Catastrophe Risk Management in a Utility Maximization Model
Borbála Szüle
Corvinus University of Budapest
Hungary
[email protected]
Climate change may be among the factors that can contribute to the changing of weather in
certain geographic regions. Sometimes extreme weather conditions are related to natural
catastrophe events, such as for example floods, that can influence the economy as well. In case
of a natural catastrophic event the reconstruction efforts may not only occur individually in the
regions affected by the catastrophe, but aid may come also from the central government.
Catastrophe insurance may also contribute to the reconstruction in case of a catastrophic event,
although the economic risks of catastrophes are not always managed by catastrophe insurance.
In this paper certain possibilies of catastrophe risk management are modeled in a theoretical
framework. In the model a natural catastrophic event may affect one of the regions of an
economy, while in other regions no catastrophe can occur. According to the assumptions, in case
of a catastrophe the central government may contribute to the reconstruction by imposing a tax
on the regions that have not been damaged by the catastrophe. Alternatively, if certain conditions
hold, a catastrophe insurance can also be bought by the region that can be affected by the natural
catastrophe. These two possibilites are compared in the paper in a theoretical model, where
regions are risk averse (have a concave wealth utility function) and catastrophe insurance
premium is calculated based on actuarial principles. In this theoretical framework, the method
with the maximum utility of the regions can be considered as the optimal catastrophe risk
management possibility.
1. Introduction
Climate change may affect the economy in several ways. Some scientists have argued for
example, that global warming can have an effect on the frequency of extreme weather events.
Analysis of management of risks associated with these changes may be interesting, since there
are numerous regions in the world that are exposed to natural catastrophes, such as for example
floods or earthquakes. Depending on the severity of the natural catastrophic event, not only can
the number of casualties be significant, for example also infrastructure can be damaged that
generally makes also the rescue and reconstruction efforts more difficult. Natural catastrophes
thus usually have far-reaching consequences also in the economy. Although among others for
example also the environmental effects of catastophes can be significant, this paper focuses on
risk management issues in connection with economic losses.
If possible, damaging effects of natural catastrophes should be restricted – preventive
measures like for example building high dams may prove to be useful (against risk of floods).
Unfortunately, only rarely can the occurrence of natural catastrophes be totally eliminated. In
case of a catastrophic event generally reconstruction is needed so that economic activity can
continue. It is also important because of the economic linkages between regions: often if one of
the regions has experienced a natural catastrophic event, the effect of damages can also influence
other regions with which the affected region has economic relationships (for example the
inhabitants of the affected region can spend less on consumption as before the catastrophe). The
full economic recovery of a region after a natural catastrophe can take a long time.
Obviously, economic recovery after a catastrophe can be quicker with higher financial
support accessible by the region that experienced the catastrophic event. Concrete solutions to
financially helping a region hit by a catastrophe may differ significantly across regions, since
also the features of catastrophes and characteristics of regions are not identical. However, there
are basically two main methods of financing at least part of reconstruction after a catastrophe:
- one of the possibilites is a catastrophe insurance solution
- the other possibility is an aid financed by a central government.
Of course, also a mix of these possibilites can help in the reconstruction of a region hit by a
catastrophe. The question of how to find the optimal catastrophe risk management solution in a
given situation, also arises. This question is not necessarily easy to answer, since usually
catastrophe losses can differ significantly across regions. This study uses a theoretical framework
to try to find the optimal catastrophe risk management option. In this theoretical model expected
utility of regions are compared in case of catastrophe insurance and tax financed governmental
aid given to a region hit by a catastrophe. This theoretical model is relatively simple compared to
the complexity of concrete empirical situations, some of the aspects of a decision about
catastrophe risk management options can however be demonstrated based on the results.
2. Catastrophe events and their economic effects
Catastrophes can basically be grouped into two categories: natural catastrophes and manmade disasters, the following theoretical parts of this study focus on the analysis on natural
catastrophes (in other parts of the study data about both types of catastrophes occures). In that
case by the way risk prevention sometimes can not be fully achieved, for example in case of
earthquakes even very strong building rules may not be enough to entirely protect buildings or
for example infrastructure in a region. Trying to find an optimal catastrophe risk management
option can thus be considered as a relevant question in case of many regions in the world.
Natural catastrophes can have different causes, the main sources of insured losses in 2010
worldwide are shown in Figure 1 (losses are measured in million USD, loss values are based on
property and business interruption, excluding liability and life insurance losses). Due to the
randomness of the occurrence of big natural catastrophes the main sources of losses in different
years are not necessarily very similar. The size of the effect of a natural catastrophe is also
influenced by (among other factors) the population growth tendencies: if a region is exposed for
example to earthquakes and population growth is relatively large in that region, then (parallel to
the growing population) an earthquake can have more serious consequences later when also
population density (and thus may be also the number of buildings) can be higher.
Figure 1: Insured losses in million USD in 2010
20 126
12 943
6 393
Storms
Earthquakes
Floods
397
10
Cold, frost
Droughts, bush
fires, heat waves
Source: Swiss Re[2011]
Catastrophe insurance is not always accessible to regions potentially exposed to natural
catastrophes, and even if theoretically insurance is available, for example not every individual
buys it in the given region. The value of estimated total economic losses and the cost to insurers
associated to these losses can differ significantly. In 2010 for example, estimated value of
economic losses of natural catastrophes and man-made disasters was approximately 218 bn USD
while the cost to insurers was approximately 43 bn USD (Swiss Re 2011). From these losses in
2010 the Asian region has the largest part, as it is shown in Figure 2:
Figure 2: Total economic loss by region in 2010
Seas/Space
9,5%
Africa
0,2%
Oceania/Australia
6,0%
Asia
34,3%
North America
9,4%
Europe
16,1%
Latin America and
Caribbean
24,5%
Source: Swiss Re[2011]
In the interpretation of catastrophe data, certain definitions also play an important role: in the
analysis of data in Figure 2 for example it should be mentioned that a (catastrophic) event is
included in the Swiss Re (Sigma) statistics if insured claims, total economic losses or the number
of casualties exceed a certain limit, for example this limit is 86,5 million USD in terms of total
economic losses (Swiss Re 2011). Of course, for example extreme weather events can also have
serious consequences on the economy, and with the adoption of other limits, the concrete loss
numbers could differ from those shown for example on the previously analysed Figure 2.
Nevertheless, information based on these limits may also be interesting: Figure 3 for example
shows the ratio of total economic loss and the GDP of given regions:
Figure 3: Total economic loss in 2010 as a percentage of GDP
1,10%
0,95%
0,28%
0,19%
0,13%
A
fr
ic
a
a
er
ic
N
or
th
A
m
si
a
A
Eu
ro
pe
ni
a/
A
us
ce
a
O
La
tin
A
m
er
ic
a
an
d
C
ar
ib
tr
al
be
an
ia
0,02%
Source: Swiss Re[2011]
Figure 3 refers also to the fact that the relative severity of the catastrophe depends not only
on the absolute value of the losses, but also on the ability (for example measured by the GDP) of
a region to help to finance at least part of the reconstruction in the region hit by the catastrophe.
In case of Asia for example (that had the largest part of total economic losses experienced in
2010, where economic losses has been caused by for example extraordinary rainfalls that were
followed by floods, typhoons and earthquakes) the ratio of catastrophe-related economic losses
relative to the GDP is not so high as for example in the Oceania / Australia region, where
economic losses were caused by for example earthquakes, floods and storms (Swiss Re 2011).
3. Modeling of insurance optimality in a utility based framework
Theoretically there are some methods for dealing with catastrophes: if possible, prevention
(for example not building on areas exposed to flood risk) or mitigation (for example a quick
reconstruction to avoid for example infections) can prove to be useful. Catastrophe insurance can
also play an important role in post-disaster financing. The availability of catastrophe insurance
can be even more widespread if local insurance companies can also rely on reinsurance
companies that can carry a part of the losses. In case of a catastrophe insurance the insurance
premium is traditionally paid in advance (before the catastrophe can occur) and if the catastrophe
event happens, a given sum is paid. This inflow of money can stimulate the economy after the
catastrophe by for example playing a role in the financing of reconstruction efforts.
Insurance does not necessarily exist for a given risk: there are some requirements that are to
be fulfilled so that insurance can be offered by private insurance companies (Banyár 1994):
-
each individual in the group of insured is exposed to the same risk
-
the group of insured is homogeneous
-
the number of insured should be sufficiently large.
In addition to this, a risk is usually considered to be insurable if the insurance event occurs
randomly and independently (in case of the insured). Independence in this context means that the
probability that one of the individuals in the group of insured experiences the insurance event
does not affect this probability in case of an other individual in the group of insured.
Based on this traditional approach insurance is best applicable in case of independent, noncorrelated risk. The law of large numbers in case of a pool of insurance policies can be
interpreted so that the larger the pool of independent risks the lower the variability of the
(financial) result of the insurance company. In case of a catastrophe however sometimes the
opposite of this relationship can be observed: if one of the individuals in the insurance pool is
damaged by the catastrophe, the probability of damages in case of other individuals in the
insurance pool is relatively high. Thus, it also means that the pooling of correlated risks increases
variability in case of an insurance pool. This phenomenon is usually observable in relatively
small insurance pools: if the insurance pool (the number of individuals in the insurance
population) is high, the risk diversification effect as a consequence of the law of large numbers
can occur.
Catastrophe insurance exists in some cases, thus these insurance calculation problems (as a
consequence of non-correlated risks) can sometimes be managed in practice. Catastrophe
insurance however is not necessarily very cheap, and the individual exposed to a catastrophe risk
can decide whether to buy catastrophe insurance. This decision can be a very complex process,
but theoretically it can be modeled based on evaluating expected wealth utilities.
Figure 4: Insurance premium in a theoretical model
utility, no catastrophe
utility, in case of a
catastrophe
expected utility
maximum
insurance premium
0
50
100
150
200
250
300
350
400
450
500
wealth
Figure 4 shows how maximum insurance premium can be calculated (if insurance is offered
by insurance companies) in a simple theoretical framework, where this calculation is done based
on wealth utility functions. Given the wealth utility function of an individual (in the
homogeneous insurance pool) one can calculate the wealth with or without the occurrence of a
catastrophe. By using the probability of the catastrophe event, expected utility level can be
determined and the maximum insurance premium is that value that can be subtracted from the
original wealth so that the resulting wealth has exactly that utility level as the expected utility
calculated with the probability of the catastrophe.
In actuarial calculations insurance premium is calculated as the sum of a net premium (that
corresponds to the expected value of the loss in this framework) and the insurance loading (for
example it should cover administrative expenses of an insurance company). In this simple
theoretical model expected value of loss as a consequence of a catastrophe is the difference
between the following two values:
-
the original wealth
-
the wealth belonging to the expected utility level.
In case of a convex utility function (that refers to a situation when individuals are risk
seeking) insurance is not offered by insurance companies, since the maximum insurance
premium that the risk seeking individual were ready to pay would not even cover the expected
loss. If the utility function however is concave (the case of risk averse individuals), insurance
contracts are theoretically possible, since individuals are ready to pay more than the expected
loss (that corresponds to the net premium). In that case the insurance company in this theoretical
model can decide whether the maximum insurance premium is enough to offer insurance. Recall
however that the insurance pool should also be relatively large (the number of individuals in the
insurance population should usually exceed a certain limit) so that insurance premium can be
calculated with a prudent actuarial method.
It is worth mentioning, that in this framework only one catastrophe can occur: it can be
interpreted so that the time period in the model is calibrated so that it allows for maximum one
catastrophe. The cumulation of effects of more than one catastrophic event is thus not analyzed
in this theoretical model.
In the following part a theoretical model is introduced in which individuals are assumed to
be risk averse (have a concave wealth utility function). In this model catastrophe insurance and
post-catastrophe central tax are analysed as two alternative catastrophe risk management options,
and conclusions are derived about the optimality of these methods based on some simple
theoretical assumptions.
4. Assumptions of the theoretical model
In this theoretical model optimal catastrophe risk management options for an economic
entity (for example a country) with numerous geographic regions are analyzed. In case of a large
economic entity (country) usually not all regions are exposed to the same type of catastrophe
risk. This feature can be modeled so that only one of the regions is exposed to a catastrophe. The
number of unexposed regions is denoted by N in the model. According to the assumptions only
one period is considered: during this period only one catastrophe event can occur. Similar to the
model mentioned in Section 2 this theoretical model does not analyze potential accumulation of
wealth effects arising from more catastrophes, either. It is also assumed that economic effects of
only one type of catastrophe are analyzed.
The wealth of individuals can have several components in practice. An important feature of
wealth is liquidity. Some components of wealth are illiquid, which means that it can not be sold
in the market, or sometimes illiquidity is also mentioned in connection with wealth components
that theoretically can be sold, but selling can not be done immediately. In contrast to this, in case
of liquid wealth components, sale of the given asset can be immediate in the markets. In the
theoretical model the wealth of regions is assumed to consist of an illiquid and a liquid part
(corresponding for example a house and the income). In the absence of catastrophe the total
wealth (W) of the homogeneous regions is:
W =H+L
where H denotes the illiquid wealth and the liquid part of the wealth is denoted by L. In case
of a catastrophe both parts of wealth of the region exposed to the catastrophe are affected. The
total damage caused by the catastrophe in the exposed region is a random variable:
d·H + (L-F), with probability p
ξ=
0, with probability (1-p)
where d is the ratio of the damage in case of the illiquid wealth and p is the probability of the
catastrophe event, and F denotes a part of liquid wealth that is not affected by the catastrophe.
In practice often also those regions are affected economically by the catastrophe that were
not directly damaged. In this model, this phenomenon is modeled so that liquid assets of regions
are not independent. According to the assumptions, the liquid wealth of the unexposed regions is
equal to the liquid wealth of the exposed region.
Individuals in the theoretical model are assumed to be the regions in the economic entity
(country). This assumption reflects the phenomenon that if the insurance pool consists of for
example individual households, then catastrophe losses can be correlated within a region. Of
course, the definition of regions can be difficult, in this model it is assumed that an adequate
determination of regions is possible, based on for example geographic features and probability of
a given type of catastrophe. If the insurance pool consists of regions, correlation between losses
belonging to regions may be lower (compared to the case when insurance pool consists of for
example individual households), thus insurance calculations may be made more easily.
Figure 5: Utility in case of the region exposed to the catastrophe
U(H+L)
U(H+L-d·H-(L-F))
expected utility
p·(d·H-(L-F))
0
50
100
150
200
250
300
350
400
450
500
wealth
Regions are assumed to be risk averse, thus utility of wealth of the regions is measured by a
concave wealth utility function. Figure 5 illustrates the expected value of the possible loss as a
consequence of a catastrophe, this expected value is p·(d·H + (L-F)). The concave utility
function of the regions is denoted by U(W). Mathematically, in case of a concave utility function
dU (W )
dU 2 (W )
> 0 and
< 0 . A possible
the derivatives of the functions have given signs:
dW
dW 2
function form for the utility function is the logarithmic one: in the following U (W ) = ln (W ) is
assumed in the model.
5. Catastrophe risk management options
According to the assumptions, a catastrophe event can affect the wealth of both exposed and
unexposed regions. There is a wide range of possible solutions how to manage the wealth effects
of a catastrophe in practice, in this theoretical model two possible catastrophe risk management
options are compared:
- if catastrophe insurance is available on the insurance market, the exposed region
could possibly buy a catastrophe insurance that could cover the total damage
- in the absence of catastrophe insurance a tax could be imposed on the regions not
hit by the catastrophe event to cover a part of the total damage.
An important difference between these options is that insurance premium is paid in advance
(at the beginning of the period), while tax is paid at the end of the period in the model.
According to the assumptions, calculation of insurance premium is based on actuarial
principles, the total insurance premium is equal to the sum of the expected value of losses and
the insurance loading:
(d ⋅ H + L − F ) ⋅ p ⋅ (1 + c )
where c refers to the insurance loading.
If the region that is exposed to the catastrophe buys a catastrophe insurance (by assuming
that this insurance is offered and the liquid wealth of the region is enough to pay the insurance
premium), then the utility of this region is:
ln (L + H − (d ⋅ H + L − F ) ⋅ p ⋅ (1 + c )) .
In this case the utility of the regions that are not exposed to the catastrophe event is:
ln (L + H ) .
The other possible catastrophe risk management option plays a role in case of a catastrophe
event. In the model it is assumed that a central government of the economic entity (country) can
impose a tax on the regions where no catastrophe event occured so that an „aid” can be paid to
the region hit by the catastrophe. The total transfer financed by the tax is part (x) of the damage
of the illiquid assets in the model. This assumption of the model reflects the phenomenon that
sometimes central government supports reconstruction of buildings and infrastructure in a region
hit by a natural catastrophe. According to the assumptions, the regions where no catastrophe
event occurs, pay the equal amount of tax:
x⋅d ⋅H
N
Given the ratio x the utility level of the region exposed to the catastrophe depends on the
random variable ξ (the value of the damage), thus the expected utility level is compared with the
utility in case of a catastrophe insurance. According to the assumptions in the model, the
expected utility in case of a „reconstruction tax”:
p ⋅ ln (F + H − d ⋅ H + d ⋅ H ⋅ x ) + (1 − p ) ⋅ ln (L + H )
The utility level of regions where no catastrophe event occurs also depends on the damage
caused by the catastrophe. The expected utility level of these regions in case of a tax which is
used for partly reconstruction of illiquid assets damaged by the catastrophe:
d ⋅H ⋅x

p ⋅ ln F + H −
 + (1 − p ) ⋅ ln (L + H )
N 

In the theoretical model it is assumed that expected utility levels can be compared (a higher
utility level can be considered as better) and in addition to this utility levels of regions can be
aggregated. It is assumed that the total utility level of the economic entity (country) can be
calculated as the sum of utility levels of the regions. Optimal catastrophe risk management
option in this theoretical model can be identified as the option with the highest total utility level.
6. Optimal catastrophe risk management
In case of an increase in the „reconstruction tax” the expected utility levels of the regions
change. This tax is only imposed after a catastrophe has hit a region.
Figure 6: Utility levels of regions as a function of the rate x
0,096
0,095
0,095
0,094
0,094
0,093
0,093
0,092
0,092
0,091
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
value of x
exposed region
one of the unexposed regions
Source: own calculations
The utility level of the region exposed to the catastrophe and the utility level of the other
regions change oppositely. Figure 6 illustrates this phenomenon (F=0.01, L=0.1, H=1, p=0.005,
d=0.5).
Optimum value of the „reconstruction” tax can be calculated based on the total utility of all
regions. With the assumption that total utility of the regions can be calculated as the sum of the
individual utility levels the optimal level of the tax is the value that maximizes the following
expression (the sum of utility values for all groups):
p ⋅ ln (F + H − d ⋅ H + d ⋅ H ⋅ x ) + (1 − p ) ⋅ ln (L + H ) +


d ⋅H ⋅x

+ N ⋅  p ⋅ ln F + H −
 + (1 − p ) ⋅ ln (L + H )
N 



Maximum is calculated by calculating the first derivative of this expression, in case of an
optimum it should be equal to zero. The optimum value of the „reconstruction” tax is:
x* =
1
1+
1
N
=
N
N +1
This is a simple expression and relatively easy to interpret: in this simple model framework
the optimal “contribution rate” to the reconstruction of damages caused by a catastrophe event
approaches 1 as the number of regions increases. This result thus means that the higher the
number of regions not exposed to the catastrophe (compared to the number of regions hit by the
catastrophe, in this model there is only one such region), the larger the optimal tax-financed
reconstruction support.
If the optimal „reconstruction” tax is imposed, total utility level of regions in case of an
imposed „reconstruction” tax is maximal. The optimality of a given catastrophe management
option can also be analyzed, since for given parameters the option with the higher total utility
level can be found. In the following a situation is analysed when the economic entity consists of
two regions, and one of the regions is exposed to a natural catastrophe.
Figure 7 illustrates a situation when total utility level in case of a „reconstruction tax”
changes with the value of x (F=0.9, L=0.95, H=1, p=0.001, d=0.5, c=0.2, N=1). It can be
observed on Figure 7 that total utility level in case of catastrophe insurance is constant, since the
value of x has no effect on utility in the absence of this type of tax. The (optimal) value of x is
equal to 0.5 where total utility in case if a „reconstruction” tax is maximal, since in this case
1
1
x* =
= . Figure 7 shows a situation where the optimal „reconstruction” tax results in a
1 2
1+
N
higher aggregate utility level than the catastrophe insurance.
The parameters on Figure 7 are not necessarily representative for practical experience, the
results illustrated on Figure 7 however indicate that theoretically there can be situations where
total utility of the regions can be higher with „reconstruction” tax than with a catastrophe
insurance.
Figure 7: Aggregate utility level in case of different catastrophe risk management options
1,3
1,3
1,3
1,3
1,3
1,3
insurance
tax
1,3
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
the value of x
Source: own calculations
Parameter values in case of Figure 7 indicate that this situation is characterized by relatively
similar values of liquid and illiquid assets, a low number of regions, and – among others – a
relatively costly insurance). The relation of the two catastrophe risk management options
changes if for example catastrophe insurance does not cost so much as in case of the parameters
of Figure 7. Figure 8 illustrates a situation with the same parameters as those in case of Figure 7
except that the insurance loading is lower (F=0.9, L=0.95, H=1, p=0.001, d=0.5, c=0.05, N=1):
Figure 8: Aggregate utility levels with a low cost catastrophe insurance
1,3
1,3
1,3
1,3
1,3
1,3
1,3
insurance
tax
1,3
1,3
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
the value of x
Source: own calculations
With lower insurance loading (illustrated on Figure 8) the better alternative is the catastrophe
insurance. These results of the theoretical model indicate that cost of catastrophe insurance is an
important factor that influences the optimality of catastrophe risk management options.
The results of the theoretical model were calculated based on the assumption that there is a
theoretical choice between catastrophe insurance and a post-catastrophe „reconstruction” tax
imposed on regions not hit by the natural catastrophe. Of course, this choice does not necessarily
exist, not even theoretically. If an insurance company can not build a large enough (and
appropriate) insurance pool, then usually no catastrophe insurance is available. In this case, in
this theoretical model only a centrally imposed tax is available for financing a post-catastrophe
reconstruction.
In the theoretical model introduced in this paper the optimal solution (that maximizes total
expected utility) can be found if for given parameters the total expected utility level is calculated
for both the catastrophe insurance alternative and the „reconstruction” tax alternative. These
alternatives differ significantly, for example in case of a catastrophe insurance the region
exposed to a catastrophe pays insurance premium before a catastrophe can occur, while in case
of a „reconstruction” tax the other (not damaged) regions pay after a catastrophe event has
happened. The optimality of these two catastrophe risk management options in the theoretical
model depends on the values of the model parameters. In addition to this, if catastrophe
insurance theoretically proves to be better than the other alternative, then it is also necessary to
analyze whether a catastrophe insurace is theoretically available. In case of an insurance the
„pooling” of individual risks is of central importance, thus for example reinsurance companies
can contribute to the availability of catastrophe insurance.
7. Conclusions
Natural catastrophes can cause large economic losses, the management of catastrophe risk
can thus contribute to the financial stability of the economy. The range of solutions is wide in
practice, but catastrophe risk management is sometimes a mix of catastrophe insurance and
government participation in the financing of reconstruction after a catastrophe. These two
alternatives are compared in a theoretical model in this paper in a model framework where the
alternative with the higher total expected utility level is considered to be the optimal option.
Based on the results of the theoretical model one of the conclusions is that the optimal rate of the
„reconstruction” tax increases as the number of regions not affected by the catastrophe increases
relative to the number of regions hit by the catastrophe. An other interesting result of the
theoretical model is that the optimality of these two alternatives depend heavily on the value of
the insurance loading: with higher insurance loading „reconstruction” tax tends to result in a
higher aggregate expected utility than catastrophe insurance. With low costs in addition to the
net premium (a lower insurance loading) however catastrophe insurance (if it is available) can be
the optimal catastrophe risk management option (that results in a higher expected utility level).
References
Banyár, J.(1994): Az életbiztosítás alapjai („Basics of life insurance”, in Hungarian)
Bankárképző – Biztosítási Oktatási Intézet, Budapest
Swiss Re (2011): Natural catastrophes and man-made disasters in 2010: a year of devastating
and costly events (authors: L.Bevere, B.Rogers, B.Grollimund)
Swiss Re, Sigma No. 1/2011.