IDENTIFYING THE RISK FROM THE AVERAGE EXPECTED ANNUALIZED LOSSES AT EARTHQUAKE

Gabriela M. ATANASIU, Genoveva PERJU 
Technical University of Iasi, Romania
gmatanasiu@yahoo.com

Abstract: The paper introduces a new method of risk identification for real estate investment projects based on the average measure of the expected annualized losses from earthquake occurrence. We present both a quantitative and qualitative decision matrix for risk identification. When the risk is considered from dynamical perspective in time we emphasize the advantage of diversification for earthquake risk mitigation.

Keywords: investment projects, earthquake, real estate, risk assessment, insurance

1.	INTRODUCTION 

Real estate investors and the insurance companies are primarily concerned with the financial aspects of earthquake consequences. However, the seismic hazard curve as recommended by the Architectural Institute of Japan (1993) or measures of the subjective utility function of the insured had little applicability in real estate investment decisions (Hoshiya, 2004). Generally, economic losses for real estate investments in case of earthquake are measured using the expected annualized losses (EAL) formula for a continuous severity distribution. In practice, real estate investment and insurance decisions are often based on average estimations of earthquake impacts. We propose an alternative way of measuring the expected natural hazard economic loss by avoiding the reliance on the scarce historical data for the extreme hazards. We introduce an average expected annualized loss (EAL) measure for the earthquake risk. This average measure of EAL facilitates the risk classification and therefore the decisional process for earthquake risk mitigation. We use the measure for the average expected annualized loss to provide a standard for earthquake risk classification and estimation.  Finally, a numerical example is provided for a real estate investment under seismic risk.

2. DEFINING LOSS ESTIMATION

The average yearly amount of loss accounting for the frequency and severity of various levels of loss can be measured using the expected annualized loss (EAL) proposed by (Porter et.all, 2004 ). The computational relationship of EAL is:

						(1)
where: 
V denotes value exposed to loss (e.g., replacement cost of the building);
s refers to some seismic intensity measure;
y(s) is the mean seismic vulnerability function, defined here as the average level of loss as a fraction of V given an occurrence of s;
v(s) is the average annual frequency of experiencing shaking intensity s.
Performing the calculus we consider a range of the most common earthquakes magnitudes as presented below in the Table 1:

Table 1. Earthquakes characterization function of their annual incidence

Earthquake magnitude	Number per year	
>8	0-1	
7-7.9	18	
6-6.9	120	
5-5.9	800	
4-4.9	6200	
3-3.9	49000	
2-2.9	350000	
1-1.9	3000000	

Using the first and second mean theorem for definite integrals Eq.1 becomes:

					(2)
where:
are the mean values of the seismic vulnerability function respectively of the annual frequency of experiencing a shaking intensity s.

Based on the transformation in Eq. (2), a medium value of the expected annualized loss (EAL) is shown below. As suggested by Porter (2004), represents the average level of loss, considered as a fraction of V for the given occurrence of s. The medium probability of loss occurrence to a medium value of the shaking magnitude is given by the relationship (3):

						(3)

Following this relationship on can consider a full range of the most common possible frequencies of earthquake occurrence and loss probabilities, as presented in Table 2 below.

Table 2. The quantitative matrix identification of the earthquake risk

Loss probability

Frequency	100%	75%	50%	33%	25%	
1/1 (1.0)	7 V	5.25 V	3.5 V	2.31V	1.75V	
½ (0.5)	3.5 V	2.65 V	0.25 V	0.17 V	0.13 V	
1/3 (0.33)	2.33 V	0.25 V	0.165 V	0.11 V	0.08 V	
¼ (0.25)	1.75 V	0.19 V	0.125 V	0.08 V	0.06 V	
1/7 (0.14)	0.98 V	0.11 V	0.07 V	0.05 V	0.04 V	
1/8 (0.12)	0.84 V	0.09 V	0.06 V	0.04 V	0.03 V	
1/10 (0.1)	0.7 V	0.08 V	0.05 V	0.03 V	0.03 V	

Table 2 is an easy to use indicator on how much loss would occur under seismic risk as proportion of the initial investment. It is straightforward a preliminary classification of the seismic risk based on EAL results. In providing the above decision matrix we have used the actual insurance rate that suggests that the insured should cover about 10% of the total value of the investment. In this case the managerial implications for the insured are highly different from what is generally suggested in the insurance literature. Since the insurer sets at least 10% to 25% deductible on earthquake insurance it turns out that average losses below 10% respectively 25% from the initial investment should not be covered. Only the geographic areas with high frequency of earthquake occurrence or very high probability of loss occurrence should be considered for inclusion into insurance programs. Using the previous results, in Table 3 is presented a qualitative matrix for the earthquake risk identification.  The qualitative analysis of earthquake risk suggests that all losses below 10% from the exposed value are low rank risk. This in turn brings the conclusion that the insured should ignore this level of risk. The only type of risk that should be decided to be transferred to the insurer company is the medium and high risk while the unacceptable risk might even involved a disinvestment decision.
In some situations a combination of low, high and medium risk might be mitigated through diversification in the case of insurer. But for investment decisional purposes, the losses caused by unacceptable high levels of risk cannot be recovered through insurance (not even close to 80% of the initial value as suggested in the literature as optimal earthquake premium (Hoshiya, Masaru et all, 2004).

3. NUMERICAL EXAMPLE

Following, the interpretation of this computational result is presented in terms of an investment. Let’s supposes an initial investment of 50,000 euro in the case of a building exposed to the possibility of earthquake occurrence. In this paper, V=50,000 Euros. If one expects a high frequency of occurrence, for example once at every 5 years in case of an earthquake of a magnitude on Richter scale of 7, that might generate a total damage of the building and therefore a total loss of 50,000 euro, the expected annualized loss might exceed that initial investment of V. Therefore the insurance path of risk mitigation looses its economic rationality above certain values of EAL higher than the initial investment.

Table 3. The qualitative earthquake risk matrix identification

Loss probability

Frequency	100%	75%	50%	33%	25%	
1/1 (1.0)	Un
acceptable risk	Un
acceptable risk	Un
acceptable risk	Un
acceptable risk	Un
acceptable risk	
½ (0.5)	Un
acceptable risk	Un
acceptable risk	Medium risk	Medium risk	Medium risk	
1/3 (0.33)	Un
acceptable risk	Medium risk	Medium risk	Medium risk	Low risk	
¼ (0.25)	Un
acceptable risk	Medium risk	Medium risk	Low risk	Low risk	
1/7 (0.14)	High risk	Medium risk	Low risk	Low risk	Low risk	
1/8 (0.12)	High risk	Low risk	Low risk	Low risk	Low risk	
1/10 (0.1)	High risk	Low risk	Low risk	Low risk	Low risk	

Table 4. Probability of loss

Fre
Quency	1	0.5	0.33	0.25	0.14	0.12	0.1	
100%	7000000	2500000	2330000	1750000	980000	840000	700000	
75%	5250000	2650000	250000	190000	100000	90000	80000	
50%	3500000	250000	165000	125000	70000	60000	50000	
33%	2310000	170000	110000	80000	50000	40000	30000	
25%	1750000	130000	4000	60000	40000	30000	30000	

Up to 100,000 euro a loss that represents 10% of the initial investment should be accepted without insurance. This is practically the part of the loss that is most commonly insured. However, the question to answer is what is to be done with that part of the risk that is unacceptable and economically not rational. 
We suppose that choosing an appropriate portfolio of investments with low or unacceptable seismic risk, the earthquake risk can be mitigated through diversification. For an investment of 1,000,000 Euro real estate portfolio of investments we can see the effects of the diversification through the following analysis.

Fig.1 The mean value of Average Expected Annualized Loss (AEAL) for a considered  earthquake

Plotting the mean value of expected annualized loss as it is shown in Fig.1, the most significant mean of the average EAL exceeding the initial value of the investment is likely to be for events of high frequency as it is the case of one earthquake at every 1 and 2 years. Therefore if a real estate investments portfolio includes investments slightly affected by seismic events (low seismic risk investments) it would be possible to achieve earthquake risk mitigation through diversification. For events with high frequency of occurrence of 1 at every one or two years, mitigation through diversification is not possible and therefore uninsurable.

4. REFERENCES

Hoshiya, M., Nakamura, T. and Mochizuki, T. 2004, Transfer of Financial Implications of Seismic Risk to Insurance, Natural Hazards Review, nr.5 pp.141-146
Kunreuther, H. 2001, Incentives for Mitigation Investment and More Effective Risk Management the Need for Public-Private Partnerships, Conference paper, February 
Porter, K. A., Beck, J. L. and Shaikhutdinov, R. 2004, Simplified Estimation of Economic Seismic Risk for Buildings, Earthquake Spectra, Volume 20, No. 4, pp 1239–1263 
Sousa O., Carlos, R., Anthony, G.  2006, Assessing and Managing Earthquake Risk, Springer, The Netherlands