Download Risk, Uncertainty, and Sensitivity Analysis

Risk, Uncertainty, and
Sensitivity Analysis
How economics can help understand,
analyze, and cope with limited
information
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.
We want to develop a way to describe
risk quantitatively by evaluating the
probability of all possible outcomes.
Attitude toward risk
Problem: Costello likes to ride his bike
to school. If it is raining when he gets
up, he can take the bus. If it isn’t, he
can ride, but runs the risk of it raining
on the way home.
Value of riding bike = $2, value of
taking bus = -$1.
Value of riding in rain = -$6.
Costello’s options & the “states
of nature”
Costello can either ride his bike or take
the bus.
Bus: He loses $0 (breaks even).
Bike: Depends on the “state of nature”
Rain: $2 - $6 = -$4.
No rain: $2 + $2 = $4.
Probabilities & risk attitude
Pr(rain)=0.5.
Costello’s expected payoffs are equal:
Bus: $0.
Bike: .5*(-4) + .5*(4) = $0.
If he:
Always bikes: he’s a risk lover
Always buses: he’s risk averse
Flips a coin: he’s risk neutral
His behavior reveals his risk preference.
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
S 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.
Example: Climate change policy
2 states of nature
High 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 loss
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.
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
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
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’s
Plot results