Cesromon

In this project we want to calculate the monthly estimated losses caused by fires that occur in Toronto. We will use the theory of Loss Distributions, in which we will try to find the severity and frequency. The intention of projects like this is to have a confidence margin, usually 95 or 99 percent, in which we are sure that 95% of the time the losses will not exceed this value and if it does exceed it, we can calculate another estimator. These estimators in Risk Theory are known as the Value at Risk (VaR) and the Conditional Value at Risk (CVaR) respectively. The database is taken from: https://open.toronto.ca/dataset/fire-incidents/

We will use the best known distributions for frequency such as the Poisson, the Negative Binomial or the Uniform and to severity, we have a range of possibilities. The first is to use the empirical distribution. Theoretically, we know that there are different known distributions, such as normal, log-normal, Gamma, Exponential, among others. Additionally, we can use distributions that are known to be heavy-tailed, such as Weibull or Generalized Pareto.

The Data

We have the advantage that the data is loaded in the R environment, therefore we will use the official package provided by open data in Toronto.

## # A tibble: 1 x 11
##   title          id       topics civic_issues publisher excerpt dataset_category
##   <chr>          <chr>    <chr>  <chr>        <chr>     <chr>   <chr>           
## 1 Fire Incidents 64a2669~ Publi~ <NA>         Fire Ser~ "This ~ Table           
## # ... with 4 more variables: num_resources <int>, formats <chr>,
## #   refresh_rate <chr>, last_refreshed <date>

There are a total of 17536 observations. We have many variables that for this particular analysis we will not take. We will only focus on “Estimated_Dollar_Loss and”TFS_Alarm_Time”, this because they adapt to the study we want to do. An important consideration is the time of the fire, because we have different variables that measure when the alarm originated, when the place was reached or when it was controlled. For practical purposes we will take the time when the alarm noticed that the fire started, in future analysis could take another direction regarding this issue.

Additionally, we have values in losses that are NA or 0, for our study, we will eliminate these observations.

## 'data.frame':    13571 obs. of  2 variables:
##  $ LossDolar: int  15000 50 1500 2000 100000 5000 500 7000 15000 60000 ...
##  $ Time     : Date, format: "2018-02-24" "2018-02-24" ...

Therefore for our project we will have a total of 13571 observations.

Frequency Distribution

To find the frequency distribution, wemake a monthly grouping of the events, this is visualized in the following graph.

From this graph we find that the distribution is very similar to a uniform,but we will use all the tools to find which is the distribution that best fits.

## Chi-squared statistic:  136.7653 4.77875 37.94487 
## Degree of freedom of the Chi-squared distribution:  8 7 7 
## Chi-squared p-value:  1.116028e-25 0.6869424 3.104186e-06 
##    the p-value may be wrong with some theoretical counts < 5  
## Chi-squared table:
##        obscounts theo 1-mle-pois theo 2-mle-nbinom theo 3-mle-unif
## <= 108        10        1.475611         10.027779       16.813187
## <= 116        10        6.005402         10.392029        8.967033
## <= 121         9        8.662435          8.652967        5.604396
## <= 125         9       10.282219          7.818023        4.483516
## <= 128         9        9.395659          6.186942        3.362637
## <= 133        11       17.355863         10.486356        5.604396
## <= 137         9       13.605855          8.150595        4.483516
## <= 146        11       22.699495         15.998408       10.087912
## <= 157        10       10.572665         13.273201       12.329670
## > 157         14        1.944796         11.013700       30.263736
## 
## Goodness-of-fit criteria
##                                1-mle-pois 2-mle-nbinom 3-mle-unif
## Akaike's Information Criterion   978.4313     897.6838   924.2153
## Bayesian Information Criterion   981.0563     902.9337   929.4653

Both graphically an the different tests, we found that the distribution that best fits our data is the Negative Binomial, so we will use this to continue with our study.

Severity Distribution

In the case of Severity, we will take 2 different methods, the first one will be try to find the best distribution of the existing ones that we mentioned at the beginning. The second we will use the Generalized Pareto distribution, which is used for events with many extreme values, which will be quite useful for this exercise.

Now just for practical purposes, we will see the behavior prior to the million dollars.

We notice how the majority of the losses events are less than $ 2,500 buy we have long tails of events with losses approaching $ 50 million. It will be difficult to find a distribution that fits our data, but we will continue with the analysis, if necessary we could use a transformation of the data in order to soften a little the effect of those values so distant. Additionally, let’s see how much estimated money is lost monthly to get an idea of what kind of final behavior we should have.

We are going to use the criteria of the AIC and BIC to find the possible best distributions

##                    distribucion df      AIC
## 1      mllnorm(datos$LossDolar)  2 284788.3
## 2    mlweibull(datos$LossDolar)  2 285367.4
## 3      mlgamma(datos$LossDolar)  2 289998.5
## 4     mlbetapr(datos$LossDolar)  2 297923.0
## 5   mlinvgamma(datos$LossDolar)  2 299226.8
## 6   mlinvgauss(datos$LossDolar)  2 313529.5
## 7        mlexp(datos$LossDolar)  1 314958.3
## 8     mlgumbel(datos$LossDolar)  2 334100.1
## 9 mlinvweibull(datos$LossDolar)  2      NaN

##                     distribucion df      BIC
## 1       mllnorm(datos$LossDolar)  2 284803.3
## 2     mlweibull(datos$LossDolar)  2 285382.5
## 3       mlgamma(datos$LossDolar)  2 290013.6
## 4      mlbetapr(datos$LossDolar)  2 297938.0
## 5    mlinvgamma(datos$LossDolar)  2 299241.8
## 6      mlpareto(datos$LossDolar)  2 307125.3
## 7    mlinvgauss(datos$LossDolar)  2 313544.6
## 8         mlexp(datos$LossDolar)  1 314965.8
## 9      mlgumbel(datos$LossDolar)  2 334115.1
## 10 mlinvweibull(datos$LossDolar)  2      NaN

From the AIC and BIC criteria, we see how the lognormal and Weibull distributions are the best to fit our data.

## Goodness-of-fit statistics
##                              1-mle-lnorm 2-mle-weibull
## Kolmogorov-Smirnov statistic  0.08387221    0.08820628
## Cramer-von Mises statistic    8.91037439   14.72768401
## Anderson-Darling statistic   51.22907343           Inf
## 
## Goodness-of-fit criteria
##                                1-mle-lnorm 2-mle-weibull
## Akaike's Information Criterion    284788.3      285367.4
## Bayesian Information Criterion    284803.3      285382.5

We noticed that there are problems with both distributions, but the normal log still stands out better.

Data Transformation

As we mentioned before, having such high values, one way to smooth these data is to take a transformation, for this case we will only use the logarithmic transformation, additionally we will filter those lost that are 1, because when using this transformation, working with 0 it gets complicated.

##                     distribucion df       AIC
## 1    mlweibull(datos2$LossDolar)  2  59840.34
## 2      mlgamma(datos2$LossDolar)  2  61439.15
## 3      mllnorm(datos2$LossDolar)  2  63077.67
## 4   mlinvgauss(datos2$LossDolar)  2  63777.18
## 5     mlbetapr(datos2$LossDolar)  2  64846.83
## 6   mlinvgamma(datos2$LossDolar)  2  65971.70
## 7 mlinvweibull(datos2$LossDolar)  2  72936.35
## 8        mlexp(datos2$LossDolar)  1  83749.92
## 9      mlgumbel(datos$LossDolar)  2 334100.06

##                      distribucion df       BIC
## 1     mlweibull(datos2$LossDolar)  2  59855.35
## 2       mlgamma(datos2$LossDolar)  2  61454.17
## 3      mlgumbel(datos2$LossDolar)  2  62750.45
## 4       mllnorm(datos2$LossDolar)  2  63092.69
## 5    mlinvgauss(datos2$LossDolar)  2  63792.19
## 6      mlbetapr(datos2$LossDolar)  2  64861.84
## 7    mlinvgamma(datos2$LossDolar)  2  65986.71
## 8  mlinvweibull(datos2$LossDolar)  2  72951.36
## 9         mlexp(datos2$LossDolar)  1  83757.43
## 10     mlpareto(datos2$LossDolar)  2 106525.73

From the AIC and BIC criteria, we see how the Weibull and Gamma distributions are the best to fit our data.

## Goodness-of-fit statistics
##                              1-mle-weibull 2-mle-gamma
## Kolmogorov-Smirnov statistic    0.06647409   0.1132842
## Cramer-von Mises statistic      5.20233377  24.1283028
## Anderson-Darling statistic     27.29902607 141.1293892
## 
## Goodness-of-fit criteria
##                                1-mle-weibull 2-mle-gamma
## Akaike's Information Criterion      59840.34    61439.15
## Bayesian Information Criterion      59855.35    61454.17

Generalized Pareto

This visually indicates that using the Generalized Pareto could work for this project.

Simulation

Now we know the behavior of the Severity and Frequency, we will continue with the simulation of the event, this uses the convulsion theory, in which it allows us to combine these 2 distributions to find a final answer, which will indicate how much money is estimated to lose for fires in the Toronto area. We will take 10 000 simulations.

set.seed(777)

m <-  10000

Set a random seed to 777 just to have equal results for loading the document.

Conclusions

Through loss distribution theory, we find how much could be estimated to lose monthly due to fires that occur in Toronto. We try to solve this problem by finding those distributions that best fit both the severity and the frequency of the event. In the case of frequency, there is no doubt that the Negative Binomial distribution is the one that best fits, then for Severity, as we mentioned, we took 2 paths, but before that we decided to transform the data or smooth them to minimize the impact of extreme values, but always being considered in our study, because these have a great importance in our study. We managed to find that the Weibull was the best to adjusted to our data, additionally we approached by the Generalized Pareto distribution, which for cases where the extreme values are so important usually manages to capture them in a good way

Table 1: Table of monthly expected losses for the Toronto Fire in Dollars

	Perdida Esperada	VaR al 95%	CVaR al 95%
Actual Monthly Data	5 361 408	11 194 679	21 824 898
Proyected Weibull	4 642 012	8 111 695	10 269 715
Proyected Generalized Pareto	5 449 758	11 048 694	16 724 907

We first clarify that the values presented with real data are only used to compare how close they are to those found, but they do not necessarily represent the VaR and CVaR in strict terms. What we see is that the Weibull underestimates the losses a little with respect to the real data. The Generalized Pareto manages to better capture these losses, reaching results that are very close to the real ones. The monthly losses are estimated to be $5,449,758 on average or it is the expected value, that is, abusing a bit of notation, in a normal average case that amount is what would expect to be the monthly fire lost.

Now, with respect to VaR, the amount is $11,048,694, this means that if we want to cover 95% of the possible cases that could arise in monthly fires, we use this value. Usually in Risk Theory we use values of 95% or 99%, in order to have greater security and cover practically all scenarios. Lastly, we use the CVaR, which is a term more linked to other areas, especially financial ones, because this value is very important in terms of losses, since it tells us, given that we have passed that 95% threshold, how much I estimate to lose in average in these cases. For our exercise, this value represents $16,724,907.

As a final argument, it is important to clarify that the terminology of VaR and CVaR, although they are more linked to other areas, it is also important to measure it in these cases, because for example, an insurer or the state itself must cover it, they must have a base or a security that with an estimated amount can cover practically 95% of the cases. Also be sure that in those months that this threshold is passed we also have defined a value that realistically represents how much money we need to cover this situation.