Sales comparison analysis of power-generating assets: the role of structure and time.
Subject:
Energy facilities (Valuation)
Industrial real estate (Valuation)
Sales (Comparative analysis)
Valuation (Analysis)
Authors:
Bode, David C.
Presto, Jennifer J.
Paez, Antonio R.
Fischbeck, Paul S.
Dean, Steve R.
Pub Date:
03/22/2006
Publication:
Name: Appraisal Journal Publisher: The Appraisal Institute Audience: Trade Format: Magazine/Journal Subject: Business; Real estate industry Copyright: COPYRIGHT 2006 The Appraisal Institute ISSN: 0003-7087
Issue:
Date: Spring, 2006 Source Volume: 74 Source Issue: 2
Topic:
Event Code: 800 Capital funds & cash flow Computer Subject: Sales analysis
Geographic:
Geographic Scope: United States Geographic Code: 1USA United States

Accession Number:
146173740
Full Text:
abstract

Valuing power-generating assets by the sales comparison methodology is complicated by the fact that these assets are often sold in heterogeneous portfolios. This article documents a difference in valuation levels between transactions comprised of homogeneous and heterogeneous portfolios. It also demonstrates that as market conditions change over time, appraisers must reassess how they interpret the historical transaction record to farm an opinion of value by the sales comparison approach. To improve the accuracy of sales comparison analysis for both retrospective and current valuations, a model is developed that integrates the influences of heterogeneity and time (market conditions) on power-generating real property.

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Estimating the market value of power-generating real property with the sales comparison approach can be problematic because these assets are often sold as part of heterogeneous portfolios. Can an opinion of value determined by the sales comparison approach for a single coal-fired power plank for example, really be derived from a simple comparison with the aggregate value of a transaction that included both coal-fired and natural gas-fired facilities? Rode et al. argue that the markedly different economic factors affecting such investments make such a comparison highly questionable (1) and that these transactions are not comparables as such.

Rode et al. use transaction data from 1997-2001 and linear regression to extract individual real property values from aggregate transaction data; their final model is as follows:

(1) VALUE = [[beta].sub.1]C + [[beta].sub.2]O + [[beta].sub.3]C[theta]G + [[beta].sub.4](1 - [theta])G

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In this model, which shall be referred to as the RDFD 2002 model, the beta coefficients are the estimated per-megawatt prices (in millions of dollars) for three different capacity types: coal (C), oil (O), and natural gas (G). Natural gas was further divided into assets located in California and all others. VALUE refers to the total monetary consideration offered in exchange for the asset in each transaction. Because of the dominant influence of fuel type on the economic viability of power generators, even this sparse model has a fairly high degree of accuracy (adjusted [R.sup.2] of 83.3%, with all variables highly significant). The regression results are presented in Table 1.

The RDFD 2002 model, however, is silent on the informativeness of the historical transaction record and the appropriate useful life of the coefficients. The power industry often encounters deficiencies and excesses of capacity because of the lead time required to add capacity and the economic inviability of storing electricity. This cyclicality is manifest in the asset values, which can fluctuate considerably across time. As The Appraisal of Real Estate notes, "changing market conditions may reduce the validity or applicability of older sales that do not reflect the change." (2) In applying the sales comparison approach, it is important to obtain transacted prices that are most representative of current market conditions. Of course, the admonition to the appraiser is not to require only perfectly contemporaneous sales, but rather to have an understanding of how changing market conditions may limit the usefulness of portions of the historical transaction record.

In this article, the RDFD 2002 model is reevaluated using an expanded data set. Using a multiple regression approach, the influence of changing market conditions on asset values is investigated as well as the role that transactional characteristics play on asset values. Specifically, three questions are considered:

1. Can a time index be incorporated into sales comparison analysis for purposes of retrospective valuation?

2. Is there a difference in values between homogeneous and heterogeneous transactions? Are buyers attributing a premium or discount to transactions in which properties of different types are bundled together?

3. Does the informativeness of transaction information decay over time? If so, at what rate does it decay?

The answers to these questions provide clear guidance to appraisers of power-generating real property. In addition to providing a quantitative measure of the impact of changing market conditions on value, this research also raises an important question for appraisers regarding the independence of asset values from transaction structures.

Data Analysis and Performance of the RDFD 2002 Model

The data used in this analysis was gathered from published accounts of transactions between 1997 and 2004. (3) Collectively, the 156 transactions represent more than 139,000 megawatts (MW) of capacity at nearly 520 plants and almost $53 billion of value. The transactions have been classified according to seven types based on the primary fuel used: nuclear, wind, geothermal, coal, natural gas, oil, and hydroelectric (hydro). Table 2 summarizes the transactions by year. As is evident, current transaction volume, while less than the 1998-1999 divestiture-driven peak, is greater than the 2003 low brought about by the post-Enron turmoil in the power sector. (4)

Figure 1 graphs the distribution of aggregate values per kilowatt ($/kw) for the transactions. As is evident, there is a significant amount of dispersion in the data. For comparison, the overnight costs of new facilities are estimated to be: $1,669/kw (nuclear), $949/ kw (wind), $2,099/kw (geothermal), $1,091/kw (coal), and $569/kw (natural gas/oil, advanced combined cycle). (5) At the other end of the spectrum, the orderly liquidation value of a combined cycle unit is typically in the range of $25/kw to $100/kw. These comparisons often suggest useful bounds on value, but the presence of unique conditions (e.g., the grandfathering of permits) can cause values for existing assets to exceed the construction costs of new capacity.

The 156 transactions occurred in various locations around the United States, with at least one transaction in every North American Electric Reliability Council (NERC) region. The NERC regions, which enjoy substantial operating autonomy under normal conditions, represent the common markets in which power generators operate. Figure 2 illustrates transaction volume in each NERC region. Most transactions have occurred in the deregulated markets of the Western Electricity Coordinating Council (WECC) and the Northeast Power Coordinating Council (NPCC).

Compared with Rode et al., the data set used here includes more transactions (a total of 156) and four additional fuel types (nuclear, wind, geothermal, and hydroelectric). However, the inclusion of assets in these four new categories is not without challenges. There are still comparatively few transactions in these categories, so they are inappropriate for many types of statistical analysis; the discussion highlights areas where appraisers should apply caution.

Before consideration is given to modifying RDFD 2002, its performance on the new data is first examined. The three panels in Figure 5 illustrate the performance of the previous model by plotting the actual versus predicted values on the additional Fifty-seven transactions that occurred between 2002 and September 2004 and involved only the three fuel types used in the original model. Qualitatively, these three years could be considered a significant transition period. The immediate post-Enron fallout made 2002 a very difficult environment for power investors; by 2004, the economic environment was markedly better.

If the predictions in Figure 3 were perfect, there would be no dispersion of points around the solid line. In a statistical sense, however, it might be expected that the dashed line, which represents the fit of predicted-to-actual values, would coincide with the solid line. Differences in slope between the dashed and solid lines represent a proportional shift in value from the predicted values that does not affect variation explained (i.e., [R.sup.2]). In presenting the results in this manner, a distinction is drawn between the ability of the model to capture factors driving the variability of asset values ([R.sup.2], a measure of proportional or relative error) and ability of the model to capture the level of aggregate asset values (a measure of absolute error). (6)

There are three notable observations with regard to the out-of-sample predictions. Table 3 lists the median absolute percentage error (APE) and percentage of common variation explained ([R.sup.2]). (7) The median APE measures the absolute value of the percentage difference between predicted and actual transaction values (represented visually by the distance between the dashed "actual fit" line and the solid "perfect fit" line). On this basis, the model's performance in 2002 was poor, but returned to more acceptable levels in 2003 and 2004. The poor performance of the model for 2002 could be attributed to the post-Enron fallout in the industry. The [R.sup.2] measure is equal to the squared correlation between predicted and actual values. In the three panels of Figure 3, this is reflected by the degree to which the dashed line fits the data points. Considered in isolation, the decline in [R.sup.2] from 2002 to 2003/2004 would suggest that performance of the model decreased over time-contrary to the apparent prediction of the median APE measure. However, both measures taken together suggest instead that although the model performed well in making relative value assessments in 2002 (as indicated by a high percentage of common variation and therefore a close fit of the data points to the dashed line), absolute value levels dropped considerably (as indicated by a high median APE and therefore a greater distance between the dashed and solid lines). The most plausible explanation, in hindsight, is that immediately after the bankruptcy of Enron, power market investors required an additional premium for any capital commitments. (8) Such an industry-wide increase in capital costs would lower the absolute values of assets (i.e., a proportional downward shift in asset values), but would tend to preserve their relative valuation levels.

[FIGURE 3 OMITTED]

By 2003, these premiums had begun to dissipate and the performance of the model improved considerably even as the percentage of common variation declined ([R.sup.2] fell from 97% to 86% in 2003 and further still to 69% in 2004). Thus, absolute errors had moderated, but the relative valuation relationships were beginning to break down over time. These observations serve to underscore the earlier point on the importance of understanding the impact of market changes on asset values (and thereby the coefficients in the models). This issue is addressed in greater detail later.

Incorporating Time for Retrospective Valuation

As a first step in expanding beyond the RDFD 2002 model, the model is reestimated with the new, larger data set. The RDFD 2002 model estimates transaction values on the basis of capacity type alone. Consequently, it remains mute as to the role of time on value. Nevertheless, it provides a useful baseline from which to proceed. Equation 3 presents a model that shall be referred to as the fuel type model. The regression results of this model are presented in Table 4; Figure 4 plots the actual transaction values against the predicted transaction values. (9)

[FIGURE 4 OMITTED]

(3) VALUE = [[beta].sub.1]N + [[beta].sub.2]W + [[beta].sub.3]T + [[beta].sub.4]C + [[beta].sub.5]G + [[beta].sub.6]O + [[beta].sub.7]H

VALUE is the total value of a transaction, in millions of dollars, and the variables represent the megawatts of capacity in the following categories: nuclear (N), wind (W), geothermal (T), coal (C), natural gas (G), oil (O), and hydroelectric (H).

Although the adjusted [R.sup.2] statistic for this model (76.8%) is lower than for the 2002 model (83.3%), this is almost certainly a consequence of the inclusion of categories containing very few transactions (wind and geothermal). As Table 4 makes evident, there were not sufficient transactions in those categories to provide statistically significant results. The lack of statistical significance for oil is more likely a result of heterogeneity within the oil category itself. Most of the oil-fired generating capacity in the United States operates only for a small portion of the year as peaking capacity. As a result, values for oil-fired assets are highly dependent on location. (10)

The time analog to the previous model differentiates transactions by date, but not by category. In estimating this time model, all of the transactions within a given year are combined as if they were comparable (except for megawatts of capacity). The nuclear, wind, and geothermal classes are excluded since they consist of entirely homogeneous transactions. That is, nuclear assets are sold only either by themselves or with other nuclear assets (and the same is true of wind and geothermal). Since homogeneous transactions are addressed in greater detail later, these transactions are excluded for now (reducing n to 130).

The time model is presented in Equations 4 and 5.

(4) VALUE = M([2004.summation over i=1997] [[beta].sub.i][D.sub.i])

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The overall transaction value (in millions of dollars) is estimated to be a function of the megawatts of capacity categorized by year. In the model, M refers to the total megawatts of capacity transacted: M = C + G + O + H. Table 5 presents the results of the regression, and Figure 5 plots the actual versus predicted values for the time model. The overall adjusted [R.sup.2] increases to 81.9%, and all of the year variables are significant. All variables except for one are strongly significant (p < 0.001); the 2002 coefficient is weakly significant (p < 0.10).

[FIGURE 5 OMITTED]

In examining the results of the time model, three transactions appear to be overly influential outliers. The 2004 value of $141,000/MW is unusually low, approaching an orderly liquidation value. Upon further investigation, two transactions with substantial leverage caused the overall coefficient to be biased downward. The sale of Duke Energy's 5,325 MW portfolio to KGen Partners LLC was both the largest and lowest-price transaction of the year ($89.2/kw). The second largest transaction of 2004 was the sale of American Electric Power's 3,800 MW Texas Central Company portfolio to a group of private equity funds for $113/kw. By themselves, these two transactions claimed 41% of the 2004 MW transaction total, causing significant downward bias on 2004's coefficient. Reestimating the model after excluding these two transactions raises the 2004 coefficient by more than 200% to $303,000/MW. Similarly, the 2003 El Paso-GS Linden transaction took place at an extremely high price ($1,123/kw). This amount is almost 90% more than the cost of building a new gas-fired facility. Removal of this transaction reduces the 2003 coefficient to 0.512. It is beyond the scope of this article to investigate these particular transactions. Nevertheless, it is clear from a statistical sense that they are not only outliers, but also extraordinarily influential ones. The results are reported here both with and without these outliers.

In both the fuel type model and time model, roughly three-quarters of the overall variability was explained. On the surface, this is odd, since one might expect fuel type and time to be orthogonal factors. If they were truly orthogonal, it would be impossible for each of them individually to explain three-quarters of the total variance (which must sum to 100%). Rather, it appears that there must be some degree of collinearity between the date of a transaction and the type of fuel characterizing the capacity in the transaction. One theory that accommodates this finding involves industry cycles. At the extreme, consider a market in which each year there were only transactions of one particular fuel type (i.e., there was a bijective mapping between the two). In this situation, a model that used only time as a factor would pick up the same variance as a model that used only fuel type. Of course, the reality is not nearly so stark; nevertheless, there is clearly some element of commonality between the two.

Figure 6 demonstrates the existence of a relationship between time and fuel type by reclassifying the transactions into two distinct economic classes: a base load class (low variable cost, high capital cost) and a peaking class (high variable cost, low capital cost). Each of these classes is influenced by different factors, including fuel costs, interest rates, and regulatory changes. As is evident, the megawatt likely to characterize a 2001 transaction is probably different than the megawatt likely to characterize a 2004 transaction. For example, the decline in high marginal cost capacity transactions to 2001 coincides with an increase in natural gas prices. Further exploration of this issue is beyond the scope of this article, but it may be concluded from this observation that it is potentially dangerous to attempt to capture the influence of fuel type or time on value independently even when apparent explanatory power is high.

[FIGURE 6 OMITTED]

In order to isolate the time and fuel type factors, a combined model is considered next (Equation 6). (11) Although useful to appraisers for retrospective valuation, (12) the combined model offers no insight as to the present value of an asset because it does not indicate how weight should be applied to the various periods in the historical transaction record to produce an indicator of current values. (The issue of current market value is discussed later.)

(6) VALUE = C([2004.summation over h=1997] [[beta].sub.h][D.sub.h]) + G([2004.summation over i=1997] [[beta].sub.i][D.sub.i] + O([2004.summation over j=1997] [[beta].sub.j][D.sub.j] + H([2004.summation over k=1997] [[beta].sub.k][D.sub.k]

The variable D is as defined in Equation 5; the variables C, G, O, and H represent the megawatts of coal, natural gas, oil, and hydroelectric capacity, respectively. Theoretically, the output of this model provides a yearly coefficient for each fuel type that may be thought of as an index of value. In practice, however, estimation of this model is subject to the limitations of the data. In estimating this model, stepwise regression is used to identify which subset of variables maximized the explanatory power of the model. (13)

Table 6 and Figure 7 contain the combined model results. As the stepwise procedure removes variables that do not attain statistical significance, individual data points are also removed (removing transactions that, for example, contain oil-fired capacity once oil has been removed as a variable). As a result, the total number of retained data points drops to 90. Nevertheless, the model's adjusted [R.sup.2] increases to 81.1%. This means that the level of inferential power in the RDFD 2002 model has been very nearly retained while extracting the influence of time and incorporating one additional variable (hydro).

[FIGURE 7 OMITTED]

In summary, the combined model provides valuable insight on the influence of both time and fuel type and suggests the early stages of an index of power asset values. The model obtains substantial explanatory power and controls for the independent influence of time and fuel type, although it is still limited by the data as to the extent to which statistical conclusions may be drawn. Figure 8a graphs the coefficients estimated in the combined model; Figure 8b graphs the coefficients from the time model and the average annual natural gas prices over the same period. In most regions of the country, natural gas is the marginal fuel used, meaning that electricity prices are heavily influenced by expectations about future natural gas prices. With both revenues and expenses heavily influenced by natural gas prices, it often serves as a useful proxy for the economic fundamentals of power generators.

[FIGURE 8 OMITTED]

As is evident, there appears to be a relationship (with a possible lag) between natural gas prices and the general trend in asset values. (14) High natural gas prices generally imply high electricity prices, providing greater profits to base load generators that do not use natural gas and increasing their market value. Second, the coefficients are generally in line with those produced by the time model--at least to the extent that they increase and decrease together. It might be expected a priori that different fuel type values may vary only by parallel shifts over time, which is the outcome observed here. Also, the influence of the outlier transactions on the 2003 and 2004 coefficients is of note. What is not apparent in these results is how they should be used in forming an opinion of current market value by the sales comparison approach. This issue is discussed in the next section.

Homogeneous vs. Heterogeneous Transactions

In developing a framework that appraisers can use for incorporating historical transactions and that includes portfolios of assets, the first step is to consider the potential for a portfolio effect itself. The motivation for the modeling approach was that many transactions for power generation assets took place as portfolios composed of different asset types. These port folios, referred to here as heterogeneous transactions, are difficult to evaluate using the sales comparison approach. Other transactions consist of either a single asset or a collection of assets with identical fuel types; these are referred to as homogeneous transactions.

It is important for the appraiser to draw a distinction between the different types of transactions for two reasons: (1) heterogeneous transactions represent a significant portion of the historical transaction record (15) for power-generating real property, and (2) there may be differences in valuation levels between homogeneous and heterogeneous transactions. Table 7 contains a summary of the homogeneous transaction data and average value levels. (16) Figure 9 graphs the average value levels, together with the minimum and maximum transaction values. To investigate the presence of a portfolio effect, a model is developed (Equations 7 and 8) using the same three capacity types as were found to be significant in the combined model.

[FIGURE 9 OMITTED]

(7) VALUE = [[beta].sub.1]CD + [[beta].sub.2]C(1 - D) + [[beta].sub.3]GD + [[beta].sub.4]G(1 - D) + [[beta].sub.5]HD+ [[beta].sub.6]H(1 - D)

(8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

As in the previous models, C, G, and H refer to the megawatts of coal, natural gas, and hydroelectric capacity in the transactions. The beta coefficients refer to the value of capacity in millions of dollars per megawatt. Table 8 displays the results of the model and Figure 10 illustrates the differences between homogeneous and heterogeneous transaction values.

[FIGURE 10 OMITTED]

As is evident, there are statistically significant differences between the values for coal and hydroelectric asset types depending on the heterogeneity of the transacted portfolio. This support for the portfolio effect hypothesis is intriguing for several reasons. First, it may suggest that real property values depend not only on physical characteristics or pure income-producing potential, but also on financial characteristics. Financial portfolio theory has long demonstrated that the aggregation of individual securities into portfolios can lower overall risk when the individual assets are less than perfectly correlated. (17) To the extent that diversified, more stable cash flow streams are more desirable to particular clienteles or are easier to finance, such portfolio benefits may appear in transaction prices.

Theoretically, such valuations may be attributed to the existence of certain clienteles in a less-than-perfectly-efficient capital market. In other words, some aspect of value may derive from the nature of the transactor, rather than from the nature of the property alone. Clearly, this is not without considerable controversy in an appraisal setting, since the "arm's-length transaction" qualification in the definition of market value seemingly requires the appraiser to abstract away from a particular transactor in determining a market value. However, the findings here indicate that the nature of the transactor is an important consideration in the use of the sales comparison approach to derive a market value.

Second, the portfolio results are interesting because they are not consistent across types. Coal facilities benefit from being part of a heterogeneous portfolio; hydroelectric facilities suffer from it. (18) This finding suggests a more strategic phenomenon. One possible explanation for this effect is that a firm attempting to divest itself of unproductive capacity may bundle that capacity with more attractive assets in order to make it saleable. Anecdotally, there is recent evidence that several transactions took place economically for the coal capacity alone. (19) Such a hypothesis is certainly consistent with the empirical findings presented here, although it would be difficult to test explicitly for such a causal explanation.

This hypothesis is significant to appraisers because it once again suggests significant variability underlying the population of transactions. Obviously, if all hydroelectric plants were worthless, there would not be any homogeneous hydroelectric transactions. The fact that there are such transactions suggests that the hydroelectric facilities sold on a standalone basis must be fundamentally different from those sold in heterogeneous transactions in terms of their economic income-producing properties. (20) Any sales comparison analysis that fails to reflect this effect is likely missing an important distinguishing feature of comparability.

The Decay of Informativeness Across Time

The previously presented combined model made clear that the value of power-generating real property changes over time. Nevertheless, "time is not the cause of the adjustment. Market conditions that shift over time create the need for an adjustment, not time itself." (21) Appraisers should select prior transactions so as to maximize the comparability between current market conditions and the conditions prevailing in prior transactions. The implication, therefore, is that a sales comparison model should be designed with respect to overall market conditions, not simply with a time index. In addition, it also suggests that transaction data may have a finite life over which it can be seen as an authoritative signal of value.

To the extent that a particular market change is hypothesized to have an impact on transaction prices, the regression approach could control for it. For example, the model could simply exclude prechange data from the sample or incorporate a dummy variable that dichotomizes the data. The presumption implicit in such an approach, however, is that the change in time can be neatly located. In reality, it is most often the case that market changes cannot be precisely identified until long after they have occurred. For example, it was only in July 2003 that the National Bureau of Economic Research concluded there has been a recession between March 2001 and November 2001-nearly nineteen months after the fact. (22) It is not simply a delay though; many market changes unfold gradually over long periods. As a result, their impact is more continuous than a simple dichotomization would allow.

Given the difficulty in identifying a market change, a better approach is to allow for a gradual increase in the influence given to recent transactions (or a gradual decrease in the influence given to more distant transactions). Weighted least squares (WLS) regression is an ideal procedure for such a task. (25) The WLS procedure allows the appraiser to place different weights on the errors from different data points or classes of transactions, thereby reducing the influence of past information on current market conditions. Performing a WLS regression to obtain the property value coefficients provides a valuation function reflective of both the asset type and the market conditions under which that transaction occurred-without forcing a (potentially arbitrary) dichotomization of the sample data.

One challenge in using WLS regression is selecting the appropriate weights. (24) Introducing weights into the regression adds variability to the model and it is generally best for the modeler to limit such variability to the extent possible. As an example, a WLS model was developed using the four most significant classes (nuclear, coal, gas, and hydro). A total of 122 transactions were possible over eight years. Instead of using 122 or even eight weights as additional free variables, however, an exponential decay specification was incorporated for the weight structure with one free parameter (a "decay factor" [alpha]). (25) This specification was capable of producing equal weights across all years of data ([alpha] = 0), larger weights on recent years ([alpha] < 0), or larger weights on more distant years ([alpha] > 0).

Figure 11 graphs the adjusted [R.sup.2] coefficient against this regression versus [alpha]. One observation from Figure 11 is that the relationship is clearly non-monotonic. This relationship suggests that the performance of the model does change over time as market conditions change. Perhaps more importantly, it also indicates when the transaction data is most conducive to a fundamentals model like this regression approach (as opposed to a speculative values model in which asset values may become detached from their core economic drivers). For values of [alpha] that produced high adjusted [R.sup.2] values, the model indicates the transaction prices are explained mostly by the fundamental factors (i.e., fuel type) included in the model. For periods during which adjusted [R.sup.2] is lower, values are influenced more by nonfundamental factors. Accordingly, it is important that appraisers understand how changing market conditions influence the explanatory power of their models so that weights in a WLS regression approach can be assigned appropriately.

It should be clear, then, that a joint optimization of the coefficients and weights should be employed to maximize explanatory power. The value coefficients should be selected to maximize explanatory power, but the weights should be selected to emphasize the most appropriate market conditions over which to estimate the coefficients. Unfortunately, weight selection in WLS is a well-known problem that typically requires a case-by-case solution because the weights themselves appear in squared-error criterion used to evaluate goodness of fit. Resolving this problem generally requires imposing additional constraints, selecting an alternative goodness-of-fit criterion, or using a weight assignment heuristic.

Application of the Methodology

To illustrate the practical application of these results, consider the valuation of an actual asset using several different modeling approaches (including the one presented in this article). The transaction in question is the 2004 El Paso-Northern Star Generation portfolio transaction in which 80 MW of coal assets and 250 MW of gas assets were sold for $125 million. One common appraisal method would ignore transaction structure and simply calculate a value based on the average transacted price of superficially similar bundles of assets. (26) On this basis, the estimated value would be $145 million-a 20% over-estimation from the actual sale price. In contrast, the fuel type model (summarized in Table 4) provides an estimated transaction value $107 million-a 15% deviation from the actual sale price. However, as has been shown, portfolio transactions (particularly those that include coal and/or hydroelectric capacity) have valuation characteristics that are statistically and significantly different from homogeneous transactions. These differences obscure the valuation process unless they are controlled for objectively. Using the portfolio effects model (summarized in Table 8), a much more accurate valuation of the asset can be provided. The regression approach, which controls for transaction structure, produces a valuation of $122 million-an error of less than 1%.

Conclusions

The sales comparison analysis is an important component of a market value appraisal. The objective of this article was to build on the RDFD 2002 framework previously set forth where transactions for power-generating real property are modeled statistically in order to extract individual asset values from aggregate transaction data. It has been illustrated that a time index could be incorporated for the purposes of retrospective valuation. Additionally, it has been demonstrated that under certain market conditions appraisers must proceed cautiously when disaggregating values from portfolios because transaction structure often influences value. Therefore, not only does transaction heterogeneity matter, but also the extent to which it matters is a function of market conditions.

This article also illustrates the inherent difficulty in identifying and controlling for market conditions and advocates the use of a weighted least squares regression approach to address the decay of transaction informativeness over time. With respect to assessing the extent to which transaction data from the past can be useful in present determinations of market value, this methodology is broadly applicable beyond power-generating assets. However, the estimation of optimal decay structures for each different class of real property remains an area for future research.

(1.) David C. Rode et al., "Inferring Individual Asset Values from Aggregate Transaction Data," The Appraisal Journal (October 2002): 417-425.

(2.) Appraisal Institute, The Appraisal of Real Estate, 12th ed. (Chicago: Appraisal Institute, 2001), 420.

(3.) A Microsoft Excel spreadsheet containing the transaction data used in this analysis is available for downloading at http://www.daimc.com/ AJ_transaction_data.xls.

(4.) The bankruptcy of the Enron energy company represents, at least qualitatively, a watershed event in the power sector. In general, this dividing line can be thought of as the fourth quarter of 2001, since Enron's $1.2 billion write-down of shareholders' equity was announced on October 16, 2001, and the company filed for Chapter 11 bankruptcy protection on December 2, 2001. Immediately after this period, the power sector was placed under severe credit constraints by lenders and investors wary of further surprises. The financial distress surrounding many industry participants prompted numerous transactions, although it was clearly a buyer's market at that point.

(6.) Consider the following example. Suppose asset values as of time 0 were V([t.sub.0]) = {2,4,6} and as of time 1 were V([t.sub.1]) = {1,2,3}, i.e., exactly half their time 0 value. The correlation between the values at the two dates is 1, implying that there has been no change in their common variability ([R.sup.2] = 100%). In spite of this preservation of their relative values, there has been a significant change in the overall level of values.

(7.) The data sample contains three outliers that are discussed in the next section. After removing the outliers, the 2003 and 2004 median APE are 22% and 34%, respectively. The 2003 and 2004 [R.sup.2] are 49% and 72%, respectively.

(8.) Perhaps the most (in)famous example of this occurred when Williams Companies, a gas pipeline and energy trading company, agreed to pay an interest rate of roughly 30% for a one-year $900 million loan that was collateralized by, among other assets, $750 million in cash, see "Williams Paying 30% Interest on 1-Year Loan as It Sells Assets," Houston Chronicle, August 21, 2002.

(9) Because transaction values range from as little as a few million dollars to several billion dollars, the figures in this section are presented on a log scale for ease of viewing.

(10.) Location can influence such factors as the magnitude and frequency of the electricity price spikes that provide revenue, as well as the degree to which such facilities may receive reliability must-run revenues from utilities in return for ancillary services they provide to the grid.

(11.) Once again, nuclear, wind, and geothermal capacity are excluded.

(12.) Appraisal institute, 54.

(13.) Santord Weisberg, Applied Linear Regression (New York: John Wiley, 1985), 210-221.

(14.) The correlation between the time model without outliers and natural gas prices is 0.03. Lagging natural gas prices by one year increases the correlation to 0.30. Statistically speaking, lagging natural gas prices by one year increases natural gas's explanatory power on asset values by almost 9%. This supports the existence of forward-looking investors, who rely on the futures markets for guidance on asset valuations.

(15.) As a proportion of the total number of transactions, there are fewer heterogeneous transactions; however, they tend to be larger in dollar value, number of megawatts included, and number of individual plants.

(16.) In aggregating transaction values, the individual values were weighted by megawatts of capacity. Calculating of the weighted average value is straight-forward. However, to construct confidence intervals around these mean values, a weighted standard deviation also must be calculated. Consider a set of n values, [x.sub.i], i = 1,..., n together with n corresponding weights [[omega].sub.i], i = 1 ,..., n. The weighted mean value is given by

[mu] = [n.summation over i=1] [[omega].sub.i][x.sub.i] / [n.summation over i=1] [[omega].sub.i].

Proceeding with calculation of the standard deviation in the traditional manner, the weighted standard deviation is obtained:

[sigma] = [square root of [n.summation over i=1] [[omega].sub.i]/[([n.summation over i=1] [[omega].sub.i]).sup.2] - [n.summation over i=1] [[omega].sup.2.sub.i] [n.summation over i=1] [[omega].sub.i][([x.sub.i] - [mu]).sup.2]]

(17.) Harry Markowitz, Portfolio Selection: Efficient Diversification of Investments (New York: John Wiley, 1959); and Edwin J. Elton and Martin J. Gruber, Modern Portfolio Theory and Investment Analysis, 5th ed. (New York: John Wiley, 1995). Of course, because investors can aggregate individual assets into portfolios on their own, the existence of a portfolio effect is evidence of some type of market failure. This is not meant to imply that this effect has normative appeal.

(18.) First, it is important to note that the hydro-heterogeneous value is not significantly different from zero. An important separate question, which is not addressed here, involves a negative value coefficient for hydroelectric facilities. Obviously, the bankruptcy shield precludes negative asset values in the extreme. In this case, it is important to realize that the estimated negative value is a conditional value--the value of a hydroelectric facility conditional on it being part of a heterogeneous portfolio. In this case, negative values relating to the asset (e.g., a costly environmental legacy) may transcend the asset itself and attach instead to the owning corporate entity. In such cases, it is not at all unreasonable to expect that a firm might actually pay to extricate itself from an ownership position.

(19.) The August 2004 sale of Texas Genco to a group of private equity funds was completed at a value that was based almost entirely on the firm's coal and nuclear base load plants, which represent less than half of its total capacity. The July 2004 sale of AEP Texas Central's gas- and coal-fired plants to a Sempra/Carlyle consortium was valued solely on the basis of Coleto Creek, the sole coal-fired project in the portfolio. In fact, the gas-fired assets were retired immediately after the transaction closed.

(20.) Sanford Grossman, "The Informational Role of Warranties and Private Disclosure About Product Quality," Journal of Law and Economics (December 1981): 461-489; and Paul Milgrom, "Good News and Bad News: Representation Theorems and Applications," Bell Journal of Economics (Autumn 1981): 380-391. Grossman and Milgrom were among the first to recognize that the very nature of transactions reveals information about the quality of the underlying assets. These results, known collectively as "unraveling theorems," suggest that in the context of the current research it would be preferable to sell a valuable hydroelectric facility on a stand-alone basis. Conversely, if one owns a valueless hydroelectricity facility, it would be preferable to bundle it with other assets. Buyers, however, would realize this strategy and adjust their behavior accordingly, hence the price differential attributable to transaction structure that penalizes hydroelectric plants in portfolios, because such plants are implicitly assumed to be valueless (otherwise they would have been sold on a stand-alone basis).

(21.) Appraisal Institute, 434.

(22.) Business Cycle Dating Committee, "The NBER's Business-Cycle Dating Procedure," National Bureau of Economic Research, July 17, 2003, http:// www.nber.org/cycles/july2003/recessions.html.

(23.) Weisberg, 81-84.

(24.) Traditionally, weighted least squares (WLS) regression is used as a procedure for correcting for heteroskedasticity in data. In such an application, theory suggests the use of a square root weighting scheme. In the current application, a WLS model is used to incorporate a belief that different portions of the sample data have varying degrees of relevance. Although an exponential decay model is used in this article (a common specification for modeling information), it is by no means the only scheme available. The view that confidence in data is tied to the abundance of that data is also common-indeed, it is the traditional Bayesian position. Along such lines, one could certainly make a case for assigning weights proportional to the number of available data points in each year.

(25.) Specifically, t is used to refer to the number of years prior to 2004 (0 refers to 2004, 3_ refers to 2003, etc.). The weights were then determined by the following equation:

[omega](t) = exp([alpha]t)/[7.summation over i = 0]exp([alpha]i)

Naturally, other specifications are possible.

(26.) "Superficially similar" means that comparability would be judged solely on the basis of plant or fuel type without respect to portfolio heterogeneity, transaction structure, or other factors. Additionally, it involves an inaccurate disaggregation of portfolio values into individual asset values.

David C. Rode is managing director of DAI Management Consultants, Inc., in Bridgeville, Pennsylvania. Rode has a BS degree in economics from the Wharton School of the University of Pennsylvania, an MS degree in behavioral decision making and economics from Carnegie Mellon University, and is currently completing his PhD in decision sciences, also at Carnegie Mellon. His principal research interests are in valuation of power generation assets, risk management, and simulation methods. Rode is a member of the Carnegie Mellon Electricity Industry Center and an adjunct professor in Carnegie Mellon's Department of Social and Decision Sciences; his research has appeared in such publications as the Journal of Economic Behavior and Organization, the Journal of Structured and Project Finance, the Journal of Psychology and Financial Markets, and The Appraisal Journal. Contact: drode@daimc.com

Jennifer J. Presto is a financial analyst with DAI Management Consultants, Inc., in Bridgeville, Pennsylvania. Presto has a BS degree in finance and an MS degree in business administration from California University of Pennsylvania. In addition to financial analysis, she performs market research for appraisal and valuation projects. Contact: jpresto@daimc.com

Antonio R. Paez is a quantitative analyst with DAI Management Consultants, Inc., in Bridgeville, Pennsylvania. Paez has a BS in economics from Carnegie Mellon University and an MA in economics from Florida International University. A former research associate with the Carnegie Mellon Electricity Industry Center, his research interests include econometrics and valuation of power generation assets. Contact: apaez@daimc.com

Paul S. Fischbeck, PhD, is professor and department head in the Department of Social and Decision Sciences at Carnegie Mellon University, where he is also a member of the Carnegie Mellon Electricity Industry Center. Fischbeck has degrees from the University of Virginia and the Naval Postgraduate School and received his PhD in industrial engineering and engineering management from Stanford University. Fischbeck's research on risk management and the valuation of industrial sites has appeared in a wide variety of publications including Science, Interfaces, the Journal of Human and Ecological Risk Assessment, and Environmental Science and Technology. Contact: pf12@andrew.cmu.edu

Steve R. Dean, ASA, PE, is managing principal of DAI Management Consultants, Inc., in Bridgeville, Pennsylvania. He has a degree from the U.S. Naval Academy and an MBA from the University of Pittsburgh. Dean is an accredited senior appraiser of the American Society of Appraisers in the specialties of public utilities and machinery and equipment, and he is a licensed professional engineer in Pennsylvania, Michigan, and Hawaii. Contact: sdean@daimc.com
Table 1 RDFD 2002 Model Results

       Regression Statistics

Multiple R                     93.4%
R Square                       87.3%
Adjusted R Square              83.3%
F-Statistic (p-value)    59.91 (0.00)
Observations                       39

                       Coefficients    Standard Error    t-Stat.

Coal MW                    0.72             0.06          11.81
Oil MW                     0.31             0.14           2.27
Gas MW (Calif.)            0.19             0.05           3.51
Gas MW (Non-Calif.)        0.51             0.08           6.71

                       p-Value

Coal MW                0.00000
Oil MW                 0.02923
Gas MW (Calif.)        0.00126
Gas MW (Non-Calif.)    0.00000

Table 2 Summary of Historical Transactions for Power-Generating Real
Property

                                             Total Megawatts
                            Total Trans.       by Fuel Type
        No. of    No. of        Value
        Trans.    Plants    (in millions)    Nuclear    Wind

1997      9         45          $3,915            0       0
1998      35       225         $15,214           33     164
1999      20        73         $11,311        2,426       0
2000      12        22          $5,532        2,424       0
2001      12        23          $5,360        4,560     160
2002      15        19          $2,333        1,530       0
2003      16        32          $3,174        1,236      91
2004      37        78          $6,461          497     106

                   Total Megawatts by Fuel Type
                                                             Total MW
        Geothermal      Coal      Gas       Oil     Hydro     Trans.

1997          0         2,781    11,397      900    1,653     16,731
1998        222        14,891    15,491    3,815    2,634     37,250
1999        722         9,320     9,643    1,587    1,339     25,037
2000          0         4,799     5,924      261        0     13,408
2001         36         3,363     2,820      985        0     11,924
2002          0         1,232     3,855      358        0      6,975
2003          0           856     3,225        0       82      5,490
2004         38           972    20,024       23      682     22,342

Table 3 RDFD 2002 Model Out-of-Sample
Performance

        Median APE    [R.sup.2]

2002       64%           97%
2003       22%           86%
2004       36%           69%

Table 4 Fuel Type Model Results

Regression Statistics

Multiple R            88.5%
R Square              78.3%
Adjusted R
  Square              76.8%
F-Statistic
  (p-value)     77.0 (0.00)
Observations            156

                Coefficients    Standard Error    t-Stat.    p-Value

Nuclear            0.408            0.081          5.001      0.0000
Wind               0.730            1.202          0.607      0.5447
Geothermal         0.673            0.569          1.182      0.2391
Coal               0.646            0.045         14.467      0.0000
Gas                0.237            0.028          8.486      0.0000
Oil                0.113            0.151          0.750      0.4543
Hydro              0.357            0.135          2.646      0.0090

Table 5 Time Model Results

Regression Statistics

Multiple R            91.4%
R Square              83.6%
Adjusted R
  Square              81.9%
F-Statistic
  (p-value)    77.84 (0.00)
Observations            130

                 Coefficients     Standard Error   t-Stat.   p-Value

1997           0.255                  0.042         6.039     0.0000
1998           0.352                  0.028        12.393     0.0000
1999           0.609                  0.035        17.342     0.0000
2000           0.514                  0.055         9.311     0.0000
2001           0.368                  0.095         3.863     0.0002
2002           0.203                  0.109         1.872     0.0635
2003           0.775 *                0.206         3.768     0.0003
2004           0.141 ([dagger])       0.040         3.542     0.0006

* Removal of the El Paso-GS Linden outlier decreases the coefficient to
0.512.

([dagger]) Removal of the outliers from the AEP/Texas Central portfolio
sale and Duke Energy portfolio sale increases the coefficient to 0.303.

Table 6 Stepwise Regression, Combined Model Results

Regression Statistics

Multiple R                        92.3%
R Square                          85.2%
Adjusted R Square                 81.1%
Standard Error             28.80 (0.00)
Observations                         90

                                       Standard
                        Coefficients     Error    t-Stat.   p-Value

1998 Coal                  0.554        0.065      8.549    0.0000
1999 Coal                  0.643        0.076      8.423    0.0000
2000 Coal                  0.658        0.114      5.791    0.0000
2001 Coal                  0.233        0.114      2.042    0.0447
1997 Gas                   0.170        0.049      3.443    0.0009
1998 Gas                   0.363        0.073      5.002    0.0000
1999 Gas                   0.298        0.071      4.223    0.0001
2000 Gas                   0.404        0.118      3.427    0.0010
2001 Gas                   0.445        0.189      2.361    0.0208
2002 Gas                   0.188        0.095      1.970    0.0526
2003 Gas                   0.515        0.270      1.907    0.0604
2004 Gas                   0.255        0.072      3.562    0.0006
1998 Hydro                 0.834        0.291      2.861    0.0055
1999 Hydro                 0.671        0.179      3.742    0.0004
2004 Hydro                 1.257        0.343      3.660    0.0005

Table 7 summary of Homogeneous Transactions

                           Number of Transactions

             1997   1998   1999   2000   2001   2002   2003   2004

Nuclear       0       1      3      2      4      2      1      1
Wind          0       1      0      0      1      0      2      1
Geothermal    0       2      2      0      1      0      0      2
Coal          1       6      2      3      2      2      3      3
Gas           5      12      5      5      2      8      8     23
Oil           0       1      1      0      0      1      0      0
Hydro         0       2      2      0      0      0      1      1
Total         6      25     15     10     10     13     15     31

                Total         Average           95% CI
             Transactions   Value ($/kw)   Range ([+ or -])

Nuclear           14           $367              $147
Wind               5           $797              $452
Geothermal         7           $696              $321
Coal              22           $444              $130
Gas               68           $302               $65
Oil                3           $291              $435
Hydro              6           $716              $337
Total            125

Table 8 Results for the Portfolio Effects Model

Regression Statistics

Multiple R                     87.8%
R Square                       77.2%
Adjusted R Square              75.8%
F-Statistic (p-value)   84.92 (0.00)
Observations                     156

                                       Standard
                        Coefficients     Error    t-Stat.   p-Value

Coal--Heterogeneous        0.834        0.070     11.921    0.0000
Coal--Homogeneous          0.488        0.065      7.496    0.0000
Gas--Heterogeneous         0.222        0.055      4.001    0.0001
Gas--Homogeneous           0.209        0.034      6.154    0.0000
Hydro--Heterogeneous      -0.088        0.174     -0.506    0.6138
Hydro--Homogeneous         0.680        0.209      3.254    0.0014
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