Risk, Uncertainty, and Sensitivity Analysis How economics can help understand, analyze, and cope...
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Transcript of Risk, Uncertainty, and Sensitivity Analysis How economics can help understand, analyze, and cope...
Risk, Uncertainty, and Sensitivity AnalysisHow economics can help understand, analyze, and cope with limited information
Generic Group Project
Land Use 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 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”?
Can be loosely defined as the “possibility of loss or injury”. 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)
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”
Rain (on way home): $4 - $4 = 0.No rain: $4 + $4 = $8.
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:If it rains, she gains 0.If it doesn’t rain, she gains $8.
Whether she is better taking the bike or bus depends on 2 things:The probability of rainHer attitude toward risk
The probability of rain
Suppose Pr(rain) = .5……Pr(no rain) = .5 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 Bus: $2 Bike: .2(8) + .8(0) = $1.60 (risky) If she is risk neutral, she rides the bus ($2 > 1.60)
Risk more generallyCoin toss pays $10 or $20
Utility
Some good (or $)10 2015
Q: Would this person ratherget 15 for sure or play coin toss?
U(15)
.5*U(10)+ .5*U(20)
This person isRISK AVERSE
Risk attitudes in general
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+…
Or ViPi
Example: Air quality regulations New air quality regulations in Santa
Barbara County will reduce ground level ozone.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
Example: Climate change policy
2 states of natureHigh damage (probability = 1%)
• Cost = $1013/year forever, starting in 100 yrs.
Low damage (probability = 99%)• Cost = $0
Cost of control = $1011
Should we engage in control now?
Control vs. no control (r=2%)
Control now: high cost, no future lossCost = $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.
Overall evaluation
Expected cost if control = $1011
Expected cost if no control = (.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”
Sensitivity Analysis
A method for determining how “sensitive” your model results are to parameter values.Sensitivity of NPV, sensitivity of policy
choice. Simplest version: change a
parameter, re-do analysis (“Partial Sensitivity Analysis”)
Climate change: sensitivity to r
0
1E+11
2E+11
3E+11
4E+11
5E+11
6E+11
7E+11
8E+11
0 0.01 0.02 0.03 0.04 0.05 0.06
Discount rate (r)
Lo
ss f
rom
no
co
ntr
ol
Sensitivity to Uncertainty on the probability of high damage
0.00E+00
2.00E+11
4.00E+11
6.00E+11
8.00E+11
1.00E+12
1.20E+12
0 0.005 0.01 0.015 0.02
p
Ben
efi
ts a
nd
Co
sts
Cost
E[Damage]
More sophisticated sensitivity The more nonlinear your model, the
more interesting your sensitivity analysis.
Should examine different combinations. Monte Carlo Sensitivity Analysis:
Choose distributions for parameters. Let computer “draw” values from distn’sPlot results
Managing Risk
Risk is a problem of its own Several tools available to reduce risk
InsuranceLiability
Insurance—fire insurance example
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
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 RegsSome regs apply to nature of tankersOther protection achieved through liability
Threat of liability reduces risky activitiesBankruptcy can be a problem