Uncertainty, Risk and Probability Analysis

In this article we will discuss about uncertainty, risk and probability analysis.

Uncertainty:

Uncertainty is a situation regarding a variable in which neither its probability distribution nor its mode of occurrence is known. For instance, an oligopolist may be uncertain with respect to the market­ing strategies of his competitors. Uncertainty as defined in this way is extremely common in economic activity.

The function of the entrepreneur is to meet those risks which are non-insurable and which are called uncertainties. Uncertainty arises when actual conditions differ from anticipated conditions.

Apart from our efforts some uncertainty will always be present.

The following reasons are important:

(i) The first is about natural laws according to which the sun rises, tide comes and seasons change.

(ii) The second is about forces working around us.

Sources of Uncertainty:

There are a few sources of uncertainty:

(1) Uncertain Pattern:

We are definite about certain events but uncertain about their pattern, for instance, there is sufficient quantum of rainfall in a particular year but its distribution over different months or days is uncertain. So there is the chance for crop failure by change in pattern of distribution of rains.

(2) Existing Facts and Future Plan:

Our belief of certainty and uncertainty about events is influenced by facts already available and future plan. As for example in constructing a dam, we face uncertainty about incoming water. But we may plan our present need with provision for future increase. The facts about past flow in volume and size reduce uncertainty to a great extent.

(3) Bias of Self-interest:

Our experience of past events are modified by our personal feeling and prejudice. It is known as bias of self-interest.

(4) Belief about an Event Either Help or Harm:

There is the maximum feeling of uncertainty when we believe that an event may either harm or help us, i.e., each one being equally likely.

Factors Determining Uncertainty:

Uncertainty bearing has been considered as a factor of production.

It has a supply price depend­ing upon:

(i) The character of the entrepreneur,

(ii) On the amount of resources possessed by him, and

(iii) On the proportion of these resources exposed to uncertainty.

State Preference Theory:

A method of examining the making of decision when there is uncertainty in the outcome. It is used primarily to analyse decisions regarding the choice of investments. The model assumes that there are several distinct possibilities as to the future economic situation.

Particular types of investment will yield various known returns, given that one of these economic states results. It is assumed that some absolutely certain form of investment exists, such as holding money in the bank at a fixed rate of interest.

This situation can be plotted given a two state world, putting the return given in state I on one axis and that given in state II on the other for any possible decision.

The results of all possible forms of investment can then be plotted with money being represented by a point on the 45° line. Joining all these points together the enclosed area represents all the possible outcomes that can be attained given the appropriate diversification of portfolio.

Next a set of indifference curves can be drawn on the graph representing those possible returns in state I or II between which the person is indifferent. Curves farther from the origin will represent a higher level of utility but shape of the curves and, in fact, whether or not they are convex will depend upon the individual’s attitudes towards risk and his assessment of the likelihood of one or another of the states resulting.

Mean Variance Analysis:

The making of decisions when there is uncertainty in the outcome. It is particularly used in examining how an investor will organise his portfolio. In this model, it is assumed that the determinants of an individual’s choice are the expected return and the variability of the return.

The individual’s choice as to how he will arrange his investments can be plotted on a graph with the expected return on the vertical axis and the variance on the horizontal.

There is usually once certain alternative for instance, holding money at a fixed interest rate. This is represented by a point on the vertical axis, that is, zero variance. The other investment possibilities are also placed on the graph.

If there is only one other possibility then the line between the certainty point and investment point will give the possibilities between which a person can choose by diversifying his portfolio. A set of indifference curves can be drawn on the diagram, their shape depending on the individual’s attitude towards risk. For a normal risk averter they will be convex towards the lower right hand side of the diagram.

Risk:

The concept ‘risk’ is a situation in which the probability distribution of a variable is known but its actual value is not. Risk is an actuarial concept. Risk may be defined as an uncertainty of financial loss on the occurrence of an unfortunate event.

A risk is an uncertainty of loss. Risk is an objectified uncertainty or a measurable misfortune. Every business involves some risk and most people do not like being involved in any risky enterprise. The greater the risk, the higher must be the expected gain in order to induce them to start the business.

Types of Risk:

Risk may be connected with either persons or properties and it can be classified as follows:

1. Pure Risk or Static Risk:

Pure risk prevails where there is a probability of loss but no chance of gain. For example, if the firm is gutted out by fire, the owner sustains financial loss. If there is no such fire accident, the owner does not gain either. Pure risks are insurable.

2. Speculative Risk or Dynamic Risk:

A speculative risk exists where there is even chance for both gain and loss. This type of risk arises from fluctuations of prices. Owners of shares and bonds will gain if the price goes up and losses if the price falls.

3. Insurable Risks:

Transferable risks are also known as insurable risks. Such risks can be predicted, estimated and measured in terms of money and so are insurable.

Non-Insurable Risk:

Those risks which cannot be calculated and insured are called non-insurable risks.

The non- insurable risks are further classified into:

(a) Competitive Risk:

The existing firms may be faced with new competitions from the newly entered firms. The new firms can enter into the industry any time. As a result of this competition, the profit of the existing firms will fall.

(b) Technical Risk:

New techniques of production may be introduced. The existing firms may not be able to follow these new techniques. As a result, they may incur loss.

(c) Risk of Government Intervention:

In the larger interest of the country, the government may nationalize a number of industries. The firms in every industry may be affected. The government may control the price of the products.

(d) Business Cycle Risk:

Depression may affect the industry as a whole. A depression in one industry may affect the other industries also.

Measurement of Risk:

The method of measuring a risk is to collect a large number of similar cases subject to risk and then divide the number of time the risk has happened by the number of such cases. For example, if there are 100 match units in a particular area and 10 units have been gutted in that year then the risk rate is 10/ 100 or 10 per cent. Such a measurement is called mathematical value of risk.

Probability Analysis:

In ordinary language the term probability refers to the chance of happening or not happening of an event. The use of the word ‘chance’ in any statement indicates that there is an element of uncertainty. Most of the managerial decisions are decisions related to uncertainty. Tomorrow is not well defined. Managers are required to make some appropriate assumptions for the ‘would be tomorrow’ and base their decisions on such assumptions.

The notion of uncertainty or chance is so common in everybody’s life that it becomes difficult to define it. We talk about or we may say, for instance, that it may rain today, or the local team will win the match or the group may fare well in statistics paper. In each of these statements there is as much uncertainty as there is certainty.

So from the above, it follows that probabil­ity is subjective and changes from person to person. We have not assigned any numerical value to these statements. If we could provide some numerical value, the statements would become more precise.

The theory of probability provides a numerical measure of the element of uncertainty. It enables the business managers to take decisions under conditions of uncertainty with a calculated risk.

Definition of Probability:

Probability may be defined as the ratio of the frequency with which a certain event occurs to the total frequency of a sufficiently long sequence of observations taken.

Chrystal gives the definition of probability as follows, “If on taking a very large number N out of a series of cases in which an event A is in question, A happens on pN occasions, the probability of the event A is said to be p”. Laplace, the French mathematician, has defined it simply as “Probability is the ratio of number of favourable cases to the total number of equally likely cases”.

If probability is denoted by P, then by this definition we have:

P = Number of favourable cases/Total number of equally likely cases

Relevance of Probability Theory:

Probability analysis is used to reduce the level of uncertainty in decision making. Let us discuss about some of the business situations characterized by uncertainty.

(i) The Individual Investor:

An investor who is engaged in buying and selling of equities is trying his maximum to optimize his output. The price behaviour of securities is subject to uncertainties. The uncertainties in the security price are due to several other factors.

Under these circumstances, the managers take business decisions on the basis of their forecast of the probable future. The ability to take better decisions need not be optimal. It is sometimes referred to as ‘business acumen’ i.e. sharpness and accuracy of judgment.

(ii) Inventory Problem:

The inventory is a complete list of the stocks of raw materials, components, work-in-progress and finished goods held by a business. The quantity of inventory depends upon various factors like demand, lead time, storage cost, ordering cost and shortage costs and the like. Some of these factors are known with certainty. Among other factors, the demand and the lead time fluctuate and are considered to be uncertain factors in inventory problems.

(iii) Investment Problem:

This relates to the spending of money for purposes other than consumption in order to earn income from it or to realise a capital gain at a later date. Large firms employ investment analysts with a view to forecasting its future profits.

This forecast will be related to the company’s present share price and the resultant ratio compared to the same ratio for other companies in the sector and for the market as a whole. The decision has to be taken on the basis of choice, the outcome of which is contingent upon the level of demand.

(iv) Introducing a New Product:

When a new product is developed by a firm the immediate problem is to decide whether or not to introduce the product in addition to the existing product mix. The decision maker may not be sure about the acceptability of the product.

The introduction of the new product is generally finalised on the basis of test marketing. If he gets contradictory results, he should drop the idea of introducing a new product is purely based on uncertainty.

(v) Stocking Decisions:

These refer to the accumulation of strategic raw materials or other commodities that are essential to run the business without any obstruction. The firm has to face the problem of stock policies.

In this context special insurance policies covering risk stock, where substantial fluctuations in the value of the risk can occur throughout in the period of policy. Therefore, insurance policies are unsuitable. To cover such risks, various policies are used. Here the businessman is not sure about the demand pattern, yet he must decide in advance how much units to stock.

Basic Concepts of Probability:

The following terms are important for the proper understanding of probability:

1. An Event:

It is said to be a possible outcome when an experiment is conducted. For instance, the head is an event and the tail is another event in the tossing of a coin.

2. Equi-Probable Event:

When two or more events are equally probable, i.e., when one event has as much chance to occur as the other, they are equal probable events. They may be also called as equally likely events. For example, when we toss a coin, we may get either the head or the tail. Both events are equally likely or have 50 per cent chance each.

3. Independent Events:

Two events are said to be independent if the occurrence of one is not or is affected by the occurrence of the other. When two coins are tossed, the result of the first toss does not affect or get affected by the second toss. Such events are called independent events.

Dependent Events:

Two events A and B are said to be dependent if the occurrence of A affects or is affected by the occurrence of the other. For example, in a pack of each, there are 52 cards. Suppose one card is withdrawn, the probability that it is a king is 4/52 or 1/13. Suppose one card is not replaced, the probability of another king is 3/51 or 1/17.

5. Mutually Exclusive Events:

By mutually exclusive events we mean that the happening of one of them prevents or precludes the happening of the other. Thus, if we toss a dice and it shows 4, then the event of getting 4 precludes the event of throwing 1,2,3,4,5,6. Therefore, the event of throwing 1,2,3,4,5,6 on tossing a dice are mutually exclusive. In other words, all simple events are mutually exclusive.

6. Collectively Exhaustive Events:

Events are also collectively exhaustive as they together constitute the set of possible events (called a sample space). Thus a set of events A₁, A₂……………. A_n is mutually exclusive of A₁OA₁ = Ø (for any i ≠ j) and collecting exhaustive E (the entire set) = A₁ OA₂ OA₃O………………. OA_n.

7. Simple Event:

In case of simple event we consider the probability of occurrence or non-occurrence of simple event. For example, in tossing a dice the chance of getting 3 is a simple event.

8. Compound Event:

When two or more events occur in conjunction with each other their simultaneous occurrence is called a compound event. In simple language, the chance of getting an odd number is a compound event.

9. Random Experiment:

It is an experiment which if conducted repeatedly under homogeneous conditions does not give the same result. The result may be any one of the various possible outcomes. Here the result is not unique. The performance of a random experiment is called a trial and outcome of an event.

Permutations and Combinations:

Permutation and combination are statistical devices employed in counting of things. Counting becomes more difficult if the number of ways of arranging a set of items is to be determined. In short, the word permutation refers to arrangements and the word combination refers to groups.

For instance, a factory owner who has received three new machines A, B and C can arrange these in 6 ways as follows:

ABC, ACB, B AC, BCA, CAB, CBA.

It may be noted that each arrangement is of three elements and no element appears twice. All the three elements are distinguishable.

Combination is a selection of objects considered without regard to in their arrangement. The number of combinations of objects all different is entirely different from the number of their permuta­tions. Thus a selection without regard to the order is called the combination. The number of combina­tions of r objects from n objects is denoted by nCr and is given by;

nCr = n!/r(n-r)!

It may be observed that nC_n =1 and nC₀ =1. One also uses the symbol (n/r)and Cⁿ_r to denote combination of n elements taken r at a time.

Kinds of Probabilities:

There are two distinct kinds of probability.

They are:

1. Aprion Probability:

We may consider the tossing of a coin. It may fall head upwards or tail upwards. Therefore, there are only 2 possible ways (head or tail) one of which is sure to happen. We can conclude that the probability of a head is 1/2 and that of tail is also 1/2.

We have arrived at this conclusion purely by reasoning or theoretical consideration. The reasoning employed here is purely deductive and we call the probability as ‘aprion’, meaning that it is determined before the event has occurred. It is otherwise known as mathematical probability.

2. Aposterion Probability:

Under the aposterion probability, the probability is determined after the result of the experiment is known. For example, out of 500 children admitted with symptoms of viral fever in a government hospital, how many survive and how many die?

The answer for this question or the probability of success can be determined only after treating the 500 cases and estimating the success of the trial. The reasoning employed here is inductive and the probability is known as ‘aposterion’, i.e., determined only after the event has occurred or after the outcome of the trial is known.