# The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation

^{1}

^{2}

^{3}

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^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Reinsurance Models

## 3. Methods

#### 3.1. Compound Poisson Process Methodology

#### 3.2. Tail Regimes

- (i)
**Cramér case:**There exists a finite positive constant ${\nu}_{0}$ such that the claim distribution F satisfies$$\lambda ({m}_{{\nu}_{0}}\left(F\right)-1)=c{\nu}_{0},$$$$f\left(x\right)=x{e}^{-x},\text{\hspace{1em}}x\ge 0.$$- (ii)
**Convolution equivalent case:**The claims distribution function F is said to be convolution equivalent with index $\alpha >0$, if its tail $\overline{F}\left(x\right):=1-F\left(x\right)$, $x>0$, satisfies$$\underset{x\to \infty}{lim}\frac{\overline{F}(x-y)}{\overline{F}\left(x\right)}={e}^{\alpha y}\text{}\mathrm{and}\text{}\underset{x\to \infty}{lim}\frac{\overline{{F}^{2\times}}\left(x\right)}{\overline{F}\left(x\right)}=2{m}_{\alpha}\left(F\right)\infty ,$$$${m}_{\alpha}\left(F\right)<\infty \text{\hspace{1em}}\mathrm{and}\text{\hspace{1em}}{m}_{\alpha +\epsilon}\left(F\right)=\infty \text{\hspace{1em}}\mathrm{for}\text{\hspace{1em}}\forall \epsilon >0.$$These distributions have “medium-heavy” tails in the sense that a convolution equivalent distribution of index α has a finite exponential moment of order α, but any larger order moment is infinite. Typical examples are distributions with tails of the form$$\overline{F}\left(x\right)\sim \frac{c{e}^{-\alpha x}}{{x}^{\rho}},\text{\hspace{1em}}\mathrm{as}\text{}x\to \infty ,$$$$f(x;a,b)=\sqrt{\frac{b}{2\pi}}{x}^{-3/2}exp\left(-\frac{b{(x-a)}^{2}}{2{a}^{2}x}\right),\text{}x0.$$Here $a>0$ is the mean parameter and $b>0$ is called the scale parameter. We denote such a distribution as $IG(a,b)$. In our simulations we choose $a=2$ and $b=1.5$.- (iii)
**Subexponential case:**When (8) is satisfied with $\alpha =0$, F is said to have a subexponential tail. Typical examples are the Pareto distributions. In our simulations we used a Pareto$(1,2)$ distribution with (power law) tail of the form$$\overline{F}\left(x\right)=\frac{1}{{x}^{2}},\text{\hspace{1em}}x\ge 1.$$These distributions have very heavy tails, giving rise to occasional extremely large jumps.

#### 3.3. Simulation Methodology

- (a)
- Neither ${C}_{t}$ nor ${C}_{t}^{R}$ transits above u in $[0,T]$. Suppose there are ${n}_{1}$ such paths among the N.
- (b)
- ${C}_{t}$ transits above u in $[0,T]$ but ${C}_{t}^{R}$ does not. Suppose there are ${n}_{2}$ such paths among the N.
- (c)
- ${C}_{t}^{R}$ transits above u in $[0,T]$ and hence ${C}_{t}$ does also. There are ${n}_{3}=N-{n}_{1}-{n}_{2}$ such paths among the N.

## 4. Results

#### 4.1. Largest Claim Reinsurance

#### 4.2. Excess of Loss Reinsurance

#### 4.3. Comparisons Across Distributions

## 5. Cost of Reinsurance

#### 5.1. Reinsurance Premium and Dividend Adjustment

#### 5.2. Choice of Parameters

#### 5.3. Proportion of Dividend Paid for Reinsurance

#### 5.4. Standard Deviation of Dividend Income

#### 5.5. Dividend Adjustment and Reinsurance Premium, Finite Horizon

## 6. Related Literature and Discussion

#### 6.1. Beveridge, Dickson and Wu Simulations

#### 6.2. Trimming More Values

#### 6.3. The “Light-Medium-Heavy” Classification

#### 6.4. Lévy Insurance Risk Models

## 7. Summary

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Laplace Transforms

**Proposition**

**A1.**

**Proof.**

## References

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^{1.}The LCR procedure can be made prospective by implementing it as a forward looking dynamic procedure in real time, from the cedant’s point of view. Designate as time zero the time at which the reinsurance is taken out. At this time, the cedant company’s assets amount to $u>0$, say. The first claim arriving after time 0 is referred to the reinsurer and not debited to the cedant. Subsequent claims smaller than the initial claim are paid by the cedant until a claim larger than the first (the previous largest) arrives. The difference between these two claims is referred to the reinsurer and not debited to the cedant. The process continues in this way so that at time t, the accumulated amount referred to the reinsurer equals the largest claim up till that time. This procedure has the same effect as referring the largest claim up till time t retrospectively to the reinsurer.^{2.}Indeed, from a theoretical perspective, very little appears to be known about the effects of trimming on an insurance risk process and the subsequent ruin quantities. A series of approximate premium calculations for LCR treaties has been made in the literature; see, for example, [15,16], and [17,18,19,20], and their references.^{3.}The work of [23] suggests that one common principle in choosing L is to keep it at “a level at which claims become very infrequent”.^{4.}There are other possibilities here, for example requiring ${V}_{T,u}^{\ast}+{F}_{T,u}^{\ast}\ge {V}_{T,u}+{F}_{T,u}$, instead of ${V}_{T,u}^{\ast}\ge {V}_{T,u}$ and ${F}_{T,u}^{\ast}\ge {F}_{T,u}$. We chose our formulation since it most clearly mirrors the infinite horizon problem. The interested reader may investigate other versions of the optimization problem.^{6.}But our particular choice of $a=2$ and $b=1.5$ makes the Inverse Gaussian heavier-tailed than the Gamma(2,1).

**Figure 1.**A schematic illustration of the largest claim reinsurance (LCR) reinsurance scheme with a Pareto(1,2) claim distribution. Black dots indicate claim amounts and red lines are the successive amounts liable for the reinsurance company.

**Figure 2.**Sample paths of the insurance risk processes without reinsurance (black line), with LCR reinsurance (red line), and with EOL reinsurance (green line), for a Pareto(1,2) claim distribution. The company’s initial reserve is $u=10$, the safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 year = 3.65 days. For comparability between the two schemes, the retention level L for the EOL scheme is chosen so that the expected values of the LCR and EOL aggregate claims are equal at maturity time $T=5000$.

**Figure 3.**Sample paths of the insurance risk processes without reinsurance (black line), with LCR reinsurance (red line), and with excess of loss (EOL) reinsurance (green line), for a Gamma$(2,1)$ claim distribution. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. L for the EOL scheme is chosen so that the expected values of the LCR and EOL claim distributions are equal at maturity time $T=5000$.

**Figure 4.**Sample paths of the insurance risk processes without reinsurance (black line), with LCR reinsurance (red line), and with EOL reinsurance (green line), for an IG$(2,1.5)$ claim distribution. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. L for the EOL scheme is chosen so that the expected values of the LCR and EOL claim distributions are equal at maturity time $T=5000$.

**Figure 6.**${v}_{\theta}\left(u\right)$ (from (32)) is the proportion of the dividend available to pay for reinsurance without reducing the value of the firm. For Pareto, Inverse Gaussian and Gamma claim distributions, initial reserve levels $u=10$, 30, 50, 70, 100, time value of money $\rho =0.0005$, and safety loadings $\theta =0$, 0.025, 0.05, 0.075, 0.1. Top panel: LCR; bottom panel, EOL with L taken as the 98th percentile of the claims distribution. Simulations are done with $N=10,000$ sample paths.

**Figure 7.**${s}_{\theta}\left(u\right)$ (from (34), obtained by approximation at $T=13,800$) is the ratio of the standard deviation of the dividend income obtained under reinsurance, to that without (infinite horizon case). For Pareto, Inverse Gaussian and Gamma claim distributions, initial reserve levels $u=10$, 30, 50, 70, 100, time value of money $\rho =0.0005$, and safety loadings $\theta =0$, 0.025, 0.05, 0.075, 0.1. Top panel: LCR; bottom panel, EOL, with L taken as the 98th percentile of the claims distribution. Simulations are done with $N=10,000$ sample paths.

**Figure 8.**${v}_{\theta}(T,u)$ (from (37)) is the proportion of the dividend available to pay for reinsurance without reduing the value of the firm. For LCR and EOL reinsurance policies, Pareto, Inverse Gaussian and Gamma claim distributions, initial reserve levels $u=10$, 30, 50, 70, 100, time value of money $\rho =0.0005$, and safety loadings $\theta =0$, 0.025, 0.05, 0.075, 0.1. Top panel: $T=100$; Middle panel: $T=500$; Bottom panel: $T=1000$. Retention level L for EOL reinsurance is the solution to (17). For $T=100$, $L\left(T\right)=5.64$, 6.89, 4.49; for $T=500$, $L\left(T\right)=12.62$, 11.27, 6.10; for $T=1000$, $L\left(T\right)=17.84$, 13.39, 6.79 for Pareto, Inverse Guassian and Gamma distributions respectively. Simulations are done with $N=10,000$ sample paths.

**Table 1.**LCR reinsurance for Pareto$(1,2)$ distributed claims. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. Simulations are done with $N=100,000$ sample paths. The $T=\infty $ case refers to the results obtained from Algorithm III in [12].

u | T | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}<\mathit{T})$ | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}^{\mathit{M}}<\mathit{T})$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}^{\mathit{M}}$ | % Changes |
---|---|---|---|---|---|---|

10 | 100 | 0.43 | 0.14 | 19.06 | 37.03 | 93.34 |

500 | 0.53 | 0.20 | 38.14 | 85.45 | 124.02 | |

1000 | 0.55 | 0.21 | 44.75 | 104.35 | 133.19 | |

∞ | $0.56\pm 0.03$ | - | - | - | - | |

30 | 100 | 0.14 | 0.02 | 35.78 | 58.97 | 64.84 |

500 | 0.26 | 0.06 | 90.65 | 164.25 | 81.19 | |

1000 | 0.28 | 0.06 | 113.18 | 214.20 | 89.25 | |

∞ | $0.32\pm 0.02$ | - | - | - | - | |

50 | 100 | 0.06 | 0.00 | 44.64 | 66.40 | 55.74 |

500 | 0.14 | 0.02 | 129.20 | 215.24 | 66.59 | |

1000 | 0.17 | 0.03 | 172.73 | 303.81 | 75.89 | |

∞ | $0.20\pm 0.02$ | - | - | - | - | |

70 | 100 | 0.03 | 0.00 | 45.66 | 73.67 | 61.37 |

500 | 0.09 | 0.01 | 157.07 | 263.60 | 67.82 | |

1000 | 0.11 | 0.01 | 221.55 | 380.55 | 71.77 | |

∞ | $0.14\pm 0.02$ | - | - | - | - | |

100 | 100 | 0.01 | 0.00 | 37.32 | 75.64 | 102.71 |

500 | 0.05 | 0.00 | 180.25 | 300.30 | 66.60 | |

1000 | 0.06 | 0.00 | 258.22 | 450.58 | 74.50 | |

∞ | $0.081\pm 0.017$ | - | - | - | - |

**Table 2.**LCR reinsurance for Gamma$(2,1)$ distributed claims. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. Simulations are done with $N=100,000$ sample paths.

u | T | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}<\mathit{T})$ | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}^{\mathit{M}}<\mathit{T})$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}^{\mathit{M}}$ | % Changes |
---|---|---|---|---|---|---|

10 | 100 | 0.43 | 0.25 | 22.22 | 34.91 | 57.11 |

500 | 0.49 | 0.32 | 44.35 | 70.81 | 61.18 | |

1000 | 0.50 | 0.32 | 47.79 | 77.57 | 63.39 | |

30 | 100 | 0.08 | 0.04 | 47.95 | 59.49 | 24.08 |

500 | 0.14 | 0.08 | 115.46 | 145.61 | 26.11 | |

1000 | 0.15 | 0.09 | 135.16 | 170.45 | 26.11 | |

50 | 100 | 0.01 | 0.00 | 62.13 | 72.35 | 16.45 |

500 | 0.04 | 0.02 | 177.90 | 208.29 | 17.08 | |

1000 | 0.04 | 0.02 | 214.04 | 254.08 | 18.71 | |

70 | 100 | 0.00 | 0.00 | 68.07 | 79.12 | 16.24 |

500 | 0.01 | 0.01 | 233.81 | 264.30 | 13.04 | |

1000 | 0.01 | 0.01 | 292.09 | 327.81 | 12.23 | |

100 | 100 | 0.00 | 0.00 | - | - | - |

500 | 0.00 | 0.00 | 315.52 | 347.89 | 10.26 | |

1000 | 0.00 | 0.00 | 393.28 | 438.94 | 11.61 |

**Table 3.**LCR reinsurance for IG$(2,1.5)$ distributed claims. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. Simulations are done with $N=100,000$ sample paths.

u | T | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}<\mathit{T})$ | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}^{\mathit{M}}<\mathit{T})$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}^{\mathit{M}}$ | % Changes |
---|---|---|---|---|---|---|

10 | 100 | 0.51 | 0.24 | 18.63 | 35.88 | 92.61 |

500 | 0.59 | 0.33 | 40.69 | 83.27 | 104.67 | |

1000 | 0.60 | 0.34 | 47.97 | 100.35 | 109.18 | |

30 | 100 | 0.16 | 0.05 | 38.97 | 56.63 | 45.33 |

500 | 0.27 | 0.13 | 101.28 | 153.44 | 51.51 | |

1000 | 0.28 | 0.14 | 122.80 | 189.76 | 54.53 | |

50 | 100 | 0.04 | 0.01 | 52.30 | 68.41 | 30.81 |

500 | 0.12 | 0.05 | 150.57 | 205.16 | 36.25 | |

1000 | 0.13 | 0.06 | 197.59 | 272.25 | 37.79 | |

70 | 100 | 0.01 | 0.00 | 61.72 | 78.22 | 26.73 |

500 | 0.05 | 0.02 | 197.71 | 253.26 | 28.10 | |

1000 | 0.06 | 0.03 | 261.72 | 339.91 | 29.88 | |

100 | 100 | 0.00 | 0.00 | 54.87 | 64.62 | 17.77 |

500 | 0.01 | 0.00 | 250.91 | 303.45 | 20.94 | |

1000 | 0.02 | 0.01 | 368.16 | 455.16 | 23.63 |

**Table 4.**EOL reinsurance for Pareto$(2,1)$ distributed claims. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. Simulations are done with $N=100,000$ sample paths. Retention level L is the solution to (17). For $T=100$, $L\left(T\right)=5.64$; for $T=500$, $L\left(T\right)=12.62$; for $T=1000$, $L\left(T\right)=17.84$.

u | T | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}<\mathit{T})$ | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}^{\mathit{L}}<\mathit{T})$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}^{\mathit{L}}$ | % Changes | No Effect |
---|---|---|---|---|---|---|---|

10 | 100 | 0.43 | 0.20 | 15.88 | 21.36 | 34.55 | 0.08 |

500 | 0.53 | 0.39 | 31.50 | 36.55 | 16.03 | 0.30 | |

1000 | 0.55 | 0.44 | 38.93 | 43.54 | 11.84 | 0.38 | |

30 | 100 | 0.14 | 0.01 | 33.48 | 49.71 | 48.47 | 0.00 |

500 | 0.26 | 0.08 | 71.37 | 92.83 | 30.08 | 0.03 | |

1000 | 0.28 | 0.12 | 87.85 | 106.86 | 21.63 | 0.07 | |

50 | 100 | 0.06 | 0.00 | 45.22 | 67.26 | 48.76 | 0.00 |

500 | 0.14 | 0.02 | 107.64 | 145.85 | 35.50 | 0.00 | |

1000 | 0.17 | 0.03 | 134.88 | 172.85 | 28.15 | 0.01 | |

70 | 100 | 0.03 | 0.00 | 70.52 | 79.83 | 13.20 | 0.00 |

500 | 0.09 | 0.00 | 144.90 | 195.31 | 34.79 | 0.00 | |

1000 | 0.11 | 0.01 | 182.04 | 243.62 | 33.82 | 0.00 | |

100 | 100 | 0.01 | 0.00 | - | - | - | 0.00 |

500 | 0.05 | 0.00 | 181.23 | 281.15 | 55.13 | 0.00 | |

1000 | 0.06 | 0.00 | 235.18 | 318.21 | 35.31 | 0.00 |

**Table 5.**EOL reinsurance for Gamma$(2,1)$ distributed claims. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. Simulations are done with $N=100,000$ sample paths. Retention level L is the solution to (17). For $T=100$, $L\left(T\right)=4.49$; for $T=500$, $L\left(T\right)=6.10$; for $T=1000$, $L\left(T\right)=6.79$.

u | T | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}<\mathit{T})$ | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}^{\mathit{L}}<\mathit{T})$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}^{\mathit{L}}$ | % Changes | No Effect |
---|---|---|---|---|---|---|---|

10 | 100 | 0.43 | 0.32 | 20.02 | 24.34 | 21.55 | 0.13 |

500 | 0.49 | 0.45 | 40.67 | 44.52 | 9.45 | 0.32 | |

1000 | 0.50 | 0.47 | 45.80 | 48.49 | 5.85 | 0.38 | |

30 | 100 | 0.08 | 0.03 | 44.45 | 53.24 | 19.77 | 0.00 |

500 | 0.14 | 0.11 | 106.87 | 116.82 | 9.31 | 0.04 | |

1000 | 0.15 | 0.13 | 127.08 | 135.13 | 6.33 | 0.07 | |

50 | 100 | 0.01 | 0.00 | 57.85 | 68.12 | 17.74 | 0.00 |

500 | 0.04 | 0.03 | 165.39 | 181.42 | 9.69 | 0.01 | |

1000 | 0.04 | 0.03 | 200.86 | 214.19 | 6.64 | 0.01 | |

70 | 100 | 0.00 | 0.00 | 66.66 | 80.33 | 20.52 | 0.00 |

500 | 0.01 | 0.01 | 219.09 | 239.44 | 9.29 | 0.00 | |

1000 | 0.01 | 0.01 | 279.28 | 294.17 | 5.33 | 0.00 | |

100 | 100 | 0.00 | 0.00 | - | - | - | 0.00 |

500 | 0.00 | 0.00 | 306.48 | 334.20 | 9.04 | 0.00 | |

1000 | 0.00 | 0.00 | 385.12 | 410.96 | 6.71 | 0.00 |

**Table 6.**EOL reinsurance for IG$(2,1.5)$ distributed claims. The safety loading is $\theta =0.1$, expected claim size is µ = 2, claim arrival rate is $\lambda =1$, and each time unit is 0.01 years. Simulations are done with $N=100,000$ sample paths. Retention level L is the solution to (17). For $T=100$, $L\left(T\right)=6.89$; for $T=500$, $L\left(T\right)=11.27$; for $T=1000$, $L\left(T\right)=13.39$.

u | T | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}<\mathit{T})$ | $\widehat{\mathbf{P}}({\mathit{\tau}}_{\mathit{u}}^{\mathit{L}}<\mathit{T})$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}$ | ${\widehat{\mathit{\tau}}}_{\mathit{u},\mathit{T}}^{\mathit{L}}$ | % Changes | No Effect |
---|---|---|---|---|---|---|---|

10 | 100 | 0.51 | 0.33 | 15.99 | 21.11 | 32.05 | 0.15 |

500 | 0.59 | 0.52 | 35.92 | 41.51 | 15.55 | 0.39 | |

1000 | 0.60 | 0.56 | 44.17 | 48.64 | 10.10 | 0.47 | |

30 | 100 | 0.16 | 0.04 | 35.16 | 47.97 | 36.44 | 0.00 |

500 | 0.27 | 0.18 | 89.10 | 106.78 | 19.85 | 0.07 | |

1000 | 0.28 | 0.22 | 109.79 | 124.45 | 13.35 | 0.12 | |

50 | 100 | 0.04 | 0.00 | 46.48 | 63.63 | 36.89 | 0.00 |

500 | 0.12 | 0.06 | 133.06 | 159.35 | 19.76 | 0.01 | |

1000 | 0.13 | 0.09 | 175.11 | 199.22 | 13.77 | 0.03 | |

70 | 100 | 0.01 | 0.00 | 55.19 | 71.45 | 29.44 | 0.00 |

500 | 0.05 | 0.02 | 177.18 | 211.36 | 19.29 | 0.00 | |

1000 | 0.06 | 0.03 | 233.06 | 267.53 | 14.79 | 0.01 | |

100 | 100 | 0.00 | 0.00 | - | - | - | 0.00 |

500 | 0.01 | 0.00 | 227.50 | 272.16 | 19.63 | 0.00 | |

1000 | 0.02 | 0.00 | 333.10 | 384.93 | 15.56 | 0.00 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fan, Y.; Griffin, P.S.; Maller, R.; Szimayer, A.; Wang, T.
The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation. *Risks* **2017**, *5*, 3.
https://doi.org/10.3390/risks5010003

**AMA Style**

Fan Y, Griffin PS, Maller R, Szimayer A, Wang T.
The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation. *Risks*. 2017; 5(1):3.
https://doi.org/10.3390/risks5010003

**Chicago/Turabian Style**

Fan, Yuguang, Philip S. Griffin, Ross Maller, Alexander Szimayer, and Tiandong Wang.
2017. "The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation" *Risks* 5, no. 1: 3.
https://doi.org/10.3390/risks5010003