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Chapter

Cost-Benefit Analysis Concepts and Practice

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

FOURTH EDITION

Dealing with

Uncertainty: Expected

Values and Sensitivity

Analysis

Seven

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Introduction

• CBA often requires us to predict the future.

• We are rarely able to make precise predictions about the

future.

• How can analysts reasonably take account of these

uncertainties in CBA?

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

In this chapter we will consider:

 Expected value as a measure reflecting risks

 Sensitivity analysis as a way of investigating the

robustness of net benefit estimates to different

resolutions of uncertainty

Introduction

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected values take into account the dependence of

costs and benefits on the occurrence of specific

contingencies or “states of the world” to which analysts

are able to assign probabilities of occurrence.

Sensitivity analysis is a way of acknowledging

uncertainty about the values of important parameters

in our predictions – it should be a component of almost

any CBA.

Introduction

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

In many situations, one might reasonably divide the future

into two contingencies: either it will or it will not rain

sufficiently to make the umbrella useful.

Other relevant contingencies can be imagined as well – it

will be a dry day. But if these additional contingencies are

very unlikely, it is often reasonable to leave them out of

one’s model in the future.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Modeling the future as a set of relevant contingencies

involves yet another narrowing of uncertainty: How likely

are each of the contingencies?

In relative simple situations, risk can be readily

incorporated into CBA through expected value analysis.

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

Modeling uncertainty as risk begins with the specification of

a set of contingencies that, within the simplified model of

the world being employed, are exhaustive and mutually

exclusive.

Contingencies can be thought of as possible events,

outcomes, or states of the world such that one and only one

of the relevant set of probabilities will actually occur.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

Contingencies capture the full range of likely variation in net

benefits of the policy.

Contingencies represent the possible outcomes between

the extremes. Each contingency can be thought of as a

scenario, which is just a description of a possible future.

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

Figure 7-1 illustrates the representation of a continuous

scale with discrete contingencies

Horizontal axis – number of inches of summer rainfall in an

agricultural region

Vertical axis – the net benefits of a water storage system,

which increases as the amount of rainfall decreases

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Figure 7-1 Representativeness of Contingencies

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

Imagine that the analyst represents uncertainty about

rainfall with only two contingencies: excessive and deficient.

Excessive contingency – assumes 22 inches of rainfall,

which would yield zero net benefits from the storage system

Deficient contingency – assumes zero inches of rainfall,

which would yield $4.4 million in net benefits.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

If the relationship between rainfall and net benefits follows a

straight line labeled A, and all the rainfall amounts between

0 and 22 inches are equally likely, then the average of net

benefits over the full continuous range would be $2.2

million.

If the analyst assumed that each of the contingencies were

equally likely, then the average over the two contingencies

also be $2.2 million, so that using the two scenarios would

be adequately representative.

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

Now, imagine that the net benefits follow the curved line

labeled B. Again assume that all rainfall amounts between 0

and 22 inches are equally likely, the average of net benefits

over the full continuous range would only be $2.2 million, so

that using only two contingencies would grossly

overestimate net benefits from the storage system.

Adding “normal” as a contingency that assumes 11 inches

of rainfall and averaging net benefits over all three

contingencies yields net benefits of $0.6 million, which is

more representative than the average calculated with two

contingencies.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

Once a tractable but representative set of contingencies is

specified, the next task is to assign probabilities of

occurrence to each of them.

To be consistent and logical, that the contingencies are

exhaustive and mutually exclusive, the probabilities

assigned must each be nonnegative and sum to exactly 1.

Thus, if there are three contingencies, C1, C2, and C3,

assigned probabilities p1, p2, and p3 should sum up to 1,

such that p1+ p2 + p3 = 1

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Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Contingencies and Probabilities

The probabilities may be based on historically observed

frequencies; on subjective assessments by clients,

analysts, or other experts based on a variety of information

and theory; or on both.

Example: The national weather service may be able to

provide data on average annual rainfall over the last century

that allows an analyst to estimate probabilities of the three

levels of precipitation (excessive, normal, deficient).

Analysts may have to base probabilities on expert opinion,

comparison with similar valleys in the region for which data

are available, or some other subjective assessment.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Calculating Expected Value of Net Benefits

The specification of contingencies and their respective

probabilities allows us to calculate the expected net benefits

of a policy.

First, predicting the net benefits of the policy under each

contingency and then taking the weighted average of these

net benefits over all the contingencies, where the weights

are the respective probabilities that the contingencies occur.

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Calculating Expected Value of Net Benefits

For n contingencies, let Bi be the benefits under

contingency i, Ci, be the costs under contingency i, and pi be the probability of contingency i occurring.

Then the expected net benefits, E[NB], are given by the

following formula:

E[NB] = p1(B1 – C1) + … + pn(Bn – Cn) (7.1)

which is the expected value of net benefits over the n

possible outcomes.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Calculating Expected Value of Net Benefits

Table 7.1 shows the analysis of alternatives for planetary

defense against asteroid collisions.

The table shows hypothetical figures.

The three actions (alternatives) are shown.

The payoffs shown as present values of net costs over the

next century, range from $30 trillion (Earth is exposed to a

collision with an asteroid larger than one kilometer in

diameter in the absence of any asteroid defense) to $0

(Earth is not exposed to any collision with an asteroid larger

than 20 meters and no defense system is built)

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Calculating Expected Value of Net Benefits

The last column of Table 7-1 shows expected values for each of

the three alternatives.

For example, the expected value of payoffs (present value of net

costs) for no asteroid defense is:

(0.001) ($30,000 billion) + (0.004) ($6,000 billion) + (0.995) (0) =

$54 billion

Similar calculations yield $69 billion and $38 billions for the other

two alternatives.

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Calculating Expected Value of Net Benefits

As the maximization of expected benefits is equivalent to

minimizing expected net costs, the most efficient alternative is

near-Earth asteroid defense.

Treating expected values as if they were certain amounts implies

that the person making the comparison has preferences that are

risk neutral. That is, when he or she is indifferent between certain

amounts and lotteries with the same expected payoff.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Expected Value Analysis

Calculating Expected Value of Net Benefits

In practice, treating expected values and certain amounts as commensurate (risk

neutral) is generally reasonable when either the pooling of risk over the collection

of policies, or the pooling of risk over the collection of persons affected by a

policy, will make the actually realized values of costs and benefits close to their

expected values.

For example, a policy that affects the probability of highway accidents involves

reasonable pooling of risk across many drivers (some will have accidents, others

will not) so that realized values will be close to expected values.

In contrast, a policy that affects the risk of asteroid collision does not involve

pooling across individuals (either everyone suffers from the global harm if there is

a collision or no one does if there is no collision), so that the realized value of

costs may be very far from their expected value.

Such unpooled risk may require an adjustment to expected benefits.

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

• The purpose of sensitivity analysis is to acknowledge

uncertainty.

• It should convey how sensitive predicted net benefits are to

changes in assumptions.

• If the sign of net benefits does not change when we consider

the range of reasonable assumptions, then our results are

robust, and we can have greater confidence in them.

Two Approaches for Sensitivity Analysis

1. Partial Sensitivity Analysis

2. Worst- and Best-Case Analysis

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Two Approaches for Sensitivity Analysis

1. Partial Sensitivity Analysis

How do net benefits change as we vary a single assumption while

holding other variables constant?

It is most appropriately applied to what the analyst believes to be

the most important and uncertain assumptions.

It can be used to find the values of numerical assumptions at

which net benefits equal to zero, or just break-even.

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Two Approaches for Sensitivity Analysis

2. Worst- and Best-Case Analysis

Does any combination of reasonable assumptions reverse the

sign of net benefits?

Analysts are generally most concerned about situations in

which their most estimates yield positive net benefits, but they

want to know what would happen in a worst case involving the

least favorable, or most conservative assumptions.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

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Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

TABLE 7-2 (continued) Base-Case Values for Vaccination Program CBA

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Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Partial Sensitivity Analysis

In partial sensitivity analysis, you select one variable, change its value

while holding other variables constant, and see how much the CBA

results in response.

In Figure 7-3, the probability of the epidemic in the current year, p1, varies

from 0 to 0.5 by increments of 0.05.

The results are displayed as the line labeled L = $3 million.

The equation of net benefits were embedded in a spreadsheet on a

computer, so it was easy to generate the points needed to draw this line

by simply changing the values of p1.

As expected, this line is upward sloping: the higher the probability of the

epidemic, the larger the net benefits of the program.

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Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Figure 7-3 Expected Net Benefits of Vaccination

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Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Partial Sensitivity Analysis

In Figure 7-3, note that for values of p1 less than about 0.11, net

benefits become negative. If the probability of the epidemic in the

current year is less than 0.11, and we are willing to accept the other

base-case assumptions, then we should not implement the program.

The probability at which net benefits switch sign is called the

breakeven value.

Finding and reporting breakeven values for various parameters is

often a useful way to convey their importance.

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Worst- and Best-Case Analysis

Best-case and worst-case scenarios establish the upper (best-case) and

the lower (worst-case) boundaries of a cost-benefit study’s results.

It shows how a broad range of a program or policy’s possible outcomes

affect the bottom line.

To perform a best-case analysis, use all of the most favorable

assumptions about the program; for the worst-case scenario, use all the

least-favorable assumptions.

To project best and worst outcomes that are plausible, look at existing

research on similar programs. Keep in mind that best- and worst-case

scenarios are extreme cases; the odds of them happening are low.

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Other Forms of Sensitivity Analysis

1. Discount rate

2. Wage rates used

3. The value of life, or risks associated with certain occupations

The researcher must determine how changes in these measures affect the

outcome of the CBA. A researcher may wish to perform additional sensitivity

analysis in any of these cases, especially if the result of the CBA is close to zero

(using NPV), or the benefit-cost ratio is close to one. Moreover, if certain

measures used are contentious sensitivity analysis may be required. Or if a

particular cost or benefit that accounts for a large proportion of the overall costs

or benefits associated with a project is calculated using a measure which is

subject to some sort of arbitrary value.

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Example: Testing Sensitivity by Varying the Discount Rate

Consider a construction project with initial costs of $1.5 million. Benefits arising

from this project are $100,000 at the end of year 1, $200,000 at the end of year 2,

$500,000 at the end of year 3, and $1 million at the end of year 4.

With a discount rate of 5% the NPV of the project is as follows:

$95,238.10 + $181,405.90 + $431,918.8 + $822,702.47

PV (B) = $1,531,265.30

NPV = $1,531,265.30 - $1.5 million

NPV = $31,265.30

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Example: Testing Sensitivity by Varying the Discount Rate

With a discount rate of 10% however the NPV of the project is as follows:

$90,909.09 + $165,289.26 + $375,657.40 + $683,013.46

PV (B) = $1,314,869.20

NPV = $1,314,869.20 - $1.5 million

NPV = -$185,130.79

In this example, with a discount rate of 5% similarly there is justification for the

project to go ahead. However, with a higher discount rate of 10% the costs

outweigh the discounted benefits by almost $200,000, suggesting the project

should not go ahead. In this case the choice of discount rate could have a

significant impact on the viability of the construction project.

8/15/2017

Upper Saddle River, New Jersey 07458

Cost-Benefit Analysis: Concepts and Practice, Fourth Edition

Boardman • Greenberg • Vining • Weimer

Sensitivity Analysis

Example: Testing Sensitivity by Varying the Discount Rate

Partial Sensitivity Analysis

Plotting the various outcomes for different discount rates can allow a

researcher to arrive at the break even discount rate. This is the rate at which

the NPV is equal to zero.