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Comparing Two Benefits Issues For Natural Hazards and Terrorism:

Ex-ante/Ex-post Valuation and Endogenous Risk

Scott Farrow

UMBC

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Outline

• Conceptual framework: Central role of preferences no matter what valuation approach pursued

• Does there appear to be distinct theory about NH and separately about terrorism? (assert no difference in the large framing--Farrow and Viscusi, P&S for Public Safety, 2010).

• Focus on two key issue– Ex-ante and ex-post valuation– Endogenous probability (intelligent adversary)

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Jump to bottom line:Five recommendations to be developed

1. Investigate evidence or conduct research regarding citizen preferences…whether terrorism induces “replaceable” or “irreplaceable” losses.

2. Consider using “ex-ante” values in place of ex-post values, perhaps simulating over various preference models conditional on findings in 1.

3. Consider implementation issues in ex-ante valuation such as the use of exceedance probabilities, appropriate asset measure of income of wealth and the expected value of the ex-ante measure.

4. Quantitatively study (perhaps done) the behavioral response of attackers to model, even if incompletely, how expenditures may alter probabilities as with “human made catastrophes”.

5. Consider estimation of “pure error” in forecast model which may expand uncertainty in anti-terrorism compared to NH.

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Valuing outcomes

• Frequent approach for both NH and terrorism

• Consider expected damages avoided based on P*D (probability times damages)– Estimate damages conditional on event

occurring– Value a change in policy as the change in

damages (mitigation) and/or change in probability (prevention)

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What is the WTP to avoid damage?Behavioral difference: Ex-ante vs. Ex-post

• Ex-ante: With risk aversion: individual WTP ex-ante based on “risk premium” to avoid exposure to risk: WTP “Z”, the difference between expected value and certainty equivalent, otherwise accept gamble

• Ex-post: expected loss• Literature on the

complicated linkages

Utility

Income μ CE

Z

Y1 Y2

U(CE)= U(EV)

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One model of the difference: ex-post

• Seek a monetary amount of damages that equates utility given the bad event, A*, occurs

V(M,A*)=V(M-CS,0) • In each period, person willing to pay

expected (prob=q) conditional loss, q*CS• For incremental events, totally

differentiate and solve to yield

WPA*=dM/dA* = -qVA*/VM*

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Partial derivation: Ex-ante

• For complete avoidance in advance, consumer is WTP up to CS which equates utility in the two states

qV(M,A*) + (1-q)V(M,0)=

qV(M-CS,A*) + (1-q)V(M-CS,0)• As with ex-post, can derive marginal

(4) WAA = dM/dA* = 0

*

** [ (1 ) ]A

M M

qVdM

dA qV q V

(3) Proportional difference= 0

*

* *(1 ) (1 )

*M

M

VWAA WPAq q

WAA V

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Freeman: Difference between Ex-ante and “Ex-Post” (SEJ, RA)

• Key Message: 1) For small probability, large consequence events, the difference can be large. 2) Value depends on specification of utility (replaceable or irreplaceable)

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Response surfaces for several utility functions (or could simulate)

Figure 1: Preliminary response surface: Ex-ante multiple of ex-post for varying risk and damage levels, risk aversion equal to 2

WAA* multiple of WPA*

Risk, q Damage as proportion of income

Figure 2: Extended surface to Losses Exceeding Fifty Percent of Income

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Recommendations on ex-ante and ex-post

1. Investigate evidence or conduct research regarding preferences and terrorism type events including whether terrorism induces “replaceable” or “irreplaceable” losses.

2. Consider using “ex-ante” values in place of ex-post values as benefits, perhaps simulating or using a response surface over various preference models conditional on findings in 1.

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Some Implementation IssuesFlood example—in progress

• Use from various utility functions to crate a prediction equation for ex-ante value as a function of q (probability) and CS/M (ex-post damages as share of income)

• Generate modeled “ex-post” damage estimates of flood using model (HAZUS-MH in this case)

• Use probabilities implicit in model based on exceedance probabilities (more later)

• Obtain damages for various sized events, e.g. 1,10,25,100,500 yr. events or comparable “time scale”

• Calculate expected (or other) value.

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Quick tangent: Damage EstimatesFEMA HAZUS MH model

• Nationwide

• Natural hazards: flood, earthquake

• GIS based for topography, building inventory to census block level

• Flood model: damage functions based on distribution of built inventory and flood height or return period.

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Building Exposure by Census block in a county

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Implementation Issue

• Risk averse over what? “income”: risk theory developed originally w.r.t. wealth, which is actually at risk here. Functions generally not well defined for CS>M which occurs.

• Exposure: is money (or wealth) only for units damaged ex-post, or for all exposed (perfect information or uninformed?) For terrorism, more likely uninformed so exposed value is large, for floods less clear.

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Expected Value and Exceedance Probability (preliminary)

• Eventually: likely want expected value of ex-ante or ex-post value for all event levels per OMB guidance

• Probability: Floods, and catastrophe, often evaluated using exceedance probability (P(X>X0), what is the prob. we will get a 9/11 or greater?...a statement about CCDF. Appears to lead to nice PDF in “return period”; PDF=1/R2; so prob. in that year of exactly 100 year flood is 10,000. Illustration: D=αR (could extend to multi-year events and probabilities)

E(Damage)= 2 12

1 1 1 1

1( ) ( )

1

R RR R R R R R

b b b

R R R R

f R D R dR R dR R dR RR b

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Recommendation 3

3. Consider implementation issues in ex-ante valuation such as the use of exceedance probabilities, appropriate asset measure of income of wealth and the expected value of the ex-ante measure.

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Endogenous Probability:Terrorism yes; NH, yes?

• For illustration, consider expected value in place of expected utility

Table 1: Probability and Consequence Representations for Natural Hazards and Terrorism

Model Hazards Terrorism

Basic: P*C(e) + u P(e)*C(e) + v

Both Endogenous P(e)*C(e) + u P(e)*C(e) + v

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Apparent practice

• Current practice: DHS break-even analysis: appears to assume only impact is on consequence (like basic NH model)

• Advanced environmental theory: endogenous risk….as people build in risky areas, the government will respond perhaps in a socially inefficient way– Differing time constants (rate of change)– Stability of response function

• Issue: probability is not exogenous and should be analytically studied (as I’m sure it is), but could be useful to link to exceedance probablity.

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Overconfidence in model• W.R.T. table, focus often on explained variation

without considering “pure error” or unexplained variation (u or v in table); creates “thin” tails.

• Modeling error likely larger in terrorism, should model. Possibility to explicitly consider model fit in forecast, simulation setting when don’t observe Y.

R2 = 1 – SSE/SST

N

SSM

R

R)

ˆ

ˆ1(ˆ

2

22

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Recommendation

4. Quantitatively integrate the behavioral response of attackers to model, even if incompletely, how expenditures may alter probabilities as with “human made catastrophes”.

5. Consider “pure error” in forecast model which may expand uncertainty in anti-terrorism compared to NH.

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Conclusions

• Still not clear to the outside that existing approaches have been exhausted,

• Core framework appears similar between NH and anti-terrorism,

• Empirically, differences matter,

• If core solidly attempted, extend into frontier behavioral responses.