Evaluation
with Risk
The previous lessons on evaluating and comparing
projects have all neglected uncertainty. We have computed the
net present worth, net annual worth and rate of return using
single values for the parameters that define the cash flow. An
example is shown below. It has a single investment of $1,800 initially,
and the revenue is uniform at $650 per year for the 5 years
of the project life. The operating cost begins at $100 in the
first year and increases as an arithmetic gradient of $50 per
year. Finally at the end of the 5 years there is a salvage
value of $400. The evaluation form constructed by the Economics addin is shown below. With an MARR of 10%, the NPW and NAW are positive
so the investment is acceptable. In fact, the IRR of the project
is almost 14%.
As with all economic evaluations
the numbers used in the model are only estimates. In this example, the magnitudes
of the cash flow components indicated in cells C10 through C14
are really not known with certainty. The life of the project is also
uncertain. Generally, we suspect that estimates of expenditures
and revenues early in the project are better than those that
occur later. How should we deal with this uncertainty in the
decision making process?
A variety of ways have been proposed to adjust
the numbers in the deterministic analysis to reflect risk. One
popular method is to increase the MARR for risky projects. Then
profits far in the future will contribute less to the NPW. Alternatively,
by arbitrarily reducing the planned life of the project,
profits beyond the reduced life are entirely neglected. Sensitivity
and breakeven analysis are methods that analyze the variability
of the parameters one at a time.
In this lesson we explicitly recognize risk by
assigning probability distributions to uncertain estimates.
We then use probability theory and Monte Carlo simulation to
draw conclusions about the statistical variability of the evaluation
measures.


Goals 



For a given cash flow, compute
the sensitivity and the breakeven point for any
single parameter of the cash flow.

Construct a spider chart showing sensitivity
to several parameters.

For a cash flow defined by
a set of cash flow components with uncertain
values, find the mean and variance of the
NPW and NAW.

Assuming normally distributed
and independent random variables make probability
statements regarding the NPW and NAW.

Use the Economics and Random
Variables addins for assigning probability
distributions to cost estimates and to estimate
the mean and variance of the NPW and NAW.

Use the Economics and Random
Variables addins to perform MonteCarlo
simulation to estimate statistics and probabilities
concerning NPW and NAW.




Text 



3.5
SENSITIVITY AND BREAKEVEN ANALYSIS 




NPW Function 

The NPW of the example is computed
by multiplying the estimated values of each cash flow component
by the appropriate equivalency factor and summing. For an MARR of
10% we have
The models of this section allow the estimates
to change, so it is useful to have a more general expression
for NPW. In the following expression, is
the equivalency factor for term k. This factor is a
constant for given values of the MARR and the project life. The quantity is
the cash flow estimate for term k.
When the life is held fixed, this expression is
easy to evaluate because all the values
are constant and the NPW value is a linear function
of the estimates. For simple analyses, we will often make
this assumption. When the life is allowed to vary, however, the
function is nonlinear and for most problems the function is not
easily written as a function of the project life N.
The Economics addin can
easily evaluate the NPW and NAW for
given values of N,
allowing numerical analysis of variations in the estimate
of life.
In our example, there are five terms in this expression.
The first is for the investment, the second for the annual revenue
and so on. The table shows the estimates, factors, the contribution
to the NPW of each component and the resultant sum that
is equal to the project NPW.
Index 
Component 
Estimate
(x) 
Equivalency
factor

Factor
value (C) 
Cx 
1 
Investment 
1800 
1 
1 

2 
Revenue 
650 
(P/A, 0.1, 5) 
3.7908 

3 
Op. Cost 
100 
(P/A, 0.1, 5) 
3.7908 

4 
Op. Gradient 
50 
(P/G, 0.1, 5) 
6.8618 

5 
Salvage 
400 
(P/F, 0.1, 5) 
0.6209 

Sum Cx to
find the NPW =

190 
Given the life of the project the NAW is
similarly calculated.
The expressions show that the measures of economic
acceptability, NPW and NAW, are
functions of the cash flow component estimates, the interest
rate used for the evaluation, and the estimate of the life. In
the following we illustrate approaches to the decision problem
when the uncertainty of the estimates is explicitly considered. 

Sensitivity Analysis 

The purpose of sensitivity analysis
is to show how changes in the individual estimates affect the
economic measures and the decision to accept or reject a project.
The method selects a base value for each estimated value. Then
each estimate is individually allowed to change above and below
the base value while holding all other estimates at their base
values. The expression below shows how the NPW varies as
a function of the variations from the base values.
The formula shows that the NPW
is equal to a constant plus a linear expression that
involves the variations. The constant term is the NPW we
computed with the original estimates and the coefficients of the
linear terms are the contributions of each component to the base
net present worth. For the example the function is:
For sensitivity analysis, we only allow one component
to change at a time, holding all other variations at zero, and
observe the variation in the NPW.
The figure below, called a spider chart,
is useful for displaying sensitivity results. Each line shows
the variation of the NPW with one of the cash flow parameters.
All lines cross at the base NPW value of 190. As expected
the lines are straight. The slope of a line indicates the
sensitivity of the NPW value to changes in a parameters. The
red line representing revenue has the greatest slope, so the NPW is
most sensitive to changes in revenue. It is least
sensitive to changes in the salvage value.
Of particular interest is the portion of the line
that falls below the NPW = 0 line. For these values of
the parameter being changed, the return on the project is less
than the MARR. 

Breakeven Analysis 

For breakeven analysis, we hold
all but one of the variations to zero and compute the value of
remaining variation that makes the NPW equal to zero. For the
cash flow component values, the linear equations are easy to
solve.
The table shows the breakeven variations and the
associated breakeven values that cause the NPW to be zero. The
columns show the factors in the equation above. If the breakeven
values are far from the values that might reasonably be expected,
the decision maker might be confident in a decision to accept
the investment. Alternatively, a value that is close to the base
case might suggest that more study should be devoted to finding
a more accurate estimate.
Component 
Base (b) 
Factor
(C) 
Variation
( ) 
Value
( ) 
Investment 
1800 
1.0000 
0.106 
1990 
Revenue 
650 
3.7908 
0.077 
600 
Operating
cost 
100 
3.7908 
0.502 
150 
Operating
gradient 
50 
6.8618 
0.554 
78 
Salvage 
400 
0.6209 
0.766 
94 


Project Life 

The sensitivity analysis for project
life is more complicated because the NPW is a nonlinear function
of the life of the project as reflected in the formula for our
example. Also our model is only valid for integer values of the
life. We include the expression for NAW because it is not meaningful
to compare values of the NPW over different lives.
The chart shows the variation of NAW with life.
It is clearly nonlinear. The breakeven point is at about four
years.


Mean and Variance
of NPW 

In this section, we model the
amounts of each component as independent random variables. The
equivalency factors for fixed values of MARR and project life
are constants.
When the component values are independent random
variables, the mean and variance of the NPW can be written
as sums of terms. We use F as an indicator of the distribution
of the values, because these results are not restricted to normal
distributions.
The linked document shows the computation of the
mean and variance of the example when the moments of the cash
flow component values are given.
With the assumed data we find the parameters of
the NPW distribution.
The net annual worth, NAW, is proportional to the
NPW so we can easily compute the moments of the NAW.
When we further assume that the component values
are normally distributed, the NPW and NAW also have normal distributions.
We then can use the Normal tables to find probability values.
A particularly useful value is the probability that the NPW is
less than zero. This is the probability that the project does
not earn the MARR.
So for the example, there is about a 21% chance
that the project will fail to meet the 10% MARR requirement.
This section has demonstrated two important results.

When
the cash flow components are independent random
variables, the mean and variance of the NPW and NAW
can be computed by weighted sums of the means and variances
of the component distributions.



When
the cash flow components are independent normally
distributed random variables, the NPW
and NAW are normally distributed.


On the next page we use simulation to evaluate systems when
the variability is not normal and when the life is allowed to
be a random variable. 

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