A Nonproportional Premium Rating Method for Construction Risks

Abstract

Correct pricing of nonproportional (primary or excess of loss) insurance for construction risks must consider not only how the insured property values build up over time, but also how the probable maximum loss (PML) changes. Conventional pricing methods for static property risks cannot be employed for construction risks, since the latter are characterized by PML patterns that change over time, as well as evolving loss exposures and perils arising from the various phases of the construction project. A further complication arises when delay in startup (DSU) is covered, because a DSU loss is triggered by a property damage loss and both losses contribute jointly to the erosion of an excess layer. This article describes a pricing method with analysis of specific cases of interest, including guidelines for creating practical excess of loss rating models.

Acknowledgments

The author expresses his sincere gratitude to Prof. Patrick L. Brockett, Gus S. Wortham Chair in Risk Management and Insurance at the University of Texas at Austin, for his painstaking review and invaluable suggestions for improvement of the article.

Discussions on this article can be submitted until July 1, 2023. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Notes

1 Construction policies typically include an “escalation allowance” that extends the policy limit above the estimated final TCV stated in the policy schedule, to cater to unexpected contract value increases that could exhaust the contingency included within the contract value.

2 Note that a primary (layer) is a special case where the attachment point is $0.

3 We ignore underlying deductibles. If a deductible $D$ applies, then $V_0$ and $V_1$ can be replaced with $V_0+D$ and $V_1+D,$ but then $P$ must be replaced with the premium $P\prime$ that would be charged with a zero deductible. This may not be easy to determine. It is not correct to write $P=\left\{1-E\left(D/M\right)\right\}P\prime$ because the curves $E\left(x\right)$ are not suitable for thin layers close to zero, as they do not account for loss frequency and reinstatement.

4 If $P\left(t\right)$ is the premium for the interval $\left[0,t\right],$ then $P\prime \left(t\right)$ is the premium density, since the premium for the interval $\sigma \le t\le \tau$ is given by $P\left(\tau \right)-P\left(\sigma \right)={\int }_{\sigma }^{\tau }P\prime \left(t\right)dt.$ Therefore, we may define $r\left(t\right)\equiv P\prime \left(t\right)/v\left(t\right).$ We are assuming that $r=r\left(t\right)$ is constant.

5 PML estimates are subjective. Uniform rules and guidelines for PML estimation do not exist, and PML estimation depends upon available underwriting information, loss estimation models and guidelines, as well as underwriter experience and judgment. In essence, the PML should be an estimate of the maximum loss at a construction site under adverse conditions with no benefit from installed fire protection systems.

6 In other words, $T_0=\inf \left\{t : m\left(t\right)>V_0\right\}$ and $T_1=\sup \left\{t : m\left(t\right)\le V_1\right\}.$

7 A more interesting example is provided by the case of a hydroelectric power plant. At the start of the project, $m\left(t\right)$ increases slowly during site preparation and infrastructure works, reaches a peak midway during the period (corresponding to collapse of a cofferdam or diversion tunnel, resulting in a catastrophic flood), decreases after the main dam is completed and the diversion tunnels are closed, and increases again during the testing period. The risk exposures change significantly during the period; therefore, $r$ is not constant. An accurate layer price requires a number of separate calculations. As $m$ is not non-decreasing, Equation (4)(4) $$L={\int }_{T_0}^{T_1}rv\left(t\right)dt-{\int }_{T_0}^{T}E\left(\frac{V_0}{m\left(t\right)}\right)rv\left(t\right)dt+{\int }_{T_1}^{T}E\left(\frac{V_1}{m\left(t\right)}\right)rv\left(t\right)dt.$$(4) would require modification; but Equation (5)(5) $$L={\int }_0^T\left\{E\left(\min \left\{\frac{V_1}{m\left(t\right)},1\right\}\right)-E\left(\min \left\{\frac{V_0}{m\left(t\right)},1\right\}\right)\right\}rv\left(t\right)dt$$(5) remains valid.

8 This may be a reasonable assumption for the construction of buildings, dams, bridges, etc.

9 This may be a reasonable assumption for the construction of “long linear” risks such as roads, railways, tunnels, pipelines, transmission and distribution lines, etc.

10 As mentioned before, $v\left(t\right)<M$ near $t=0$ is unrealistic. If desired, we could choose

$$m\left(t\right)=\{ \begin{array}{cc}v\left(t\right)& 0\le t<\tau \\ M& \tau \le t\le T\end{array}$$where $\tau$ is chosen so that $v\left(\tau \right)=M.$ However, we ignore this complication by assuming $M\ll V.$

11 The Lloyd’s curve is used for $E\left(x\right).$

12 Independent means that a loss cannot arise from more than one peril or exposure. The various phases of the project (contract works, testing, maintenance) are independent because they do not overlap. Similarly, natural catastrophe perils (earthquake, windstorm, flood) are independent of one another and independent of any other peril or exposure.

13 In the case of delay in startup, the PML is sometimes called the maximum probable delay (MPD).

14 These produce the same function $\widehat{m}$ if $m\propto v.$

15 If $m\left(t\right)$ is constant (e.g., for roads, railways, tunnels, etc.) then $\widehat{m}\left(t\right)=\left(\widehat{M}/M\right)m\left(t\right)$ would not be appropriate, because $\widehat{m}\left(t\right)$ should still be increasing.

16 Replacing $v\left(t\right)$ with any multiple $\alpha v\left(t\right)$ leaves Equation (31)(31) $$L_{+}=\frac{P_{+}}{{\int }_0^{T}v\left(t\right)dt}\underset{0}{\overset{T}{\int }}\left\{E_{+}\left(\min \left\{\frac{V_1}{m_{+}\left(t\right)},1\right\}\right)-E_{+}\left(\min \left\{\frac{V_0}{m_{+}\left(t\right)},1\right\}\right)\right\}v\left(t\right)dt.$$(31) unchanged, because the constants cancel in the two integrals. Therefore, we can replace $v\left(t\right)$ with ${\left(1+\widehat{V}/V\right)v\left(t\right)=v\left(t\right)+\widehat{v}\left(t\right)\equiv v}_{+}\left(t\right)$ to obtain the perfectly symmetrical formula

$$L_{+}=\frac{P_{+}}{{\int }_0^T{v}_{+}\left(t\right)dt}{\int }_0^T\left\{E_{+}\left(\min \left\{\frac{V_1}{m_{+}\left(t\right)},1\right\}\right)-E_{+}\left(\min \left\{\frac{V_0}{m_{+}\left(t\right)},1\right\}\right)\right\}{v}_{+}\left(t\right)dt .$$

17 The underwriter may not know $P_{i+}$ (the combined PD + DSU premium for the exposure ${\mathcal{E}}_i$)! He should be able to supply $P_i,$ but perhaps not ${\widehat{P}}_i$ (which is needed to specify $P_{i+}=P_i+{\widehat{P}}_i$), since DSU premiums are usually not broken down by phases and perils. It is not uncommon for DSU to be priced as a multiple of the PD price or in some other coarse fashion. If ${\widehat{P}}_i$ is not provided, we can make the simple assumption that ${\widehat{P}}_i=\left(P_i/P\right)\widehat{P},$ and hence $P_{i+}=P_i+{\widehat{P}}_i=P_i+\left(P_i/P\right)\widehat{P}=P_i\left(1+\widehat{P}/P\right).$ The policy DSU premium $\widehat{P}$ should be known.

18 This occurs because $\beta \left(c\right)=e^{3.1-0.15\left(1+c\right)c}=1$ in Equation (9)(9) $$E_c\left(x\right)=\frac{\ln \left[\frac{\left(\alpha -1\right)\beta +\left(1-\alpha \beta \right){\beta }^x}{1-\beta }\right]}{\mathrm{ln\hspace{0.17em}}\alpha \beta }$$(9) when $c=\left(\sqrt{753}-3\right)/6.$