Does insurance fraud in automobile theft insurance fluctuate with the business cycle?

Abstract

Financial institutions face various cyclical risks, but very few studies have analyzed the cyclicality of operational risk. External fraud is an important operational risk faced by insurers. In this research, we analyze the empirical relationship between insurance fraud and the business cycle and we concentrate our study on two insurance contracts that may create an incentive to defraud. We find that residual insurance fraud exists both in the contract with replacement cost endorsement and the contract with no-deductible endorsement in the Taiwan automobile theft insurance market. These results are consistent with previous literature on the relationship between fraud activity and non-optimal insurance contracting. We also show that the severity of insurance fraud is countercyclical. Fraud is stimulated during periods of recession and mitigated during periods of expansion. Although this last result seems intuitive, our contribution is the first to measure its significance.

Appendices

Appendix A

Table 6 The relationship between Taiwan’s auto theft insurance loss ratio of non-commercial vehicles and GDP

Appendix B: Insurance fraud under replacement cost endorsement

Assume that the consumer is risk averse. The expected utility of an insured is equal to

$$ gU\left(W+\theta \frac{A}{h(t)}+A-D,\omega \right)+\left(1-g\right)U\left(W,\omega \right). $$

where U is the utility function for the risk averse individual, W is the individual’s wealth, not including the value of the vehicle, A is the market value of the vehicle at the beginning of the policy year, h(t) is the depreciation rate for the market value of the vehicle, which is an increasing function of time t, and t denotes the month of the policy year. θ is the discount rate when the fraudulent individual starts his fraud activity, 0 ≤ θ ≤ 1. g is the probability of the fraud being successful. D is the deductible designed in the contract. The model assumes that the market value of the vehicle will be totally expropriated if the individual’s fraudulent behavior is discovered by the insurance company. This causes the wealth level of the fraudulent individual who is caught to be limited to W. ω is the moral cost of fraud. As in Dionne et al. (2009), we assume that $\raisebox{1ex}{$\partial U$}\!\left/ \!\raisebox{-1ex}{$\partial \omega $}\right.<0$. We also assume that $\raisebox{1ex}{${\partial}^2U$}\!\left/ \!\raisebox{-1ex}{$\partial W\partial \omega $}\right.<0$. The individual will defraud if

$$ gU\left(W+\theta \frac{A}{h(t)}+A-D,\omega \right)+\left(1-g\right)U\left(W,\omega \right)\ge U\left(W+\frac{A}{h(t)},0\right) $$

In the absence of moral cost of fraud (ω = 0), there is a critical successful fraud probability $\tilde{g}$ at which there is indifference between being honest and dishonest:

$$ \tilde{g}U\left(W+\theta \frac{A}{h(t)}+A-D,0\right)+\left(1-\tilde{g}\right)U\left(W,0\right)=U\left(W+\frac{A}{h(t)},0\right) $$

$0<\tilde{g}<1$. When the successful fraud probability $p>\tilde{g}$, the individual would defraud. Furthermore, this critical value of the probability of the fraud being successful is:

$$ \tilde{g}=\frac{U\left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)-U\left(W,0\right)}{U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,0\right)-U\left(W,0\right)}. $$

The expected utility function of an individual who has a probability (α) of engaging in fraudulent behavior can therefore be written as:

$$ EU=\alpha \left[\tilde{g}U\left(W+\theta \frac{A}{h(t)}+A-D,0\right)+\left(1-\tilde{g}\right)U\left(W,0\right)\right]+\left(1-\alpha \right)U\left(W+\frac{A}{h(t)},0\right) $$

The individual has a probability equal to 1 of engaging in fraud when the probability of the fraud being successful is above $\tilde{g}$. On the contrary, the individual has a probability equal to 0 of engaging in fraud when the probability of the fraud being successful is below $\tilde{g}$. In addition, there is a probability of fraud of between 1 and 0 when the probability of the fraud being successful is equal to $\tilde{g}$. Consequently, the probability of an individual engaging in fraud decreases with $\tilde{g}$. Intuitively, $\tilde{g}$ should be very low to defraud, which means that when there is no moral cost for an individual, the threshold for the individual to defraud is very low.

When there is no moral cost for the individual, as time increases:

$$ \frac{d\tilde{g}}{ dt}=\left\{\frac{-\left[U\left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)-U\left(W,0\right)\right]U\prime \left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,0\right)\left[-\left(\theta {\scriptscriptstyle \frac{A}{h{(t)}^2}}\right)\right]}{{\left[U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,0\right)-U\left(W,0\right)\right]}^2}+\frac{U\prime \left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)\left(-{\scriptscriptstyle \frac{A}{h{(t)}^2}}\right)}{\left[U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,0\right)-U\left(W,0\right)\right]}\right\}h\prime (t) $$

We rewrite (den) as the denominator of the above equation second term and obtain:

$$ \frac{d\tilde{g}}{ dt}=\frac{\left({\scriptscriptstyle \frac{A}{h{(t)}^2}}\right)}{(den)^2}\left\{\left[U\left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)-U\left(W,0\right)\right]U\prime \left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,0\right)\theta -\left[U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,0\right)-U\left(W,0\right)\right]U\prime \left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)\right\}h\prime (t) $$

Because the incentive for an individual to engage in fraud is higher when $\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D\ge {\scriptscriptstyle \frac{A}{h(t)}}$, the first set of square brackets is smaller than the second set in the above equation, and the first derivative of the utility under $W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D$ is also smaller than the corresponding derivative under $W+{\scriptscriptstyle \frac{A}{h(t)}}$. In addition, θ is between 0 and 1. All of these factors make the value inside the braces negative. Moreover, h′(t) > 0, hence, the sign of the above equation ($\frac{d\tilde{g}}{ dt}$) is negative.

The above analysis infers that $\tilde{g}$ will decrease with t, and that the probability α will increase with t. Hence, the probability (α) of an individual engaging in fraud is higher near the end of the policy year. If the audit probability of the insurance company is flat over the whole policy year, the equilibrium rate of fraud could also be higher near the end of the policy year.

When there is a moral cost (ω > 0), there is also a critical successful fraud probability $\overset{\frown }{g}$ at which there is indifference between being honest and dishonest:

$$ \overset{\frown }{g}U\left(W+\theta \frac{A}{h(t)}+A-D,\omega \right)+\left(1-\overset{\frown }{g}\right)U\left(W,\omega \right)=U\left(W+\frac{A}{h(t)},0\right) $$

We can derive the critical value of the probability of the fraud being successful $\overset{\frown }{g}=\overset{\frown }{g}\left(\omega \right)$. For each ω level, there is a corresponding $\overset{\frown }{g}\left(\omega \right)$. When the successful fraud probability is $p>\overset{\frown }{g}$, the individual would defraud. We can verify that $0<\tilde{g}<\overset{\frown }{g}<1$. When the successful fraud probability is $p>\overset{\frown }{g}$, the individual would defraud, where $\overset{\frown }{g}$ solves:

$$ \overset{\frown }{g}\left(\omega \right)=\frac{U\left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)-U\left(W,\omega \right)}{U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,\omega \right)-U\left(W,\omega \right)} $$

As time increases:

$$ \frac{d\overset{\frown }{g}}{ dt}=\left\{\frac{-\left[U\left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)-U\left(W,\omega \right)\right]U\prime \left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,\omega \right)\left[-\left(\theta {\scriptscriptstyle \frac{A}{h{(t)}^2}}\right)\right]}{{\left[U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,\omega \right)-U\left(W,\omega \right)\right]}^2}+\frac{U\prime \left(W+{\scriptscriptstyle \frac{A}{h(t)}},0\right)\left(-{\scriptscriptstyle \frac{A}{h{(t)}^2}}\right)}{\left[U\left(W+\theta {\scriptscriptstyle \frac{A}{h(t)}}+A-D,\omega \right)-U\left(W,\omega \right)\right]}\right\}h\prime (t) $$

We also find that $\frac{d\overset{\frown }{g}}{ dt}<0$, which means that the critical successful fraud probability would decrease over time. We can also analyze the effect of moral cost on $\frac{d\overset{\frown }{g}}{ dt}$. We do not present this relationship here because we do not have data to test its sign. The theoretical result is available from the authors.

Appendix C: Insurance fraud under the no-deductible contract

First, we examine whether the incentive of insurance fraud is higher under a no-deductible endorsement. We still assume the individual is risk-averse,

$$ gU\left(W+\theta \frac{A}{h(t)}+\frac{A}{k(t)}-D,\omega \right)+\left(1-g\right)U\left(W,\omega \right). $$

The individual’s insurance contract is indemnified with depreciation. The depreciation rate (k(t)) increases with time t. Furthermore, the depreciation rate from the indemnity of the insurance company (k(t)) is more stringent than that in the market (h(t)), i.e., k(t) > h(t) and ${\scriptscriptstyle \frac{1}{k{(t)}^2}}k\prime (t)>{\scriptscriptstyle \frac{1}{h{(t)}^2}}h\prime (t),\forall t$. The definitions of g, W, θ, A, and ω are the same as those in the model in Appendix B. The totally expropriated constraint is also the same as in Appendix B.

There exists a critical value of the probability ($\tilde{g}$) of the fraud being successful at which there is indifference between being honest and dishonest:

$$ \tilde{g}U\left(W+\theta \frac{A}{h(t)}+\frac{A}{k(t)}-D,\omega \right)+\left(1-\tilde{g}\right)U\left(W,\omega \right)=U\left(W+\frac{A}{h(t)}\right). $$

The expected utility function of an individual who has the probability (α) of engaging in fraud is therefore expressed as follows:

$$ EU=\alpha \left[\tilde{g}U\left(W+\theta \frac{A}{h(t)}+\frac{A}{k(t)},\omega \right)+\left(1-\tilde{g}\right)U\left(W,\omega \right)\right]+\left(1-\alpha \right)U\left(W+\frac{A}{h(t)}\right). $$

The individual has a probability of 1 of engaging in fraud when the probability of the fraud being successful is above $\tilde{g}$. Conversely, the individual has a probability of 0 of engaging in fraud when the probability of the fraud being successful is below $\tilde{g}$. Finally, the individual has a probability of fraud of between 1 and 0 when the probability of the fraud being successful equals $\tilde{g}$. To summarize, the probability of the individual’s engaging in fraud decreases with $\tilde{g}$.

Furthermore, this critical value of the probability of the fraud being successful is:

$$ \tilde{g}=\frac{U\left({W}_2\right)-U\left({W}_3,\omega \right)}{U\left({W}_1,\omega \right)-U\left({W}_3,\omega \right)}=\frac{\varDelta_2}{\varDelta_1}. $$

where ${W}_1=W+\theta \frac{A}{h(t)}+\frac{A}{k(t)}-D,{W}_2=W+\frac{A}{h(t)},{W}_3=W$; $\tilde{g}$ is the relative utility of no cheating. It is the utility difference between no cheating and being caught cheating, compared with the utility difference between cheating and being caught cheating.

As the deductible increases:

$$ \frac{d\tilde{g}}{ dD}=\frac{\varDelta_2}{\varDelta_1}\left[-\frac{1}{\varDelta_1}U\prime \left({W}_1,\omega \right)\left(-1\right)\right], $$

we find that $\tilde{g}$ is affected by the comparative utility of no cheating ($\frac{\varDelta_2}{\varDelta_1}$), and the relative marginal effect of deductible ($\frac{1}{\varDelta_1}U\prime \left({W}_1,\omega \right)$). The above derivative is positive, which means that when the deductible increases, the critical value of the probability of successful fraud also increases, and the incentive to defraud decreases. Hence, people who purchase no-deductible contracts have a stronger incentive to defraud.

We now discuss the relative claim timing pattern for the contract with depreciation and no-deductible endorsement in contrast to the reference contract. We consider the impact of timing on the incentive induced by the no-deductible endorsement. In other words, we consider the impact of t on $\frac{d\tilde{g}}{ dD}$. Let $H=\frac{d\tilde{g}}{ dD}$, then

$$ \begin{array}{l}\frac{ dH}{ dt}=\frac{d}{ dt}\left(\frac{\varDelta_2}{\varDelta_1}\right)\left[-\frac{1}{\varDelta_1}U\prime \left({W}_1,\omega \right)\left(-1\right)\right]+\left(\frac{\varDelta_2}{\varDelta_1}\right)\frac{d}{ dt}\left[-\frac{1}{\varDelta_1}U\prime \left({W}_1,\omega \right)\left(-1\right)\right]\hfill \\ {}\hfill =\frac{1}{{\left({\varDelta}_1\right)}^2}\left({\varDelta}_1U\prime \left({W}_2\right)\left(-\frac{A}{h{(t)}^2}h\prime (t)\right)-{\varDelta}_2U\prime \left({W}_1,\omega \right)\left(-\theta \frac{A}{h{(t)}^2}h\prime (t)-\frac{A}{k{(t)}^2}k\prime (t)\right)\right)\left[-\frac{1}{\varDelta_1}U\prime \left({W}_1,\omega \right)\left(-1\right)\right]+\left(\frac{\varDelta_2}{\varDelta_1}\right)\left[\frac{1}{\varDelta_1}U\prime \prime \left({W}_1,\omega \right)-\frac{1}{\varDelta_1{}^2}{\left(U\prime \left({W}_1,\omega \right)\right)}^2\right]\left(-\theta \frac{A}{h{(t)}^2}h\prime (t)-\frac{A}{k{(t)}^2}k\prime (t)\right)\hfill \end{array} $$

The first term in the above equation means that the relative utility of no cheating is varying with time. Regardless of whether cheating occurs, the utility will decrease because of depreciation. If the difference in wealth level between no deductible and deductible contracts is sufficient, this first term will be negative. This means that over time, the utility of the no deductible contract will decrease more under no cheating, but it will decrease less under cheating. The incentive of fraud induced by no-deductible contracts would decrease over time.

The second term means that the marginal utility affected by the deductible is varying over time. It is positive, and it contrasts with the first term whereby the fraud incentive from a no-deductible contract increases over time, because the marginal utility of wealth is higher when the wealth level depreciates over time. However, if the difference in wealth level between no deductible and deductible contract is sufficient, the whole equation would still be negative. This means that the critical value of the fraud probability induced by no deductible contract is mitigated over time. This corresponds to the second part of our second hypothesis, which states that the probability of fraud (for no-deductible endorsement contracts) is higher at the beginning of the policy year. We can also explore the theoretical relationship between moral cost of fraud and $\frac{ dH}{ dt}$. Results are available from the authors.

Appendix D

Table 7 Complete empirical results of the 12th policy month and the 1st policy month of Eqs. (1) and (2)

Appendix E

Table 8

Table 8 Complete empirical results of the 12th policy month and the 1st policy month of Eq. (3)