The calculation of optimal premium in pricing ADR as an insurance product

Transcription

1 icccbe 2010 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building Engineering W Tizani (Editor) The calculation of optimal premium in pricing ADR as an insurance product Xinyi Song & Feniosky Peña-Mora Columbia University, New York, NY, USA Carlos A. Arboleda University of Illinois, Urbana-Champaign, IL, USA Abstract From the perspective of transferring risk, pricing Alternative Dispute Resolution (ADR) techniques as an insurance product is worth being considered in order to shift the uncertainty of future ADR cost from project participants to the insurance company. In this process, insurance company reimburses any costs incurred related to dispute resolution, and in return it receives a premium. Thus an appropriate premium acceptable to both project participants and the insurance company is the most important condition for a mutually advantageous insurance policy is. The purpose of this paper is to propose a model on how to determine this premium. First, we use a flow chart to present the process of premium calculation for ADR insurance. Then drawing analogy from seismic risk insurance, Event Tree Analysis (ETA) is used to simulate scenarios of dispute resolution process and to determine the probability mass function of ADR cost. These probabilities are then employed to calculate the expected dispute resolution cost based on which we derive the policy premium. In addition, subjective loss is presented to justify the feasibility of this premium. Finally, a numerical example is carried out to illustrate how the process works. Keywords: ADR, insurance, optimal premium, event tree analysis 1 Introduction In order to deal with future unexpected disputes, one way is for project participants to purchase Alternative Dispute Resolution (ADR) insurance. It not only has the function of reducing the likelihood of dispute occurrence, but could also reduces project participants financial risk by transferring potential risk to insurance company. The key is to determine an appropriate gross premium that is acceptable to both parties. The Expected Value perspective suggests that the gross premium charged by the insurance company would always be larger than the Expected Loss Value of the project participants because insurance company needs cover underwriting expenses and targeted profit. However, Expected Value basis fails to acknowledge the risk-averse attitude of project participants when facing the uncertainty of potentially high ADR cost. Using Subjective Loss Function from Utility Theory could quantitatively incorporate this attitude and thus provide an opportunity for a mutually advantageous insurance policy (Song et al 2010). The purpose of this paper is to propose a model on how to determine an attractive premium. We first include in the problem statement a flow chart to present the process of premium calculation for ADR insurance. Then each step of the process is elaborated. Finally, a numerical example is carried out to illustrate the whole process.

2 2 Problem Statement Figure 2 shows the flow chart on how to determine the premium acceptable to both project participants and insurance company: Figure 1. Analytic flow of determining the premium Each term of Figure 1 will be evaluated based on past experience, statistical data and unique characteristics of a project. Specifically, we use Poisson distribution to model total dispute occurrence. Drawing analogy from seismic risk insurance, we use Event Tree Analysis (ETA) to simulate scenarios of dispute resolution process and to determine the probability mass function of ADR cost. After we calculate the expected dispute resolution cost, the premium is derived with Expense Loading Factor (ELF). This premium is then compared with the maximum fixed cost derived from subjective loss to determine whether it is acceptable to the project participants. Later chapter will provide a detailed illustration on how to implement this system in determining the premium. 3 Calculation of optimal premium 3.1 Expected number of disputes occurrence The first step of the process is to simulate the occurrence of ADR disputes. According to Touran (2003), disputes occur randomly over time and the number of disputes in construction can be approximated with a Poisson distribution. Thus we have: where N is a random variable representing the number of disputes, α is the mean rate of occurrence per unit of time, and T is the estimated project duration. According to Hoshiya et al (2004), the Expected Loss E(C) for dispute resolution cost can thus be determined by: (2) where E(N) is the expected number of disputes occurrence; c i is the ADR cost in scenario i; and p(x i y) is the conditional probability of the ADR cost being c i, given event of dispute occurrence Y. In the next subsection, we borrow ETA method to determine this conditional probability. 3.2 Event Tree Analysis (ETA) An Event Tree Analysis is a graphical representation of logic model that identifies and quantifies all possible outcomes resulting from an accidental (initiating) event, taking into account whether installed safety barriers are functioning or not, and additional events and factors. It provides an inductive approach to reliability assessment as they are constructed using forward logic (Rausand and Høyland 2005). By studying all relevant accidental events, ETA can be used to identify all potential accident scenarios and sequences in a complex system. (1)

3 To determine the frequencies of outcomes, let P(y) denote the frequency of the initiating event; let P(x i ) denote the probability of event x i. Once the initiating event Y occurred, according to Conditional Probability (Ang and Tang 2006), the probability of Outcome X is: P (Outcome X initiating event Y) = P (x1 x2 x3 xn) = P (x1) *P (x2 x1) * P (x3 x1 x2) *P (xn x1 x2 xn-1) Then the frequency of Outcome 1 is: P(y)* P (x1 x2 x3 xn) The frequencies of the other outcomes can be determined in a similar way. In seismic risk analysis, ETA can be utilized to identify the sequential damage and their probabilities to a concerned structure (Hoshiya et al 2004; U.S. Nuclear Regulatory Commission 1975). In this paper, ETA is used to identify a scenario of dispute resolution process and quantitatively determine the probability of corresponding ADR implementation cost. First, ETA sets up event of dispute occurrence as a specified condition. The contractual DRL has m stage in the ladder: ADR1, ADR2, ADRm. For the ith stage, we assume the effectiveness of ADRi is ki, and the average cost for ADRi is ci. For example, k1= 0.5 means 50% of the disputes can be resolved on the first stage. When a dispute occurs, it goes to ADR1, the first stage of the contractual DRL. If dispute resolution does not come to a satisfied settlement by both parties, it will go to the next stage ADR2, and so on. The whole process is shown in Figure 2. Conditional Prob. P(x i y) k 1 ADR Cost (ci) c1 (1-k 1 )*k 2 c2 (1-k 1 )*(1-k 2 )*k 3 c3 Figure 2. ETA of ADR cost 3.3 Subjective Loss Function (SLF) Utility theory attempts to provide insights into decision making in the face of uncertainty by inferring subjective value, or utility, from choices (Bowers et al 1997). Generally a utility function u(w) is used to indicate the value or utility that project participants attach to a certain wealth of amount w. Different from this classic positive utility, subjective loss employed in this paper is quantified by a subjective loss function (SLF) to indicate the negative utility u(c) that project participants attach to a given loss amount of ADR cost c resulting from dispute resolution. According to Song et al (2010), risk-averse project participants should have a Subjective Loss Function (SLF) with u'(c)>0 and u"(c)>0, meaning their SLF is a strictly convex upward function. Project participants evaluate the uncertainty with expected subjective loss (utility) rather than expected loss ($). Then this expected subjective loss is translated to obtain the maximum acceptable fixed loss ($)P for assuming loss C:

4 This maximum fixed loss is then compared with the gross premium to determine whether an insurance is justified. Figure 3 shows the illustrative relationship between expected total loss, gross premium and maximum acceptable fixed loss. As expected total loss increases, maximum acceptable fixed loss increases faster than the gross premium. After a certain point, this maximum fixed loss becomes larger than the gross premium thus investing in insurance is attractive. Song et al (2010) provides a detailed illustration of how to obtain a SLF and calculate the subjective loss, this paper will follow the methodology and show project participants whether it is applicable for them to purchase ADR insurance with their specified SLF. (3) Figure 3. Illustrative Relationship between expected total loss, gross premium and maximum acceptable fixed loss 4 Illustrative example Assume there is a highway bridge project where project participants decide to include a three-stepped DRL in the contract for dispute resolution, m = 3. In this DRL, a dispute goes through Architect/ Engineer or Supervising Officer (ADR1) to mediation (ADR2), then arbitration (ADR3). If the DRL fails to provide a satisfactory settlement, dispute resolution will eventually escalate to litigation. Details are showed in Figure 4: Figure 4. Project DRL (Adapted from Menassa et al 2009)

5 The estimated duration of this project is T = 1440 days from Notice To Proceed (assume there are 30 days in each month, T = 48 months). Assume that the rate of dispute occurrence α = 1 dispute per month. According to Eq.(1), the expected total number of disputes is determined by the mean value of: To determine the Probability Mass Function of ADR cost, Figure 5 shows the ETA for dispute occurrence and ADR cost that are quantified in monetary terms: Conditional Prob. P(x i y) ADR Cost (ci) k 1 = 0.5 $15,000 (1-k 1 )*k 2 = 0.25 $52,500 (1-k 1 )*(1-k 2 )*k 3 = $165,000 (1-k 1 )*(1-k 2 )*(1-k 3 )= $255,000 Figure 5 Project ETA of ADR cost According to Eq. (2), the total expected loss will be: Add an Expense Loading Factor of 30%, the Gross Premium is: Following the methodology of constructing the SLF of project participants in Song et al 2010, assuming project participants have a SLF u(c) (where u'(c)>0 and u"(c)>0 to represent risk-averse project participants) as follows: u(c) = *c^ * c And the corresponding subjective loss would be 815,320,100 (in utils ). Then according to Eq.(3), to make insurance attractive to a participant, there should be a maximum fixed loss P* which satisfies u(p*) E[u(C)] or

6 This means project participants are willing to pay up to $5,199,870 to transfer the risk from themselves to the insurance company. For a GP of $4,705,714, insurance would be an attractive option for project participants. In other words, for this specific project participant, the insurance company can potential charge an ELF up to 36.5%. Therefore, the gross premium for the ADR insurance is feasible and mutually advantageous to both the project participants and the insurance company. 5 Conclusions Pricing ADR insurance is a complex process that involves many factors. This paper proposed a model on how to calculate a gross premium that is acceptable to both project participants and the insurance company. Through the study, the following conclusions can be obtained: first, using Subjective Loss Function makes insurance policy possible for risk-averse project participants despite the fact that the gross premium is higher than the Expected Loss; second, Event Tree Analysis can serve as an effective tool to find the probability for each step on the DRL and to obtain the Expected Loss. For future research, we will first introduce a deductible limit into the model to prevent moral hazard. Moreover sensitivity analysis will be conducted on parameters such as dispute occurrence rate (α) and Expense Loading Factor (ELF) to show how different factors affect the determination of optimal premium. References ANG, A.H. and TANG, W.H., 2006 Probability Concepts in Engineering: Emphasis on Application to Civil and Environmental Engineering. New Jercy: John Wiley &Sons, Inc. BOWERS, N.L., GERBER, H.U., HICKMAN, J.C., JONES, D.A. and NESBITT, C.J., (1997) Actuarial Mathematics. Society of Actuaries. HOSHIYA, M., NAKAMURA, T. and MOCHIZUKI, T., 2004 Transfer of Financial Implications of Seismic Risk to Insurance. Natural Hazards Review, ASCE, 5(3), MENASSA, C., PENA-MORA, F., and PEARSON, N., 2009 An Option Pricing Model to Evaluate ADR Investments in AEC Projects. Journal of Construction Engineering and Management, 135 (3): RAUSAND, M. and HøYLAND, A., 2005 System reliability theory: models, statistical methods, and applications. New Jercy: John Wiley &Sons, Inc. SONG, X., PENA-MORA, F., ARBOLEDA, C., CONGER, R., and MENASSA, C., 2010, The Application of Utility Theory In The Decision-Making Process for Investing In ADR Insurance. In: Construction Research Congress (CRC),2010 Banff, Canada. (Accepted) TOURAN, A., 2003 Calculation of Contingency in Construction Projects. In: IEEE Transactions on Engineering Management, IEEE Engineering Management Society, Piscataway, NJ, 50 (2), UNITED STATES NUCLEAR REGULATORY COMMISSION, An assessment of accident risk in U.S. commercial nuclear power plants. Appendix I. Accident definition and use of event tree, WASH-1400, NUREG-75/ 014, USNRC, Gaithersburg, Md.