Risk and Uncertainty Revisited: A Clarification of Theory and Application

In addition to the obvious public confusion and lack of distinction between the terms, risk and uncertainty and other related concepts, the interchangeable use seems to seep into the academic and professional research. According to a 2018 study by De Groot and Thurik, it was reported that 88.3% of articles in this topic, across the related fields, did not adhere to the distinction between risk and uncertainty, rendering all the undesirable theoretical and empirical consequences. This paper is intended to revisit the concepts of risk and uncertainty, not only clarifying the meaning and use of the terms, but also shedding a light on differentiating all the related concepts. The focus is on risk, being the core element directly related to the success and failure of all financial and managerial decision making. The approach is not only conceptual, but also supported by mathematical and numerical applications.


Introduction
How certain can we be of the nature and direction of the consequences to any decision we make? The rational answer would most likely be that we cannot be one hundred percent sure! but some degree of certainty can be discerned, analyzed, and estimated, along with some uncertainty, ambiguity, and risk. Some of our decisions are made under the right circumstances that allow for an excellent degree of certainty, while other decisions are made under less fortunate circumstances, allowing different degrees of uncertainty and risky conditions. However, life experience shows that there has always been an atmosphere of uncertainty and risk surrounding all decisions, no matter how well-suited the circumstances, how well-deliberated the process, and how much checks and calculations were made. Experience has also shown that success and failure can be determined by how all potential factors of risk and uncertainty are recognized and accounted for. There has been a considerable amount of research done on this very subject, namely the process of decision making under the condition of risk and uncertainty, but surprisingly most of the published studies, not only did not distinguish between risk and certainty, but also neglected to recognize the other related concepts and constructs, and therefore underestimate their role in determining the outcome. Groot and Thurik (2018) reported that, "88.3% of articles in this topic does not adhere to the distinction between risk and uncertainty" (p. 4)! Not to mention the distinction among other related terms that are assumed to be interchangeable. The authors continue to declare that not distinguishing between these closely related terms would "contribute to the contamination of the concepts that currently dominate the literature and make research prone to confusion and may lead researchers to erroneous conclusions" (p. 5), and undesirable theoretical and empirical consequences.
The objective of this study is to revisit and clarify major concepts and important theoretical constructs that have been sometimes used in an equivocal way and applied in a misleading manner. It shed a light on differentiating risk and uncertainty and all other sub-terms and related constructs. The focus is specifically on risk, being the core construct directly related to the financial and managerial decisions. The approach is conceptual and supported by mathematical and numerical applications.

Sources of Risk
Many possible sources can introduce certain conditions of risk into the decision-making process. Most of these sources are external to the firm. We can group the most common sources into three categories: Economic Sources, which are related to the economic environment of a country. The fluctuations in the financial market pose a credible risk to the value of assets in the current and future periods. Such a risk is known as "market risk". Major economic factors such as inflation and interest rate pose, yet other significant impact on prices and value of lending and borrowing and their effects on earnings. Changes in the credit obligations, and in the state of liquidity can also introduce what are called credit risk and liquidity risk in addition to the currency risk which can stem from changes in the exchange rate between the domestic and foreign currencies. Also, the state of competition in the same industry or region poses another type of economic risk. Political Sources are related to the government policies, domestically and internationally, that may introduce certain risk on a certain industry, or on the entire economy. Changes in tax policies is a typical example, and expropriation risk is another example. This risk arises where a government abroad seizes a property, restricts the rights, or remove the privileges of the hosted firm. Terrorism and cybercrimes nowadays constitute a significant political risk on business activities of all firms, domestically and globally. Social Sources are related to cultural or religious reasons or to certain social norm or trend that affect consumer preferences and demand. Certain food or clothing items or weather-related products may not have any chance to be marketed in certain countries, which is a risk to be accounted for. Even domestically, consumer taste and preferences are subject to change and any business that cannot respond and keep up with those changes would face the risk of being outdated or off-trend and may lose its market share. International Sources are related to commercially or politically competitive reasons among countries (Alhabeeb, 2013;Baltussen et al., 2016).

Measurement of Risk
As it involves calculable multiple outcomes, risk can be defined in terms of the variability of those outcomes and to what extent they are dispersed. The relationship between risk and variability and dispersion of outcomes is direct. Large variability and wide dispersion would mean higher risk and small variability and tight distribution of outcomes indicates lower risk. For example, if an investment opportunity earns 5% fixed and guaranteed rate of return, and another opportunity may earn anywhere between -10% and 30%, we can easily discern that it would be considered risk free in the first opportunity and risky in the second opportunity. Such a realization of that risk is based on the wide range of possibilities of the earned return in the second opportunity. Ironically, in considering this example, we can also vividly see that the only possibility of earning the high return of 30% would be available only with the risky package, hence the direct relationship between risk and return! higher return is associated with higher risk. This is to say that seeking higher return means the willingness to deal with higher risk and seeking security means accepting a modest return. Risk, therefore, can be measured by the classic statistical measurement of dispersion. That is variance or standard deviation. We can classify risk measures into an absolute and relative measure of risk. The objective of the absolute measure is to see how the actual outcome is deviating from the expected value. Can we guess how risky some assets by only looking at their returns? Let's contemplate the range of returns for X and Y assets (Table 1) and take it as a hint to the dispersion of returns, and let's assume that there are three returns for each. We can see, in the following table, that the difference between the highest and lowest return for each would refer to more dispersion for asset Y (range: 15 -5 = 10), than for asset X (range: 11 -9 = 2). This may indicate that asset Y is riskier than asset X for having higher variability of returns. This simple variability notion can better be represented by the probability distribution of returns. The tighter the probability distribution, the more likely to have the actual return be close to the expected value, and therefore the lower the risk for that asset, and vice versa. The statistical variance (σ 2 ) would provide a measure of variability or dispersion for it is the weighted average of the squared deviations from the mean: where x is the mean or the expected value of outcomes? Risk as expressed by variability or dispersion of outcomes can also be measured by the standard deviation (σ) as it is the squared root of variance (σ 2 ): If we assume the probabilities of the returns to X and Y assets are 25%, 50%, and 25% respectively (Table 2), we can calculate the standard deviations for the three returns.
The standard deviation of .71 means that the returns on asset X are much closer to their own expected value than the returns on asset Y which has a standard deviation of 3.5, indicating how wide the dispersion of returns.
In the long run, asset risk would be an increasing function of time.
ijef.ccsenet.org International Journal of Economics and Finance Vol. 14, No.1; 2022 The variability of returns gets wider, and the risk gets greater as time goes by. Practically, this would be translated such that the longer the life of an investment asset, the higher the risk involved. Suppose that two investment project proposals were submitted to a firm for funding, with their own estimations of the profits (in hundreds of thousands of dollars) in the next five years (Table 3 and Table 4). We can expect that the financial advisors/managers would make their assessment and choice based on some measures such as the one described above.
Since the two projects will yield the same expected value, the next crucial criterion would be which of them is safer or riskier than the other. The answer would be clear at the calculation of variance (σ 2 ) and standard deviation (σ). The calculated results show that Project I had a smaller standard deviation (658) than Project II (1,710). Project I would win for being less risky than Project II. It is clear on Figure 1. how data of Project II are dispersed over horizontally, forming a widely spread curve while they are much tighter in Project I, which shows how the outcomes are generally close to the expected value or mean.  Assuming the distribution is normal, it would mean that: There is a 68.26% chance that the actual outcome is within one standard deviation from the expected value. Based on the symmetry of the normal distribution, this chance is divided equally between a negative 34.13% and a positive 34.13% (Alhabeeb, 2013;Kilka & Weber, 2001). So, if the standard deviation is 1,710, for example, there would be a 34.13% chance that the actual value is 5,000 + 1,710, and a 34.13% chance that it is 5,000 -1,710. So, the general range would be from 3,290 to 6,710. The chance would increase to 95.44% within two standard deviations (2 x 1,710), which is also split equally on both sides of the mean. In this case, the chance would be 47.7% that the range of the actual outcome would be between 1,580: [5,000 -(2 x 1,710)] and 8,420: [5,000 + (2 x 1,710)].
If we deal with a smaller standard deviation such as the 658, the ranges of the actual outcome would be closer to the expected value, rendering more security and less risk. This interpretation is, of course, not limited to the discrete one, two, or three standard deviations from the mean. it would apply to any range in between. Therefore, we can find the probability of a specific outcome (x i ) such as 5,500, for example. We can calculate how much of a standard deviation from the mean (x) this value would reveal by calculating the value of Z and looking up the statistical . In this sense, the measure of risk would be translated into a measure of standard deviation per unit of the expected value. The relationship between the coefficient of variation and risk is still as positive.
So, the criteria would be "the lower the value of the coefficient the lower the risk, and vice versa. In our absolute measure in the last example, Project II (σ = 1,710) was riskier than Project I (σ = 658) while both would yield the same expected value (x 5, 000).
 Suppose now that Project II has an expected value of $6,000. It would still be riskier than Project I if we compare their coefficient of variation (V): .13 x 5, 000 .28 x 6, 000 where Project II revealed a higher coefficient of variation reflecting a higher risk. ijef.ccsenet.org International Journal of Economics and Finance Vol. 14, No.1; 2022

Risk Aversion
Risk aversion can be translated into people's general tendency to avoid, or at least minimize, all sorts of risk and uncertainty when they make decisions. Decision makers are described based on three major attitudes they take towards risk. A Risk Averter who would choose no risk, or at best the lowest possible level of risk. A Risk Taker who prefers to venture and gets involved in risky situations and in conditions that require a higher level of speculation in pursuit of the highest possible payoff. A Risk Neutral who is indifferent to risk, and only focuses on expected returns much more than to pay any attention to the way those returns are dispersed. (Jarrow & Li, 2021;Rabin, 2001). Although it has been very well established in the business world that the highest return is usually associated with the highest risk, most people, and specifically managers, are naturally risk averters, especially when larger potential losses are involved. It has been observed that even risk neutrals would turn into risk averters when large amounts of money are at stake (Barber & Odean, 2001;Battalio et al., 1990).
Suppose that a group of people in a club decided to play a coin gamble, and since many wanted to play, the following rules were put forward: If a head turns up, the player wins $200. If a tail turns up, the player loses $100, and because of the competition to play, $10 is offered to the player who gives up his turn, or basically pledges not to play. According to the attitudes towards risk, a risk averter would have no problem leaving the game and take the $10 for doing nothing. For him, it would be an easy gain, although it is at the expense of foregoing a possibility of gaining $200. For a risk neutral, the focus would be on the weighted average that would come out of this game. He would make his decision since in reality there would be an average gain since the amount for gain is higher than the amount for loss while both stand the same probability (50%). In this case, the risk neutral would calculate the expected value: x 1 (p 1 ) + x 2 (p 2 ) = x ; (200)(.5) + (-100)(.5) = 50 A risk taker would be the most enthusiastic to play, focusing on the highest win and he may not hesitate to play again in pursuit of that $200.

Risk Attitudes and Utility of Money
Risk attitudes can be explained by the utility of the earned or lost money. Each attitude can be represented more accurately by the change in total utility or what we call the marginal utility. Marginal utility is generally decreasing for the risk averter, increasing for the risk taker, and constant for the risk neutral. Table 5 contains data on five possible payoffs and both total utility TU and marginal utility MU derived from them as subjectively determined by the three types of decision makers. In Figure  . We can observe further differences among these three managers based on their attitudes towards winning and losing money. If, for example, we consider an event that would increase the payoff from $50,000 to $75,000 (Table 6)

Expected Utility of Money vs. Expected Monetary Return
Suppose a manager wants to invest in oil drilling, and he would face the following possibilities: 1) If no oil turns up, he will lose all the investment of $25,000. The probability of this outcome is 80%. 2) If oil is found, the payoff would be $100,000, but this outcome is probable at only 20%.  Therefore, it is expected that the risk averter manager would decide not to drill because of the negative expected utility (-3.15) as compared to the expected utility of zero in no drilling. On the contrary, the risk taker would decide to drill based on his positive expected utility of (1.25). The risk neutral manager ended up with no expected utility and therefore it is as good as no drilling.
If we know the function of the utility of money U, such as U = 400 m .25 And we know the initial amount of money (m), then we can test whether the function is increasing or decreasing. This one is an increasing function of money since the first derivative is positive: which confirms that the marginal utility function is decreasing, and the decision maker would be described as a risk averter. Now, let's consider the impact of winning $500 as well as losing $500 on an initial amount of money of $1,000 (m = 1,000). U 1 = 400 m .25 = 400(1,000) .25 = 2,249.36 If the person wins $500, m would be 1,500: U 2 = 400(1,500) .25 = 2,489.33, and if the person loses $500, m would be 500: U 3 = 400(500) .25 = 1,891.48, ∆U 1-2 = U 2 -U 1 = 2, .33 -2,249.36 = 239.97, ∆U 1-3 = U 3 -U 1 = 1,891. 48 -2,249.36 = -357.88 If this game is a coin flipping game, the expected value of utility would be obtained by: E(U) = ∑∆UP i = ∆U 1 (P 1 ) + U 2 (P 2 ) = (239.97) (.5) + (-357.88)(.5) = -58.95 Since the expected value of utility turns out to be negative, the decision would be not to get into this gamble.

Risk Discount and Certainty Equivalent
The person who received $10 as a reward for giving up the gamble is certainly a risk averter. This amount of $10 is called certainty equivalent (CE). It is defined as the compensation which renders the player indifferent to a risky gamble. In that scenario, the expected value of the game was $50, as the risk neutral player has considered it. The person who accepted a significantly less outcome ($10) is for sure a player with a definite risk aversion attitude. This would further define the risk averter as one whose certainty equivalent limit is less than the expected value of a certain risk. The difference between the expected value E(v) and the certainty equivalent (CE) is called the Risk Discount (RD): RD E(v) CE

 
Risk discount shows the extent to which the expected value for a given risk is reduced to avoid such a risky prospect. In our previous example, risk discount was $40: RD = E(v) -CE = 50 -10 = 40 Figure 3 shows the certainty equivalent and risk discount as we recall the shapes of the curves for the risk averter (the diminishing marginal utility curve DMU), and for the risk neutral (the constant marginal utility curve CMU).

Figure 3. Certainty equivalent and risk discount
Point A represents the expected utility of the game, the level that would generate two points, B and C, on the risk averter and risk neutral curves, DMU and CMU respectively. From those points, we can drop verticals to see the amounts of payoff for both players. For the risk averter, point P 1 would represent the certainty equivalent, and for the risk neutral, point P 2 would represent the expected value. The difference (P 2 -P 1 ) would be the risk discount RD.

Risk Impact on the Valuation Model
When a firm wants to evaluate the worthiness of an investment project, risk factor should be among the priorities to be considered, as it affects the actual net present value NPV of the project. There are two common ways to adjust the valuation model for risk:

Risk Premium Adjustment
Risk premium is defined as the difference between the expected rate of return on a risky investment and the risk-free rate (Adusei, 2019;Alhabeeb, 2010;Chalamandaris & Rompolis, 2020;Gagliardini, 2016). Let's consider three managers or financial advisors with three different attitudes about risk, as represented by the three curves on Figure 4 where risk is on the x-axis, as it is measured by the standard deviation (σ), and rate of return is on the vertical axis. Let's think of these curves in a way similar to the indifference curves. They depict the tradeoff between risk and return from three different attitudes towards risk. The first, RR1, represents the least risk-averse among the three. The top RR 3 , represents the most risk-averse, and RR 2 stands in between. ijef.ccsenet.org International Journal of Economics and Finance Vol. 14, No.1; 2022

Figure 4. Three financial advisors with three different attitudes towards risk
Point a is on all curves and it shows a 5% risk-free rate (risk = 0). RR 1 shows a manager who is indifferent between accepting a 5% rate with no risk or taking a 1.5 σ risk to get a 7.5% return. In other words, for the added risk (from 0 to 1.5σ), his risk premium becomes 2.5% (7.5% -5%). For the more risk-averse manager on RR 3 , the move to accept the additional risk of 1.5σ would not be satisfactory unless there is a higher risk premium of 11% so that the required rate of return becomes 16%. Not only that, but if the next opportunity happens to come with an additional risk of .5σ (such as moving from 1.5σ to 2σ), the risk-averse manager on RR 3 would want his return to be as high as 25% where his risk premium goes to 20% (25% -5%). The same level of risk (2σ) would make the manager of RR 1 happy to accept only 10% return making his risk premium 5% this time (10% -5%).
As for the moderate manager on RR 2 , he would accept moderate levels of risk for reasonable rates of return. He would be indifferent between risk-free rate of 5% and 10% rate with 1.5σ risk or 17% rate with 2σ risk. His risk premium would be 7% (12% -5%) at point d, and 12% (17% -5%) at point e. Table 7 shows a comparison between the three positions on risk and return, and risk premium.
The different attitudes by managers towards risk would produce various risk premiums and that would be reflected on the valuation model of a firm as the risk-adjusted rate (k) would replace the risk-free rate (r) that is normally used in the evaluation model: where V is the value of the asset, i  is the expected profit per year, r is the risk free rate of return so that the value is equal to the present value of the future returns or cash flow. When the firm faces the prospect of a risky project, the valuation would be adjusted to the expected risk by incorporating the firm's risk premium (R p ). In this case, the net present value NPV of the project would be: ijef.ccsenet.org International Journal of Economics and Finance Vol. 14, No.1; 2022 where k is the risk-adjusted rate of return, which is equal to the risk-free rate (r) used previously, plus the firm's risk premium R p : k = r + R p , and C 0 is the initial cost of the project. The criteria would remain such that an investment is worthwhile when the net present value (NPV) is either equal or larger than zero. NPV > 0 Suppose a managerial/financial team must decide on capital allocation for two proposed investment projects, each of which will yield profits for the next 5 years as shown in Table 8. They require initial investments of $420,000 and $500,000 respectively. Although the firm's cost of capital is 6%, further investigation revealed certain risk elements associated with both projects. The decision makers found it necessary to adjust for risk by assigning risk premiums (R p ) of 2½ % and 3½ % to both projects respectively. The classic criterion for granting an investment has to utilize calculating the net present value using the risk-adjusted rate of return: First we calculate the net present value of the cash inflows for both projects at the time of their yields using the firm's interest rate (r -6%). Then we calculate the same net present values, using the risk-adjusted rate (  At the normal interest rate of 6%, Project B would win the approval of the financial/managerial team since its net present value ($51,511), is larger than that of Project A ($45,440). However, after considering the expected risk involved in both projects, the decision makers would give its approval to Project A due to its larger adjusted net present value of ($17,423) as compared to that of Project B ($14,440).

Certainty-Equivalent Adjustment
As it was explained before, certainty equivalent (CE) is the sure sum that is equal to the expected value E(v) of the risky project. The equivalency is in the utilities of both to the manager or investor, and not necessarily in their monetary values. Let's assume there is a proposal that requires the company to invest $30,000 in a project, where the probabilities of its success and failure are 50/50 between earning $100,000 and earning nothing, respectively. The expected value for such a project would be: E(v) = 100,000(.5) + 0(.50) = 50,000 If the company approves the funding, it will mean that it is trading off the certainty of $30,000 for a risky expected return of $50,000. In fact, it means that a sure risk-free capital of $30,000 is yielding the same utility of a risky $50,000, hence, the term certainty equivalent to the amount of $30,000 that would make the decision maker indifferent between the two prospects. The certainty equivalent coefficient (α) is the ratio between the certainty equivalent (CE) and its expected risky return E(v).

CE E(v) 
The certainty equivalent is subjectively determined by the decision maker and, therefore, it would be a product of how risk averse or risk taker is that decision maker. Figure 5 shows that three different attitudes towards risk would produce three certainty equivalent amounts for the same expected value of $1,000 for a specific risky project: The most risk-averse manager on RR 3 would assign $870, the least risk-averse would assign $220, and the moderate manager among the three would assign $460. These cases would produce three different certainty equivalent coefficients: The risk-averse manager would value the sure risk-free money more. That is why his alpha is higher. An alpha of .87 means that each dollar of the certain money would be worth 87¢, as compared to the 22¢ for the risk taker. This is one reason to see why the risk taker dares to take a high risk. Alternatively, each dollar of the expected risky return is valued less ($1.15) for the risk-averse than for the risk taker who values his expected dollar at $4.55. Generally speaking, the criterion for alpha is as follows: -When α = 0: It is an indication that the probability of getting the expected return does not exist, and therefore the project is too risky to be pursued.
-When α = 1: It refers to the equality between the certainty equivalent (CE) and the expected value of return of the risky project. When the manager or investor gets his return equal to what he assigns as a certainty equivalent, the project is considered risk free.
-When 0< α <1: It is an indication that there is some level of risk. The project is riskier as α value is closer to zero, and less risky if it is closer to 1. It would depend on how smaller the certainty equivalent (CE) is, as compared to the expected value of the risky return E(v).
The valuation model would be adjusted for risk by introducing α to the numerator of the formula as a multiplier to the expected return or profit or cash flow, while the bottom of the formula would keep the risk-free rate (r): Let's assume that the manager assigned certainty equivalent sums to each annual return of the five years in both projects of the last example. Table 9 shows α values as it is calculated by dividing the assigned certainty equivalent by the corresponding expected return.  (227,170 + 89,391 + 71,787 + 60,199 + 50,739) -500,000 = 499,286 -500,000 = -714 Considering the expected risk for both projects in terms of estimating the certainty equivalent and calculating α for each return in every year revealed that Project A is more worthwhile for yielding a positive value of $32,532 while Project B went into a negative net value.

Conclusion
The study stated its objective as to revisit and clarify major concepts and important theoretical constructs that have been, sometimes, used in a confusing way and misleading applications with erroneous results. Therefore, it was necessary to differentiate between risk and uncertainty and shed a light on sub-terms such as total and partial uncertainty and related constructs such as risk aversion, uncertainty aversion, and ambiguity aversion. The focus was on risk for its practical value, being the core construct directly related to the financial and managerial decisions. For that, risk was detailed by its sources, measurement, people's attitudes towards risk, and the impact on expected return and economic utility. Discussed also was the risk discount and risk premium, certainty ijef.ccsenet.org International Journal of Economics and Finance Vol. 14, No.1;2022 equivalence, and the impact on valuation model. Clear definitions were given, and numerical examples were solved, with graphical illustration.