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The payouts are still zero (for a 1), $2 (for 2â€“5) and $4 (for a 6). The most likely

payout4 is $2 but we know not to take the simple arithmetic average of the

payouts because the $4 will have a greater chance of occurring. If the game is

tries example I am referring to the price paid divided by the number of dice that are rolled). This

approach is suggested because people understand that it is more likely that the answer will end up

close to the average if you have lots of tries. Markets, however, give people the opportunity to invest in

lots of risky ventures. You do not have to invest in exactly the same game to spread your bets, you just

have to invest in a range of games, each with their own risks. We will return to this when we consider

the portfolio effect in the next part.

This payout will happen on average in 320 out of every 600 times the game is played. This simplifies to

4

eight out of every 15 times.

146 The five financial building blocks

played lots of times we know that we get a payout of $4 one third of the time5

and a payout of zero, two fifteenths of the time. For the remaining occasions

the payout is $2. The fair market value of this game is five fifteenths of $4 plus

eight fifteenths of $2 plus two fifteenths of nil. This is $2.40.

Now what happens if you do pay $2.40 to play this final game? The most

likely outcome is that you will lose $0.40 and there is a small chance that you

will lose the lot. The thing, though, is that you do not mind risking these

outcomes because of the one chance in three of winning $1.60. The price is

exactly at the margin of what a rational person would be prepared to pay.

A rational person would rather pay less and, on balance, feel that they were

winning but at least they would not feel that, over time, they were losing if

this was typical of the decisions that they would take. The rational person

would know that if they set their bid to play the game at a level that gave them

a profit then they would probably be outbid.

One can think of this price as being an equilibrium price. If there were

a market in the right to play this game the equilibrium price is the price at

which it would be reasonable to believe that buyers and sellers would balance

out. If the price was a little higher then buyers would go away and other peo-

ple who were about to play the game would rather sell their rights. If the price

were lower then sellers would go away and more buyers would arrive.

So we have seen that what sets fair market value is the probability-weighted

outcome. Fair market value is the sum of the possible outcomes times their

individual probabilities. This probability-weighted outcome has been given a

special name. It is called the expected value. Once again, I have to say that I

do not like the name that has been chosen but I am stuck with the jargon of

my subject. In the context of statistical calculations the expected value is the

name for the probability-weighted value.6 So, fair market value is the expected

value, i.e. probability-weighted outcome given the possible outcomes.

There is one minor and one major condition to attach to this rule. The

minor condition concerns transaction costs. If for any reason it costs

money simply to make a purchase or a sale then this must be factored into

the calculation. We will ignore this effect at this time. The major condition

concerns an availability of money that people are prepared to risk. Suppose

200 Ã· 600 = â…“.

5

I cannot escape the thought that, to the general public, an expected value will mean what you expect

6

to happen and what you expect to happen is the most likely outcome. I understand why statisticians

used the term expected value. It is because they think in a particular way, but I do wish that the term

was not used! I always worry about a possible misunderstanding by those who are not aware of the

definition of the term expected value. It is for this reason that I recommend when using the term

expected value it should always be qualified by the expression, i.e. probability-weighted outcome,

until it is clear that all are aware of its meaning.

147 Building block 5: Risk

that the sums of money in our game were not measured in dollars but were

measured in millions of dollars. Very few individuals would be willing to pay

$2.4m to play a game where they would lose at least $400,000 two thirds of

the time. Individuals simply cannot afford risks as big as this. The logic that

they would do OK over time would not apply if they lost all their money (and

more!) on the first throw.

So our expected value rule does depend on the availability of a suitably

large pool of risk capital. We will return to this question in the second part

of this building block. For the present we will simply assume that decisions

are being taken by investors that are financially strong enough to take the

implicit downside risks. In this situation it is the expected (i.e. probability-

weighted) NPV that sets the fair market value of an asset and that should be

used as the decision rule for deciding on investment opportunities.

Probability distributions

We have now learned that we want to establish expected value NPVs. This was

easy for simple situations such as the rolling of dice. It is more difficult in most

real-life situations. We can, however, gain some useful clues as regards what an

expected value will be if we understand a little about probability distributions.

I assume that the concept of a probability distribution is already familiar

to readers. It gives a depiction of the range of possible outcomes for an event

and of the relative likelihood of different outcomes. One way to depict this

information is called a frequency distribution. This plots the range of out-

comes on the x axis and the probability of the outcome on the y axis. An

example is given below.

Probability

Outcome

Expected Value

Fig. 5.1 Example of a frequency distribution

The area under the curve is, by definition, equal to 1 because all the possi-

ble outcomes have been depicted. The curve is my freehand version of what

is called a normal distribution. The distribution of many outcomes fits this

148 The five financial building blocks

particular shape. Of particular importance is the fact that the shape of the

curve is symmetrical about its mid-point. This means the expected value of

a normal distribution is the same as the mid-point in the full range and also

the most likely outcome.7

Frequency distributions for actual events will have different shapes

depending on what is causing the underlying variability in outcomes. So,

for example, the frequency distribution for the throw of a dice will be flat

because each of the six possible outcomes is equally likely. Some distribu-

tions will cover a wide range while others will be closely bunched around

the mid-point. There are also what are called bi-modal distributions where

there are two main possibilities, each subject to its own minor uncertainties.

The easiest distribution to draw is the so-called triangular distribution. This

will comprise two straight lines, one joining the lowest point and the most

likely outcome and the second joining the most likely outcome to the highest

possible outcome. This may be easy to draw but is very unusual in real-life

situations.

In my opinion there is one particular feature of probability distributions

that is of the greatest importance to the calculation of economic value. This

concerns the question of skews. A distribution is said to be skewed when it is

not symmetrical. An example follows. This could well be a plot of the range

Probability

Outcome

Fig. 5.2 Example of a skewed distribution

of possible capital costs for a major project. Experience tells us that cost over-

runs are very likely but that there is little chance of the project coming in

under budget.8 With this distribution the most likely outcome and also the

mid-point of the range are both lower than the expected value.

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