# Direct and Hierarchical Models for Aggregating Spatially Dependent Catastrophe Risks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Copula Trees

#### 2.1. Problem Statement

#### 2.2. Direct Model

#### 2.3. Hierarchical Model

#### 2.4. Implementation of Risk Aggregation at Branching Nodes

#### 2.4.1. Split-Atom Convolution

Algorithm 1: Split-Atom Convolution: 9-products |

Input: Two discrete pdfs ${p}_{X}$ and ${p}_{Y}$ with supports:$x=\{{x}_{1},{x}_{2},{x}_{2}+{h}_{x},{x}_{2}+2{h}_{x},\dots ,{x}_{2}+({N}_{x}-3){h}_{x},{x}_{{N}_{x}}\}$; $y=\{{y}_{1},{y}_{2},{y}_{2}+{h}_{y},{y}_{2}+2{h}_{y},\dots ,{y}_{2}+({N}_{y}-3){h}_{y},{y}_{{N}_{y}}\}$; and probabilities: ${p}_{X}(x)={\sum}_{i=1}^{{N}_{x}}\delta (x-{x}_{i}){p}_{X}({x}_{i})$; ${p}_{Y}(y)={\sum}_{i=1}^{{N}_{y}}\delta (y-{y}_{i}){p}_{Y}({y}_{i})$ ${N}_{s}$ - maximum number of points for discretizing convolution grid - 1:
- ${x}^{(1)}=\{{x}_{1}\}$; ${p}_{X}^{(1)}(x)=\delta (x-{x}^{(1)}){p}_{X}({x}_{1})$
`// split the left atom of ${p}_{X}$` - 2:
- ${x}^{(2)}=\{{x}_{{N}_{x}}\}$; ${p}_{X}^{(2)}(x)=\delta (x-{x}_{i}^{(2)})\xb7{p}_{X}^{(2)}({x}_{i}^{(2)})$
`// split the right atom of ${p}_{X}$` - 3:
- ${x}^{(3)}=\{{x}_{2},{x}_{2}+{h}_{x},{x}_{2}+2{h}_{x},\dots ,{x}_{2}+({N}_{x}-3){h}_{x}\}$; ${p}_{X}^{(3)}(x)={\sum}_{i=1}^{{N}_{x}-2}\delta (x-{x}_{i}^{(3)}){p}_{X}^{(3)}({x}_{i}^{(3)})$
- 4:
- ${y}^{(1)}=\{{y}_{1}\}$; ${p}_{Y}^{(1)}(y)=\delta (y-{y}^{(1)}){p}_{Y}({y}_{1})$
`// split the left atom of ${p}_{Y}$` - 5:
- ${y}^{(2)}=\{{y}_{{N}_{y}}\}$; ${p}_{Y}^{(2)}(y)=\delta (y-{y}_{i}^{(2)})\xb7{p}_{Y}^{(2)}({y}_{i}^{(2)})$
`// split the right atom of ${p}_{Y}$` - 6:
- ${y}^{(3)}=\{{y}_{2},{y}_{2}+{h}_{y},{x}_{2}+2{h}_{y},\dots ,{y}_{2}+({N}_{y}-3){h}_{y}\}$; ${p}_{Y}^{(3)}(y)={\sum}_{i=1}^{{N}_{y}-2}\delta (y-{y}_{i}^{(3)}){p}_{Y}^{(3)}({y}_{i}^{(3)})$
- 7:
- ${h}_{{s}^{\u2605}}=\frac{{x}_{{N}_{x}}+{y}_{{N}_{y}}-({x}_{1}+{y}_{1})}{{N}_{{s}^{\u2605}}-1}$
`// corresponding step size of the main part of convolution grid s` - 8:
- ${h}_{s}=max({h}_{x},{h}_{y},{h}_{{s}^{\u2605}})$
`// final step size of the main part of convolution grid s` - 9:
- ${N}_{s}=\lfloor \frac{{x}_{{N}_{x}}+{y}_{{N}_{y}}-({x}_{1}+{y}_{1})}{{h}_{s}}\rfloor $
`// corresponding number of points` - 10:
`// set irregular convolution grid:`- 11:
**if**${h}_{s}\ge ({x}_{2}+{y}_{2}-{x}_{1}-{y}_{1})$**then**- 12:
- $s=\{{x}_{1}+{y}_{1},{x}_{2}+{y}_{2},{x}_{2}+{y}_{2}+{h}_{s},\dots ,{x}_{2}+{y}_{2}+({N}_{s}-3){h}_{s},{x}_{{N}_{x}}+{y}_{{N}_{y}}\}$
- 13:
**else**- 14:
- $s=\{{x}_{1}+{y}_{1},{x}_{2}+{y}_{2}-{h}_{s},{x}_{2}+{y}_{2},\dots ,{x}_{2}+{y}_{2}+({N}_{s}-4){h}_{s},{x}_{{N}_{x}}+{y}_{{N}_{y}}\}$
- 15:
**end if**- 16:
- ${x}^{{(3)}^{\prime}}=\{{x}_{2},{x}_{2}+{h}_{s},{x}_{2}+2{h}_{s},\dots ,{x}_{2}+({N}_{x}-3){h}_{s}\}$
`// discretize ${x}^{(3)}$ with ${h}_{s}$` - 17:
- ${p}_{X}^{{(3)}^{\prime}}(x)={\sum}_{i=1}^{{N}_{x}^{\prime}}\delta (x-{x}_{i}^{{(3)}^{\prime}}){p}_{X}^{{(3)}^{\prime}}({x}_{i}^{{(3)}^{\prime}})$
`// regrid ${p}_{X}^{(3)}$` - 18:
- ${y}^{{(3)}^{\prime}}=\{{y}_{2},{y}_{2}+{h}_{s},{y}_{2}+2{h}_{s},\dots ,{y}_{2}+({N}_{y}-3){h}_{s}\}$
`// discretize ${y}^{(3)}$ with ${h}_{s}$` - 19:
- ${p}_{Y}^{{(3)}^{\prime}}(y)={\sum}_{i=1}^{{N}_{y}^{\prime}}\delta (y-{y}_{i}^{{(3)}^{\prime}}){p}_{Y}^{{(3)}^{\prime}}({y}_{i}^{{(3)}^{\prime}})$
`// regrid ${p}_{Y}^{(3)}$` - 20:
- ${\mathcal{B}}^{(1)}={p}_{X}^{(1)}\oplus {p}_{Y}^{(1)}$
`// Brute Force convolution` - 21:
- ${\mathcal{B}}^{(2)}={p}_{X}^{(2)}\oplus {p}_{Y}^{(1)}$
`// —”—` - 22:
- ${\mathcal{B}}^{(3)}={p}_{X}^{{(3)}^{\prime}}\oplus {p}_{Y}^{(1)}$
`// —”—` - 23:
- ${\mathcal{B}}^{(4)}={p}_{X}^{(1)}\oplus {p}_{Y}^{(2)}$
`// —”—` - 24:
- ${\mathcal{B}}^{(5)}={p}_{X}^{(2)}\oplus {p}_{Y}^{(2)}$
`// —”—` - 25:
- ${\mathcal{B}}^{(6)}={p}_{X}^{{(3)}^{\prime}}\oplus {p}_{Y}^{(2)}$
`// —”—` - 26:
- ${\mathcal{B}}^{(7)}={p}_{X}^{(1)}\oplus {p}_{Y}^{{(3)}^{\prime}}$
`// —”—` - 27:
- ${\mathcal{B}}^{(8)}={p}_{X}^{(2)}\oplus {p}_{Y}^{{(3)}^{\prime}}$
`// —”—` - 28:
- ${\mathcal{B}}^{(9)}={p}_{X}^{{(3)}^{\prime}}\oplus {p}_{X}^{{(3)}^{\prime}}$
`// —”—` - 29:
- Regrid ${\mathcal{B}}^{(1-9)}$ onto convolution grid s
Output: Discrete probability density function ${p}_{{S}^{\perp}}$ of the independent sum ${S}^{\perp}=X+Y$ with the support$s=\{{s}_{1},{s}_{2},{s}_{2}+{h}_{s},{s}_{2}+2{h}_{s},\dots ,{s}_{2}+({N}_{s}-3){h}_{s},{s}_{{N}_{s}}\}$ and the associated probabilities ${p}_{{S}^{\perp}}(s)={\sum}_{k=1}^{{N}_{s}}\delta (s-{s}_{k}){p}_{{S}^{\perp}}({s}_{k})$, where ${s}_{{N}_{s}}-[{s}_{2}+({N}_{s}-3){h}_{s}]\le {h}_{s}$, ${s}_{2}-{s}_{1}\le {h}_{s}$, ${h}_{s}\ge max({h}_{1},{h}_{2})$. |

Algorithm 2: Brute force convolution for supports with the same span |

Input: Two discrete probability density functions ${p}_{X}$ and ${p}_{Y}$, where the supports of X and Y aredefined using the same span h as: $x=\{{x}_{1},{x}_{1}+h,\dots ,{x}_{1}+({N}_{x}-1)h\}$, $y=\{{y}_{1},{y}_{1}+h,\dots ,{y}_{1}+({N}_{y}-1)h\}$ and the associated probabilities as ${p}_{X}(x)={\sum}_{i=1}^{{N}_{x}}\delta (x-{x}_{i}){p}_{X}({x}_{i})$, ${p}_{Y}(y)={\sum}_{j=1}^{{N}_{y}}\delta (y-{y}_{i}){p}_{Y}({y}_{j})$ - 1:
- $s=\{{x}_{1}+{y}_{1},{x}_{1}+{y}_{1}+h,{x}_{1}+{y}_{1}+2h,\dots ,{x}_{1}+{y}_{1}+({N}_{x}+{N}_{y}-2)h\}$
`// compute convolution support` - 2:
- ${p}_{{S}^{\perp}}(s)=\mathbf{0}$
`// initialize` - 3:
- ${p}_{{S}^{\perp}}({s}_{i+j-1})\leftarrow {p}_{{S}^{\perp}}({s}_{i+j-1})+{p}_{X}({x}_{i}){p}_{Y}({y}_{j})\phantom{\rule{1.em}{0ex}}1\le i\le {N}_{x},1\le j\le {N}_{y}$
`// compute probabilities`
Output: Discrete probability density function ${p}_{{S}^{\perp}}$ of the independent sum ${S}^{\perp}=X+Y$ with the support$s=\{{s}_{1},{s}_{2},\dots ,{s}_{{N}_{x}+{N}_{y}-1}\}$ and the corresponding probabilities ${p}_{{S}^{\perp}}(s)={\sum}_{k=1}^{{N}_{x}+{N}_{y}-1}\delta ({s}_{k}){p}_{{S}^{\perp}}({s}_{k})$. |

#### 2.4.2. Regriding

Algorithm 3: Linear regriding |

Algorithm 4: 4-point regridding, Stage I |

Algorithm 5: 4-point regridding, Stage II |

#### 2.4.3. Comonotonization and Mixture Approximation

Algorithm 6: Distribution of the comonotonic sum |

#### 2.5. Order of Convolutions and Tree Topology

## 3. Results

## 4. Conclusions

## 5. Future Research Directions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Algorithm A1: Split-Atom Convolution: 4-products |

Input: Two discrete pdfs ${p}_{X}$ and ${p}_{Y}$ with supports:$x=\{{x}_{1},{x}_{1}+{h}_{x},{x}_{1}+2{h}_{x},\dots ,{x}_{1}+({N}_{x}-2){h}_{x},{x}_{{N}_{x}}\}$; $y=\{{y}_{1},{y}_{1}+{h}_{y},{y}_{1}+2{h}_{y},\dots ,{y}_{1}+({N}_{y}-2){h}_{y},{y}_{{N}_{y}}\}$; and probabilities: ${p}_{X}(x)={\sum}_{i=1}^{{N}_{x}}\delta (x-{x}_{i}){p}_{X}({x}_{i})$; ${p}_{Y}(y)={\sum}_{i=1}^{{N}_{y}}\delta (y-{y}_{i}){p}_{Y}({y}_{i})$; ${N}_{s}$ - maximum number of points for discretizing convolution grid - 1:
- ${x}^{(1)}=\{{x}_{1},{x}_{1}+{h}_{x},{x}_{1}+2{h}_{x},\dots ,{x}_{1}+({N}_{x}-2){h}_{x}\}$; ${p}_{X}^{(1)}(x)={\sum}_{i=1}^{{N}_{x}-2}\delta (x-{x}_{i}^{(1)}){p}_{X}^{(1)}({x}_{i}^{(1)})$
- 2:
- ${x}^{(2)}=\{{x}_{{N}_{x}}\}$; ${p}_{X}^{(2)}(x)=\delta (x-{x}_{i}^{(2)})\xb7{p}_{X}^{(2)}({x}_{i}^{(2)})$
`// split the right atom of ${p}_{X}$` - 3:
- ${y}^{(1)}=\{{y}_{1},{y}_{1}+{h}_{y},{y}_{1}+2{h}_{y},\dots ,{y}_{1}+({N}_{y}-2){h}_{y}\}$; ${p}_{Y}^{(1)}(y)={\sum}_{i=1}^{{N}_{y}-2}\delta (y-{y}_{i}^{(1)}){p}_{Y}^{(1)}({y}_{i}^{(1)})$
- 4:
- ${y}^{(2)}=\{{y}_{{N}_{y}}\}$; ${p}_{Y}^{(2)}(y)=\delta (y-{y}_{i}^{(2)})\xb7{p}_{Y}^{(2)}({y}_{i}^{(2)})$
`// split the right atom of ${p}_{Y}$` - 5:
- ${h}_{{s}^{\u2605}}=\frac{{x}_{{N}_{x}}+{y}_{{N}_{y}}-({x}_{1}+{y}_{1})}{{N}_{{s}^{\u2605}}-1}$
`// corresponding step size of the main part of convolution grid s` - 6:
- ${h}_{s}=max({h}_{x},{h}_{y},{h}_{{s}^{\u2605}})$
`// final step size of the main part of convolution grid s` - 7:
- ${N}_{s}=\lceil \frac{{x}_{{N}_{x}}+{y}_{{N}_{y}}-{x}_{1}-{y}_{1}}{{h}_{s}}\rceil +1$
`// corresponding number of points` - 8:
`// set irregular convolution grid:`- 9:
- $s=\{{x}_{1}+{y}_{1},{x}_{1}+{y}_{1}+{h}_{s},\dots ,{x}_{1}+{y}_{1}+({N}_{s}-2){h}_{s},{x}_{{N}_{x}}+{y}_{{N}_{y}}\}$
- 10:
- ${N}_{{x}^{{(1)}^{\prime}}}=\lceil \frac{{x}_{1}+({N}_{x}-2){h}_{x}-{x}_{1}}{{h}_{s}}\rceil +1$
`// size of the regridded main part` - 11:
- ${x}^{{(1)}^{\prime}}=\{{x}_{1},{x}_{1}+{h}_{s},{x}_{1}+2{h}_{s},\dots ,{x}_{1}+({N}_{{x}^{{(1)}^{\prime}}}-1){h}_{s}\}$
`// discretize ${x}^{(1)}$ with ${h}_{s}$` - 12:
- ${p}_{X}^{{(1)}^{\prime}}(x)={\sum}_{i=1}^{{N}_{x}^{\prime}}\delta (x-{x}_{i}^{{(1)}^{\prime}}){p}_{X}^{{(1)}^{\prime}}({x}_{i}^{{(1)}^{\prime}})$
`// regrid ${p}_{X}^{(1)}$` - 13:
- ${N}_{{y}^{{(1)}^{\prime}}}=\lceil \frac{{y}_{1}+({N}_{y}-2){h}_{y}-{y}_{1}}{{h}_{s}}\rceil +1$
- 14:
- ${y}^{{(1)}^{\prime}}=\{{y}_{1},{y}_{1}+{h}_{s},{y}_{1}+2{h}_{s},\dots ,{y}_{1}+({N}_{{y}^{{(1)}^{\prime}}}-1){h}_{s}\}$
- 15:
- ${p}_{Y}^{{(1)}^{\prime}}(y)={\sum}_{i=1}^{{N}_{y}^{\prime}}\delta (y-{y}_{i}^{{(1)}^{\prime}}){p}_{Y}^{{(1)}^{\prime}}({y}_{i}^{{(1)}^{\prime}})$
- 16:
- ${\mathcal{B}}^{(1)}={p}_{X}^{{(1)}^{\prime}}\oplus {p}_{Y}^{{(1)}^{\prime}}$
`// Brute Force convolution` - 17:
- ${\mathcal{B}}^{(2)}={p}_{X}^{(2)}\oplus {p}_{Y}^{{(1)}^{\prime}}$
`// —”—` - 18:
- ${\mathcal{B}}^{(3)}={p}_{X}^{{(1)}^{\prime}}\oplus {p}_{Y}^{(2)}$
`// —”—` - 19:
- ${\mathcal{B}}^{(4)}={p}_{X}^{(2)}\oplus {p}_{Y}^{(2)}$
`// —”—` - 20:
- Regrid ${\mathcal{B}}^{(1-4)}$ onto convolution grid s
Output: Probability mass function ${p}_{S}$ of the independent sum $S=X+Y$ with the support$s=\{{s}_{1},{s}_{1}+{h}_{s},{s}_{1}+2{h}_{s},\dots ,{s}_{1}+({N}_{s}-2){h}_{s},{s}_{{N}_{s}}\}$ and the associated probabilities ${p}_{S}(s)={\sum}_{k=1}^{{N}_{s}}\delta (s-{s}_{k}){p}_{S}({s}_{k})$, where ${s}_{{N}_{s}}-[{s}_{1}+({N}_{s}-2){h}_{s}]\le {h}_{s}$. |

Algorithm A2: Modified local moment matching |

Input: Discrete pdf ${p}_{X}$ with fine scale support $x=\{{x}_{1},{x}_{1}+h,\dots ,{x}_{1}+(N-1)h\}$ and associated probabilities ${p}_{X}(x)={\sum}_{i=1}^{N}\delta (x-{x}_{1}-(i-1)h)\xb7{p}_{X}({x}_{1}+(i-1)h)$; the support of coarse scale probability mass function ${p}_{{X}^{\prime}}$: ${x}^{\prime}=\{{x}_{1}^{\prime},{x}_{1}^{\prime}+{h}^{\prime},\dots ,{x}_{1}^{\prime}+({N}^{\prime}-1){h}^{\prime}\}$. Requirement: ${x}_{1}^{\prime}\le {x}_{1}$, ${x}_{{N}^{\prime}}^{\prime}\ge {x}_{N}$, $h<{h}^{\prime}$.
- 1:
- For convenience, let $\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}\leftarrow x$ and $\{{x}_{1}^{\prime},{x}_{2}^{\prime},\dots ,{x}_{{N}^{\prime}}^{\prime}\}\leftarrow {x}^{\prime}$
- 2:
- Initialize ${p}_{{X}^{\prime}}({x}^{\prime})\leftarrow \mathbf{0}$
- 3:
**if**${x}_{1}^{\prime}={x}_{1}$**then**- 4:
- ${p}_{{X}^{\prime}}({x}_{1}^{\prime})\leftarrow {p}_{{X}^{\prime}}({x}_{1}^{\prime})+{p}_{X}({x}_{1})$
- 5:
**end if**- 6:
**for**$i<{N}^{\prime}$, $j\le N$, ${x}_{i}^{\prime}<{x}_{j}\le {x}_{i+2}^{\prime}$, ${x}_{j}-{x}_{i}^{\prime}\le {x}_{i+3}^{\prime}-{x}_{j}$**do**- 7:
- $$\left[\begin{array}{c}{p}_{{X}^{\prime}}({x}_{i}^{\prime})\\ {p}_{{X}^{\prime}}({x}_{i+1}^{\prime})\\ {p}_{{X}^{\prime}}({x}_{i+2}^{\prime})\end{array}\right]\leftarrow \left[\begin{array}{c}{p}_{{X}^{\prime}}({x}_{i}^{\prime})\\ {p}_{{X}^{\prime}}({x}_{i+1}^{\prime})\\ {p}_{{X}^{\prime}}({x}_{i+2}^{\prime})\end{array}\right]+{\left[\begin{array}{ccc}1& 1& 1\\ {x}_{i}^{\prime}& {x}_{i+1}^{\prime}& {x}_{i+2}^{\prime}{{x}^{\prime}}_{i}^{2}\\ {{x}^{\prime}}_{i}^{2}& {{x}^{\prime}}_{i+1}^{2}& {{x}^{\prime}}_{i+2}^{2}\end{array}\right]}^{-1}\left[\begin{array}{c}{p}_{X}({x}_{j})\\ {x}_{j}{p}_{X}({x}_{j})\\ {x}_{j}^{2}{p}_{X}({x}_{j}^{2})\end{array}\right]$$
- 8:
**end for**
Output: Pdf ${p}_{{X}^{\prime}}({x}^{\prime})$ |

## Appendix B

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**Figure 1.**Aggregation of five risks using copula trees. Direct model (upper panel), hierarchical model with sequential topology (middle panel) and hierarchical model with closest pair topology (lower panel). The leaf nodes represent the risks whose aggregate we are interested in. The branching nodes of direct tree represent a multivariate copula model for the incoming individual risks while the branching nodes of hierarchical trees represent a bivariate copula model for the incoming pairs of individual and/or cumulative risks.

**Figure 2.**Illustration of hypothetical correlation matrices: (

**A**) exchangeable, (

**B**) nested block diagonal, and (

**C**) unstructured correlation matrix.

**Figure 3.**A discrete loss pdf represented as a mixture of two “spikes” (atoms) at minimum and maximum x damage ratio (red) and the main part (blue). Damage ratio is discretized on 64-point grid.

**Figure 4.**An example of RNN approach for determining topology of hierarchical risk aggregation tree for six risks with zero minima. The maxima and cumulative maxima characterizing losses for the six risks are presented in the upper panel. (

**A**) The algorithm takes the largest cumulative max and halves it to obtain the number c. Then, it binary searches for the number closest to c except for the last element in the sequence. This number (showed in bold) becomes the cumulative maximum of the new subsequence. The search is repeated until the subsequence consists of two elements. (

**B**) The resulting hierarchical aggregation tree.

**Figure 5.**An example of recursive nearest neighbor (RNN) approach for determining topology of direct risk aggregation tree for six risks shown in Figure 4. Note that the order of comonotonic aggregation follows the order of independent aggregation.

**Figure 6.**MC (red line) and convolution/comonotoinization based (blue bars) distributions of the total risk for 29,139 locations affected by hurricane peril using different aggregation models with linear regriding (upper row) and 4-point regriding (lower row). No tail truncation was applied. For consistency, the losses are plotted in [0; $100 MM] interval.

**Figure 7.**(

**A**–

**E**) Percentage errors in statistics of the total risk relative to the corresponding values obtained from MC simulations. Risk aggregation was performed for 29,139 locations affected by hurricane peril using sequential (blue) and RNN (red) models with 4-point regriding and maximum support size varying from 64 to 6400; (

**F**) shows the average time cost of five runs for each maximum support size.

**Table 1.**Mean ($\mu $), standard deviation ($\sigma $) and tail Value-at-Risk (TVaR

_{%}) at levels 1%, 5% and 10% of the total hurricane risk for 29,139 locations for (A) direct, (B) hierarchical sequential and (C) hierarchical RNN aggregation models compared to the average values from 30 realizations of MC runs. The losses are in [$MM]. Numbers in brackets represent percentage errors relative to MC simulations.

MC | Linear Regriding No Truncation | Linear Regriding Tail Truncation | 4-Point Regridding | ||
---|---|---|---|---|---|

(A) | $\mu $ | 44.3 | 44.3 (0.00%) | 44.3 (0.00%) | 44.3 (0.00%) |

$\sigma $ | 7.1 | 10.9 (53.02%) | 7.1 (0.00%) | 7.1 (0.00%) | |

TVaR_{1%} | 51.4 | 2550.9 (4864.2%) | 2550.8 (4864.0%) | 2550.8 (4863.9%) | |

TVaR_{5%} | 48.5 | 502.1 (934.7%) | 501.4 (932.8%) | 501.9 (933.8%) | |

TVaR_{10%} | 47.6 | 252.0 (429.7%) | 250.7 (427.1%) | 251.0 (427.6%) | |

(B) | $\mu $ | 44.3 | 44.3 (0.00%) | 44.3 (0.00%) | 44.3 (0.00%) |

$\sigma $ | 7.1 | 7.6 (6.11%) | 7.1 (0.00%) | 7.1 (0.00%) | |

TVaR_{1%} | 93.7 | 81.2 (−13.3%) | 84.7 (−9.6%) | 88.0 (−6.1%) | |

TVaR_{5%} | 66.3 | 62.5 (−5.8%) | 68.5 (3.4%) | 66.8 (0.8%) | |

TVaR_{10%} | 58.6 | 67.3 (15.0%) | 60.5 (3.3%) | 59.8 (2.1%) | |

(C) | $\mu $ | 44.3 | 44.3 (0.00%) | 44.3 (0.00%) | 44.3 (0.00%) |

$\sigma $ | 7.1 | 7.2 (1.55%) | 7.1 (0.00%) | 7.1 (0.00%) | |

TVaR_{1%} | 75.9 | 151.0 (98.9%) | 95.0 (25.2%) | 76.0 (0.1%) | |

TVaR_{5%} | 62.9 | 184.4 (193.4%) | 98.0 (55.8%) | 63.5 (0.9%) | |

TVaR_{10%} | 58.4 | 95.8 (63.9%) | 87.0 (48.9%) | 59.1 (1.1%) |

**Table 2.**Processing times [s] for different risk aggregation models. MC run is a single realization with 1 MM samples. The mixture method implementation for hierarchical trees was optimized for performance with nested block diagonal correlation structure in Figure 2B. Intel i7-4770 CPU @ 3.40 GHz architecture with 16 GB RAM was used.

Aggregation Model | MC | Linear Regridding No Truncation | Linear Regriding Tail Truncation | 4-Point Regriding |
---|---|---|---|---|

Direct | 1539 | 0.25 | 0.26 | 0.33 |

Hierarchical, sequential | 12,769 | 0.34 | 0.35 | 0.44 |

Hierarchical, RNN | 10,512 | 0.40 | 0.41 | 0.52 |

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**MDPI and ACS Style**

Wójcik, R.; Liu, C.W.; Guin, J.
Direct and Hierarchical Models for Aggregating Spatially Dependent Catastrophe Risks. *Risks* **2019**, *7*, 54.
https://doi.org/10.3390/risks7020054

**AMA Style**

Wójcik R, Liu CW, Guin J.
Direct and Hierarchical Models for Aggregating Spatially Dependent Catastrophe Risks. *Risks*. 2019; 7(2):54.
https://doi.org/10.3390/risks7020054

**Chicago/Turabian Style**

Wójcik, Rafał, Charlie Wusuo Liu, and Jayanta Guin.
2019. "Direct and Hierarchical Models for Aggregating Spatially Dependent Catastrophe Risks" *Risks* 7, no. 2: 54.
https://doi.org/10.3390/risks7020054