Chapter 5 Expected Monetary Value of Alternatives PDF

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Expected Monetary Values of Alternatives Decisions Recognizing Risks

Risk •

The concept of risk can be linked to uncertainties associated with events. Within the context of projects, risk Is commonly associated with an uncertain event or condition that, if it occurs, has a positive or a negative effect on the objectives of a project.

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Risk originates from the Latin term risicum, which means the challenge presented by a barrier reef to a sailor. The Oxford Dictionary defines risk as the chance of hazard, bad consequence, loss, and so on, or risk can be defined as the chance of a negative outcome. Risk should be associated with a system and commonly defined as the potential loss resulting from an uncertain exposure to a hazard or resulting from an uncertain event that exploits the system’s vulnerability.

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Risk should be based on identified risk events or event scenarios.

Risk Assessment •

is an overall process of (1) risk identification, (2) risk analysis, and (3) risk evaluation

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is a systematic process for identifying risk sources and quantifying and describing the nature, likelihood, and magnitude of risks associated with some situation, action, or event that includes consideration of relevant uncertainties

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Can require and/or provide both qualitative and quantitative data to decision makers for use in risk management – providing additional risk-related terminology and provides a typical risk-informed methodology for analyzing a system.

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Subsequent sections offer details on the different steps involved in a typical methodology.

Risk Studies •

require the use of analytical methods at the system level that take into consideration subsystems and components when assessing their event probabilities and consequences

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Systematic, quantitative, qualitative, or semi-quantitative approaches for assessing event probabilities and consequences of engineering systems are used for this purpose

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A systematic approach allows an analyst to evaluate expediently and easily complex systems for safety and risk under different operational and extreme conditions.

Estimate Variability and the Expected Value •

Engineers and economic analysts usually deal with estimate variation and risk about an uncertain future by placing appropriate reliance on past data, if any exist. This means that probability and samples are used. Actually the use of probabilistic analysis is not as common as might be expected.

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The reason is not that the computations are difficult to perform or understand, but that realistic probabilities associated with cash flow estimates are difficult to assign.

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Experience and judgment can often be used in conjunction with probabilities and expected values to evaluate the desirability of an alternative.

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The expected value can be interpreted as a long-run average observable if the project is repeated many times. Since a particular alternative is evaluated or implemented only once, the expected value results in a point estimate. However, even for a single occurrence, the expected value is a meaningful number. The expected value E(X) is computed using the relation 

  =   ( ) 

Where  =   ℎ        

  =  ℎ  fi    ! 

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Probabilities are always correctly stated in decimal form, but they are routinely spoken of in percentages and often referred to as chance, such as the chances are about 10%. When placing the probability value in the given equation or any other relation, use the decimal equivalent of 10%, that is, 0.1. In all probability statements the P(Xi) values for a variable X must total to 1.0. 

 () = 1.0



We may frequently omit the subscript i on X for simplicity. If X represents the estimated cash flows, some will be positive and others will be negative. If a cash fl ow sequence includes revenues and costs, and the measure of worth is present worth calculated at the MARR, the result is the expected value of the discounted cash flows E(PW).

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If the expected value is negative, the overall outcome is expected to be a cash outflow. For example, if E(PW) = $-1500, this indicates that the proposal is not expected to return the MARR.

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Example 1

ANA airlines plans to offer several new electronic services on flights between Tokyo and selected European destinations. The marketing director estimates that for a typical 24-hour period there is a 50% chance of having a net cash fl ow of $5000 and a 35% chance of $10,000. He also estimates there is a small 5% chance of no cash flow and a 10% chance of a loss of $1000, which is the estimated extra personnel and utility costs to offer the services. Determine the expected net cash flow.

Solution •

Let NCF be the net cash fl ow in dollars, and let P(NCF) represent the associated probabilities. E(NCF) = 5000(0.5) + 10,000(0.35) + 0(0.05) + 1000(0.1) = $5900

Although the “no cash flow” possibility does not increase or decrease E(NCF), it is included because it makes the probability values sum to 1.0 and it makes the computation complete.

Expected Value Computations for Alternatives •

The expected value computation E(X) is utilized in a variety of ways. Two prime ways are to:

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Prepare information for use in an economic analysis.

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Evaluate the expected viability of a fully formulated alternative.

Example 2 •

There are many government incentives to become more energyefficient. Installing solar panels on homes, business buildings, and multiple-family dwellings is one of them. The owner pays a portion of the total installation costs, and the government agency pays the rest. Nichole works for the Department of Energy and is responsible for approving solar panel incentive payouts. She has exceeded the annual budgeted amount of $50 million per year in each of the previous 2 years. Disappointed with this situation, Nichole and her boss decided to collect data to determine what size increase in annual budget the incentive program needs in the future. Over the last 36 months, the amount of average monthly payout and number of months are shown in the table. She categorized by level the monthly averages according to her experience with the program. Provided the same pattern continues, what is the expected value of the dollar increase in annual budget that is needed to meet the requests?

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Table 5.1 – Solar Panel Incentive Payouts

Solution: •

Use the 36 months of payouts PO, (j = low. . . . .very high) to estimate the probability P(Poj) for each level, and make sure the total is 1.0.

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The expected monthly payout is calculated. In $ million units, E[PO] = 6.5(0.417) + 4.7(0.278) + 3.2(0.194) + 2.9(0.111) = 2.711 + 1.307 + 0.621 + 0.322 = $4.961 ($4,961,000)

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The annual expected budget need is 12 x 4.961 million = $59.532 million. The current budget of $50 million should be increased by an average of $9.532 million per year.

Example 3 •

Lite-Weight Wheelchair Company has a substantial investment in tubular steel bending equipment. A new piece of equipment costs $5000 and has a life of 3 years. Estimated cash flows (Table 5.2) depend on economic condition classified as receding, stable, or expanding. A probability is estimated that each of the economic conditions will prevail during the 3-year period. Apply expected value and PW analysis to determine if the equipment should be purchased. Use a MARR of 15% per year.

Solution: •

First determine the PW of the cash flows in Table 18–4 for each economic condition, and then calculate E(PW) using Equation [18.2]. Define subscripts R for receding economy, S for stable, and E for expanding. The PW values for the three scenarios are PWR = -5000 + 2500(P/F,15%,1) + 2000 (P/F,15%,2) + 1000 (P/F,15%,3) = -5000 + 4344 = $ -656 PWS = -5000 + 5708 = $ -708 PWE = -5000 + 6309 = $ -1309

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Only in a receding economy will the cash flows not return the 15% to justify the investment. The expected present worth is *  % =  %& [ ( ] &+,-,.

= −656 0.4 + 708 0.4 + 1309 0.2 = $283 At 15%, E(PW) = 0; the equipment is justified, using an expected value analysis. Comment: It is also correct to calculate the E (cash fl ow) for each year and then determine PW of the E(cash flow) series, because the PW computation is a linear function of cash flows. Computing E(cash flow) first may be easier in that it reduces the number of PW computations. In this example, calculate E(CFt) for each year, then determine E(PW).

E(CF0) = $ -5000 E(CF1) = 2500(0.4) + 2500(0.4) + 2000(0.2) = $2400 E(CF2) = $2400 E(CF3) = $2100 E(PW) = -5000 + 2400(P/F,15%,1) + 2400(P/F,15%,2) + 2100( P/F,15%,3) = $283

Example 4 Regional Express is a small courier service company. At the central office, all parcels from the surrounding area are collected, sorted, and distributed to the appropriate destinations. Regional Express is considering the purchase of a new computerized sorting device for its central office. The device is so new—in fact, it is still under continuous improvement—that its maximum capacity is somewhat uncertain at the present time. The company has been told that the possible capacity can be 40 000, 60 000, or 80 000 parcels per month, regardless of the size of the parcels. It has estimated the probabilities corresponding to the three capacity levels. Table 5.3 shows this information. What is the expected capacity level for the new sorting device? Regional Express is growing steadily, so such a computerized sorting device will be a necessity in the future. However, if Regional Express currently deals with an average of 50 000 parcels per month, should it seriously consider purchasing the device now or should it wait?

If the discrete random variable X denotes the capacity of the device, then the expected capacity level E(X) is Table 5.3 - Probability Distribution Function for Capacity

Solution:

E(X) = (40 000)p(x1) + (60 000)p(x2) + (80 000)p(x3) = (40 000)(0.3) + (60 000)(0.6) + (80 000)(0.1) = 56 000 parcels per month

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The expected capacity level exceeds the average monthly demand of 50, 000 parcels per month, so according to the expected value analysis alone, Regional Express should consider buying the sorting device now. However, by studying the probability distribution, we see that there is a 30 percent probability that the capacity level may fall below 50 000 parcels per month. Perhaps Regional Express should include this information in its decision making, and ask itself whether a 30 percent chance of not meeting its demand is too risky or costly if it decides to purchase the sorting device....