Transcript Risk, Uncertainty, and Sensitivity Analysis

Risk, Uncertainty,
and Sensitivity
Analysis
How economics can help
understand, analyze, and cope with
limited information
Generic Group Project
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Land Use
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Habitat conservation plan calls for acquisition of
100 acres of land in coastal area. Cost uncertain.
Maybe $1,000,000 (30% chance), maybe
$3,000,000 (70% chance). NPV of benefits are
$2,000,000 (for sure!). Good idea?
Marine Reserves
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Proposal to add 10 km2 to the CIMR at cost
(reduced harvest) of $2 million (NPV); ecological
benefits uncertain: $4 million with probability 0.7;
$0 with probability 0.3
What is “risk”?
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Can be loosely defined as the “possibility of
loss or injury”.
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Should be accounted for in social projects
(and regulations) and private decisions.
Think of there being different “states of
nature” that can emerge, and we are
uncertain about which we will end up with.
We want to develop a way to describe risk
quantitatively by evaluating the probability
of all possible outcomes.
Attitude toward risk
Problem: Dean Haston likes to ride her
bike to school. If it is raining when she
gets up, she can take the bus. If it isn’t,
she can ride, but runs the risk of it
raining on the way home.
 Value of riding bike (no rain) = $4 each
way
 Value of riding bike in rain = -$4 (each
way)
 Value of taking bus = $1 (each way)
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Dean Haston’s options & the
“states of nature”
The Asst. Dean can either ride her bike or
take the bus.
 Bus: She gains $1 each way: $2
 Bike: Depends on the “state of nature”
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Rain (on way home): $4 - $4 = 0.
 No rain: $4 + $4 = $8.
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Which does she prefer?
If the Asst. Dean takes the bus, she knows
she’ll gain $2 (no uncertainty).
 If the Dean rides her bike:
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If it rains, she gains 0.
 If it doesn’t rain, she gains $8.
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Whether she is better taking the bike or bus
depends on 2 things:
The probability of rain
 Her attitude toward risk
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The probability of rain
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Suppose Pr(rain) = .5……Pr(no rain) = .5
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Bus: $2 (certain)
Bike: .5(8) - .5(0) = $4 (risky)
If she is risk neutral, she takes her bike ($4 > $2)
If she is a risk lover, she takes her bike
If she is sufficiently risk averse, she may bus
Suppose Pr(rain) = .8….Pr(no rain) = .2
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Bus: $2
Bike: .2(8) + .8(0) = $1.60 (risky)
If she is risk neutral, she rides the bus ($2 > 1.60)
Risk more generally
Coin toss pays $10 or $20
Q: Would this
person rather
get 15 for sure or
play coin toss?
Utility
U(15)
This person is
RISK AVERSE
.5*U(10)
+ .5*U(20)
10
15
20
Some good (or $)
Risk attitudes in general
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Generally speaking, most people risk averse.
Diversification can reduce risk.
Since gov’t can pool risk across all taxpayers, there is
an argument that society is essentially risk neutral.
Most economic analyses assume risk neutrality.
Note: may get unequal distribution of costs and
benefits.
Expected payoff more
generally
Suppose n “states of nature”.
 Vi = payoff under state of nature i.
 Pi = probability of state of nature i.
 Expected payoff is: V1p1+V2p2+…
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Or
S ViPi
Example: Air quality
regulations
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New air quality regulations in Santa
Barbara County will reduce ground level
ozone.
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Reduce probability of lung cancer by
.001%; affected population: 100,000.
How many fewer cases of lung cancer
can we expect?…about 1
.00001*100,000 = 1.
 We don’t know who will get sick but this
is our expectation of the number of
cases
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Example: Climate change
policy
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2 states of nature
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High damage (probability = 1%)
• Cost = $1013/year forever, starting in 100
yrs.
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Low damage (probability = 99%)
• Cost = $0
Cost of control = $1011
 Should we engage in control now?
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Control vs. no control (r=2%)
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Control now: high cost, no future loss
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Cost = $1011
Don’t control now: no cost, maybe
high future loss:
If high damage = 1013[1/(1.02100) +
1/(1.02101) + 1/(1.02102) + … ]
= (1013/(.02))/(1.02100) = $7 x 1013
 If no damage = $0.
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Overall evaluation
Expected cost if control = $1011
 Expected cost if no control =
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(.01)(7 x 1013) + (.99)(0) = $7 x 1011
By this analysis, should control even
though high loss is low probability
event.
Value of Information
The real question is not: Should we engage
in control or not?
 The question is: Should we act now or
postpone the decision until later?
 So there is a value to knowing whether the
high damage state of nature will occur.
 We can calculate that value…this is “Value
of information”
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Sensitivity Analysis
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A method for determining how
“sensitive” your model results are to
parameter values.
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Sensitivity of NPV, sensitivity of policy
choice.
Simplest version: change a
parameter, re-do analysis (“Partial
Sensitivity Analysis”)
Climate change: sensitivity to r
Loss from no control
8E+11
7E+11
6E+11
5E+11
4E+11
3E+11
2E+11
1E+11
0
0
0.01
0.02
0.03
0.04
Discount rate (r)
0.05
0.06
Sensitivity to Uncertainty on the
probability of high damage
1.20E+12
Benefits and Costs
1.00E+12
8.00E+11
Cost
6.00E+11
E[Damage]
4.00E+11
2.00E+11
0.00E+00
0
0.005
0.01
p
0.015
0.02
More sophisticated
sensitivity
The more nonlinear your model, the
more interesting your sensitivity
analysis.
 Should examine different
combinations.
 Monte Carlo Sensitivity Analysis:
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Choose distributions for parameters.
 Let computer “draw” values from
distn’s
 Plot results
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Managing Risk
Risk is a problem of its own
 Several tools available to reduce risk
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Insurance
 Liability
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Insurance—fire insurance
example
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Probability of loss: 0.001; Loss=$100,000
Expected annual loss: $100
No insurance
 Most years: no loss; some years $100,000 loss
1000 houses pool $100 each/yr ($100,000/yr)
 Most years—one loss
 Sometimes no losses, sometimes 2-3 losses
 Much less variability in annual losses
Fire is amenable to risk pooling
 Risks uncorrelated
 Earthquake insurance in Cal NOT amenable to risk pooling
• Most years no loss; some years enormous loss
Conditions for insurability of
risks
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Loss must be amenable to risk pooling
There must be a clear loss
Loss must be in well-defined period of time
Frequency of loss must allow a premium calculation
Moral hazard must not be severe (eg, hazardous
waste insurance causes folks to be sloppy)
Adverse selection must not be severe (eg, only high
risk folks take out insurance)
Liability – a way of regulating risk
For firms/individuals engaged in risky
activities
 Rather than regulate risk, hold parties
responsbible for negative outcomes
 Eg, Oil Tanker Regs
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Some regs apply to nature of tankers
 Other protection achieved through
liability
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Threat of liability reduces risky
activities
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Bankruptcy can be a problem