Insurance and Endogenous Bankruptcy Risk: When is it Rational to Choose Gambling, Insurance and Potential Bankruptcy?

Transcription

1 Insurance and Endogenous Bankruptcy Risk: When is it Rational to Choose Gambling, Insurance and Potential Bankruptcy? Lisa L. Posey Pennsylvania State University 369 Business Building University Park, PA Vickie Bajtelsmit Colorado State University 321 Rockwell Hall Fort Collins, CO July 2015

2 1 INTRODUCTION It is well established in the literature that potential bankruptcy can play an important role in the decision of whether and how much to insure. In the absence of bankruptcy costs, the effect of limited liability is to reduce the demand for liability insurance by risk averse individuals when damages can exceed net worth (Sinn, 1982; Huberman, Mayers and Smith, 1983). In this case, the floor on wealth created by limited liability allows the potential insured to externalize some of the costs of damages to third party victims if bankruptcy occurs. This dampens insurance demand. As Sinn (1982) and Winter (2006) show under a standard expected utility model with actuarially fair premiums, if a loss of fixed size may occur that is greater than initial wealth, then either full insurance or no insurance will be purchased. The greater the potential loss is, the more likely it is that insurance will not be purchased. A parallel literature considers the coexistence of gambling and insurance, where gambling refers to accepting a risky prospect that has a non-positive expected value. In their classic paper, Friedman and Savage (1948) hypothesized that expected utility maximizers have wealth preferences that are convex over some ranges and concave over others. Others, like Markowitz (1952), Yaari (1965), Machina (1982) and Quiggin (1991) criticized the Friedman and Savage model on several grounds including the fact that it is not applicable to all wealth levels. Markowitz (1952) showed that the Friedman and Savage model could lead to unusual predictions, like a preference for unfair over fair gambles, and suggested that the ranges for the convex and concave portions of the utility function be altered. Yaari (1965) presented experimental evidence that was inconsistent with the Friedman and Savage model and postulated that subjective probabilities that overestimate low probabilities and underestimate high probabilities could explain the coexistence of gambling and insurance. Like Yaari, Quiggin (1991) turned to transformations of the probabilities to explain the phenomenon of insurance with gambling, introducing the rank-dependent utility model with overweighting of low probability, extremely favorable events. In contrast, Conlisk (1993) added a term for the utility of gambling to the standard expected utility of wealth model and was able to explain demand for very small payoff gambles and demand for insurance for large losses. In each of these studies, the models developed were used to explain the decision to gamble and the decision to insure, but

3 2 situations where these decisions were made in conjunction with one another were not explicitly considered. Similar to large uninsured losses, gambles may place sufficient wealth at risk such that the occurrence of an unfavorable outcome can have a significant effect on individual or business wealth. Although limited liability protections under bankruptcy law have been separately shown to increase the incentives for risk-taking and to decrease the incentives for insuring, the combined effects have not been formally addressed in the literature. 1 This issue has practical relevance for small business and entrepreneurial decisions. Despite the relatively high risk of failure, 2 an entrepreneur often invests his or her own wealth in the high-risk gamble of a new venture and yet simultaneously purchases insurance against various risks. Because risk premiums for new venture investments are not particularly high (Vereshchagina and Hopenhayn, 2009), it is often assumed that entrepreneurs must be risk loving. However, there is also evidence of risk averse behavior, 3 such as investment in risk management and business insurance. A major contribution of our paper is that we show that over certain ranges of gambled wealth and insurance, the decision to both gamble and insure is rational and consistent with expected utility maximization and risk aversion. Limited liability 4 plays an important role in this outcome and, to the extent that public policy favors new business formation, the protections of bankruptcy law provide important incentives for entrepreneurial investment. 5 However, because bankruptcy law provides limited liability to individuals as well as businesses, our results are not limited to the business context. We develop a model in which a risk averse individual, in an environment of limited liability, decides whether to gamble on a zero expected value risky project and/or insure a 1 Posey (1993) considers the impact of limited liability on the investment in risky activities and self-protection activities. This paper shows that the impact on both of these types of activities is ambiguous. In particular, individuals may either over-invest, or under-invest in both selfprotection activities and risky activities relative to socially optimal levels. 2 According to SBA (2012), about half of small businesses survive 5 years or more and one-third survive 10 years or more. These survival rates have been consistent over the last two decades. Although corporate entities often have the protection of limited liability, a new venture entrepreneur s personal wealth is usually at stake in the event of business failure. 3 See for example, Forlani and Mullins (2000). 4 For a review of the history and impact of limited liability rules, see Leebron (1991). 5 Limited liability protection in bankruptcy laws has been shown to be an important factor entrepreneurial activity (Jia, 2010) and a factor explaining jurisdictional differences in new business formation. See Fossen (2014), and Athreya (2006).

4 3 potential liability loss that cannot, in and of itself, lead to bankruptcy. 6 Unlike Sinn (1982), Huberman, Mayers and Smith (1983) and Winter (2006), the status quo does not involve any bankruptcy risk. In addition, the occurrence of the liability loss is not certain to lead to bankruptcy even if the risky investment is undertaken without insurance. The individual who may be an entrepreneur, for example, has a riskless component of earnings and a nonbankrupting potential liability loss. 7 The entrepreneur may use some of the earnings to purchase actuarial fair insurance. The amount of money that is not used to purchase insurance, minus a small guaranteed floor on wealth, is available to invest in a risky project that has a zero expected value and is independent of the liability loss. The entrepreneur can invest any percentage of the available funds from 0% to 100%. If full insurance is purchased, the risky project cannot bankrupt the individual because only the initial investment will be lost if the outcome of the project is unfavorable. But in this case, the risky investment is not attractive due to risk aversion and the expected return of zero. Previous research indicates that introduction of an independent, zero expected value risk will not decrease the demand for insurance under any of the familiar DARA utility functions in the absence of bankruptcy risk (Gollier and Pratt, 1996; Guiso and Jappeli, 1998; Pratt and Zeckhauser, 1987; Kimball, 1993). But consider what will happen if insurance is not purchased and the risky project is undertaken. The risky project cannot lead to bankruptcy if there is no liability loss, similar to the case with full insurance. But without insurance, bankruptcy will occur if there is a liability loss, the outcome of the risky investment is unfavorable, and the amount invested is large enough. If this strategy of no insurance with a relatively large risky investment were to be chosen, it would imply that the risk of bankruptcy was endogenously created and the effect of limited liability was a lack of demand for insurance. Two questions arise: can endogenously chosen bankruptcy risk be optimal, and is it possible that some insurance may be purchased at the same time the entrepreneur gambles on the risky investment? The answer to these questions is yes. In the current paper, if both gambling and insurance are chosen, it will be in a range where a sufficiently large amount is gambled and a sufficiently small amount is insured that 6 There are no bankruptcy costs explicitly included in the model. This allows us to abstract from the role of bankruptcy costs in the insurance and investment decisions and focus on the other effects we are trying to uncover. 7 Although we have an entrepreneur in mind, the model can easily be applied to any individual or to a risk averse, closely held corporation.

5 4 bankruptcy will occur if both a liability loss and a gambling loss occur. Rather than avoiding bankruptcy, the individual wants to assure that it is possible because in doing so the expected value of the gamble is changed from zero to positive. This is because the full loss of the gamble is not borne by the individual in the event of bankruptcy since part of the uninsured liability loss will simply go unpaid. Even in light of this, the individual may still choose to purchase insurance since it will be valuable when the gamble is not lost and a liability loss occurs. The remainder of the paper is organized as follows. The next section presents the formal model for the situation of the individual described above. The decisions to insure and to gamble on the risky investment under limited liability are explored. Then an example is presented with a range of parameter values that lead to endogenously chosen bankruptcy risk with a demand for both insurance and investment in the risky project. A final section provides some concluding remarks. A MODEL OF GAMBLING WITH POTENTIAL BANKRUPTCY We begin with a simple mathematical model of an individual s demand for insurance. The individual, with non-random earnings W, faces a potential liability loss of size L with probability p. The loss is zero with probability (1-p). The individual is risk averse with utility function u(x) where x is final wealth, u!(x) > 0 and u!! (x) < 0. Insurance is available at an actuarially fair premium: coverage of α L costs α pl where α [0,1]. An assumption that is key to the analysis below is the existence of limited liability with no bankruptcy costs. Wealth is limited to a minimum of F. The term bankruptcy is used to refer to the situation where wealth would fall below F in the absence of limited liability. It is assumed that L W-F, so that bankruptcy cannot be caused solely by a liability loss. Under these conditions, the individual will buy full insurance, α = 1. In effect, purchasing less than full insurance would be equivalent to participating in a zero expected value gamble. In addition to the insurance decision, the individual has the opportunity to invest in (i.e., gamble on) a risky project that has zero expected value in the absence of limited liability. The investment decision is made immediately following the insurance decision. The money that is left after any insurance has been purchased, minus the small guaranteed floor on wealth, is the most that can be invested (i.e., there is no borrowing). Therefore, up to [W α pl F] can be

6 5 invested in the risky project and the individual must choose the proportion, β [0,1], of that amount to risk. If the project has an unfavorable outcome, the individual loses the full amount invested, β[w α pl F]. The project fails with probability π. If the project succeeds, which it does with probability ( 1 π ), the individual receives ( π (1 π ))β[w α pl F]. Since the investment is restricted to be less than the amount left over after insurance has been purchased, bankruptcy can only occur if the individual suffers a liability loss and an unfavorable outcome for the project, and has purchased a sufficiently small amount of insurance and risked a sufficiently large amount in the project. The individual maximizes the following expected utility function: EU = π p u(max{w α pl β[w α pl F] (1 α)l, F}) +π (1 p) u(w α pl β[w α pl F]) +(1 π )p u(w α pl + ( π (1 π ))β[w α pl F] (1 α)l) +(1 π )(1 p) u(w α pl + ( π (1 π ))β[w α pl F]). (1) This can be rewritten as EU = π p u(max{w 1, F}+ π (1 p) u(w 2 )+ (1 π )p u(w 3 )+ (1 π )(1 p) u(w 4 ) (2) where W 1, W 2, W 3 and W 4 are all functions of α and β. As the first term in EU shows, with probability π p, both a liability and gambling loss occur and final wealth is: Max{W α pl β[w α pl F]+ (1 α)l, F}. (3) Bankruptcy occurs whenw α pl β[w α pl F]+ (1 α)l < F, or equivalently, when (1 α)l β >1 [W α pl F]. (4) Therefore, the probability of bankruptcy is either π p, if the chosen α and β satisfy inequality (4), or 0 otherwise. Let the potential bankruptcy line be defined as (1 α)l β 0 (α) =1 [W α pl F]. (5) Figure 1 depicts the potential bankruptcy line for α, β combinations, where 0 β 1and 0 α 1. For combinations on or below the potential bankruptcy line, bankruptcy is not

7 6 possible even if both types of losses occur. For combinations above the potential bankruptcy line, bankruptcy definitely will occur if both types of losses occur, but will not occur if both types of losses do not occur. The area above the bankruptcy line can be called the feasible bankruptcy range and the area on or below the bankruptcy line can be referred to as the feasible nonbankruptcy range. 1 Figure 1 Feasible Bankruptcy Range β (1 α)l β! (α) = 1 W αpl F Feasible Non-Bankruptcy Range Consider the expected value of the risky project to the individual in light of the effects of limited liability. If the chosen expected value of the bet is α, β combination is below the potential bankruptcy line, the (1 π ){(π / (1 π ))β[w α pl F]} π{β[w α pl F]} = 0. (6) On the other hand, if the chooses an expected value of the risky project is E[Wealth(α, β)] E[Wealth(α, 0)] = π pf + π (1 p){w α pl β[w α pl F]} α, β combination is above the potential bankruptcy line, the +(1 π )p{w α pl + (π / (1 π ))β[w α pl F] (1 α)l} +(1 π )(1 p){w α pl + (π / (1 π ))β[w α pl F]} {W pl} α = π p{w α pl β[w α pl F] (1 α)l} (7)

8 7 = π p{w 1 } > 0. The expected value of the project becomes positive because, in the event the project fails, the full loss will be incurred only if a liability loss does not occur. If a liability loss does occur, then the amount of the uninsured liability paid out to the victim will be reduced to leave the individual with wealth F. So investing in the potential bankruptcy range creates a default put option that transforms a zero expected profit project into one with a positive expected value. The expected utility function in (1) can be redefined as follows. Let EU β represent the expected utility applicable in the feasible non-bankruptcy range and let EU β + represent the expected utility applicable in the feasible bankruptcy range. The difference between EU β + is the first term: in β EU β and EU the first term is πp u W ) which varies with α and β, while in EU β +, the first term is π p u(f) which does not vary with α and β. Below the expected ( 1 bankruptcy line the risky project impacts expected utility as follows: EU β β = π[w α pl F]{p[ u #(W 3 ) u #(W 1 )]+ (1 p)[ u #(W 4 ) u #(W 2 )]} < 0. (8) Here, limited liability plays no role and expected utility is decreasing in β since the investment has zero expected value and the individual is risk averse. So, β = 0 is optimal in this range. When β = 0, expected utility reduces to EU β = p u(w α pl (1 α)l)+ (1 p) u(w α pl). (9) This is increasing in α since insurance is actuarially fair so there is a corner solution for the optimal α. The optimal α, β pair in the range below the bankruptcy line is α = 1 and β = 0, i.e., full insurance and investment in the risky project. It is never optimal to purchase full insurance and invest in the risky prospect, simultaneously; i.e., an outcome of ˆ α = 1 and ˆ β > 0 is never optimal. This follows straightforwardly from the fact that all α, β pairs such that α = 1 are in the feasible non-bankruptcy range where β = 0 is optimal. If an optimal outcome is to be found with both insurance and investment in the risky project, it must be in the feasible bankruptcy range. To see if such a possibility exists, it is necessary to find the levels of α and β that maximize EU β+ within the feasible bankruptcy range

9 8 and to determine whether they provide a level of expected utility above the full insurance/no investment alternative. EU β+ = π p u(f) +π (1 p) u(w α pl β[w α pl F]) +(1 π )p u(w α pl + ( π (1 π ))β[w α pl F] (1 α)l) +(1 π )(1 p) u(w α pl + ( π (1 π ))β[w α pl F]) (10) has the following partial derivatives which, set equal to zero, give the first order conditions for the maximization problem: EU β+ β = π[w α pl F]{p u #(W 3 )+ (1 p)[ u #(W 4 ) u #(W 2 )]} = 0, (11) EU β+ α = π (1-p) u "(W 2 )(β pl pl)+ (1 π )p u " (W ) $ L pl β $ π ' ' 3 & & ) pl) % % 1 π ( ( # # π & & (1 π )(1 p) u "(W 4 )% pl + β % ( pl( = 0. (12) $ $ 1 π ' ' Manipulation of (11) and (12), gives the following conditions for an interior solution to the maximization of (10): uʹ ( W uʹ ( W 3 ) 1 = ) 1 p 2 π uʹ ( W2 ) 1 π =. (14) uʹ ( W4 ) 1 π p The second order condition is " $ # 2 EU β+ β 2 %" ' $ &# 2 EU β+ α 2 % ' & " 2 EU β+ % $ # α β ' & 2 (13) > 0. (15) Defining ˆα and ˆβ as theα and β that solve (13) and (14), the following conditions also must hold for this solution to be in the interior of the feasible bankruptcy range 0 < ˆα <1 (16a) β 0 ( ˆα) < ˆβ <1 EU β+ ( ˆα, ˆβ) u(w pl) > 0. (16b) (16c)

10 9 Remark: For an interior solution to exist with partial insurance and partial investment (i.e., ˆβ <1), it must be true that 1 π p > 0, i.e., the sum of the probabilities of the project failing and of having a liability loss must be less than 1. This follows from condition (14) since u ʹ ( W ) > 0. i A Solution with Power Utility To obtain further insight into the incentives to both insure and invest in the risky project in the presence of bankruptcy risk it will be assumed that the individual has a power utility " 1 % function, u(w i ) = $ ' W i # 1 r & ( ) 1 r, with r > 0. This utility function, commonly used in both empirical and theoretical studies in economics and finance, exhibits decreasing absolute risk aversion (DARA) and constant relative risk aversion (CRRA). If the individual has a power utility function then the conditions (13) and (14) are:! # " W 2 W 3 and! # " W 2 W 4 $ & % $ & % r r! = 1 π $ # & (17) " 1 p %! 1 π $ = # &. (18) " 1 π p % The optimal levels of insurance and investment for an interior solution are: ˆα = L(π + (1 π )b) W (b a) L(π + (1 π )b) pl(b a) (19) and ˆβ = (W pl)(1 π )(b 1) (W pl F)(π + (1 π )b)+ Fp(b a). (20) " where a = 1 π % $ ' # 1 p & 1 r " 1 π % and b = $ ' # 1 π p & 1 r. In order for this α, β combination to be in the feasible bankruptcy range and to be preferred to full insurance without investment in the risky project, (16a) through (16c) must hold as well.

11 10 With the power utility function, r W represents the Arrow-Pratt measure of absolute risk aversion, so risk aversion varies directly with r. The derivative of ˆα with respect to r in the feasible bankruptcy range is obtained by differentiating (19): ˆα r = # (W pl)l (π + (1 π )b) a & (π + (1 π )a) b $ % r r ' ( L(π + (1 π )b) Lp(b a) [ ] 2 > 0. (21) This follows because b > a > 0 and b r < a r < 0. The derivative of ˆβ with respect to r can be derived using the fact that the first order conditions (17) and (18) give the following relationship between ˆα and ˆβ : ˆβ = (b 1)(1 π )(W ˆα pl) (W ˆα pl F)(π + b(1 π )) and then differentiating this with respect to r, ˆβ r = (22) # (1 π ) (W ˆα pl F)(W ˆα pl) b ˆα & + F(b 1)(π + b(1 π ))pl $ % r r ' ( < 0. (23) [(W ˆα pl F)(π + b(1 π ))] 2 These derivatives have the expected signs, i.e., an increase in risk aversion increases the amount of insurance purchased and decreases the amount gambled on the risky project when an interior solution has been obtained. A Numerical Example Next, an example is given to show that an interior solution is possible for the individual s decision regarding whether to insure and gamble on the risky project. In this example, the potential loss from the liability risk is just short of bankrupting if the risky project is rejected, i.e., L = W F. An actuarially fair premium will lead to full insurance in the absence of the risky investment. Recall that after the insurance premium has been paid, the individual can choose to invest any percentage of the wealth remaining above the floor F, but the zero expected profit project is not attractive if full insurance has been purchased. The most that can be lost with the risky investment is the amount invested, so bankruptcy will not occur if no liability loss is

12 11 incurred or if it is fully insured. The floor on wealth is set at F = 1, simply to avoid division by zero in the analysis. A relatively low level of risk aversion is assumed with r =.075. Given these parameters, the second order condition in (15) for an interior solution to the maximization of EU β+ holds for all W 100,000 when.35 π p 0. Therefore, the optimal insurance-gambling choice will be determined for all probability pairs where the sum of the probability of an insurable loss and the probability that the project fails is less than or equal to.35. The results are identical up to at least four decimal places for all wealth levels above 100,000. Figure p π

13 12 Figure 2 shows the values of ˆα from equation (19), which solve the first order conditions for maximizing EU β+ from equation (10) if the second order condition holds. The origin in this figure is in the rear, right-hand corner to allow a more clear view of the contours of the function. For any π, p pair, the value of ˆα depicted will be the optimal fraction of the liability loss to be insured if there is an interior solution in the feasible bankruptcy range. The dark portion of the surface is where ˆα 0 and the lighter portion is where ˆα < 0 ; therefore, only in the region corresponding to the dark surface is an interior solution possible. Figure 3 depicts this latter information from Figure 2 as viewed from below with the origin at the bottom left. The lower left region of Figure 3, below the line where.35 π p = 0, represents the area of focus where the second order condition holds. Note that as depicted in Figure 2, ˆα decreases both as p increases and as π increases. So, the higher the probability of a liability loss, and the higher the probability that the investment will fail, the less insurance is purchased if there is an interior solution. π 0.35" 0.3" Figure " 0.2" 0.15" 0.1" 0.05" α! < 0 α! 0 3(4" 2(3" 1(2" 0(1" 0" 0" 0.05" 0.1" 0.15" 0.2" 0.25" 0.3" 0.35" p Figure 4 depicts the values of ˆβ derived from equation (20). For any π, p pair, this will be the optimal fraction of the wealth to be invested in the risky project if there is an interior solution for both α and β in the feasible bankruptcy range. Note that ˆβ increases as p increases

14 13 and increases more gradually as π increases. As with ˆα, the only region that is relevant for our example is where.35 π p 0 since that is where the second order condition holds. If ˆα < 0 in Figure 3, then for those π, p pairs, α must be replaced with zero in EU β+ in equation (10) and the first order condition for β in (11) must be solved with α = 0 to obtain the optimal fraction of wealth to be invested in the risky asset if an interior solution exists for β in the absence of insurance. Figure β π p Figure 5 depicts three regions in the part of π, p space where π + p.35. In region 1, the optimal decision for the individual is to buy full insurance and to invest nothing in the risky project. This outcome results from the fact that condition (16c), EU β+ ( ˆα, ˆβ) u(w pl) > 0, is violated, or if α < 0, condition (16c) is violated when α is replaced by 0 and β is replaced by the value of β which maximizes (10) when α = 0. For the probability pairs in region 2, it is optimal for the individual to both insure and gamble on the risky project, since α and β comprise an interior solution for the maximization of EU β+ in (10) and at that solution, conditions (16a) - (16c) are not violated. In this region, the higher the probability of a liability loss, and/or the higher the probability that the investment will fail, the more wealth is invested in the risky

15 14 project and the less insurance is purchased. In region 3, the optimal choice for the individual is to go uninsured and to invest in the risky project. Since α is negative in this region, the optimal β is determined by the maximization of EU β+ while α is set to zero. In this range, the proportion of wealth invested in the risky asset increases as p increases and as π increase. In both regions 2 and 3, the proportion of wealth put at risk approaches but remains under 1 as each of the two loss probabilities increases. Figure 5 π 0.35" 0.3" 0.25" 0.2" " 0.1" 0.05" 1 2 0" 0" 0.05" 0.1" 0.15" 0.2" 0.25" 0.3" 0.35" p Bankruptcy is not possible in region 1 of Figure 5, but in regions 2 and 3 the individual is sure to experience bankruptcy if both the liability loss occurs and the risky project fails. This bankruptcy risk is endogenously chosen and makes investment in the risky project desirable. Importantly, in region 2, even while bankruptcy risk is willingly chosen, the individual chooses to purchase insurance. As the level of risk aversion as measured by r decreases, regions 2 and 3 expand toward the south and west such that full insurance is chosen under fewer p, π pairs and the region where the risky project is undertaken correspondingly expands. This also implies that exposure to bankruptcy risk is chosen for this growing set of p, π pairs as well. The opposite is true when the risk aversion parameter r increases, with more p, π pairs corresponding to full insurance and a smaller region of probability pairs associated with endogenously chosen bankruptcy risk.

16 15 CONCLUSION The model and analysis in this paper show that for a large set of parameters, the decision to both gamble on a zero expected profit project and purchase insurance is rational and consistent with expected utility maximization and risk aversion. The affect of limited liability in altering the expected benefits and costs of both the insurance and risky investment is the driving force behind the results presented here. A risk averse individual may choose to purchase less than full insurance, even though premiums are actuarially fair, and simultaneously gamble on a risky project with an otherwise zero expected value, because limited liability both decreases the expected benefit of insurance and increases the expected benefit of the project. Gambling and insurance will not occur simultaneously unless the amount invested is sufficiently large and the amount insured is sufficiently small so that bankruptcy is certain to occur if both an insurable loss and a gambling loss occur. Without the possibility of bankruptcy, the expected benefits of insurance and the risky investment are unaltered and full insurance without investment is preferred. These results offer an explanation for why entrepreneurs and closely held firms may purchase insurance while engaging in a go-for-broke investment strategy. If that type of behavior is optimal for the entrepreneur, a full insurance requirement can stop it. The purchase of partial insurance is not a signal that go-for-broke investing in zero expected value projects will not take place. To the extent that neither the insured loss nor the gamble alone can bankrupt the individual, this type of strategy is distinct from the case where a preexisting chance of default induces excessive risk-taking. Here, the possibility of default is created along with the incentives that typically come along with it.

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