# Rating a Wildfire Mitigation Strategy with an Insurance Premium: A Boreal Forest Case Study

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

_{2}) and A2 (3 × CO

_{2}) climate scenarios, fire cycles should lower to around 254 and 79 years respectively, values lying either in between the current and historical fire cycles [44] or below the historical fire cycle.

#### 2.2. Timber Supply Model

^{3}/period) (Equation (1)). The first constraint provided an even flow of harvest volume over time (Equation (2)). For harvest planning purposes, the study area was divided into different spatially organized compartments (operating areas between 30 km

^{2}and 150 km

^{2}) as a function of canopy closure and species composition [45] to emulate fire size distribution [46]. These operating areas are open to harvest when more than 30% of their timber productive area is eligible to harvest (Equation (3), [47]). Planting of jack pine after a clear-cut was limited to less than the actual plantation level (7500 ha per period (Equation (4))). A forest age structure was also targeted with a minimal abundance of three age classes (0–150 years: 63%, 150–275 years: 21% and more than 275 years: 16%) (Equation (5)) [41]. Two harvesting systems were implemented, careful logging around advanced regeneration [48] and irregular shelter-wood cuts (50% removal of merchantable volume; [49]). The areas planned to be harvested must be positive (Equation (6)).

- $o:\text{operationalarea}\left(1\dots 107\right)$
- $s:\text{successionalpathway}\left(1\dots 3\right)$
- $p:\text{period}\left(1\dots 30\right)$
- $a:\text{standage}$
- $h:\text{harvesttype}\left(1\dots 2\right)$
- $c:\text{cohortnumber}\left(1\dots 3\right)$

- ${X}_{op}=\{\begin{array}{c}\text{}1,\text{}\mathrm{if}\text{}\mathrm{operating}\text{}\mathrm{area}\text{}\mathrm{is}\text{}\mathrm{open}\text{}\mathrm{to}\text{}\mathrm{harvest}\\ 0,\text{}\mathrm{otherwise}\end{array}\forall \text{}o,p$
- ${A}_{ashop}\text{}:\text{Areaharvested}\left(\text{ha}\right),\text{}\forall \text{}a,s,h,o,p$
- ${e}_{ashop}:\text{Areaeligibleforharvest}\left(\text{ha}\right),\text{}\forall \text{}a,s,h,o,p$
- ${C}_{cp}:\text{Areabelongingtocohortc}\left(\text{ha}\right),\text{}\forall \text{}c,p$

#### 2.3. Interaction between Harvest Scheduling and Stochastic Processes

^{3}·ha

^{−1}) not prescribed in the harvest plan until it reached the targeted timber supply level. Disturbance-specific changes in forest composition and age structure drive the interactions between fire, succession and harvest. Natural succession was modelled as a semi-Markov process [53] with probabilities of transition estimated from the proportions of each stratum by stand age class (20-year interval) observed in the forest map. The spatial resolution of the model was 10 ha per pixel and the temporal resolution five years. We performed 100 replications of each scenario, which provided stable estimates of indicators, especially VaR [56]. We used the technique of common random numbers [57] to reduce the variability generated by random effects between the scenarios [58]. Simulation outputs allowed us to quantify loss likelihood distributions and estimate insurance premiums as detailed below.

#### 2.4. Risk Management

#### 2.4.1. Risk Characterization

_{p}= 0, $\forall $ p), one for which risk is the highest while respecting all the constraints of the timber supply model and one for which PTH is equal to the median realized harvest level. We used the landscape simulation model for this purpose by decreasing the PTH originally estimated with the timber supply model by steps of 10% until no risk occurred anymore for three fire cycles. We then estimated the parameters of a piecewise linear model with one knot between realized harvest and planned timber supply values (Equation (8)):

#### 2.4.2. Insurance Premium

#### 2.4.3. Timber Supply Reduction

_{th}in Equation (8)) and the one for which disruptions do not occur anymore (i.e., VaR

_{p}= 0, $\forall $ p).

#### 2.5. Comparison of Risk Management Strategies

_{p}and median VaR values were computed from these loss distributions. We also used these simulation results to estimate the parameters of piecewise linear models (Equation (8)) in order to find the PTH value equal to the median realized harvest level for each considered fire cycle. Premium insurance was then computed (Equation (9)) for a range of interest rates used for public investments (0%, 1%, 2% and 4% [61]) at the threshold PTH value equal to the median realized harvest level. Finally, we compared the timber supply reductions required to cancel risk to insurance premiums for each fire cycle.

## 3. Results

#### 3.1. Risk Assessment

^{3}period

^{−1}when not considering fire risk (Equation (1)). However, a blind implementation of such a strategy will not enable the procurement of expected timber levels (Figure 2) and timber supply disruptions caused by fire are expected (Figure 3). Despite the likely occurrence of such disruptions, the median rate of planning success reaches 97% (3.7 Mm

^{3}period

^{−1}) of the optimal solution provided by the timber supply model with a fire cycle of 400 years and decreases only up to 73% (2.8 Mm

^{3}period

^{−1}) with a fire cycle of 100 years (Figure 2). The chances of obtaining such a rate of success are, however, threatened by infrequent but possibly very significant disruptions. Timber supply disruptions may start to occur as soon as the 6th planning period (30 years) and, depending on the considered fire cycle, either tend to disappear after 50 years (and occur again approximately after one mean stand rotation) or maintain themselves for the rest of the planning horizon (Figure 3).

^{3}period

^{−1}) and represent 55% to 74% of the periodic timber harvest, depending on the fire regime that is considered. Median VaR across the planning horizon with the longest fire cycle (400) years is substantially lower than the maximum VaR (0.9 vs. 2.2 Mm

^{3}period

^{−1}) when compared to that resulting from a fire cycle of 100 years (2.4 vs. 2.9 Mm

^{3}period

^{−1}), indicating more frequent occurrences of important timber supply disruption throughout the planning horizon with a higher burn rate (Figure 3c).

^{3}period

^{−1}, depending on the considered fire cycle (Figure 4, Table 1). At the threshold PTH value beyond which the implementation success decreases, median VaR values (0.02 to 1.12 Mm

^{3}period

^{−1}depending on the fire cycle) are much lower than those induced by the implementation of the entire optimized timber supply solution. They are in fact reduced by a factor varying between two and 20. Maximum VaR values are less reduced, by a factor between 1.6 (for a fire cycle of 100 years) and 1.9 (for a fire cycle of 400 years). This means that ignoring fire in the timber supply model and assuming that fire risk is totally controlled (planning optimism) increased the risk of supply disruptions by almost one order of magnitude, even with a fire cycle of 400 years. Such increased risk is, however, accompanied by an increase in realized harvest level, the rate of which varies between 0% and 60% (= slope of the second segment of the piecewise regression) (Figure 4, Table 1). This increase is only significant with a fire cycle of 400 years (Table 1).

#### 3.2. Risk Management Strategies

_{p}= 0, $\forall $ p) with the current fire regime (400 years, Figure 3), and such reductions increase to 48% and 56% for fire cycles of 200 and 100 years, respectively. Harvest reductions therefore seem to increase non-linearly with an increase of the fire cycle (i.e., +6%/100 years between 200 and 400 years and +8%/100 years between 100 and 200 years). In fact, the sensitivity of maximum VaR to a timber harvest reduction decreases as maximum VaR tends to zero (Figure 5). Depending on the interest rate and the fire cycle, we looked at the changes in the amount of insurance premium an insurer should hold against unexpected losses as a function of fire risk (Figure 6). Insurance premiums represent between 5% and 7% of the level of supply for a fire cycle of 400 years (Figure 6), which are noticeably lower than for a timber harvest reduction strategy. With a change of fire cycle between 200 and 400 years, premium increases are also lower than those of a timber supply reduction strategy (between 5% and 11%/100 years depending on the interest rate), and increase less between 100 and 200 (between 2% and 7%/100 years). Such premium increases are more directly related to an increase in median VaR rather than to an increase in fire cycle (Figure 7).

## 4. Discussion

^{3}period

^{−1}) in interaction with fire had a success rate of 97%. The maximum PTH value that could be totally implemented with the landscape simulation model was 2.8 Mm

^{3}period. This means that a timber harvest reduction of 36% would be required to increase the success rate up to 100%, which is clearly very expensive [15,63]. In fact, since absolute protection against losses cannot be guaranteed, some level of acceptable loss expressed as a risk tolerance must be established, which can widely vary based on knowledge of exposures and proposed risk management solutions [27]. This points to the importance of choosing a level of tolerance to risk when facing a relatively low fire cycle, as already noted by [60]. Increasing tolerance to risk requires the availability of other mitigation strategies, such as the diversification of procurement sources [64], when supply disruptions occur.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Location of study area Forest Management Unit 085-51. Grey polygons correspond to operating areas [45].

**Figure 2.**Box and whiskers plots representing the probability distributions of the simulated implementation of the optimal solution (3.8 Mm

^{3}period

^{−1}) provided by the timber supply model (Equations (1)–(6)) under current (400 years) and probable interval for future fire regimes (100 and 200 years).

**Figure 3.**Probability distributions of the success rate of the simulated harvest schedule implementation under current ((

**a**) 400 years) and probable future fire regimes ((

**b**) 200 years; (

**c**) 100 years). From bottom up, broken and bold lines represent the 5th, median and 95th percentiles. One hundred percent represents the target (continuous line) and ninety percent correspond to a cutoff value below which a timber supply disruption was considered to occur [57].

**Figure 4.**Relationship between planned timber harvest and its simulated implementation (median value) when considering the risk of fire for three fire cycles (100, 200 and 400 years). Parameters of the segmented linear models are provided in Table 1. Error bars represent the 5th and 95th percentiles of the probability distribution of simulated harvest levels.

**Figure 5.**Relationship between planned timber harvest and maximum value at risk for three fire cycles.

**Figure 6.**Distribution of the planned timber harvest into: a part that is not entirely feasible (planning optimism, in white, see Figure 4), a part that should be used to build a buffer stock of timber (

**dark gray**) (with a timber harvest reduction—THR, which should not be harvested, or with an insurance premium, which should be harvested and set apart, with an interest rate between 0% and 4%) (protection strategy), and a part available for harvest (

**light grey**), considering three possible fire cycles.

**Figure 7.**Relationship between median value-at-risk (VaR) and insurance premium as a function of interest rates (0% to 4%) and present (400 years) or probable fire cycles (100 and 200 years).

**Table 1.**Parameter values of piecewise linear models with one knot (threshold planned timber harvest (PTH)) (Equation (8)) relating PTH and periodic median realized harvest levels implemented with the landscape simulation model under current (400 years) and probable interval for future fire regimes (100 and 200 years).

Fire Cycle | Threshold PTH (Mm^{3} Period^{−1}) | β |
---|---|---|

100 | 2.77 (0.18) | 0.04 (0.26) ^{a} |

200 | 3.23 (0.15) | −0.01 (0.39) ^{a} |

400 | 3.34 (0.18) | 0.60 (0.25) |

^{a}Not significantly different from 0 at α = 0.05 (p = 0.78 and p = 0.96, respectively). Numbers in parentheses represent a half-confidence interval.

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**MDPI and ACS Style**

Rodriguez-Baca, G.; Raulier, F.; Leduc, A.
Rating a Wildfire Mitigation Strategy with an Insurance Premium: A Boreal Forest Case Study. *Forests* **2016**, *7*, 107.
https://doi.org/10.3390/f7050107

**AMA Style**

Rodriguez-Baca G, Raulier F, Leduc A.
Rating a Wildfire Mitigation Strategy with an Insurance Premium: A Boreal Forest Case Study. *Forests*. 2016; 7(5):107.
https://doi.org/10.3390/f7050107

**Chicago/Turabian Style**

Rodriguez-Baca, Georgina, Frédéric Raulier, and Alain Leduc.
2016. "Rating a Wildfire Mitigation Strategy with an Insurance Premium: A Boreal Forest Case Study" *Forests* 7, no. 5: 107.
https://doi.org/10.3390/f7050107