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Those who have studied statistics should remember the three terms mean, median and mode. The

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mean is another term for the expected value. The median is the mid-point in the range with half of the

possible outcomes above it and half below. The mode is the single most likely outcome.

Unless, of course, the budget has been padded in order to allow for this effect.

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149 Building block 5: Risk

Clearly it is a lot easier to estimate expected values when distributions

are not skewed. All that is needed is the most likely outcome or the out-

come that will be exceeded half the time and you have your expected value.

Unfortunately, many of the real life situations that we need to model as we

calculate economic value are subject to skews of one type or another and this

makes our work that much harder. In fact the difficulty in establishing good

expected value assumptions goes to the heart of this book and we will need

to work through it all as we seek to build our ability to calculate an expected

value NPV.

For the present, and as a good place to start, we will simply note that we

should seek to make our valuation assumptions expected values. In doing

this we should take due account of the shape of a distribution and allow

for any anticipated skews. The greater the skew, the greater the difference

between the expected value and the most likely value. We should avoid

excessive analysis as we seek to estimate expected values and should usu-

ally be satisfied just with the application of informed judgement. So, for

example, informed judgement for a capital investment estimate would be

that the cost would be greater than the most likely outcome because of the

upside skew.

Cumulative probability curves

There is a second way that probability distributions can be presented. This is

in the form of a cumulative probability distribution. An example is shown

below. This is based on the same capital investment estimate. Instead of plot-

ting the probability distribution we plot the chance that the capital cost will

be no more than the specified number. This chart will start at the bottom left

with no chance of the cost being lower than the minimum possible number.

It then rises quite steeply for a while then slowly all the way to the highest

possible cost.

Probability

100%

0%

Outcome

Fig. 5.3 Example of a cumulative probability distribution

150 The five financial building blocks

The advantage of this presentation style is that it is easy to use it to support

statements like â€˜There is a 90% chance that the cost will be no more than

dollars x million.â€™ Or â€˜There is a fifty-fifty chance of the cost being less than

dollars y millionâ€™.

The jargon that is used for conveying information such as this is as follows.

We say that a P50 number has a 50% probability of being exceeded. We also

use terms such as P10, P80, etc. These correspond to the 10% and 80% points

on the curve. These points, however, can be ambiguous. Do we mean only

a 10% chance of being bigger than the quoted number or a 10% chance of

being smaller? The problem is that some people will draw the curve one

way (starting at the lowest number and working upwards) while others will

draw it the other way (starting high and working down). A third approach

involves making the judgement about what is good or bad and plotting that

way. Regrettably, there is no agreed convention here so all I can do is warn

readers to beware and always make it clear what they mean when they use

terms such as P10.

We will return to the concept of P10s and P90s when discussing which

sensitivities to show later in this part.

Doing calculations with expected values

Expected values have one hugely useful property. In what are termed linear

functions they behave in a linear manner. If you add up the expected value of

one outcome and the expected value of a second outcome then the result will

be the expected value of the sum of the two outcomes. The same is true for

the product of two variables. The expected value of a times b is equal to the

expected value of a times the expected value of b.

Take for example the roll of a dice. Roll once and the expected value is

three and a half. Roll twice and the expected score is seven. That is easy! Now

what is the expected value of the product of the number on the first roll and

the number on the second roll? This is more complicated. The long way to do

the calculation is to calculate all 36 possible answers,9 add them up (you will

get 441), and divide by 36. The answer is 12.25. The simple way is to square

3.5 to get 12.25.

In most financial situations where we are simply adding, subtracting, mul-

tiplying and dividing, if we make expected value assumptions then our NPV

The possible outcomes are 1,2,3,4,5,6,2,4,6,8,10,12,3,6,9,12,15,18,4,8,12,16,20,24,5,10,15,20,25,30,6,

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12,18,24,30 and 36.

151 Building block 5: Risk

will also be an expected value. This gives us a much simpler way of calculating

value. Instead of having to calculate all possible outcomes, if we simply want

to know the expected value NPV we only need to do one calculation with all

individual assumptions set at their expected value.

We can contrast this way that expected values â€˜workâ€™ with the way other

parameters such as the P10 or the P50 work. With these numbers you

cannot assume that, for example, a set of P10 assumptions plugged into a

financial model will produce a P10 NPV. I can illustrate this through a few

examples.

We know, for example, that the chance of getting two 1s when rolling a

dice is just one in 36. So if you make two assumptions, each of which have

a one in six chance of happening, your result has only a one in 36 chance of

happening. As a second example consider our dice rolling and multiplying

game developed above. The P50 outcome for this game is ten. This can be

established by looking at the full range of possible outcomes.10 The square

root of ten is 3.16 but 3.16 is not the P50 assumption for a single dice.

What this means is that if we prepare our financial evaluations with

expected value assumptions our calculated NPV will also be an expected

value. If we use any other way of setting assumptions we will not be able to

say for sure what the statistical meaning of the result is. So this serves as a

big reason to try to use assumptions that are expected values. This is not,

however, a â€˜knockout blowâ€™ in favour of expected value assumptions for vari-

ous reasons including the practical difficulty of establishing what a correct

expected value is.

Decision trees

The previous few sections have focussed on dealing with variables where

outcomes will typically fall within a range. These techniques are ideal for

dealing with things like the uncertainty associated with cost estimates or

selling prices/volumes. Another type of risk concerns what can be termed all-

or-nothing events; for example, you win an auction or you donâ€™t. These are

best dealt with via what are called decision trees. A decision tree will have

one or more nodes and at each node there is the potential to follow more than

one path into the future. A very simple decision tree concerning a sealed bid

property action follows.

There are 17 ways of getting a number lower than ten and 17 ways of being higher. Ten is the score in

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just two of the 36 possible cases.

152 The five financial building blocks

Outcome:

Spend $500,000

Win

Own the property

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