Comparing Two Benefits Issues
For Natural Hazards and Terrorism:
Ex-ante/Ex-post Valuation and
Endogenous Risk
Scott Farrow
UMBC
1
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)
2
Jump to bottom line:
Five recommendations to be developed
1.
2.
3.
4.
5.
Investigate evidence or conduct research regarding citizen
preferences…whether terrorism induces “replaceable” or
“irreplaceable” losses.
Consider using “ex-ante” values in place of ex-post values,
perhaps simulating over various preference models conditional on
findings in 1.
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.
Quantitatively study (perhaps done) the behavioral response of
attackers to model, even if incompletely, how expenditures may
alter probabilities as with “human made catastrophes”.
Consider estimation of “pure error” in forecast model which may
expand uncertainty in anti-terrorism compared to NH.
3
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)
4
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
U(CE)
=
U(EV)
Z
Y1
CE
μ
Y2
Income
5
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*
6
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* =
dM
dA *
(3) Proportional difference=

qV A *
[ qV M *  (1  q )V M 0 ]
W A A * W P A *
W AA *
 (1  q )  (1  q )
VM 0
VM *
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Freeman: Difference between Exante 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
Figure 2: Extended surface to Losses Exceeding Fifty Percent of Income
WAA* multiple
of WPA*
Risk, q
Damage as
proportion 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 expost 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 Issues
Flood 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 Estimates
FEMA 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 expost 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)
RR
RR
E(Damage)=  f ( R ) D ( R ) dR  
R 1
R 1
1
R
RR
 R dR    R
2
b
R 1
b2
dR 

b 1
RR
R
b 1
R 1
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Recommendation 3
3. Consider implementation issues in exante 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
ˆ
2
2
ˆ
1 R
SSM
 (
)
2
ˆ
N
R
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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 antiterrorism 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.
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