1. Incertidumbre y riesgo.ppt

8/12/2019 1. Incertidumbre y riesgo.ppt

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RISK AND UNCERTAINTY

Uncertainty: Sensibility studies.

The approach used for analyzing different scenariosbased on possible changes in income and expenditurefrom the initial expected scenario.

Risk: Probability studies.

Frequently not feasible due to a lack of time series dataconcerning public projects.


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SENSITIVITY ANALYSIS


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UNCERTAINTY


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UNCERTAINTY


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Risk analysis involves a fuller assessment of possible variation.Its purpose is to provide a probability estimate of how likely a projectdecision is to be wrong.

Risk analysis begins from the best estimates contained in the initialresource flow and from the effect of variation given by sensitivityanalysis; but now different variables are considered simultaneously.

The actual outcome for a project may vary from the original bestestimates.

A range of values above and below these best estimates can bedefined: for example, in the illustration below values 10 and 20 percent above and below the best estimates are used.

This is a relatively conservative range for variation.


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Some project items can be estimated with greater certainty than others.

Although it is convenient to use the same range of variation for eachvariable considered in risk analysis, the probability of the differentvalues in the range occurring will differ.

For example, given the optimism with which projects are often prepared,some items like investment costs are more likely to vary upwards ratherthan downwards from the best estimate, whilst others like revenue aremore likely to be below rather than above the best estimate.

A probability should be attached to the best estimate and each variation,to reflect the likelihood with which the different values in the range willoccur.

The sum of these probabilities must total 1.0 for each variable.


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The effect of varying values within a range can be calculated through

sensitivity analysis.

It is the additional probability estimates associated with each variationthat represent the essential feature of risk analysis.

Where do these 'probability estimates come from? For some variablesthey may come from past evidence, for example, of fluctuations in prices,outputs, or of material ratios in different production processes.

For other variables, intuitive guesses may have to be made on the basisof experience.

For this example, the following table summarizes the effect on the NPVof variation in each variable, presents the probabilities associated witheach variation, and also converts these to a set of two-digit randomnumbers.


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The random numbers are a means of making random

choices within the range for each variable.

The probabilities for a particular variable, which sumto 1.0, are associated with numbers between 00 and

99.

For investment cost in previous table, for example,the probability of 0.1 for investment cost being 20 percent less than the best estimate is associated withthe ten values 00-09; the probability of 0.1 forinvestment cost being 10 per cent less than the bestestimate is associated with the ten values 10-19, andso on.


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These 100 estimates in each case are distributed around the bestestimate of the NPV (and around their own mean value).

The results of the risk analysis are presented in the following table.

At market prices, the expected NPV by this process is larger than

the best estimate and positive; at shadow prices, it is still negativeand well below the best estimate.

These expected values confirm the earlier decision that the projectcould be accepted at market prices but rejected at shadow prices.

However, there is a very large range of values around the bestestimate and the expected NPV, implying considerable risk.

Owing to the selection process in risk analysis, both distributions ofthe NPV values are approximately normal, and because of the largerange both distributions contain many negative and positive values.


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The proportion of negative and positive values can be

calculated.

At market prices, 39 per cent of the NPV values are negative.

If the project is accepted because of its positive NPV at bestestimates, there is a 39 per cent chance that the decision willturn out wrong.

At shadow prices, 31 per cent of the NPV values are positive.

If the project is rejected because of its negative NPV at bestestimates, there is a 31 per cent chance that the decision willturn out to be wrong.


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What is the probability that VAN < 100?

= - 0.3453

Area asociated = 0.1368 + 0.5 = 0.64There is a 64% probability that VAN > 100

There is a 36% probability that VAN < 100

779

369100


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These are the essential results of the risk analysis.

They contain more information than simply the bestestimate, but also provide a dilemma for those whohave to decide.

For this project, whether market prices or shadowprices are chosen as the basis of decision, there is asubstantial change of being wrong.

It may be thought unacceptable to risk the 39 percent chance of a negative NPV at market prices; thiswould reinforce the shadow price calculation thatyields a negative NPV anyway, although there is

considerable uncertainly with this calculation also.


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Risk analysis is most important for marginal projects,with a rate of return just above the discount rate.

For projects with a much larger rate of return theprobability of a negative NPV with variation in the majorvariables is likely to be small.

The main effect of risk analysis is thus likely to be ondecisions among alternative marginal projects.

If all project analyses include a risk analysis conductedon similar lines to that in the illustration above, then theless risky among the marginal projects can be chosen.


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REDUCING RISK


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Identifying the effects of variation in major variables, and investi-gating the likelihood of their combined variation, provides con-siderable information on the risks associated with a project.

It indicates where the risk might be reduced.

Risk can be reduced at both the analysis and implementation stagesof a project.

An analysis carried out using the most pessimistic estimates foreach variable shows the amount that has to be available as a

contingency reserve in the worst case.

Reducing risk should aim at improving the project results at least tothis extent so the reserve is no longer necessary.


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The project results in many cases can be improved by aredesign of the project.

Alternative technologies, locations, output mixes and scalesshould be investigated.

Redesign can also include the phasing of investment so thatthe production lessons learnt in early phases can be appliedin later phases, improving the overall performance.

Each course of action will have its own risks; lower expectedNPV results will probably have to be traded off against lowerrisks.


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Risk reduction can also be achieved by choosing projects for whichthere are well-known precedents.

Replication of small-scale projects rather than commitment to a fewlarge-scale projects can reduce risk.

Investment decisions can be seen in terms of programs of investmentsrather than one-off projects.

Copying and adapting from imported technologies under supply andmanagement contracts can also reduce production and marketing risks.

These forms of dealing with risk, however, cannot easily beincorporated into project analysis calculations or decisions on individualprojects.

Moreover, such approaches lose the potential benefits of developingnew technologies or learning-by-doing.


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For projects where there are several sources of risk, a different approachmay have to be adopted.

The illustration above revealed a project highly sensitive both to changesin revenue and in the materials ratio.

Where considerable uncertainty attaches both to output sales andtechnology then the best approach may be to initiate pilot production ona small scale.

Pilot production will allow the technical ratios and the output to be testedand improved.

It will allow a re-estimation of the main project variables and a reductionin risk for a full investment.

However, this does not mean that the full investment will necessarily

become acceptable; a reduction in risk may be associated with a smallerestimate of the project net benefits.


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A better design for the project may result from these actions; but inpractice there will still be considerable uncertainty about project effects.

A major tool to use in project implementation to reduce risk iscontracting.

Generally, the longer for which a contract runs, the more certain theelements contained in the contract.

Long-term contracting can be applied both to the purchase of inputsand to the distribution of outputs.

It can be applied in management and technical and marketingagreements, to encompass pricing, profit sharing and exchange ratecalculations.

Again, however, achieving greater stability, or less uncertainty, inproject effects may be purchased at a lower level for the net benefits.


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RISK IN ECONOMIC

ANALYSIS


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Risk is defined as a hazard or peril; as anexposure to harm; and, in business, as achance of loss. Thus, risk refers to thepossibility that some unfavorable event willoccur. For example, if one buys a $1 millionshort-term government bond priced to yield 9,can be estimated precisely, and we say thatthe investment is risk free. If, however, the $1million is invested in the stock of a companybeing organized to prospect for natural gas in

the Gulf of Mexico, the return on theinvestment cannot be estimated precisely.The return could range from minus 100percent (a complete loss) to some extremelylarge figure.


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Because of its significant danger of loss, we saythat the project is risky. Similarly, sales forecastsfor different products may exhibit differentdegrees of risk. For example, The Dryden Pressmay be sure that sales of a fifth editionintroductory finance text will reach the projectedlevel of 30,000 copies, but the company may beuncertain about the number of copies that it willsell of a new first edition statistics text. Thegreater uncertainty associated with the sales

level of the statistics text increases the chancethat the firm will not profit from publishing thatbook. Thus, that project's risk is greater than therisk of revising the finance text.


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Probability Distributions

The probability of an event is defined as the chance, orodds, that the event will occur. For example, a salesmanager may state, "There is a 70-percent chance thatwe will get an order from Delmarva Corporation and a30-percent chance that we will not." If all possible eventsor outcomes are listed, and if a probability of occurrenceis assigned to each event, the listing is called aprobability distribution. For our sales example, wecould set up the following probability distribution:


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The possible outcomes are listed in Column 1,and the probabilities of each outcome,expressed both as decimals and percentagesappear in Column 2. Notice that the probabilitiessum to 1.0, or 100 percent, as they must if the

probability distribution is complete.Risk in this very simple example can be read fromthe probability distribution as a 30-percentchance that the undesirable event (the firm not

receiving the order from Delmarva Corporation)will occur. For most managerial decisions,however, the relative desirability of alternativeevents or outcomes is not so absolute.


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Here we see that both projects will provide a $5,000 profitin a normal economy, higher profits in a normal economy,higher profits in a boom economy, and lower profits if arecession occurs.

Notice also that the profits from Project B vary far more widelyunder the different states of the economy than do the profitsfrom Project A. In a normal economy, both projects return$5,000 in profit. Should the economy be in a recession nextyear, Project B will produce nothing whereas Project A willstill provide a $4,000 profit. On the other hand, if theeconomy is booming next year, Project B's profit willincrease to $12,000, but profit for Project A will increase onlymoderately to $6,000.


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How, then, is one to evaluate these alternatives?Project A is clearly more desirable if the economy isin a recession, whereas Project B is superior in aboom economy. (In a normal economy the projectsoffer the same profit potential, and we would notfavor one over the other.) To answer the question,we need to know how likely a boom, a recession, ornormal economic conditions are. If we haveprobabilities for the occurrence of these events, wecan develop probability distributions of profits for the

two projects and from these obtain measures of boththe expected profits and the variability of profits.These measures enable us to evaluate the projectsin terms of their expected profit and the risk that theprofit will deviate from the expected value.


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To continue the example, assume that economicforecasts of current trends in economic indicatorsindicate chances of two in ten that a recession willoccur, six in ten of a normal economy, and two in tenof a boom. Redefining chances as probability, wefind that the probability of a recession is 0.2, or 20percent; the probability of normal economic activity is0.6, or 60 percent; and the probability of a boom is0.2, or 20 percent. Notice that the probabilities addup to 1.0: 0.2 + 0.6 + 0.2 = 1.0, or 100 percent.These probabilities have been added to the Payoffmatrix to provide the probability distributions of profitfor Projects A and B shown in the following ExpectedValues table.

If we multiply each possible outcome by its probabilityof occurrence and then add these products, we have

a weighted average of the outcomes.


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The expected-profit calculation can also be expressedby the equation:

Here, is the profit level associated with the ith outcome, P i isthe probability that outcome i will occur, and N is the number

of possible outcomes or states of nature. Thus, E( ) is aweighted average of possible outcomes (the i values), witheach outcomes weight being equal to its probability of itsoccurrence.

Using the data for Project A, we can obtain its expected profitas follows:


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We can graph the results in the table ofExpected Values to obtain a picture of thevariability of actual outcomes; this is shownas a bar chart. The height of each barsignifies the probability that a given outcomewill occur. The range of probable outcomesfor Project A is from $4,000 to $6,000, with an

average, or expected value, of $5,000. Thevalue for Project B is $5,400, and the range ofpossible outcomes is from $0 to $12,000.


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Relation between State of theEconomy and Project Returns


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Probability Distributions Showing Relation betweenState of the Economy and Project Returns

The actual return from Project A is likely tobe close to the expected value. It is lesslikely that the actual return from Project Bwill be close to the expected value.


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Note: The assumptions about the probabilities ofvarious outcomes have changed from those inthe bar charts. We no longer assume that theprobability is zero that Project A will yield lessthan $4.000 or more than $6.000 and that

Project B will yield less than $0 or more than$12.000. Rather we have constructed normaldistributions centered at $5.000 and $5.400 withapproximately the same variability of outcome

as in the bar chart. Although the probability ofobtaining exactly $5.000 was 60 percent in thebar chart, in the probability distribution it is muchsmaller.


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The number of possible outcomes is infinite

instead of just three. With continuousdistributions, it is generally more appropriateto ask the cumulative probability of obtaining

at least some specified value than to ask theprobability of obtaining exactly that value.This cumulative probability equals the areaunder the probability distribution curve up tothe point of interest.


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Measuring Risk Risk is a complex concept, and a great deal of

controversy has Surrounded attempts to defineand measure it. However, a common definitionand one that is satisfactory for many purposes isstated in terms of probability distributions suchas those presented in the ProbabilityDistributions. This notion of risk is conveyed bythe observation that the tighter the probabilitydistribution of possible outcomes, the smallerthe risk of a given decision, because there is alower probability that the actual outcome willdeviate significantly from the expected value.

According to this definition, Project A is lessrisky than Project B.


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To be most useful, our measure of risk should havesome definite value - we need a measure of thetightness of the probability distribution. One suchmeasure is the standard deviation, the symbol forwhich is , read sigma. The smaller the standard

deviation, the tighter the probability distribution and,accordingly, the lower the riskiness of thealternative. To calculate the standard deviation, weproceed as follows:

1.Calculate the expected value or mean of thedistribution:


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Here is the profit or return associated with the ith

outcome; p i is the probability that the ith outcomewill occur; and E( ), the expected value, is aweighted average of the various possibleoutcomes, each weighted by the probability of itsoccurrence.

2. Subtract the expected value from each possibleoutcome to obtain a set of deviations about theexpected value:


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Calculation of the standard deviation of profitfor Project A illustrates this procedure. (Thecalculation of the expected profit was shownpreviously and is therefore not repeated.)

Deviation*i E()+

Deviation 2 *i E()+2

Deviation 2 x Probability*i E()+2 x pi

$4,000 - $5,000 = -$1,000 $1,000,000 $1,000,000(0.2) = $200,000

$5,000 - $5,000 = 0 0 0(0.6) = 0

$6,000 - $5,000 = $1,000 $1,000,000 $1,000,000(0.2) = $200,000

Variance = 2 = $400,000

Standard deviation = = = = $632.46


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Using the same procedure, we can calculate thestandard deviation of Project B's profit as$3,825.23. Since Project B's standard deviation islarger, it is the riskier project.

This relation between risk and standard deviationcan be clarified by examining the characteristics ofa normal distribution as shown in the followinggraph of Probability Ranges. If a probability

distribution is normal, the actual outcome will liewithin 1 standard deviation of the mean orexpected value about 68 percent of the time.


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That is, there is a 68-percent probability that theactual outcome will lie in the range "ExpectedOutcome 1 ." Similarly, the probability thatthe actual outcome will be within two standarddeviations of the expected outcome isapproximately 95 percent, and there is a greaterthan 99-percent probability that the actual eventwill occur in the range of three standarddeviations about the mean of the distribution.Thus, the smaller the standard deviation, the

tighter the distribution about the expected valueand the smaller the probability of an outcomethat is very far from the mean or expected valueof the distribution.


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We should note that problems can arise when the

standard deviation is used as the measure ofrisk. Specifically, in an investment problem, ifone project is larger than another-that is, if ithas a large cost and larger expected cash

flows-it will normally have a larger standarddeviation without necessarily being riskier.For example, if a project has expected returns of

$1 million and a standard deviation of only$1,000, it is certainly less risky than a projectwith expected returns of $1,000 and a standarddeviation of $500; the relative variation for thelarger project is much smaller.


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Probability Ranges for a NormalDistribution

When returns display a normal distribution, actual outcomeswill lie within 1 standard deviation of the mean 68.26percent of the time, within 2 standard deviations 95.46percent of the time, and within 3 standard deviations99.74 percent of the time.


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Notes: a. The area under the normal curve equals 1.0, or 100 percent. Thus, the areas under anypair of normal curves drawn on the same scale, whether they are peaked or flat, must beequal.

b. Half of the area under a normal curve is to the left of the mean, indicating a 50-percent

probability that the actual outcome will be less than the mean and a 50-percent probabilitythat it will be greater than the mean.c. Of the area under the curve, 68.26 percent is within 1 of the mean, indicating that the odds

are 68.26 percent that the actual outcome will be within the range (mean - 1 ) to (mean +1 ).

d. For a normal distribution, the larger the value of , the greater the probability that the actualoutcome will vary widely from, and hence perhaps be far below, the most likely outcome.Since we define "risk" as the odds of having the actual results turn out to be bad, and since measures these odds, we can use as a measure of risk.


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One way of eliminating this problem is to calculate a measure

of relative risk by dividing the standard deviation by theexpected value, E( ), to obtain the coefficient of variation:

In general, when comparing decision alternatives with costsand benefits that are not of approximately equal size, thecoefficient of variation measures relative risk better thanthe standard deviation does.


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Use of the Standard Normal Concept

Probability distribution can be viewed as a seriesof discrete values represented by a bar chart,or as continuous function represented by asmooth curve. Actually, there is an importantdifference in the way these two graphs areinterpreted: The probabilities associated with theoutcomes are given by the heights of the bars, the

probabilities must be found by calculating thearea under the curve between points of interest.


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Suppose, for example, that we have thecontinuous probability distribution shown in thenext figure. This is a normal curve with a mean of20 and a standard deviation of 5; x could bedollars of sales, profits, or costs; units of output;percentage rates of return; or any others units. Ifwe want to know the probability that an outcomewill fall between 15 and 30, we must calculate the

area beneath the curve between these points, theshaded area in the diagram.


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The area under the curve between 15 and 30 can bedetermined by painstaking graphic analysis of thisinterval or, since the distribution is normal, byreference to tables of the area under the normal

curve.Continuous Probability Distribution

Area under the Normal Curve


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Area under the Normal Curve

"z is the number of standard deviations from the mean. Some area tables are set up to indicatethe area to the left or right of the point of interest; in this book. we indicate the area betweenthe mean and the point of interest.

The distribution to be investigated must first be transformed orstandardized. A standardized variable has a mean of zeroand a standard deviation equal to one. Any distribution ofrevenue, cost, or profit data can be standardized with thefollowing formula:


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For our example, we are interested in the probability that anoutcome will fall between 15 and 30. We first normalizethese points of interest using the z statistic formula:

These areas associated with these z values are found in theprevious z statistics table to be 0.3413 and 0.4773. Thismeans that the probability is 0.3413 that the actual outcomewill fall between 15 and 20, and 0.4773 that it will fallbetween 20 and 30. Summing these probabilities shows thatthe probability of an outcome falling between 15 and 30 is0.8186, or 81.86 percent.


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Suppose that we had been interested indetermining the probability that the actualoutcome would be greater than 15. Here wewould first note that the probability is 0.3413that the outcome will be between 15 and 20,then observe that the probability is 0.5000 ofan outcome greater than the mean, 20. Thus

the probability is 0.3413 + 0.5000 = 0.8413,or 84.13 percent, that the outcome willexceed 15.


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Some interesting properties of normal probability

distributions can be seen by examining the z statistictable with the Probability Ranges graph, which is agraph of the normal curve. For any normal distribution,the probability of an outcome falling within plus orminus one standard deviation from the mean is 0.6826,or 68.26 percent (0.3413 x 2.0). The probability of anoccurrence falling within two standards of the mean is95.46 percent, and 99.74 percent of all outcomes willfall within three standard deviations of the mean.

Although the distribution theoretically runs from minusinfinity to plus infinity, the probability of occurrencesbeyond three standard deviations is very near zero.


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An example will illustrate the use of the standardnormal concept in managerial decision making.Suppose that Hastings Realty is considering aboost in advertising in an attempt to reduce alarge inventory of unsold homes. The firm's

management plans to make its media decisionusing the data shown in the Return Distributionstable on the expected success of televisionversus newspaper promotions. For simplicity,

assume that the returns from each promotionare normally distributed. If the televisionpromotion costs $4,000 while the newspaperpromotion costs $3,000, what is the probabilitythat each will generate a profit?


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The negative sign on z , is ignored, since the normal curve is symmetrical around themean; the minus sign merely indicates that the point lies to the left of the mean.


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To calculate the probability that each promotion

will generate a profit we must calculate theportion of the total area under the normalcurve that is to the right of (greater than)each breakeven point. Using methods:described earlier, we find that E(R TV) =$6,000, TV = $2,828.43. E(R N) = $6,000,and N = $1,414.21. For the television

promotion, we note that the breakevenrevenue level of $4,000 is 0.707 standarddeviations to the left of the expected revenuelevel of $6,000 because:


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= - 0.707

The standard normal distribution function value for z= -0.707 is between that for z = - 0.70 andz = - 0.71

To find the precise probability value for z = - 0.707,we must interpolate where:


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and the probability value for z = - 0.707 is 0.2580 + 0.0022This means that 0.2602, or 26.02 percent, of the total area

under the normal curve lies between X TV and E(R TV). and itimplies a profit probability for the television promotion of0.2602 + 0.5 = 0.7602, or 76.02 percent.

In calculating the newspaper promotion profit probability, we

find:

= - 2.121


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and the probability value for z = - 2.121 of0.4830 + 0.0000 = 0.4830. This means that0.483, or 48.3 percent, of the total area underthe normal curve lies between X N and E(R N),

and it implies a profit probability for thenewspaper promotion of 0.483 + 0.5 = 0.983,or 98.3 percent. In terms of profit probability,

the newspaper advertisement is obviouslythe less risky promotion alternative.

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1. Incertidumbre y riesgo.ppt