More on Uncertainty and Expected Value

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1 More on Uncertainty and Expected Value As we saw last time, expected value can be a powerful tool for estimating the financial impact of events that we can t predict with certainty. For example, we used expected value to examine the value of a used car in the future. (If you ve forgotten how expected value works, take a quick look at the sidebar entitled Understanding Expected Value. ) Using Expected Value in P2 Calculations As you can see, expected value calculations go well beyond just pollution prevention applications. However, we often have to deal with uncertainty in environmental situations. For example, assume you are analyzing the financial impact of a material substitution proposal that substitutes a nonhazardous material in a manufacturing process. One of the benefits you should take into consideration is the fact that the nonhazardous material should limit employee injuries, lost time, and other problems. But how do you calculate the value of preventing something that may or may not have happened? Most companies keep records regarding accidents, spills, injuries, and other mishaps. In addition, there is a great deal of data available from the Occupational Safety and Health Administration (OSHA) and other government agencies, as well as a continually growing number of potential Internet resources. To use these data in analyzing a material substitution pollution prevention project, you first identify a benefit, such as the potential to reduce employee injuries. Then, to define the value of the benefit, you determine both the probability of injury and the cost of injury. The cost is relatively simple to obtain from company records by searching for a value such as the average number of days of work missed. Alternatively, if your company does not keep such records, you could possibly locate values in the literature or from trade associations. The probability calculation could be as simple as dividing the number of recorded accidents by the total amount of time employees have spent working with the material for which you are considering a substitute. For example, if there have been six accidents reported in the past year and the nine employees that used the material spent a total of 13,000 hours at that job, the probability of an accident would be 6 accidents/13,000 hours, or accidents/hour. The net benefit of eliminating accidents would be the expected value of the benefit connected with not having injuries-that is, the cost per injury times the probability of injury. In other words, you would multiply the benefit of not sustaining an injury by the probability of not sustaining an injury and add the cost of incurring an injury times the probability of incurring an injury. While this may sound complex, as we can see from the example in the sidebar explaining expected value, it is common for one of the financial impacts in an expected value calculation to be zero. James R. Aldrich John Wiley & Sons, Inc. Pollution Prevention Review /Winter 2000 / 111

2 In the case we are considering here, there is no cost associated with not sustaining an injury. So only the second term (the probability of injury times the cost of the injury) need be considered. Quantifying the P2 Downside In some cases, a P2 project can create new problems at the same time that it solves old ones. Expected value can help you calculate the cost of these potential risks. Consider a company that has installed a distilling unit to recycle/reuse solvent, thereby saving both hazardous waste and reducing its costs for new solvent purchases. This sounds like a fine investment from an environmental standpoint. However, given that the still must be heated, you have to assume that, in addition to the benefits, there is the potential for employees to be injured maintaining or operating the still. Since these burns can be particularly nasty, and they require the employee to be checked by a doctor for worker s compensation requirements, the financial result of such a burn injury can be three days missed work. A review of the literature reveals that roughly 7 out of 50 employees are injured in this type of job each year. Assume you have ten full-time employees working on the line with the still, and they are being paid $20/hour. The probability of one of these employees being injured would be (7/50) or 14 percent. This implies that the annual injury cost of using the still would be: (8 hourdday) x(3 days) x($20/hour) x(.l4/yr) = approximately $67/year/operator or $670/year total While this may seem like a trivial amount, it is generally better to show such costs whenever possible to lend credibility to your financial analysis. As we ve shown, the concept of expected value is not complicated. However, the calculations can become somewhat involved. For instance, in the distilling unit example, even though each individual s chance of injury may be small, given the number of employees, their individual costs for time missed, the va rious probabilities for each employee depending upon the degree of involvement with the still, and other factors, arriving at the answer could require a number of separate calculations. Expected Value and long-term Environmental liability While the cost involved with lost workdays may be relatively small, the expenses associated with an environmental lawsuit or a site remediation project under either the Resource Conservation and Recovery Act (RCRA) or the Comprehensive Environmental Response, Compensation, and Liability Act (CERCLA) can be quite large. Expected value can be used to estimate liability in such cases. The key is to be able to assign a probability and a cost for each outcome. Here, two rules of thumb are useful. First, you have to assume that the length of a product warranty is representative of the product s useful life. To accept this concept, you have to think in terms of averages. Everyone knows of a particular product, such as a refrigerator, that had a threeto five-year warranty, but that is still going strong after 20 or 25 years of use. But this has to be the exception to the rule. If refrigerators routinely lasted 25 years, some manufacturer would extend the warranty period for that length of time. That would certainly increase the manufacturer s market share and, consequently, its profits. Remember that Lee Iacocca advertised Chrysler with the slogan, If you want to know who builds them better, look to see who backs them better. From the manufacturer s viewpoint, the estimated life of any product, on average, must be just a bit longer than the warranty period. The second rule of thumb involves how to compute a probability of failure. Fortunately, 112 / Winter 2000 / Pollution Prevention Review James R. Aldrich

3 Exhibit 1. Cumulative Probability of Product Failure Occurring over a Six-Year Period (between 19 and 25 years) Years to Failure items generally do not stop functioning the day after the warranty runs out. To understand how probable failure is at any given time, we use a probability curve. Here is how the probability curve works: Assume that an item has a 20-year warranty. In such a case, the probability of.failure would effectively be zero for the first 19 years. It would probably creep up in year 20, and then increase each year beyond that until it reached 100 percent. This increase in each year beyond the warranty period can be estimated by a standard bell curve. Exhibit 1 shows the probability of failure expressed cumulatively. In this case, the probability would slowly rise in year 20 and increase to 100 percent over a six-year period. To use the curve, you simply look at the probability of failure in any given year times the cost of the event in that year. Spreading the probability of failure out over time does add an additional factor to the expected value calculation. Because the probability varies as time passes, the cost of failure can also change. For example, the cost of failure may increase by X percent each year. As a result, you must take the expected value (i.e., the probability of the event times the financial impact of the event) for each individual year. Then each of those future costs are brought to a common time with the present value equation, which we have discussed in previous installments of this column. Expected Value in Action: Estimating long- Term Hazardous Waste liability To make expected value easier to understand, consider the following example, in which we estimate the liability associated with placing hazardous waste in a lined landfill. Since we will be using net present value as our decision criterion, the first consideration must be when the liability expense will occur. (If you ve forgotten what net present value entails, refer to previous installments of this column.) P2 and the Bottom Line Pollution Prevention Review /Winter 2000 I 113

4 Understanding Expected Value To refresh your memory on how expected value works, consider the following example from the game of Craps: Assume your point is 8 and you laid down a $10 bet. What is the expected value of your next roll? ~3 There are 36 possible combinations when you roll two dice. Remember that our formula says to multiply all of the probabilities times all of their financial outcomes and then combine them. When you do, you would get the following: Combinations To Win (roll an 8) Probability 5 of 36 = 13.9%) +$lo Financial Impact To Lose (roil 7 or snake-eyes) Other Combinations 7 of 36 = 19.4% 24 of 36 = 66.7% -$lo $0 Expected Value = (+$ 0)(.139) + (-$10)(.194) + ($0)(.667) = -$0.55. Since each roll of the dice is an independent event, your odds never change. So on the average, each roll will cost you 55 cents. To determine when liability is likely to arise, we need to know how long the landfill will last. Under the assumption that a failed liner will mean a failed landfill, we also need to estimate how long the liner will last. Using our warranty equates to useful life rule of thumb, we need to look at landfill liner warranties. We find that, even though the market for these liners is very competitive, the standard landfill liner warranty is only 20 years. As discussed above, a manufacturer will tend to offer as long a warranty as possible. Thus, since no- manufacturer has offered a warranty beyond 20 years, it seems fair to expect a 20-year useful life for a landfill liner. This assumption is substantiated by the relevant literature on landfills. For instance, EPA reportedly expects most landfill liners to fail in approximately 20 years. See: McHugh, R.T. (1990). The economics of waste minimization. In Freeman, H.M. (ed.). Hazardous Waste Minimization (p. 132). New York: McGraw-Hill. Next, we spread the probability of failure over time. In this case, we ll use the curve presented in Exhibit 1 to predict the probability of failure in any given year beyond the warranty. If you think that the post-warranty time period in this example is too short, or that the probability should not start increasing until a later time, the curve can be adjusted in any manner you choose. As represented in Exhibit 1, the probability of landfill liner failure would be.02 in year 20,.14 in year 21,.34 in year 22,.34 in year 23,.14 in year 24, and.02 in year 25. (Remember that these values are cumulative and add up to 1.00, or 100 percent, by 114 / Winter 2000 / Pollution Prevention Review James R. Aldrich

5 ~ How Much k It Worth to Seal Out Spills? In this example, assume you are working for a company that generates hazardous waste which could contaminate groundwater if it escapes. Even though your main job is pollution prevention, you find yourself joining your company s compliance inspection team on a regular basis. Not only does this help your compliance-oriented compatriots, it also allows you to get out on the shop floor to look for pollution prevention opportunities. While you are inspecting the hazardous waste storage area at one of your facilities, you notice the floor in the building looks funny. Not cracked, not worn, but funny. You find that a special sealer was applied to the floor when the building was opened four years earlier. The sealer prevents spills from soaking into the concrete floor, where they could leach through to the ground. Now the appearance of the sealer is telling you that something is wrong. You do some checking and find out that the contractor gave only a five-year warranty on the sealant job. With no warranty, any spill cleanup that occurs after next year will be the financial liability of your company. It seems logical to replace the sealer within the next year and get another five-year warranty, but the sealer is very expensive. Your company s environmental committee knows about the potential problem, but replacing the sealer will cost $187,000, and there is no money in the budget for the foreseeable future. The question is whether it would be cheaper to rep.lace the sealer or take a chance on having to clean up a spill. You do some further checking and find out that there have been nine spills in the building during the four years it has been in use. Fortunately, the spills were easy and inexpensive to clean up because of the floor sealer. If the floor sealer fails, a spill could easily be adsorbed into the concrete. Were this to occur, cleanup would require tearing out the floor of the building. This would mean losing the building, the replacement cost of which is $980,000. You decide that the failure probability follows a standard bell-shaped distribution and should have a failure window of about ten years. The probability of the floor sealer failing once the warranty is expired is shown below. I 1 Year Prob If your company uses a 6-percent discount rate, the annual spill cleanup costs would be calculated as follows: We are now in year 4 (i.e., the building has been in use for four years). The sealer warranty will expire in year 5, and the probability of failure will kick in starting in year 6. You assume that the $980,000 replacement cost of the building would include taking care of any spilled material. The expected value expression will be the probability of failure in each year times the cost of failure. Year (i.e., building age): a Failure Probability Expected Value ($ x 1000) - constant replacement cost at $980,000: Pollution Prevention Review / Winter 2000 I 115

6 ~ How Much Is It Worth to Seal Out Spills? (Continued) This information can be shown on a cashflow timeline, as in Exhibit 2. Rather than computing a net present value, let s see how many years must pass until the net present value of a spill would justify the cost of resealing the floor. This is most easily accomplished in a table%ith the cashflow being converted to a present value and being cumulatively summed until the net present value is greater than the cost of the sealer ($187K). Age: Years in future: Expected Value ($ x 1000): 0 0 $23 $45 $88 $147 $187 $187 $147 $88 $45 $23 NPV ($ x 1OOo): 0 0 $20 $38 $70 $110 $132 $124 $92 $52 $25 $12 Cumulaiive NPV ($ x 1000): 0 0 $20 $58 $128 $238 $369 $494 $586 $638 $663 $675 As shown in the table, the expected value of the cleanup (cumulative) would exceed the current price of the sealer ($187,000) within four years after the warranty expires. The present value of the total failure scenario over the ll-year period would be $675K. If the price of replacing the building increased at even a modest 2.5-percent inflation rate, the results are even more dramatic, as shown below. Year: Probability of Failure: Inflated Cost of New Building ($ x 1OOO): 980 1,005 1,030 1,055 1,082 1,109 1,136 1,165 1,194 1,224 1,254 1,286 Years in future: Expected Value ($ x 1OOO): NPV ($ x 1000): 0 0 $21 $41 $77 $124 $153 $148 $112 $65 $32 $16 Cumulative NPV ($ x 1000): 0 0 $21 $62 $139 $263 $416 $561 $677 $742 $774 $790 Although inflating the cost of buildi replacement did not ch ge the time frame in which the net present,slue of failure will eclipse the cost of the sealer (still in year 4), it certainly had a drastic effect on the total expected value of failure. 116 / Winter 2000 / Pollution Prevention Review

7 the time the liner reaches year 25. For purposes of our calculation, however, we must determine the expected value for each year separately.) If you were considering using a landfill that was built five years ago, failure probabilities would be zero for the next 15 years (20-year life minus 5- year age). Then the potential liability would increase in accordance with the annual probabilities given above. Because we are using net present value as the criterion, the failure probability and the discounting factor for an expense in the future would both be used in determining the financial impact. The next factor that we need to address is the magnitude of the liability if the landfill fails. There could be any number of costs involved in cleaning up the landfill site-including monitoring, chemical analyses of groundwater, and other charges. Depending upon your company, you may have historical cleanup cost data based on experience with Superfund and/or spill cleanup actions. Unfortunately, these data are normally very sitespecific and, if used to predict future cleanup expense, could inflate the cost of liability. It s a good idea to keep the cleanup cost estimate as conservative as possible. (That should be rule of thumb number three!) Given that your company, as waste generator, will remain legally responsible for the waste as long as it exists, we can safely say that if the landfill fails, you will have to gather up the waste and either put it into a new landfill or treat it in some manner. Hence, a conservative estimate would reflect the cost of either waste treatment or relandfilling. Using treatment or landfill disposal cost also allows us to express this cost on a per-unit basis, which makes the model easier to apply in a number of different situations. In addition, because we are ignoring the costs for excavation, analysis, and the like, our estimate is conservative. At this point, all that is left to be done is to estimate what it will cost to treat or landfill a unit of waste 20 years in the future. Unless you are con- vinced that the cost of landfilling or treating a waste is going to decrease, we ll stay on the conservative side and treat the cost of waste destruction in the future as a constant. This brings us to the point where we know all the variables needed for the expected value calculation. We can read the probability from the graph. The present value term can be readily computed for any given interest rate. And the cost of failure will simply be the cost of relandfilling the waste (Le., the cost of destruction ). The equation for a five-year-old landfill would be as follows (remember, since the cost of not failing is zero, we can drop that part of the equation): Expected Value = (Cost of Destruction) x (Probability of Failure) x (Present Value Factor) = (Cost of Landfilling) C 9 x =(p,,) t ( 1 t T) (1 +r) (1 tr)i6 1 (l+r) i which, if you we re using a &percent discount rate, would reduce to: If the current charge for landfilling hazardous waste were $300 per ton, the liability incurred from sending that waste to a landfill would be $300 x.361, or $108. This means that the actual cost of landfilling the waste (that is, the cost with liability included) is actually 36 percent higher than the tipping fee. P2 and the Bottom Line 1 Pollution Prevention Review /Winter 2000 / 117 1

8 Exhibit 2. Cashflow Timeline (Sealant Failure leading to Hazardous Waste Spill) 2oo s 0 F X 1 * t Tt, Year If you increase any of our conservative estimates, the percentage would be even higher. For instance, if you predict that landfill fees will rise, the cost of landfilling would not be constant and would have to be computed for each individual year. As you can see, the basic equation never changes. However, any of the variables can be altered to fit any situation. For example, if you feel that the landfill will not fail until year 25 or 30, the present value factor will change. If you feel that the probability curve in Exhibit 1 should cover 8 or 10 or 12 years, all that needs be done is to change the probability. If your company records show that, because of the daily cover added to a landfill, each ton of waste turns into two tons, that information can easily be included. Finally, instead of holding the costs constant, we can increase or decrease the landfill costs. If you d like another example of how the expected value calculation works in an environmental context, see the sidebar entitled How Much Is It Worth to Seal Out Spills? Conclusion This brings us to the end of our discussion of expected value, but not to the end of cost estimating techniques. Just like production, your waste generation probably fluctuates. Next time we ll look at two methods to predict waste generation: indexing and through-put ratio. James R. Aldrich, Ph.D., is senior engineer with EarthTech in Warner Robins, Georgia. 18 / Winter 2000 / Pollution Prevention Review James R. Aldrich