# Proper definitions for Risk and Uncertainty

### Including a definition of relative risk, and an application to insurance

Thomas Cool, February 25 & July 8 1999
Scheveningen, Holland
http://thomascool.eu
99-02 update
JEL A00, C00, G00

Summary

The currently commonly adopted definitions of risk and uncertainty generate conceptual problems and inconsistencies, and they are a source of confusion in general. However, alternative and proper definitions are:

(1) First there is the distinction between certainty and uncertainty.
(2) Uncertainty forks into known (assumed) and unknown probabilities.
(3) Ignorance, or unknown probabilities forks into known categories and unknown categories.
(4) Known categories forks into 'including the uncertainties in the probabilities by explicitly assuming a uniform distribution' (Laplace) or neglect (or use other non-probabilistic techniques)

Note that the term 'risk' has not been used in the 4 points above, so that an independent definition is possible. 'Risk' can be defined as the absolute value of probable loss, i.e. as = -E[x < 0]. Also, relative risk is the probable loss with respect to a target t, giving (t) = t -E[x < t].

The definitions provided here are directly in line with the Oxford English dictionary. It turns out that economics textbooks generally can keep their mathematics but will best rewrite their texts to these definitions. Not only the students and the general public will benefit from this sudden clarity, but eventually also economic theory itself.

Note: This paper updates a February 1999 version with: (a) better notation, (b) relative risk.

### Introduction: Distinguishing uncertainty and risk

This discussion will present proper definitions for risk and uncertainty. Such definitions are required since the common definitions in current use are rather erroneous and generate conceptual problems.

The New Palgrave (1998:III:358) gives the current common view. The Palgrave states: "The most fundamental distinction in this branch of economic theory, due tot Knight (1921), is that of risk versus uncertainty. A situation is said to involve risk if the randomness facing an economic agent can be expressed in terms of specific numerical probabilities (these probabilities may either be objectively specified as with lottery tickets, or else reflect the individual's own subjective beliefs). On the other hand, situations where the agent cannot (or does not) assign actual probabilities to the alternative possible occurences are said to involve uncertainty."

Indeed, most economic texts use this distinction in this manner (at least, up to now). However, I cannot disagree more.

(a) Certainty and uncertainty are binary. So, if a situation is not uncertain, then we have certainty, and there is no assigning of probabilities.
(b) If I am uncertain about a situation and assign equal probabilities to all cases - the Laplace suggestion - then according to "Knight" this would no longer be uncertainty !

The following definitions are proper, and solve the predicaments. They are also clarified in the subsequent diagram in figure 1.

(1) First there is the distinction between certainty and uncertainty.
(2) Uncertainty forks into known (assumed) and unknown probabilities.
(3) Ignorance, or unknown probabilities forks into known categories and unknown categories.
(4) Known categories forks into 'including the uncertainties in the probabilities by explicitly assuming a uniform distribution' (Laplace) or neglect (or use other non-probabilistic techniques such as minimax)

##### Figure 1: A diagram of the new definitions

For comparison, figure 2 contains a diagram of the current, and objectionable, use of terms. The diagram clarifies the inbreach on the binary character of certainty/uncertainty, the curious treatment of "Laplace", and the over-use of terms by introducing the term 'risk' where there already is the qualification that the probabilities are known.

##### Figure 2: A diagram of the current but objectionable use of terms

A.S. Hornby's (1985) "Oxford Advanced Learner's Dictionary of Current English" defines 'uncertain' as: "1 changeable; not reliable: ~ weather; a man with an ~ temper. 2 not certainly knowing or known: be/feel ~ (about) what to do next; a woman of ~ age, one whose age cannot be guessed". Here the distinction is not between known and unknown probabilities, but the distinction is between objects and events and human thought and knowledge. This definition of uncertainty is consistent with the four points above.

Let us now turn to risk. Note, specifically, that the above four points don't define 'risk' yet. This term is reserved for something else below. The four points only use certainty, knowledge and the distinction about categories (category-uncertainty). We don't need the term 'risk' if we already have these terms to do the work.

(Note, p.m., that having known probabilities also assumes knowledge of the categories. Thus the 'known categories' in the subpath of 'probabilities unknown' is conditional to that path.)

Risk can be said to deal with the case of known probabilities or when unknowns are assumed to be uniformly distributed over known categories. It is not customary to use the term 'risk' for unknown categories. For example, it is uncommon to say, or write economics papers about this, that "all our lives are at risk of a suddenly imploding universe, or black hole hitting Earth, or waking up as a cockroaches". Such real 'Acts of God' are commonly neglected.

For 'risk', there is the additional problem that the current view often takes the variance as a measure of risk. Also this practice has undesirable effects. A prospect with a high variance but with only profits would be considered, at least in current academic economics papers, to be riskier than a prospect with a smaller variance with the possible losses ! This is also at variance with common notions of what risk amounts to.

### Definition of risk

#### The proper definition

Hornby's Dictionary op. cit. defines 'risk' as: "(instance of) possibility or chance of meeting danger, suffering loss, injury, etc." Also: "at the ~ risk of / at ~ of, with the possibility of (loss etc.)".

Thus, if there are possible outcomes O = {, , ..., }, then the situation is risky if at least one of the o's represents a loss. The risks themselves are the events or the that are those losses. The risks factors are the dimensions or positions of the risky outcomes, the i's (or the causes that make such positions to be filled).

We will use the term 'valued risk' when a risk is valued with money or utility. When all risks have been made comparable by valuing them, then we can add them, and we will use the term expected risk value for the expected value of the 'valued risks'. Then, crucially, once these definitions are well understood, then we may also use 'the risk' for the the expected risk value.

Note:
(A) The risk (the expected value of the valued risks) is single (i.e. non-plural).
(B)  The risks (thus plural) gives the list of the . For a single outcome, we would have the difference between and } (element and singleton).

#### Money or utility valued risk

With a list of outcomes {, .., } we also have lists of prices P = {, ..., } and probabilities Pr ={, , ..., }, and a utility function u.

The money valued risks are X = {, .., }  = O * P = {, ..., }.

The utility valued risks are U = {, .., }. The expression U' = u(, .., ) is less appropriate since the outcomes are mutually exclusive. However, since one might consider cases where one has some utility about 'the whole situation', the U' might still be useful.

#### Mathematical expectation

Let us work with X. We will look at prospects, i.e. money valued outcomes with attached probabilities.

As stands for the expected value and for the standard deviation (spread), we can use to stand for the risk. (PM: Then, use R for the coefficient of correlation. PM Note that the use of 'spread' facilitates translation from learned journals to popular audiences that are less familiar with the standard deviation. Authors that use the word 'spread' for the difference between a futures and a spot price, should relabel to 'time premium'.)

A general definition is: = -E[x < 0], or the absolute value of probable loss. The term 'probable value' denotes the product of an outcome and its probability. E.g. the probable value of an (money valued) outcome with probability is: . Note: The expression "E[x < 0]" is not to be read as "the expectation that x < 0" but as "the expectation for x < 0". See the further discussion of the notation below.

Note that it still remains possible to say that a situation is risky even though one cannot put a number to it. Above expectation may be indeterminate since one may lack knowledge about the probability distribution or even the categories.

Return is revenue minus costs, and the rate of return is given as: return / costs - 1. Let profit ≥ 0 stand for the positive return of a prospect, and - loss ≤ 0 for the negative return of a prospect, where loss is the absolute value of that negative return. The probability of a profit is p, the probability of a loss is (1 - p). We shall be using the following definitions:

ExpectedValue = = p profit + (1 - p) (-loss) = probable profit - probable loss
Risk = risk value = expected value of the risks = probable loss = (1 - p) loss
RiskRatio = Risk / (ExpectedValue + Risk) = (1 - p) loss / (p profit)
ExpectedValue = p profit (1 - RiskRatio)
RiskProbability = cumulative probability of all losses

Risk is the (absolute value of the) down side of a bet. A venture is judged to be risky if the probable loss is large. Note that this notion still is somewhat vague. A probable loss can be large because of the probability or because of the sum of money involved. This vagueness is unfortunate, in some respects, but here is little to be done about it, since this vagueness is inherent in working with probabilities. In fact, this vagueness is an essentially positive aspect of working with probabilities. For, when we have different prospects, then we can order and evaluate them on risk, neglecting differences in losses and probabilities.

#### Example formulas

The following formulas are from a Mathematica notebook that will be in Cool (1999), "The Economics Pack".

Note that one might define the utility U[x] = Log[W - x] for some initial level of wealth W (as just an example).

### What risk is not

#### Risk is not the variance

The finance literature often uses the term 'risk' for the variance or spread (standard deviation) of the distribution of the rates of return of investments. This would be an improper use of the term.

Suppose that one has a very profitable venture without the possibility of a loss. Suppose that the rate of return of this venture has a large variance, from mildly profitable to highly profitable. Is this a risky venture ? No, not in the usual understanding of the term.

#### Risk is not the negative of expected revenue

In mathematical statistics, some authors, like Ferguson (1967), define 'risk' as 'expected loss'. However, it appears that they actually regard 'loss' as the negative of total returns (i.e. - returns), so the definition actually used is -(p profit + (1-p) (-loss)), which is the negated expected value. This use of the term 'risk' is inappropriate. My proposal is to use the word "due" to stand for the negative of expected value, so that the standard statistical decision theory (with the game against nature) can be described as minimising due.

### Definition of relative risk

Rather than just wanting to stay out of trouble, investors often have a more ambitious preoccupation: to do as well as the market. Some investors even are willing to pay a price to stay in line with the market: they are worried that they can lose, but a little bit less worried when everybody else loses too. These situations give rise to the concept of relative risk.

Relative risk will now be defined as (t) = t - E[x < t] for some target level t. Risk (or absolute risk) takes t = 0, and relative risk would allow for a different target level.

An interesting application of the latter measure is to take t = r for r the certain rate of return, and with (r) = r - E[x < r], where x is the stochastic rate of return. This relative risk answers the question: What would one expect to lose with respect to r, if earnings underperform and fall below r. (Indeed, r - (r) would give your expected return when underperforming.)

In a numeric example, if the certain rate of return is 5% and the relative risk with respect to that is 3%, then this means that one can expect a return of 2% when it is given that the market underperforms.

Since relative risk is relative, and may defined on any target, it can be useful to have some standard of reference. The absolute risk measure itself already provides some perspective. Another point is of reference is t = E[x] = . The relative risk generated by that, () = - E[x < ], will be called the relative risk.

#### Where 'relative' is absolute; Fixed and moving target

Consider the situation of an investor and the market, so that there are two probability density functions f(x) and g(y) respectively. Then h(z) is the density of z = x - y, and = (0) = -E[z < 0] would give the expected relative loss if the investor would do worse than the market. We find that the relative risk for the investor is an absolute risk in terms of z.

Note that the latter result requires some calculus though. A common rough estimate that avoids that calculus is E[y] - E[x]: If I make 6% and the market makes 7% then I have a relative loss (not risk) of 1%.

Note that the relative risk (E[y]) = E[y] - E[x | x < E[y]] for the investor takes the middle ground between the relative loss E[y] - E[x] and the calculus of using h(z). The first takes a fixed target, the latter is a moving target. To avoid confusion it is best that one always specifies what the target t is doing.

PM. A crucial point concerns the kind of densities that we are considering. (1) If g(y) gives the density of market returns, and if the investor selects a market portfolio, then not only f = g, but also z = 0 by definition for the whole domain. (2) Alternatively, the investor indeed does not have a market portfolio, and the above procedure has real meaning. This situation can for example arise when there is inadequate information about the market portfolio.

PM. Note the difference between A = t - (t) = E[x < t] and B = (t) - t = -E[x < t]. There may be confusion about what to substract from what, and this confusion will be fueled by risk being a nonnegative number like B for t = 0. The suggestion is to use B only for absolute risk, and to use A only for relative risk. The consideration is that A gives the proper expected value without any additional change of sign.

### Note on the notation

In the former version of this paper I used the notation "- E[x | x < t]". It appears that this notation is better reserved for the real conditional density.

For example, if we have a distribution f(x) and consider underperformance till target t, then we might take the new pdf as f*(x) = f(x) /, and determine the expected value with this. This procedure would be the right track if indeed new information arrives that the returns will remain below that target level.

For (relative) risk we don't want to use the conditional distributions. For example, if there would be a small loss with a small probability p, the conditional might turn this in a large 'risk', since 1/p is would be a large number. So for risk we have a proper measure in the 'probable value' (loss * probability).

Risk is concerned with one's worry that bad information might arrive while it may not arrive.

However, the conditional notation is better reserved for real conditionals. Therefor I changed the notation to "E[x < t]" and hope that it is clear now that this is shorthand for E[ I(x, x < t) ] where I(x, x < t) is the identity function defined as x in x  < t and 0 elsewhere.

Tajuddin (1996) discusses proper conditional expectations (from the viewpoint of skewness), and his discussion may help one to understand the difference in perspective. (While skewness is no measure for risk, for skewness doesn't say anything about the values below 0.)

### Risk and uncertainty aversion

From the above it follows that also terms like 'risk aversion' have to adjusted to mean their proper meaning.

Generally 'risk aversion' reflects what risk averse people resent, and thus the power in would be a measure of risk aversion.

Similarly for 'spread aversion', and the power in would be a measure of spread aversion.

The term 'uncertainty aversion' can best be the general. Commonly, the aversion is not directed at the Acts of God referred to above (though indeed, some people appear to be averse to life). In normal conditions we can limit attention to known categories. Then, 'uncertainty aversion' might indicate an aversion that one, in itself, has to neglect categories or assign uniform probability. This would be a constant factor in the aversion that has to do with uncertainty itself, and that does not change over uncertainties. What changes from one case to another are spread and risk. It remains uncertain whether this aversion is directed at spread in particular or risk in particular, or combinations of these. Hence, a general measure is    .

### Prospect examples

Let us define 5 example prospects, using an object programming approach, with PM a function that completes a probability measure.

``pr[1] = Prospect[{5, -5, 10, 4}, {0.1, 0.1, 0.2, 0.6}]; pr[2] = Prospect[{0.5, -15, 10, -0.4}, {0.2, 0.1, 0.2, 0.5}]; pr[3] = Prospect[{0, 1.1, -3, 2}, PM[{0.21, Rest, 0.4, 0.3}]]; pr[4] = Prospect[{13.8, -1., 0.6, 1}, PM[{0.2, 0.2, 0.3, 0.3}]]; pr[5] = Prospect[{5, 1, 1, 2.}, PM[{0.1, Rest, 0.2, 0.06}]]; ``

### Prospect plots

We can plot the various example prospects into the various spaces. Figure 3 gives these plots. The finance community is familiar with the upper right kind of plot, the other plots are novel.

The common finance analysis in the {} space will regard the line A-E as the efficiency border (that dominates the below and right). The point D would be considered to be dominated. However, in the {, } plot, the D takes a dominant position.

### Prospects with joint probabilities

We can further illustrate the meaning of the new definition of risk by reproducing a small part of the common analysis of the Markowitz efficiency frontier in the {, } space of two assets.

For simplicity, assume a two dimensional world, with different outcomes of the weather and of the oil price. Let us also assume two persons, who have different revenues depending upon different outcomes of the weather and the oil price. This is basically a game situation, but with the players not in control of the probabilities. Let us for simplicity assume that there are three classes for the weather and similarly for oil, so that we get 3 by 3 matrices of possible states of the world, their probabilities and revenues for either person. We collect all this information in a JointProspect object, that lists the revenue matrices and the probability matrix.

``jp = JointProspect[{{100, -10, 5}, {1, 0, 0}, {-10, -10, 20}},                    {{-10, -10, 1}, {0, 0, 100}, {100, 10, 0}},                    {{2, 6, 6}, {6, 6, 12}, {8, 12, 12}}/70]``

We can find the various prospects statistics by creating separate (full) prospects:

An investor with a budget will allocate that budget over the various prospects, and will be interested in the optimal mix. Allocate share S to one prospect, and 1 - S to the other.

``prs = JointToProspects[jp, {S, 1 - S}] // Simplify``

The following shows only some of the statistics results. The key point is that these results are parametric functions of S, and that we can plot in the various spaces for S in the [0, 1] range (no borrowing).

Figure 4 gives these plots. The finance community is familiar with the upper right plot, the other plots are novel.

##### Figure 4: Plots of the Joint Prospect for 0 ≤ S ≤ 1
``ProspectPlot[prs, S];``

### Continuous example of relative risk

#### Take an arbitrary Beta density

Let us take an arbitrary continuous probability density (pdf) of the Beta distribution.

Check on pdf:

Let us look at the cumulative distribution just for completeness.

From the cumulative plot we can already note that the median is close to 5.

With reference to the graph of the pdf, it is useful to point out that the expectational integrand x f(x) on the left is negative, so that, in computing the expected value, this area has to be compensated on the right before positive net outcomes start to show up.

#### Risk and Relative Risk

The following statements give us the Expected Value, Risk and Relative Risk (with respect to the expected value).

If one would take the target higher, e.g. at 8, then we get a relative risk of 4.6. This means that one can expect a return of 3.4 if the worry is that the market stays below 8.

#### Reference to the normal distribution

It may be useful to compare these plots to the normal distribution. Let us first define the standard normal distribution and plot it, for ease in the domain [e, e].

Again, the expectational integrand x f(x) is negative on the left hand side, so that, in computing the expected value, this area has to be compensated on the right before the final expectation of 0 is reached.

Given this 'imbalance' in the expectational integrand x f(x) we may now better understand why people have developed an alertness for 'risk'.

We can calculate the risk in the standard normal distribution as about .4 while the risk ratio of course is 1.

### Application to insurance

We may use these concepts to determine an acceptable level of insurance within a very simple model.

#### Restating some of the insurance basics

In the following an Insurance is just an object, containing variables and parameters. These settings will be input for the client and the insurance company, and they will react to them differently.

The basic insurance object is:

An insurance situation can be looked at from the viewpoint of the client or the firm.
The client's prospect is this:

``ic = ToClient[ins]``

Note that the client's prospect can be simplified as follows.

The clients expected value and expected utility value are:

``ProspectEV[ic] // Simplify``

The firm that sells the insurance has the following prospect and expected revenue.

``ToFirm[ins]``
``ProspectEV[%] // Simplify``

#### Looking for an acceptable level of insurance

The following is an example when the loss can take different levels with different probabilities. The insurance policy is select people by risk class, then to set a maximum level of coverage, and to pay out only the damage (that can be less than coverage).

The firms expected value is:

``Simplify[ProspectEV[ToFirm[ins2]]]``

Some numerical values are useful. Let us take arbitrary values, and now a set of five.

Let us assume that the insurance company has a low risk class with a 5% premium and a high risk class with a 10% premium. (So that we do not have to make a 3D plot.) Once the class of the client has been established, the company controls its costs by determining the allowable coverage.

The plot below shows that the client clearly is in the high risk class: the 5% premium does not give a positive expected value for any level of coverage.

Also, the expected value starts to become positive at higher risk values. From a certain level the risk remains constant. At that point any more coverage is plain profit for the firm.

Note that the indifference curves are concave (with a similar shape as the above RHS curve). The firm will have a preference for high expected value and low risk. In that respect we can conclude that there are no dominated sections. The firm would set a range for minimal and maximal coverage.

Some numerical data are:

#### With relative risk

Let us now consider relative risk.

The firms revenue has a maximum value of 10 (upper coverage of 100 and premium 10%). The minimum value is -90.

A fair target value might be 7. The question is: What might one expect to lose with respect to 7, when revenues would remain below 7 ?

The relative risk plot looks like this:

We extend the data print with 7 - rr[coverage, 10%], i.e. the expected revenue when revenue would not rise above 7.

This result can be clarified by looking at some additional plots. Note that discrete probabilities and Min conditions create discontinuities. The upshot however is that the selection of a target level would be important for one's idea about the optimal coverage and the related expected revenue. If there indeed is a serious worry that revenue will remain below 7, then the expected value of 4.7 of the coverage of 100 will be less believable, and one should fear for an expected result of -4.3. Optimal then would be a coverage of 60, with 1.8 for both expected values (absolute and relative). Of course, this is conditional on choosing the target of 7. If there is no information about what target to worry about, then the choice in the {EV, Risk} space like above would seem to be the best.

#### Subconclusion

Looking back at this application to insurance, it would seem valid to conclude that these definitions of risk and relative risk are useful for decision making, and provide more information than the use of a spread.

### Note on Bernstein's "Against the gods"

I came across Bernstein (1996) "Against the gods", and found it equally entertaining as his "Capital Ideas" that I already referred to op. cit..

One comment is that Bernstein indeed emphasises Knight's and Keynes's statements on "uncertainty". My answer to that is, again, that unknown probabilities or even unknown categories indeed are serious cases of uncertainty, so that earlier writes on the subject were right in emphasising that seriousness. However, we should not be tempted to reserve the word "uncertainty" to only those cases. So with all due respect to Knight and Keynes, the definitions provided in this paper are the proper ones.

### Conclusion

1) We identified logical problems and sources of confusion for the currently commonly adopted definitions for risk and uncertainty.

2) We designed useful categories for uncertainty that do not involve the term risk. And independently from these, we designed a proper definition of risk.

3) It appeared possible to extend the definition of risk with a definition of relative risk, that captures situations of people dealing with relative positions.

The definitions provided here are directly in line with the Oxford English dictionary. It turns out that economics textbooks can keep their mathematics but will best adapt their texts to these definitions. Not only the students and the general public will benefit from this sudden clarity, but eventually also economic theory itself.

### Selected literature

Bernstein (1996), "Against the gods", Wiley
Cool (1999), "The Economics Pack, User Guide", published by the author with ISBN 90-804774-1-9
Eatwell c.s. (1998, "The New Palgrave", Macmillan (entries risk, risk aversion, CAPM)
Luenberger (1998), "Investment science", Oxford
Ferguson (1967), "Mathematical statistics", AP
Hornby (1985), "Oxford advanced learner's dictionary of current English", Oxford
Tajuddin (1996), "A simple measure of skewness",  Statistica Neerlandica pp 362-366

Converted by Mathematica      & adapted on mu, rho and sigma by TC by hand, July 8, 1999