ñòð. 156 |

What happens is that the company will lose the potential to produce at rates

which are above the pipelineâ€™s capacity. The probability distribution will con-

tain a spike at this point reflecting the relatively likely outcome that produc-

tion will be limited by the pipelineâ€™s maximum capacity. Indeed, this outcome

may well finish up as the highest probability outcome. Now since the higher

production was needed, in statistical terms, to offset the possibility of lower

production, the overall expected value must fall. What one needs to do is to

calculate the effect of losing access to the tail of a statistical distribution and

replacing it with the constrained production. I call the technique which does

this â€˜valuing the tailâ€™, because this is what it is doing.

What I will do first is develop some rules of thumb for a very simple statis-

tical distribution. I will then show how a more accurate approach can be used

if this is felt desirable. For this first stage I will assume that the statistical dis-

tribution takes the shape of a triangle. Instead of the usual bell-shaped curve

which typifies a normal distribution I will assume a straight line between

the absolute maximum and the most likely value. If the distribution is not

skewed then the downside triangle will be a mirror image of this.

The following diagram illustrates a triangular distribution for this hypo-

thetical oil field. I have assumed a range of Â±20%.

The triangle at the top of the diagram shows what would be the situation

if there were no pipeline constraint. This would be the case as long as the

Probability

Initial view:

Production potential

for oil field

100 Â± 20 kb/d

80 kb/d 100 kb/d 120 kb/d

Production

Oil field: effect of 100 kb/d pipeline constraint

Half the time production:

is limited to 100 kb/d

Expected value falls

100 kb/d

Production

Fig. 12.4 Valuing the tail: triangular distribution

505 Second view: Valuing flexibility

maximum pipeline throughput was at least 120 kb/d. The lower half of the

diagram shows what the effect will be if the pipeline has a capacity of only

100 kb/d. I have not shown the distribution on the right-hand side of this

lower chart because it would be a vertical line at exactly 100 kb/d.

The key question concerns what the loss of upside potential does to the

expected value. The answer can be obtained from the useful fact (which I do

not propose to prove here) that a statistical distribution with the shape of a tri-

angle can, for the purposes of calculating expected value, be replaced with a

single point one third of the way along the lower axis. In this case, the upside

of between 100 and 120 kb/d which happens half of the time can be considered

equivalent to a production of 106.67 kb/d also happening half of the time. This

is now replaced with a production of just 100 kb/d happening half of the time.

So the effect on the overall expected value of losing access to all of the potential

upside is that one suffers a reduction in expected value of one sixth of the max-

imum range. In this particular case, the expected value of the oil volume which

will be available to sell will be 96.7 kb/d rather than the original 100 kb/d.

Now up to now I have assumed that the stated range of uncertainty of oil

production was the full range. In a lot of instances people do not state the

full range because the top and the bottom ends cannot actually happen. It is

quite common for ranges to be stated to the P10/P90 range of statistical con-

fidence. This would mean that the range would be exceeded 10% of the time

and would be too optimistic 10% of the time. The full range is always much

bigger than just the P10/P90 range. For a triangular distribution the range

is, in fact, almost doubled. A P10/P90 range of Â±20% indicates an absolute

maximum range of Â±36% (trust me!11). So if, as I suspect would be the case,

the oil reservoir engineer who provided the original 100 kb/d estimate and

stated that it was Â±20% meant that this covered the P10/P90 range, then

the expected value of oil sales that would be lost in the peak year through

building a 100 kb/d pipeline would be 6 kb/d. We could, if we wanted to, use

this simple rule of thumb to see if the extra cost of building a pipeline with a

capacity of 136 kb/d was justified.

If you want to know how to do the maths, read this footnote. This is actually a useful piece of learning

11

should you wish to do any other calculations such as working out the impact of, say, losing production

in excess of 110 kb/d. If 120 kb/d has just a 10% chance of being beaten there must be a 40% chance of

the resultâ€™s being in the range 100â€“120 kb/d. If the actual peak production is p kb/d we know that the

area of the triangle with a base that is from 100 to p is 50% while the triangle with a base from 120 to

p has an area of 10%. The areas of the two triangles are proportional to the square of the size of the base.

We know the length of each base (one is p units long while the smaller triangle is (p â€“ 20) units long.

This means that p2 must be five times bigger than (p â€“ 20)2. We can solve by trial and error to find that

p = 36.2 or revert to algebra and the formula to solve quadratic equations if we want to be really clever!

506 Three views of deeper and broader skills

Now so far I have only dealt with a simple situation where the constraint

was set at the expected value and where the statistical distribution was

triangular. What I hope is evident is that the generic approach can be applied

to a wide range of situations where a company gains or loses access to the

tail of a statistical distribution. The constraint need not be set at exactly the

expected value. It can be set at any level and one then must investigate the

economic impact of losing the upside above this constraint while still suffer-

ing from the downside.

One could, for example, investigate various sizes of pipeline and also the

effect of various shapes of statistical distribution. Then, since a larger pipe-

line must cost more money, in principle each set of assumptions will have

an optimal solution. If pipelines were very expensive in relation to the cost

of producing the oil, one could even discover that it paid to build a line that

was below the expected value of oil production in order to ensure that there

was a high probability that it would be fully utilised. The reality for pipelines,

however, is usually that the incremental cost of a slightly larger line is well

below the average cost and it is generally beneficial to oversize rather than

undersize.

My personal flexibility toolkit contains a spreadsheet model which allows

me to specify a range of possible types of statistical distributions, an expected

value and a constraint which can be anywhere between the minimum and

the maximum possible production. The model then generates for me the

revised expected value in view of the constraint. I did, however, need to enrol

the assistance of a PhD student in order to build it.

I will leave it to readers to decide whether they wish to build such a model

for themselves. It is, I believe, quite useful when one is dealing with the effect

of a single constraint and one wants to go beyond simple assumptions con-

cerning triangular distributions. Once multiple constraints are involved,

one in any case needs to switch to Monte Carlo simulation, which I will be

explaining in the following section.

There are two further steps which I need to cover in order to complete this

example. The first is to explain that in the case of an oil field, production that

might be lost in one year owing to a pipeline constraint should be recoverable

at a later stage once the fieldâ€™s production starts to fall. For most manufac-

turing plants, however, a sale lost in one year owing to a physical constraint

will be lost for good.

The second step is to consider how one might also value the potential for

an oversized pipelineâ€™s being used on further oil fields. The answer is that

one applies the growth value technique as outlined in section 8C above. One

507 Second view: Valuing flexibility

makes estimates of when other oil fields might wish to utilise the pipeline,

what fee they would pay and what the associated probabilities were.12

I have a particular enthusiasm for the valuing the tail technique because

I believe that a good case can be made for oversizing many assets. There are

two factors which could justify building bigger assets than would be sug-

gested by the simple rule of build to cover expected value demand. The first

is that there must be uncertainty about demand. Without this there is clearly

no point in oversizing. Most situations, however, have demand uncertainty.

ñòð. 156 |