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Recapitalization Investment Analysis

by Terry B. Armentrout¹

Abstract

Capital investment strategies involve a whole spectrum of investment options ranging from new ventures to securities. Included in the continuum, is the re-investment to the already existing plant. When does the return potential of a replacement exceed the return potential of investments in other areas? Comparing such investment options is difficult because the probable return of re-investment in capital plant is difficult to access. This paper presents a possible method to access re-investment options.

Introduction

Equipment survivor curves have existed since the 1930s. The University of Iowa developed an analysis for the railroad companies concerning the life expectancy of rolling stock. Survivor curves describe only the classes of equipment as a whole. Bayes' theorem of conditional probability, allows tempering of general survivor curve information with information of the specific individual apparatus. This combination gives a good indication of the probability of failure for individual apparatus.

All equipment is not equally important to the operation of an energy system. Reliability Centered Maintenance, (RCM) attempts to address this problem by analyzing the relative impact of items to the system. Some items are not necessary to reliable operation of a system, thus, intuitively should rank low as re-investment options. Other items would be just the opposite.

Equipment in different markets, however, have varying economic impacts. The failure of a transformer in New York City has a different financial impact than the failure of an identical transformer in the Pacific Northwest. The market with the higher priced energy will weight re-investments more than lower priced markets, thus, skew the Survivor curves, conditional probability, importance to reliability operation of a system and different markets need consideration when developing the priority of investment options. This paper will explore the combination of these four methods into a single procedure for identifying the productive investment options.

Equipment Survival Curves

R. Winfrey, of the Iowa State University, published a pioneer study in 1935 entitled "Statistical Analysis of Industrial Property Retirements"⁶(Winfrey, '35). This study developed curves such the one shown below. The curve indicates the relative proportion of specific classes of equipment which survive to a given age. Further, the survival of all equipment displays similar relationship. That is, a family of curves exist. By selecting a particular curve which fits the characteristics of specific equipment, one can reasonably estimate the probability of survival to a given age⁴(Rodenburg, '95).

figure 1

Winfrey's research process was first to accumulate data on the specific life spans of specific pieces of equipment. Plotting of this data produced a discrete curve. The final survival curve is the result of some sort of curve fitting process which condenses sets of data into a single curve point. The single data point then represents a most probable outcome from the given data set.

figure 2

Averages or any type of statistical condensation describes the data set. It does not provide information about a single data value. For instance we can say humans have a high probability of living longer than a day, and zero probability that anyone will live to see 200 years. Drawing a conclusion about the life span of a single person from the obvious facts is difficult. Likewise, drawing a conclusion about the probability of failure of a single piece of equipment from the survival curves is difficult.

By realizing a survival curve is not a single point but instead a probabilistic range of possibility, allows us to move from the general case to a specific instance. The single data point is the expected result of some data distribution. The figure below shows how one can superimpose any distribution over single survivor curve points. A standard normal distribution, a binomial distribution or a skewed distribution such as chi squared distribution can fit. We can now temper our generalized information about a class of equipment with information specific to a single piece of equipment.

figure 3

Bayes Theorem of Conditional Probability

Bayes Theorem deals with determining the probability given some information about the situation. For instance, the probability of drawing a Jack from a standard deck is 4 in 52, or 1 in 13. The probability of the card being a Jack, given the draw of a face card is 1 in 3. The latter example is similar to determining the probability of failure of a specific piece of equipment.

Mathematically, Bayes Theorem is²(Lindgren, '69):

equation 1

Equation 1 in non-mathematical language, the probability of a failure equals the product of the probability of a failure given a certain test result and the probability of the test result divided by the probability of a test result given a failure has occurred. The first term in the numerator of the equation, is the chance of a test predicting a failure. The second numerator term is the chance of a piece of equipment having that test result. The numerator is the probability that a failure occurred but the test result did not predict it.

Dr. Paul F. Mlakar, treats this process more vigorously in his "Reliability of Hydropower Equipment"³(Mlakar, '94), a report prepared for the US Army Corps of Engineers. He uses a composite index generated from several different tests to modify data in a survival curve. Any test of individual equipment which can be the basis for developing an individual probability of failure distribution will satisfy the requirements of the above process. Of course the more accurate, that is the smaller the dispersion of test results, the more useful the test will be in tempering the general survivor curve data.

We have taken specific data, which relates to the condition of specific equipment to temper data, which describes the general class of equipment. Now we can investigate the collective probability of systems.

Reliability Centered Maintenance:

Energy delivered to the high voltage bus of the power house is the result of a system of many individual components. Turbines, shafts, governors, pumps, generators, circuit breakers combine in unique ways to define a specific power house system. Some components are critical to the operation of the system while others are all but irrelevant. Some components are very reliable, while others are not dependable. In the above sections, we have accessed the probability of failure of separate components. Now we turn our attention to accessing the probability of failure of the system given these combination components.

figure 4

Individual powerhouses are just components in larger transmission and distribution systems. In a similar fashion, powerhouses react as individual components in a specific power house system. Depending on size, water supply, configuration, location and perhaps a thousand other considerations, an individual powerhouse can affect the larger system more or less than another. We can extend our systems thinking to the broader picture.

figure 5

From the graphically depicted relationships shown in the above two charts, we can construct a tree diagram. In this diagram we can make more clear the parallel and serial relationship between stations. Our diagram would look like this:

figure 6

Now the relationship of each component to the system is evident. Further we can make the diagram as detailed or as general as we desire. Its level of complexity would depend on the scope of our interest.

Following the mathematical laws of probability for the appropriate serial or parallel combination of components, we can now determine the associated probabilities of failure. Reliability dependency flows from left to right on figure 6. Raw materials enter the system on the left side of the diagram. Services and products leave the system on the right. The probability of failure for components X and Y depend on the reliable operation of component N. Components U, W and Z are parallel to X, Y and N and thus alternatives to each other. Now we can develop a clear picture of the scope and effect of individual component or individual station failures.

Opposite of reliability and service, financial effects of failure flow from right to left. Those components on the right have a very limited effect on the delivery of products and services. Thus the revenue lost from failure is small in comparison. A pole mounted distribution transformer failure may affect only one or two households and thus the revenue lost is energy sales to the households. On the left side of the diagram are components such as major generating stations and system inter ties. Failure in these components can cause major regional outages.

In figure 6, single components can be the compilation of lesser components. For example component N can be the representation of a more complex system such as a powerhouse. Moving about the analysis in this fashion allow investigation of the scope and effect on what might appear to be irrelevant components in a larger system. For instance the affect of trees growing in the right of way of a major system inter tie line would appear to some to be incidental and postponable maintenance. Analysis of relays in the context of a single station might never justify replacement with state of the art equipment. However malfunction of right-of-way maintenance and powerhouse generator relays can precipitate the failure of the power system in the Western United States.

Financial Analysis:

In traditional investing, the pertinent question is what return is probable from a given investment. In re-capitalization of existing plant the relationship is not quite so clear. In reinvesting in capital plant an increase in return does not occur. The return is totally in loss avoidance. In original investments an entrepreneur purchases a widget machine so in a certain time he can obtain an even greater return. In re-capitalization the same investment is necessary but the return does not increase. The return is the avoidance of lost opportunity associated with failed equipment and systems.

Our previous analysis of probability of failure becomes the statistics necessary for analyzing risky investments. Figure 6 terminates at the right with delivery of services and products to a customer. The failure of the system to make that delivery has associated with it a lost revenue. The failure data describes the set of possible returns from plant re-investment.

Carried forward through the failure analysis is the median and dispersion data. In risk based analysis of investments, the average of the possible pay back alternatives is the return. Risk is the dispersion of that data set. In mathematical symbols the expected return is:

and the risk is:
equation 2

In our analysis thus far we have generated a range of possibilities through a tree diagram. Each with an associated probability of failure. This entire collection is the set of investment opportunities complete with associated risks.

All investments in the set are not independent of each other. For instance replacing the winding in one powerhouse generator correlates directly to replacing the winding of an adjacent unit. In contrast the correlation of replacing a hydro generator winding for a unit in the center of the electrical system with replacing the winding of a small coal fired unit on the fringe is far less direct. Like wise the investment in right-of-way cleaning on a major transmission inter tie line to replacement of protective relaying in a small hydro plant on the fringe of the system is less direct. In general the equation is⁵(Van Horne, '77):

equation 3

where rjk is the correlation coefficient between investment opportunities j and k, is the standard deviation for investment opportunity n. This relationship will produce the risk associated with a combination of possibilities.

We now have the expected return and associated dispersion or risk for a full spectrum of possible investments in revitalized plant. This data set reflects the general survival characteristics of classes of equipment tempered by knowledge of specific equipment. The equipment is a cog in the larger mechanism of the energy system and by the method described above we can investigate the scope and affect of singular components on the entire system. The expected return and associated risk reflect the effect of specific components on the delivery capabilities of the entire system.

Market Area

The above analysis depends on a value for the lost opportunity of equipment failure. This opportunity loss is not consistent across market boundaries. A wide energy price disparity exists from market area to market area. Consolidated Edison of New York has a vastly different price structure than Bonneville Power Administration. The lost opportunity in the more expensive market will deliver greater possible return than in an inexpensive market. Following this analysis will favor options in the more expensive market. The result will be a highly developed and well-maintained system in a high price market and a decrepit system in a lower price market.

Renewable energy sources such as hydro power are less expensive. Systems dependent on high cost fossil or nuclear fuels are more expensive. Following the above method will skew the capabilities toward the more expensive and less renewable systems. The end result will be far less than prudent use of available energy resources.

This method breaks down across market boundaries. The results are valid only within a given market area.

Conclusion

Here is a model for reliability based re-capitalization. It begins with taking general data from survivor curves then tempering the general situation with specific data. The second step is placing opportunities into the perspective of the entire system. This probability data is the input for further analysis for risky investment. The results are valid for specific market areas and loose meaning across market boundaries.

References

¹ Manager, The Dalles/John Day/Willow Creek Project, Portland District, US Army Corps of Engineers

² Lindgren, B., & McElrath G., Introduction to Probability and Statistics, The Macmillan Company, Toronto, 1969

³ Mlakar, P., "Reliability of Hydro Power Equipment", US Army Corps of Engineers, April 1994

⁴ Rodenburg, R., Replacements: Units, Service Lives, Factors, US Department of energy, July 1995

⁵ Van Horne, J., Financial Management and Policy, Prentice Hall, Englewood Cliffs, NJ, 1977

⁶ Winfrey, R., "Statistical Analysis of Industrial Property Retirements", Bulletin 125, Engineering Research Institute, Iowa State University, 1935