The Role of Uncertainty on Alternative Disaster Management Strategies
Patricia Grossi and Howard Kunreuther
Department of Operations and Information Management
The Wharton School
Philadelphia, PA 19104-6366
Abstract
The work presented in this paper is an outgrowth of a multi-year study at the Wharton School of
the University of Pennsylvania on Managing Catastrophic Risks. We focus on the role of homeowners
and insurance companies in managing the hazard from earthquake risk. Specifically, we consider
alternative earthquake disaster management strategies for a typical homeowner and a small insurance
company in the Oakland, California region. These strategies involve the adoption of mitigation measures
and the purchase of earthquake insurance by the homeowner and the purchase of an indemnity contract
(e.g., excess-of-loss reinsurance) by the insurer.
We focus on how uncertainty impacts these disaster management strategies. Specifically, we
illustrate the impact of structural mitigation and risk transfer mechanisms on the insurer’s performance
when there is uncertainty in the company’s risk profile. This risk profile is captured through a loss
exceedance probability (EP) curve, representing the probability that a certain level of monetary loss will
be exceeded on an annual basis. Parameters considered in the sensitivity analysis that will shift the loss
EP curve include: earthquake recurrence, ground motion attenuation, soil mapping schemes, and the
exposure and vulnerability of the residential structures. The paper demonstrates how uncertainty in these
parameters impacts the cost-effectiveness of mitigation and reinsurance on the insurer’s profitability and
chances of insolvency, as well as the number of policies the insurer is willing to issue.
Introduction
An important part of the Wharton Managing Catastrophic Risks project has been evaluating alternative
strategies for natural disaster risk management in the private sector (Grossi et al., 1999). These risk
management strategies include: structural mitigation, insurance and other risk transfer mechanisms.
Evaluation of these strategies requires quantification of the risks through the use of advanced catastrophic loss
modeling tools. Accomplishing this quantification has been a central issue for the project. In the work to date,
four main questions have been addressed: (1) What role does uncertainty play in the estimates of losses from
natural disasters? (2) How can insurers bear risk against events that may create large losses of surplus for
them? (3) How can insurers transfer portions of their risk to other sources? (4) What impact will specific loss
reduction measures have on the ability of insurers to bear risk and their need to utilize risk transfer contracts?
1
A central tool in our analyses of the impact of disasters on the insurance sector has been the
construction of loss exceedance probability (EP) curves. This paper describes how loss EP curves are
developed and how they can serve as an integrating framework for evaluating alternative risk management
strategies for low probability-high consequence events. To make these ideas concrete, we turn to the
challenges facing an insurer who has issued earthquake insurance policies in the Oakland, California region
and is concerned with protecting itself against the possibility of suffering catastrophic losses from future
earthquakes.
Loss Exceedance Probability Curves
A loss exceedance EP curve depicts the probability that a certain level of loss will be exceeded on an
annual basis. Using probabilistic risk analysis (PRA), one combines the set of events that could produce a
given dollar loss and determines the resulting probability of this loss occurring. Based on these estimates, one
can construct the mean exceedance probability curve, as depicted in Fig. 1. By its nature, the mean EP curve
inherently incorporates uncertainty in event occurrence probability and magnitude of dollar losses. In essence,
the loss EP curve is a key element for evaluating a set of risk management strategies. The accuracy of the EP
curves depends upon the ability of natural hazard experts (e.g., geologists and engineers) to estimate the
impact of events of different probabilities and magnitudes on the performance of the firm.
Integrating Framework: Certain elements are necessary if one is to construct a loss EP curve as an
integrating framework for linking risk assessment with risk management strategies for dealing with
catastrophic risks. These include: defining the nature of the problem, characterizing the risk, and specifying
the relevant parties at risk.
First, to begin the analysis, it is important to specify the nature of the problem one is studying and to
indicate why it is considered important. Normally, the problem will involve a natural hazard where there is
some likelihood that there will be a high impact event which produces severe economic consequences. The
problem we are considering below is how an insurer can protect itself against the financial losses from a
severe earthquake. Its goal is to reduce its chances of insolvency while still being profitable.
Next, the risk must be sufficiently characterized. One needs to rely on expert judgment to estimate the
chances of certain events occurring and their potential consequences. In utilizing these estimates, it is valuable
to indicate the degree of uncertainty surrounding these figures. Below, we will examine risk uncertainty by
focusing on a series of earthquake events, their probabilities, and the economic consequences to residential
structures in the Oakland, California region of the United States.
Finally, one needs to specify the relevant parties affected by the risk. Suppose the analysis is being
undertaken for a commercial company concerned with potential losses from a natural hazard. Then, the loss
EP curve will depict the potential losses to the firm from a series of events. If other stakeholders are involved
2
in helping to cover these losses (e.g., an insurer or reinsurer), then one will need to construct a loss EP curve
for these parties. Below, we will construct a loss EP curve for an insurer providing financial protection against
earthquakes to homeowners in the Oakland, California region.
With these three elements, the decision processes of the relevant parties and the strategies for risk
management available to the parties can be evaluated through a loss EP curve. First, the use of the loss EP
curve depends on the type of information collected by the relevant parties and the types of decision rules that
are utilized by them. For example, a company concerned with the possibility of insolvency will turn to its loss
EP curve to estimate the probability that its losses will exceed some given magnitude. In our example we will
utilize the loss EP curve to estimate the probability that earthquakes in the Oakland, California region will
create losses to an insurer that exceed a certain dollar value.
The loss EP curve also enables one to construct and evaluate the impact of alternative strategies for
dealing with the consequences of natural disasters. These strategies incorporate a set of policy tools that
include the following: risk bearing (e.g., having enough surplus on hand and diversification of activities to
cover potential losses), mitigation (e.g., investing in a measure to reduce future disaster losses), and risk
transfer (e.g., use of insurance or reinsurance to shift some of one’s risk to others). Some of these strategies
can be implemented through the private sector in a financially attractive manner (e.g., long-term loans for
mitigation coupled with lower insurance premiums and/or lower deductibles) while others may require public
sector intervention (e.g., well enforced building codes). In the example below, we evaluate the impact on the
insurer’s performance of a requirement that homeowners invest in a cost-effective mitigation measure as a
condition for insurance. Furthermore, we examine the insurer’s profitability and probability of insolvency
with and without the use of reinsurance as a risk transfer mechanism.
Risk Aversion
The recent literature in economics suggests that firms are generally risk averse and hence, they are
concerned with non-diversifiable risks such as catastrophic losses from natural disasters (Mayers and Smith,
1982). Suppose a firm is risk averse and there are transaction costs associated with bankruptcy. Then, the firm
will want to purchase protection against catastrophic losses at higher premiums than implied by their expected
losses, even when their shareholders can eliminate this risk through their own portfolio diversification. Firms
are also likely to be ambiguity averse. In other words, they are concerned with the uncertainty regarding the
probability of a loss occurring (Kunreuther et al., 1993).
One way to reflect the firm’s risk aversion and ambiguity aversion is a safety-first model, proposed by
the French economist Roy (1952). Such a model can be contrasted with a value maximization approach to
firm behavior. A safety-first model explicitly concerns itself with insolvency when making a decision
regarding how much protection the insurer would like to have against catastrophic losses and how much it is
3
willing to pay to protect itself against this. More specifically, a safety first model implies that there is some
target insolvency probability α (e.g., α = 0.01), which guides its decisions on what premiums to set and how
much coverage to offer against a specific risk. In fact, Stone (1973) used this type of constraint when
characterizing insurer underwriting decisions.
Mitigation and Risk Transfer: Risk-averse firms, in their quest to reduce probability of bankruptcy,
can use risk reduction and risk transfer strategies. One way for risk-averse firms to reduce its insolvency
probability would be to adopt loss reduction or mitigation measures. By incurring an upfront cost today, one is
able to avoid much larger losses in the future from a disastrous event. To reflect this lower probability of loss
in the future, the mitigation measure shifts the loss EP curve downwards from where it would be if no loss
reduction measures were in place.
Risk transfer mechanisms complement mitigation measures in enabling a risk-averse firm to survive
financially should its suffer a catastrophic loss. More specifically, the firm would use its loss EP curve to
determine how much protection it would need to reduce its probability of insolvency to a given target ruin
probability α. for a given book of business (BOB). The more risk averse the firm is, the more it will need to
use risk transfer instruments to protect itself against large losses and the more it is willing to pay for this
protection.
For example, risk-averse firms may want to purchase an indemnity contract to cover losses above a
certain amount. For a corporation, the normal indemnity contract is an insurance policy; for an insurer, the
typical indemnity contract is an excess-of-loss reinsurance policy that provides coverage against unforeseen or
extraordinary losses. A typical excess-of-loss reinsurance contract requires the primary insurer to retain a
specified level of risk and then covers all losses between an attachment point (LA) and exhaustion point (LE)
on the loss exceedance probability curve (Fig. 2). In other words, the indemnity contract is of the following
form: the reinsurer pays all of the loss in the interval LA to LE with a maximum payment of LE – LA. In return
for this protection, the insurer pays the reinsurer a premium.
The Oakland, California region
We now illustrate the above framework by turning to the case of the Oakland, California region.
Consider the case of an insurer who has offered financial protection to residential property owners against
damage from an earthquake in this seismically active region of the United States. The loss EP curve depicts
the chances that aggregate claims by homeowners exceed certain monetary levels, based on earthquake
damage to a set of insured structures in the company’s portfolio.
To develop a loss EP curve, it is necessary to have credible estimates of the relevant hazard and
vulnerability of the portfolio. With respect to the earthquake hazard, scientists agree that there is uncertainty in
4
the location, severity, and frequency of future event occurrences. The physical manifestation of future
earthquake events (e.g., ground motion or ground failure) is estimated by examining geologic records and
historical events and by conducting experiments on how the ground responds to earthquake waves. In general,
limited scientific information or lower quality data results in greater variability of the expected hazard and
subsequent loss.
Structural engineers have focused on the nature, distribution, and level of damage (i.e. vulnerability) of
structures from earthquake events. Damage is estimated from historical data (e.g., from the Loma Prieta or
Northridge earthquakes), expert opinion of structural engineers proficient in post-disaster investigation of
structures, and experimental testing of structural components subject to lateral (e.g., earthquake) loading. As
in the case of hazard analysis, limited engineering information results in greater variability of expected
vulnerability.
The hazard and damage/vulnerability analyses are integrated to estimate losses using probabilistic risk
assessment (PRA). The recent use of geographic information systems (GIS) for incorporating hazard and
vulnerability information for a region has enabled scientists to more efficiently estimate the potential damage
and loss from different earthquake scenarios. The data for the region is stored in the form of GIS maps of
ground motion estimation, maps of secondary seismic hazards such as liquefaction, landslide and fault
rupture, and maps of damage to structures in the region (King and Kiremidjian, in press).
Catastrophic risk models, which use GIS and other advances in information technology to estimate
losses from natural disasters, have been developed in the United States in the public and private sectors. This
study uses the public regional loss estimation model, HAZUS (NIBS, 1997). In conjunction with this model,
pre-processing and post-processing software modules are utilized to estimate direct economic losses to
residential structures in the Oakland, California region (Grossi, 2000).
The DTE Insurance Company
Consider a hypothetical insurance company that is providing financial protection to residential homes
in the Oakland, California region. The DTE (Down to Earth) Insurance Company has a book of business
(BOB), which consists of residential wood-frame homes in different parts of the region. Homes constructed
prior to 1940 are assumed to have unbraced cripple walls (i.e., the walls between the foundation and first floor
diaphragm) and to be inadequately bolted to the foundation. The damage to these homes from an earthquake
in the future can be reduced if the cripple walls are braced and the structure is bolted adequately to its
foundation. Therefore, the structural mitigation technique chosen for analysis is the structural retrofit of the
homes with unbraced cripple walls.
DTE insures 5,000 homes, chosen randomly from the 108 census tracts of the cities of Oakland and
Piedmont, California. Additionally, based on information from the Alameda county tax assessor’s office, it is
5
assumed that 38% of the 5,000 homes the company insures were built prior to 1940. A set of loss exceedance
probability curves is developed for this company under various assumptions of uncertainty in the loss
estimation process. The HAZUS analysis uses losses from forty-six earthquake events on seven right-lateral
strike-slip faults in the region to develop the curves.
Five areas of uncertainty are considered: earthquake recurrence, ground motion attenuation, soil
mapping schemes, inventory exposure, and fragility of residential building stock with and without mitigation.
In Fig. 3, the four steps of the earthquake loss estimation process (define the seismic hazard, define the
inventory characteristics, estimate the inventory damage, and calculate the economic loss) are shown with
these five areas of uncertainty highlighted. The uncertainty is assumed to be epistemic in nature. In other
words, the uncertainty is due to the lack of knowledge of the parameter or model considered. Epistemic
uncertainty can be reduced through the incorporation of new information. For details on the default
assumptions (i.e. without new information) and updated assumptions (i.e. with new information), see Grossi
(2000).
Default Loss EP Curve: The default loss EP curve for the DTE insurance company without
mitigation is depicted in Fig. 4. The default loss curve is defined as the most likely scenario of events using
the default information without any structural mitigation. As shown in Fig. 4, at an exceedance probability
level of 0.01 (1%), the mean estimate of loss is approximately $179.6 Million. Based on its concern with
insolvency, DTE will want to limit the number of earthquake policies that it writes in this region due to the
high correlation among losses should a major earthquake occur in the area. More specifically, we assume that
DTE focuses on a worst-case loss (WCL) in determining their book of business (BOB). A worst-case loss is
defined as a disaster where the probability of exceeding a certain dollar amount is at a predetermined target
ruin probability, α, that reflects its safety-first concern.
Table 1 specifies the parameters for the DTE insurance company. We assume that DTE has $60
Million in assets at its disposal with full insurance coverage against damage from the disaster available to its
clients. DTE imposes a 10% deductible on its residential clients. Furthermore, the premium charged on an
annual basis is proportional to the expected loss per year for the property covered and then multiplied by a
loading factor to reflect the administrative costs associated with marketing and claims settlement. In our case,
the loading factor is 1.0. In other words, for this analysis, property owners are charged premiums that are
twice the expected losses to the insurer. The premiums for the default case if DTE insured all 5,000 homes
(i.e. had a full BOB) are $7.8 Million, as listed in Table 1.
Suppose that DTE sets a “target ruin probability”, α, equal to 1% (or 1 in 100 year event). This implies
that it would like to limit its book of business (BOB) so that there is a 99 percent chance of being able to pay
6
insured losses from assets and premiums. If it has sufficient assets and/or premiums, then DTE’s actual
probability of insolvency may be less than the target level. Based on its assets and premiums for a full BOB
without structural mitigation, DTE would be insolvent if its earthquake claims exceeded $67.8 Million. Using
DTE’s loss exceedance probability curve, the chance of this happening to the DTE insurance company is
1.99% (i.e. 0.0199), as indicated in Fig. 4.
If DTE was utilizing a safety-first strategy with α = 0.01, then it would either have to raise its
premiums, reduce its book of business (BOB), encourage structural mitigation among its residential clients,
diversify its BOB into other geographic areas, and/or purchase an indemnity contract to cover a portion of its
catastrophic losses. In our case, we are assuming that premiums are fixed due to competitive pressures and/or
regulatory constraints, so it has to rely on some combination of the other four strategies listed above. . If DTE
had the freedom to change its rates, then it could also raise its premiums to achieve its target ruin probability
under its safety-first concern.
Structural Mitigation: DTE could reduce its future claims by requiring, as a condition for insurance,
that the owners of wood-frame homes built prior to 1940 must bolt their house adequately to its foundation
and brace its cripple walls. This type of structural mitigation measure has been shown to be cost-effective for
homes of this type given the range of earthquakes that are possible in the Oakland, California region. In effect,
DTE would be imposing a more recent building code on pre-1940 homes and would enforce it from their
perspective by refusing to issue insurance unless the measures were adopted. If banks required earthquake
insurance as a condition for a mortgage, then DTE would have some leverage in making mitigation a
requirement for coverage.
Fig. 5 depicts DTE’s default loss exceedance probability curve, as well as the revised one when all
pre-1940 homes in DTE’s book of business adopt these mitigation measures. At the exceedance probability
level of 0.01 (1%), the mean estimate of loss is approximately $92.9 Million with structural mitigation. Since
premiums are based on expected losses, the total amount of premiums collected is now $3.7 Million for a full
BOB. Given that its assets are $60 Million, DTE will be insolvent if their claims exceed $63.7 Million. The
likelihood of this happening is 1.36% rather than 1.99% as originally shown in Fig. 4. Hence, even if all the
pre-1940 homes in the BOB for DTE bolted the structure to the foundation and braced its cripple walls, the
chance that the firm will become insolvent exceeds its pre-determined level of α of 0.01. Therefore, DTE will
also have to look toward risk transfer mechanisms to reduce its insolvency risk.
Risk Transfer Mechanisms: DTE can reduce its catastrophic risk potential through the use of one
type of risk transfer mechanism called excess-of-loss reinsurance. Reinsurance protects the insurer from losses
within a loss layer that extends from the attachment point, denoted LA, to the exhaustion point, denoted LE
7
(Fig. 2). In our case, the maximum reinsurance available is set at $30 Million. In other words, the maximum
difference between LA and LE is $30 Million. Additionally, we attach the reinsurance layer to the loss
exceedance probability curve at a fixed level of loss. Specifically, the attachment point, LA, is set at 10% of the
worst-case loss (i.e. the 1 in 100 year event defined above). Finally, the premium charged for reinsurance on
an annual basis is proportional to the expected loss per year for reinsurance and then multiplied by a loading
factor to reflect the costs associated with this risk transfer mechanism. In our case, the loading factor is 0.8.
Fig. 6 depicts the default exceedance probability curve for DTE when the pre-1940 homes were not
mitigated. The layer of reinsurance attaches at the loss level of $17.96 Million, which is 10% of the worstcase loss of $179.6 Million. The point of exhaustion of the reinsurance layer is at $47.96 Million (i.e. extends
to the maximum level of $30 Million). This transfer of risk to another entity through the use of reinsurance
allows the insurer to increase its book of business while still satisfying its safety-first concern (i.e. achieving
target ruin probability of 0.01). In this case, in order to satisfy the safety-first constraint, the BOB without
reinsurance is approximately 35%. In other words, the insurer would only cover 35% of the original 5,000
structures. With reinsurance, the book of business can be increased to 52% of the full BOB.
Uncertainty and Effectiveness of Strategies: Now that the effects of mitigation and risk transfer
through reinsurance are shown in a representative loss exceedance probability curve for the DTE insurance
company, the next step is to determine the effectiveness of these strategies. The effectiveness is measured in
terms of the insurer’s profitability, chances of insolvency, and maximum book of business (BOB) while
satisfying the target ruin probability of 1%.
First, the profitability of the firm without any structural mitigation is shown in Fig. 7. Specifically, the
profitability of DTE is shown for the various cases of epistemic uncertainty tested in the sensitivity analysis
considering only the risk transfer strategy of reinsurance. The profits are defined as the insurance premiums
collected by the company minus the expected loss to the insurer and any other fixed costs. In the default case
(i.e. default loss EP curve in Fig. 4), the profits are $1.55 Million with reinsurance and drop to $1.36 Million
without reinsurance. Therefore, based on the cost of reinsurance, if DTE wishes to maximize its profits, the
company should transfer some of the risk to another entity (i.e. a reinsurer). Furthermore, considering
uncertainty in the profitability of DTE, the profits range from $1.32 Million to $1.85 Million. The lower
bound on the profits is seen where there is new information incorporated on the fragility or vulnerability of the
residential structures (without reinsurance); the upper bound on the profits is achieved when new information
on the annual recurrence of earthquake events is incorporated into the analysis (with reinsurance). The profits
increase with reinsurance because it is more cost effective to transfer the risk to the reinsurer (as modeled
here).
8
Next, the chances of insolvency of the DTE insurance company, without the use of a risk transfer
mechanism and under the various assumptions of uncertainty in the model, are presented in Fig. 8.
Specifically, the probabilities of insolvency of DTE are shown for the various cases of epistemic uncertainty
tested in the sensitivity analysis considering only the risk reduction strategy of structural mitigation. The
insolvency probability is defined as the point on the loss exceedance probability curve where the loss equals
the assets available to DTE plus the premiums collected by the insurer. As already noted, in the default loss
EP curve, this point is where the loss is $67.8 Million (without structural mitigation) and $63.7 Million (with
structural mitigation). Considering uncertainty in the analysis, the range on insolvency probabilities is 1.15%
to 2.72%. The insurer can achieve an insolvency probability of 1.15% when one considers new information on
the annual earthquake recurrence in the HAZUS analysis and the pre-1940 homeowners mitigate their
structures. The insurer will achieve an insolvency probability of 2.72% when one considers new information
on the attenuation (ground motion) relationship in the analysis and none of the pre-1940 homeowners mitigate
their residential structures. The insolvency probability decreases when new information on earthquake
recurrence is incorporated into the model because the recurrence probabilities of the earthquake events
considered are less (on average). On the other hand, the insolvency probability increases when new
information on the ground motion attenuation is incorporated into the model because the peak ground
acceleration expected at distances close to the fault are greater (For more information, see Grossi, 2000).
Finally, the percentage of the book of business (BOB) that DTE is willing to cover, while still
satisfying its safety-first concern, is shown in Fig. 9. As already noted, this safety-first concern is satisfied
when the chances of insolvency to the firm is at 0.01 on the loss EP curve (i.e. the 1 in 100 year event).
Structural mitigation (i.e. a risk reduction strategy) and reinsurance (i.e. a risk transfer strategy) are both
incorporated in these results, along with the uncertainty in the HAZUS analysis. There are four case presented:
(1) no risk reduction or risk transfer strategy is taken by DTE (no reinsurance and no mitigation); (2) DTE
requires its residential clients to mitigate their homes but does not transfer any of the risk (no reinsurance and
mitigation); (3) DTE transfers a portion of its risk only (reinsurance and no mitigation); and (4) both a risk
reduction and a risk transfer strategy is taken by DTE (reinsurance and mitigation).
In case 1, if the insurer considers no risk management strategy and DTE chooses to insure all 5,000
structures in the Oakland, California region, the probability that DTE will become insolvent is greater than
1%. More specifically, when one considers uncertainty in the analysis, the probability of insolvency ranges
from 1.82% (with new information on the soil in the region) to 2.72% (with new information on the
attenuation of ground motion). If DTE must satisfy its safety-first concern without turning to any risk
management strategy, it will only be able to provide coverage between 35% and 52% of the original 5,000
structures in its BOB.
9
Turning to case 2, where mitigation is in place but no risk transfer strategy is considered, DTE will be
willing to insure between 58% and 96% of the 5,000 structures. Therefore, structural mitigation is very
effective and the insurer can increase its book of business significantly in some cases. In the default case, the
percentage BOB that DTE will be willing to cover increases from 35% to 67%. In case 3, with a risk transfer
strategy but no mitigation in place, DTE will increase the number of insurable structures from 47% to 76% of
the full BOB. Therefore, while reinsurance is effective for DTE to manage its catastrophic risk, it is not as
effective as structural mitigation.
Finally, in case 4, where both mitigation and reinsurance are utilized, the insurer will be willing to
cover between 85% and 100% of the original 5,000 structures. With the incorporation of both a risk reduction
and a risk transfer strategy, in the default case, DTE has a 99.7% probability of remaining solvent when it
covers all the structures in its full book of business. When one considers new information on the ground
motion attenuation in the model, the percentage of the book of business that the insurer will be willing to
cover drops to 85%.
Conclusions
In this illustrative example of a small insurance company in an earthquake prone region, it is clear that
disaster management plays an important role in the company’s strategy. Specifically, for the DTE insurance
company described in this paper, profitability of the firm increases with the use of reinsurance as a risk
transfer mechanism. Furthermore, the chances of insolvency of the firm decrease significantly if the woodframe structures in its book of business mitigate their homes for the earthquake risk. Finally, the number of
policies DTE will be willing to issue increases with the risk reduction strategy of structural mitigation and the
risk transfer strategy of reinsurance.
Uncertainty also needs to be taken into account in evaluating alternative disaster management
strategies for a risk-averse company. In regard to the incorporation of epistemic uncertainty in the analysis, the
largest estimate of worst-case loss (i.e. loss at target ruin exceedance probability of 0.01) is for the case where
new information on ground motion attenuation is incorporated into the model. In this situation, the company
will have to reduce its BOB to the smallest level and/or rely most heavily on the use of risk transfer
mechanisms to reduce its probability of insolvency to an acceptable level.
The above analysis indicates that it is quite important for companies in earthquake prone regions to not
only understand the alternative disaster management strategies available to them, but to understand the
influence of various uncertainties on the earthquake loss estimation process on its performance.
References Cited
10
Grossi, P., Kleindorfer, P. R. and Kunreuther, H., 1999, The impact of uncertainty in managing seismic risk:
the case of earthquake frequency and structural vulnerability: Risk Management and Decision Processes
Center Working Paper 99-03-26, The Wharton School, Philadelphia, PA.
Grossi, P., 2000, Quantifying the uncertainty in seismic risk and loss estimation: Dissertation, Department of
Systems Engineering, University of Pennsylvania, 264p.
King, S., and Kiremidjian, A., in press, Use of GIS for earthquake hazard and loss estimation, In Geographic
Information Research: Bridging the Atlantic: Taylor & Francis, London.
Kunreuther, H., Meszaros, J., Hogarth, R., and Spranca, M., 1995, Ambiguity and underwriter decision
processes: Journal of Economic Behavior and Organization, v. 26, p. 337-352.
Mayers, D., and Smith, C., 1982, On corporate demand for insurance: Journal of Business, v. 55, p. 281-296.
NIBS, 1997, HAZUS: Hazards U.S.: Earthquake Loss Estimation Methodology: National Institute of Building
Sciences Document Number 5200, Washington, D.C.
Roy, A.D., 1952, Safety-first and the holding of assets: Econometrica, v. 20, p. 431-449.
Stone, J., 1973, A theory of capacity and the insurance of catastrophe risks: Part I and Part II: Journal of Risk
and Insurance, v. 40, p. 231-243 (Part I) and p. 339-355 (Part II).
Figures
Fig. 1. Example of a mean loss exceedance probability curve, emphasizing importance of uncertainty in
exceedance probability and monetary loss to develop curve.
Fig. 2. Example of an insurer’s loss exceedance probability curve, showing layer of loss transferred through
the purchase of reinsurance.
Fig. 3. Steps in the earthquake loss estimation process, showing the areas of epistemic uncertainty considered
in the analysis.
Fig. 4. Default mean loss exceedance probability curves for insurance company, showing worst-case loss at
exceedance probability level of 0.01, as well as exceedance probability level with loss equal to assets
plus premiums.
Fig. 5. Loss exceedance probability curves for insurance company, showing default case without mitigation
and case with the incorporation of structural mitigation.
Fig. 6. Default loss exceedance probability curve for insurance company, showing reinsurance layer attaching
at exceedance probability level where loss is equal to ten percent of the worst-case loss.
Fig. 7. Profitability of insurance company, showing the effects of epistemic uncertainty on the firm’s
profitability with and without reinsurance in place.
Fig. 8. Probability of insolvency of insurance company, showing the effects of uncertainty on the firm’s
chances of insolvency with and without mitigation in place.
11
Fig. 9. Percentage of book of business of insurance company, showing the effects of uncertainty on the firm’s
decision to cover only a percentage of its full book of business to satisfy its target ruin probability of
0.01.
Probability that loss
will exceed L
Uncertainty in
Exceedance
Probability
Uncertainty in
Loss
Loss, L (in Dollars)
12
Probability that loss
will exceed L
Reinsurance
Layer
LA
Loss, L (in Dollars)
LE
Define Earthquake Hazard
Define Inventory Characteristics
(Seismic Sources, Recurrence,
Attenuation, Soils)
(Structure Location, Value, Year
Built, Construction Class, etc. )
Earthquake
Recurrence
Estimate Inventory Damage
Through Historical Loss Data,
Engineering Data, Expert Opinion
Attenuation
Relationship
Calculate Economic Loss
Soil
Mapping
Schemes
Expected Loss or WCL
to Insurer and Homeowner
13
Fragility
(Before and
After
Mitigation)
Inventory
Exposure
0.06
0.05
0.04
0.03
0.02
0.01
$92.9 M
0
$0
$100,000,000
$200,000,000
Loss
No Mitigation
0.06
0.05
0.04
Reinsurance
0.03
0.02
$30 M
0.01
0
$0 $17.96 M
$100,000,000
Loss
14
$200,000,000
$2.0
No Reinsurance
Reinsurance
$1.0
$0.0
Parameter/Model
3.0%
Mitigation
No Mitigation
2.5%
2.0%
1.5%
1.0%
0.5%
0.0%
Parameter/Model
15
No Reinsurance, No Mitigation
Reinsurance, No Mitigation
No Reinsurance, Mitigation
Reinsurance, Mitigation
100%
80%
60%
40%
20%
0%
Recurrence Attenuation
Soils
Exposure
Parameter/Model
Table 1. DTE Insurance Company Parameters
Parameter
Value
Company Assets
$ 60 Million
Deductible
10%
Insurance Loading Factor
1.0
Premiums for full BOB
$ 7.8 Million
16
Fragility
Default