Year loss table

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A Year Loss Table (YLT) is a list of historical or simulated years, with financial losses for each year. [1] [2] [3] YLTs are widely used in catastrophe modeling, as a way to record and communicate historical or simulated losses from catastrophes. The use of lists of years with historical or simulated financial losses is discussed in many references on catastrophe modelling and disaster risk management, [4] [5] [6] [7] [8] [9] but it is only more recently that the name YLT has become standard. [1] [2] [3]

Contents

Overview

Year of interest

In a simulated YLT, each year of simulated loss is considered a possible loss outcome for a single year, the year of interest, which is usually in the future. In insurance industry catastrophe modelling, the year of interest is often this year or next year, due to the annual nature of many insurance contracts. [1] However, the year can also be defined to be any year in the past or the future.

Events

Many YLTs are event based i.e., they are constructed from historical or simulated catastrophe events, each of which has an associated loss. Each event is allocated to one or more years in the YLT and there may be multiple events in a year. [4] [5] [6] The events may have an associated frequency model, that specifies the distribution for the number of different types of events per year, and an associated severity distribution, that specifies the distribution of loss for each event.

Events in an event-based YLT may all be of one peril-type (such as hurricane) or may be a mixture of peril-types (such as hurricane and earthquake).

Period Loss Tables (PLTs)

YLTs represent the possible losses in a period of one year, but can be generalized to represent the possible losses in any length of time, in which case they may be referred to as Period Loss Tables (PLTs).

Use in insurance

YLTs are widely used in the insurance industry, [1] [2] because they are a flexible way to store samples from a distribution of possible losses. Two properties, in particular, make them useful:

Examples of YLTs

YLTs are often stored in either long-form or short-form.

Example of a long-form YLT

In a long-form YLT, [1] each row of the YLT corresponds to a different loss-causing event. For each event, the YLT records the year, the event, the loss, and any other relevant information about the event.

YearEvent IDsEvent Loss
1965$100,000
17$1,000,000
2432$400,000
3--
.........
100,0007$1,000,000
100,000300,001$2,000,000
100,0002$3,000,000

In this example:

Example of a short-form YLT

In a short-form YLT, [3] each row of the YLT corresponds to a different year. For each event, the YLT records the year, the loss, and any other relevant information about the year.

The same YLT as above, but condensed to short-form, would look like:

YearAnnual Total Loss
1$1,100,000
2$400,000
3$0
......
100,000$6,000,000

Frequency models

Poisson distribution

The most commonly used frequency model for the events in a YLT is Poisson distribution with constant parameters. [6]

Mixed poisson distribution

An alternative frequency model is the mixed Poisson distribution, which allows for temporal and spatial clustering of events. [10]

Weighted YLTs (WYLTs)

YLTs can be generalized to weighted YLTs (WYLTs) by adding weights to the years. [11] The weights would typically sum to 1.

YearEvent LossWeight
1$1,100,0000.0001
2$400,0000.0002
3$00.00001
.........
100,000$6,000,0000.0003

Stochastic parameter YLTs

When YLTs are generated from parametrized mathematical models, they may use the same parameter values in each year (fixed parameter YLTs), or different parameter values in each year (stochastic parameter YLTs). [3] In a stochastic parameter YLT the parameters used in each year would typically themselves be generated from some underlying distribution, which could be a Bayesian posterior distribution for the parameter. Varying the parameters from year to year in a stochastic parameter YLT is a way to incorporate epistemic uncertainty into the YLT.

As an example, the annual frequency of hurricanes hitting the United States might be modelled as a Poisson distribution with an estimated mean of 1.67 hurricanes per year. The estimation uncertainty around the estimate of the mean might considered to be a gamma distribution. In a fixed parameter YLT, the number of hurricanes in every year would be simulated using a Poisson distribution with a mean of 1.67 hurricanes per year, and the distribution of estimation uncertainty would be ignored. In a stochastic parameter YLT, the number of hurricanes in each year would be simulated by first simulating the mean number of hurricanes for that year from the gamma distribution, and then simulating the number of hurricanes itself from a Poisson distribution with the simulated mean.

In the fixed parameter YLT the mean of the Poisson distribution used to model the frequency of hurricanes, by year, would be:

YearPoisson mean
11.67
21.67
31.67
......
100,0001.67

In the stochastic parameter YLT the mean of the Poisson distribution used to model the frequency of hurricanes, by year, might be:

YearPoisson mean
11.70
21.62
31.81
......
100,0001.68

Adjusting YLTs and WYLTs

It is often of interest to adjust YLTs, to perform sensitivity tests, or to make adjustments for climate change. Adjustments can be made in a number of different ways.

Resimulation with different frequencies

If a YLT has been created by simulating from a list of events with given frequencies, then one simple way to adjust the YLT is to resimulate but with different frequencies.

Incremental simulation

Resimulation with different frequencies can be made much more accurate by using the incremental simulation approach. [12]

Weighting

YLTs can be adjusted by applying weights to the years, which converts a YLT to a WYLT. An example would be adjusting weather and climate risk YLTs to account for the effects of climate variability and change. [11] [13] By putting more weights on some years and less on others, the implied distribution of events changes, and the distributions of event loss and annual loss change accordingly.

Adjusting an existing YLT to represent a different view of risk, as opposed to rebuilding the YLT from scratch, may have the benefit that it avoids having to resimulate the events, the positions of the events in the years, and the losses for each event and each year. This may be more efficient.

YLT importance sampling

One general and principled method for applying weights to YLTs is importance sampling [11] [3] in which the weight on the year is given by the ratio of the probability of year in the adjusted model to the probability of year in the unadjusted model. Importance sampling can be applied to both fixed parameter YLTs [11] and stochastic parameter YLTs. [3]

Repeat and delete

WYLTs are less flexible, in some ways, than YLTs. For instance, two WYLTs, with different weights, cannot easily be combined to create a single new WYLT. For this reason, it may be useful to convert WYLTs to YLTs. This can be done using the method of repeat-and-delete, [11] in which years with high weights are repeated one or more times, and years with low weights are deleted.

Calculating metrics from YLTs and WYLTs

Standard risk metrics can be calculated straightforwardly from YLTs and WYLTs. [1] Examples would be:

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References

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  2. 1 2 3 Yiptong, A; Michel, G (2018). "Portfolio Optimisation using Catastrophe Model Results". In Michel, G (ed.). Risk Modelling for Hazards and Disasters. Elsevier. p. 249.
  3. 1 2 3 4 5 6 Jewson, S. (2022). "Application of Uncertain Hurricane Climate Change Projections to Catastrophe Risk Models". Stochastic Environmental Research and Risk Assessment. 36 (10): 3355–3375. doi:10.1007/s00477-022-02198-y. S2CID   247623520.
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  5. 1 2 Friedman, D. (1975). Computer Simulation in Natural Hazard Assessment. University of Colorado.
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  7. ↑ Woo, G. (2011). Calculating Catastrophe. Imperial College Press. p. 127.
  8. ↑ Edwards, T; Challenor, P (2013). "Risk and Uncertainty in Hydrometeorological Hazards". In Rougier, J; Sparks, S; Hill, L (eds.). Risk and Uncertainty Assessment for Natural Hazards. Cambridge. p. 120.
  9. ↑ Simmons, D (2017). "Qualitative and Quantitative Approaches to Risk Assessment". In Poljansek, K; Ferrer, M; De Groeve, T; Clark, I (eds.). Science for Disaster Risk Management. European Commission. p. 54.
  10. ↑ Khare, S.; Bonazzi, A.; Mitas, C.; Jewson, S. (2015). "Modelling Clustering of Natural Hazard Phenomena and the Effect on Re/insurance Loss Perspectives". Natural Hazards and Earth System Sciences. 15 (6): 1357–1370. Bibcode:2015NHESS..15.1357K. doi: 10.5194/nhess-15-1357-2015 .
  11. 1 2 3 4 5 Jewson, S.; Barnes, C.; Cusack, S.; Bellone, E. (2019). "Adjusting Catastrophe Model Ensembles using Importance Sampling, with Application to Damage Estimation for Varying Levels of Hurricane Activity". Meteorological Applications. 27. doi: 10.1002/met.1839 . S2CID   202765343.
  12. ↑ Jewson, S. (2023). "A new simulation algorithm for more precise estimates of change in catastrophe risk models, with application to hurricanes and climate change". Stochastic Environmental Research and Risk Assessment. doi:10.1007/s00477-023-02409-0.
  13. ↑ Sassi, M.; et al. (2019). "Impact of Climate Change on European Winter and Summer Flood Losses". Advances in Water Resources. 129: 165–177. Bibcode:2019AdWR..129..165S. doi:10.1016/j.advwatres.2019.05.014. hdl: 10852/74923 . S2CID   182595162.
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