(4.1) 
where is some loading factor. The expected value principle is popular due to its simplicity and transparency. When , one can consider to be the pure risk premium (Book:Albrecher2017). We assume the basic model parameters are unchanged by the introduction of microinsurance coverage.
The stochastic capital process of a household covered by a microinsurance policy is denoted by . We differentiate between all variables and parameters relating to the original uninsured and insured processes by using the superscript in the latter case.
Since the premium is paid from a household’s income, the capital growth rate is adjusted such that it reflects the lower rate of income generation resulting from the need for premium payment. The premium rate is restricted to prevent certain poverty, which would occur should the premium rate exceed the rate of income generation. The capital growth rate of the insured household is lower than that of the uninsured household, while the critical capital is higher.
In between jumps, where , the insured stochastic growth process behaves in the same manner as (2.2), with parameters corresponding to the proportional insurance case of this section, making particular note of the increased critical capital :
(4.2) 
For , the process is given by
(4.3) 
By enrolling in a microinsurance scheme, a household’s capital losses are reduced to . Considering the case in which losses follow an exponential distribution with parameter , the structure of (3.5) remains the same. However, acquisition of a proportional microinsurance policy changes the parameter of the distribution of the random variable of the losses (). Namely, we have that for , where . We can therefore utilise the results obtained in Section 3 to obtain the Laplace transform of the trapping time for the insured process, which is given by
(4.4) 
where . Figure 3(a) displays the Laplace transform for varying values of .
Remarks.

[label=()]

The trapping probability of the insured process , displayed in Figure 3(b), is given by
(4.5) 
When the household has full microinsurance coverage, the microinsurance provider covers the total capital loss experienced by the household. On the other hand, when , no coverage is provided by the insurer i.e., .

We are interested in studying significant capital losses, since lowincome individuals are commonly exposed to this type of shock. Hence, throughout the paper, the parameter should be considered to reflect the desired loss behaviour.
Figure 4 presents a comparison between the trapping probabilities of the insured and uninsured processes. As in Article:Kovacevic2011, households with initial capital close to the critical capital (here, the critical capital ), i.e. the most vulnerable individuals, do not receive a real benefit from enrolling in a microinsurance scheme. Although subscribing to a proportional microinsurance scheme reduces capital losses, premium payments appear to make the most vulnerable households more prone to falling into the area of poverty. In Figure 4, the intersection point of the two probabilities corresponds to the boundary between households that benefit from the uptake of microinsurance and those who are adversely affected.
5 Microinsurance with Subsidised Constant Premiums
5.1 General Setting
Since microinsurance alone is not enough to reduce the likelihood of impoverishment for those close to the poverty line, additional aid is required. In this section, we study the costeffectiveness of government subsidised premiums, considering the case in which the government subsidises an amount , while the microinsurance provider claims a lower loading factor (Article:Kovacevic2011). The following relationship between premiums for the nonsubsidised and subsidised microinsurance schemes therefore holds
(5.1) 
Naturally, we assume governments are interested in optimising the subsidy provided to households. Governments should provide subsidies to microinsurance providers such that they enhance households’ benefits of enrolling in microinsurance schemes, however, they also need to gauge the costeffectiveness of subsidy provision. Households with capital very close to the critical capital will not benefit from enrolling into the scheme even if the entire loading factor is subsidised by the government, however, more privileged households will. One approach to finding the optimal loading factor for households that could benefit from the government subsidy is to find the solution of the equation
(5.2) 
where and denote the trapping probabilities of the insured subsidised and uninsured processes, respectively, since all loading factors below the optimal loading factor will induce a trapping probability lower than that of the uninsured process through a reduction in premium. This behaviour can be seen in Figure 5(a), while the “richest” households do not need help from the government since the nonsubsidised insurance lowers their trapping probability below the uninsured case, the poorest individuals require more support. Moreover, as highlighted previously, there are households that do not receive any benefit from enrolling in the microinsurance scheme even when the government subsidises the entire loading factor (when households pay only the pure risk premium, this could occur if the government absorbs all premium administrative expenses). Note that Figure 5(b) illustrates the optimal loading factor for varying initial capital. Initial capitals are plotted from the point at which households begin benefiting from the subsidised microinsurance scheme, i.e. the point at which the dashed line intersects the solid line in Figure 5(a). Additionally, Figure 5(b) verifies that, from the point at which the dasheddotted (insured household) line intersects the solid line in Figure 5(a), the optimal loading factor remains constant, with , i.e. the “richest” households can afford to pay the entire premium.
5.2 Cost of Social Protection
Next, we assess government costeffectiveness for the provision of microinsurance premium subsidies to households. Let be the force of interest for valuation, and let denote the present value of all subsidies provided by the government until the trapping time such that
(5.3) 
We assume a government provides subsidies according to the strategy introduced earlier, i.e. the government subsidises an amount , while the microinsurance provider claims a lower loading factor .
For , where denotes the critical capital of the insured subsidised process, let be the expected discounted premium subsidies provided by the government to a household with initial capital until trapping time, that is,
(5.4) 
Since , we can define , the Laplace transform of the trapping time with rate and critical capital , using the Laplace transform for the insured process previously obtained in (4.4) to compute when losses are exponentially distributed with parameter . This yields
(5.5)  
where . We now formally define the government’s cost of social protection.
Definition 2.
Let be the trapping probability of a household enrolled in a subsidised microinsurance scheme with initial capital . Additionally, let be a constant representing the cost to lift households below the critical capital out of the area of poverty. The government’s cost of social protection is given by
(5.6) 
Remarks.

[label=()]

For uninsured households, the government does not provide subsidies, i.e. . Furthermore, we consider their trapping probability to be .

The government manages selection of an appropriate force of interest and constant . For lower force of interest the government discounts future subsidies more heavily, while for higher interest future subsidies almost vanish. The constant could be defined, for example, using the poverty gap index introduced by Article:Foster1984, or in such a way that the government ensures with some probability that households will not fall into the area of poverty. Thus, higher values of will increase the certainty that households will not return to poverty.
Figure 6 displays the government cost of social protection. Observe that in this particular example, we consider high values for both the force of interest and the constant . The choice of is motivated by Figure 4, which shows that from , the trapping probability for uninsured households is very close to zero. Note that a high value of hands a lower weight to future government subsidies whereas a high value of grants higher certainty that a household will not return to the area of poverty once lifted out.
It is clear that governments do not benefit by entirely subsidising the “richest” households, since they will subsidise premiums indefinitely, almost surely (dashed line for highest values of initial capital). Hence, as illustrated in Figure 5(b), it is favourable for governments to remove subsidies for this particular group since their cost of social protection is even higher than when uninsured (solid line for highest values of initial capital). Conversely, governments perceive a lower cost of social protection when fully subsidising the loading factor for households with initial capital lying closer to the critical capital . The cost of social protection when households pay only the pure risk premium is lower than when paying the premium entirely for values of initial capital in which the dashed line is below the dotted, in which case the government should support premium payments. However, due to the fact that they will almost surely fall into the area of poverty, requiring governments to pay the subsidy in addition to the cost of lifting a household out of poverty, it is not optimal to fully subsidise the loading factor for the most vulnerable, since the cost of social protection is higher than that for uninsured households. Note that, from the point of view of the governmental cost of social protection, Figure 6 confirms earlier statements asserting the inefficiency of providing premium support to the most vulnerable, i.e. neither individual households nor governments receive real benefit under such a scheme. Thus, alternative risk management strategies should be considered for this sector of the lowincome population.
6 Microinsurance with Subsidised Flexible Premiums
6.1 General Setting
Since premiums are generally paid as soon as microinsurance coverage is purchased, a household’s capital growth could be constrained. It is therefore interesting to consider alternative premium payment mechanisms. From the point of view of microinsurance providers, advance premium payments are preferred so that additional income can be generated through investment, naturally leading to lower premium rates. Conversely, consumers may find it difficult to pay premiums up front. This is a common problem in lowincome populations, with consumers preferring to pay smaller installments over time (Book:Churchill2006). Collecting premiums at a time that is inconvenient for households can be futile. Flexible premium payment mechanisms have been highly adopted by informal funeral insurers in South Africa, where policyholders pay premiums whenever they are able, rather than at a specific time during the month (WorkingPaper:Roth2000). Similar alternative insurance designs in which premium payments are delayed until the insured’s income is realised and any indemnities are paid have also been studied. Under such designs, insurance takeup increases, since liquidity constraints are relaxed and concerns regarding insurer default, also prevalent in lowincome classes, reduce (Article:Liu2016).
In this section, we introduce an alternative microinsurance subsidy scheme with flexible premium payments. We denote the capital process of a household enrolled in the alternative microinsurance subsidy scheme by . Furthermore, as in Section 4, we differentiate between variables and parameters relating to the original, insured and alternative insured processes using the superscript . Under such an alternative microinsurance subsidy scheme, households pay premiums when their capital is above some capital barrier , with the premium otherwise paid by the government. In other words, whenever the insured capital process is below the capital level , premiums are entirely subsidised by the government, however, when a household’s capital is above , the premium is paid continuously by the household itself. This method of premium collection may motivate households to maintain a level of capital below in order to avoid premium payments. Consequently, we assume that households always pursue capital growth. Our aim is to study how this alternative microinsurance subsidy scheme can help households reduce their probability of falling into the area of poverty. We also measure the costeffectiveness of such scheme from the point of view of the government.
The intangibility of microinsurance makes it difficult to attract potential clients. Most clients will never experience a claim and so cannot perceive the real value of microinsurance, paying more to the scheme (in terms of premium payments) than what they actually receive from it. It is only when claims are settled that microinsurance becomes tangible. The alternative microinsurance subsidy scheme described here could increase client value, since, for example, individuals below the barrier may submit claims, receive a payout and therefore perceive the value of microinsurance when they suffer a loss, regardless of whether they have ever paid a single premium. Other ways of increasing microinsurance client value include bundling microinsurance with other products and introducing Value Added Services (VAS), which represent services such as telephone hotlines for consultation with doctors or remote diagnosis services (for health schemes) offered to clients outside of the microinsurance contract (Article:Madhur2019).
Under the alternative microinsurance subsidy scheme, the Laplace transform of the trapping time satisfies the following differential equations:
(6.1) 
As in Section 3, use of the change of variable leads to Kummer’s Confluent Hypergeometric Equation and thus,
(6.2) 
for arbitrary constants . Under the boundary condition with asymptotic behaviour of the Kummer function as presented in Section 3, we deduce that . Also, since , we obtain .
Due to the continuity of the functions and at and the differential properties of the Confluent Hypergeometric Functions
(6.3) 
(6.4) 
upon simplification,
(6.5) 
and
(6.6) 
where
(6.7) 
and
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