Convexity of Valuation Accuracy Function: Empirical Evidence for the Canadian Economy

Convexity of Valuation Accuracy Function:Empirical Evidence for the

Canadian Economy*

KARIM JAMAL University of Alberta

SHYAM SUNDER Camegie Mellon University

Abstract. We investigated the benefits (more accurate valuations) obtained from the useof more specific measures, such as the use of specific pdce indexes, rather than a generalpdce index, to approximate the current value of histodcal cost accounting numbers. Wefound the valuation accuracy function for the Canadian economy to be highly convex; thatis, marginal gains in accuracy decline sharply as the specificity of pdce indexes increas-es. A few broad pdce indexes yield a relatively accurate valuation of assets, and little accu-racy is gained by using more specific pdce indexes. These results are consistent withempidcal studies conducted in the United States (Hall 1982; Sunder and Waymire 1983;Shdver 1986) and Australia (Tippett 1987).

These results .suggest that Canadian standard setters may be able to achieve theirobjective of getting relatively accurate valuations for industdal assets by adopting a sim-ple and vedftable pdce index methodology, rather than resorting to costly and complicat-ed computations such as those required by Canadian Institute of Chartered Accountants(CICA) Handbook Section 4510.

Resume. Les auteurs 6tudient les avantages (evaluations plus exactes) de I'utilisation plusgrande de mesures specifiques - comme celles d'indices des prix specifiques plutot qued'un indice des pdx gen6ral - pour obtenir la valeur actuelle approximative des donnfiesproduites par la comptabilite au cout d'odgine. Ils constatent que la fonction d'exacdtudede revaluation pour l'&onomie canadienne presente une grande convexite, c'est-^-direque les gains marginaux en exactitude declinent abruptement lorsque la specificity desindices de pdx augmente. Quelques grands indices des pdx livrent une Evaluation rela-tivement exacte de l'actif, et l'on gagne peu d'exactitude en utilisant des indices des pdxplus sp&ifiques. Ces rdsultats sont conformes aux etudes empidques r^alis6es aux Etats-Unis (Hall, 1982 ; Sunder et Waymire, 1983 ; Shdver, 1986) et en Australie(Tippett, 1987).

* Accepted by Michael Cibbins. Comments from two anonymous referees, financial supportfrom the Faculty of Business, University of Alberta, to Karim Jamal for this research andsupport from Richard M. and Margaret Cyert Family Funds to Shyam Sunder are greatlyappreciated.

Contemporary Accounting Research Vol. i I No. 2 (Spring 1995) pp 961-972 ®CAAA

962 Contemporary Accounting Research

Les resultats portent h croire que les responsables canadiens de la normalisationpourraient etre en mesure d'atteindre leur objectif d'6valuations relativement exactes,dans le cas des actifs industriels, en adoptant une methodologie dMndice des prix simpleet verifiable plut6t que de recourir k des calculs on6reux et complexes comme ceux qu'ex-ige le chapitre 4510 du Manuel de I'l.C.C.A.

Accounting practitioners, researchers, users of accounting reports, stan-dard setting bodies, tax authorities, and securities regulators have beenconcemed about the effects of changing prices on historical cost financialstatements (Mattessich 1984; Lemke and Page 1992). The level of inter-est and concem about accounting for changing prices in both the acade-mic and the practitioner communities rises and falls with the prevailingrate of price change. However, the time required to conduct academicresearch or develop generally acceptable accounting standards is so longthat, by the time specific standards are developed or research studies pub-lished, the rate of price changes as well as interest in accounting for suchchanges often subsides.

From the mid-seventies to the early eighties, standard setters inCanada, the United States, and the United Kingdom conducted experi-ments to develop accounting standards for adjusting historical cost finan-cial statements to incorporate the effects of changing prices. The resultsof these experiments indicate that standard setters in all three countrieswere unsuccessful in developing standards that incorporated accurate,verifiable, and easy to implement measurement methods (see, for exam-ple. Price Waterhouse's Review and Perspective on AccountingDevelopments 1992-93).

In the present paper, we investigate the accuracy of using price index-es to obtain a current valuation of assets in the Canadian economy. The useof price indexes to adjust the historical cost of assets is a relatively inex-pensive and verifiable method of approximating the current value of firms'assets. How accurate this approximation is depends not only on how manyand which price indexes are used, but also on the magnitude, variability,covariability, and errors in the price data used to constmct the indexes.

A fimdamental issue of concem to accounting theoreticians, standardsetters, and practitioners is to understand how much cost and effort theyshould invest to obtain accurate measures for approximating the currentvalue of historical cost accounting numbers. At one extreme, we coulduse a very coarse index (such as the producer price index or consumerprice index ) to adjust historical cost numbers. The use of such an indexis inexpensive, simple, and easily verifiable. At the other extreme, wecould hire an appraiser to determine the ctirrent value of each specificasset (or ask Statistics Canada to provide finer partitions of price indexesso that, in the extreme case, we could have a price index for each specificgood in the economy). The use of an appraiser (or constmction of very

Convexity of Valuation Accuracy Function 963

fine indexes) may be very expensive, complex, and highly subjective (ifan appraiser is used).

It is reasonable to expect that, as the measures used for adjusting his-torical cost numbers become more specific (finer), a more accurate valu-ation will be obtained. However, we do not know what the marginal gainsare from increasing the specificity (fineness) of the measures used toadjust historical cost numbers. An understanding of the benefits (moreaccurate valuation) obtained from use of finer measures will providesome guidance regarding the vatue of investing additional resources toconstmct such measures. Insights from portfolio theory in finance suggestthat the marginal gains in accuracy of valuation are likely to declinerapidly as more specific measures are developed.

Prior research in the United States (e.g.. Sunder and Waymire 1983;Hall and Shriver 1990) and Australia (e.g., Tippett 1987) indicate that thestructure of prices in the United States and Australia are such that even asmall number of coarse (i.e., highly aggregated) price indexes can yieldrelatively accurate estimates of the cunent value of assets. In technicalterms, this means that the accuracy function for U.S. and Australianprices is highly convex. A convex accuracy function means that marginalgains in accuracy decline as the specificity of price indexes increases. Afew broad price indexes yield a relatively accurate valuation of assets, andlittle additional accuracy is obtained by using more specific price index-es. The practical implication of this finding is that investment of resourcesto constmct finer measures (or hire an appraiser) is not necessary. In thispaper, we analyze Canadian price data from the same high inflation peri-od studied in the United States and report that these data, too, share theconvexity property.

The convexity of the accuracy function for the Canadian economyshould be of interest to standard setters, preparers, and users of account-ing reports. If a relatively small number of price indexes can adjust his-torical costs to closely approximate the total current value of a firm'sassets, specific price index adjustment methods become serious con-tenders against the more costly and less verifiable "direct" cunent valua-tion methods in which cunent price quotations for individual assets areused (such as CICA [Canadian Institute of Chartered Accountants]Handbook Section 4510). After summarizing the theoretical stmcture andthe empirical evidence from the extant literature in the following section,we describe the Canadian data and estimate the accuracy function fromthis data. The paper concludes with a discussion of our results and theirimplications for accountants and standard setters.

Statistical Theory of ValuationValuation methods, such as historical cost, general price level, and cunentvaluation, have traditionally been thought of as discrete, mutually exclu-


sive, and competing altematives. Detailed arguments on the medts of var-ious approaches for adjusting histodcal cost data have been provided byEdwards and Bell (1961), Chambers (1966), and Sterling (1970). In con-trast to this discrete approach, Ijid (1967, 1968) and Sunder (1978) havedeveloped a statistical approach to valuation. In this statistical approach,methods of valuation constitute a broad spectmm, extending from histor-ical cost accounting at one end to current value accounting at the other.

The statistical approach makes it possible to quantitatively comparethe precision attained by vadous valuation mles in any specified environ-ment. This reframing of the problem makes it possible to use a vadety ofintermediate valuation mles as well as the two extreme end points.Theoretical, as well as empidcal studies within this new framework (e.g..Hall 1982; Sunder and Waymire 1983; Shdver 1986; Lim and Sunder1990, 1991; and Hall and Shdver 1990), have generated importantinsights into, and data about, the statistical properties of inflation pdcedata and valuation mles.

The statistical theory of valuation is based on the idea that the assetsof a firm can be represented as portfolios of goods in the economy. Howaccurately (in the sense of smaller mean square enor) the pdce indexadjusted histodcal costs approximate the cunent value of collections ofassets is a function of the mean vector and covadance matdx of pdcechanges for goods in the economy. Sunder (1978) showed that the meansquared enor (MSE) associated with a valuation mle can be given by:

MSE(R,i) = \(<o'(cr + (i)-P

Where e = vector of unit elements of appropdate length;CO = E(w)', n - vector of relative weight of n assets in the economy,/I = E(r), n - vector;/i = n - vector of squared elements of//;S = E(r - fj)(r - / / ) ' , n X n covadance matdx of relative pdce changes forn assets;(T = n- vector of diagonal elements of S;k = number of pdce indexes used in the valuation mle. The set ofn assetsis panitioned into k nonempty subsets and a pdce index is constmcted foreach subset; cOy,// , and Sya are the subvectors and submatdx, respective-ly corresponding to the M-th of the k subsets;p = number of multinomial tdals by which the bundle of assets for indi-vidual firms is randomly drawn from the economy-wide bundle definedby (o.

For each value of k, let Lj denote the number of distinct ways inwhich a set of n elements can be partitioned into k nonempty subsets. Let

denote the most accurate of the L^ valuation mles (i.e., Rj^* has the


smallest economy-wide average of mean squared error):

Let the partition of the n assets that corresponds to R^* be denoted by11;.*. Thus, for every value of k, there exists a partition n^* which yieldsthe best accuracy (i.e., smallest mean squared error) through the valuationrule R^*. Sunder and Waymire (1983) define H{k) to be the accuracy func-tion that gives the accuracy of the most accurate A;-index valuation ruler

H{k) = MSE{Rt*\ k = 1,2,...«.

We present empirical estimates of H{k) for the Canadian economy.Specifically, we are concerned with whether Hik) is convex in k, as is thecase for U.S. data.

Estimation of accuracy function H(k)Two problems must be solved to estimate H(k) from data. First, we needunbiased estimates of fi and S, the mean vector and covariance matrix,respectively, of relative price changes for the n assets in the economy.Second, we need to identify the index configuration with the best accura-cy for each value ofk = 2,...n-\. The total number of alternative A;-par-titions is very large, and thus, a search algorithm is required to obtain anapproximation of the most accurate H{k), given limited computationalresources. In the present paper, we use an unbiased estimator and a sys-tematic search algorithm from Shih and Sunder (1987) to deal with thesetwo estimation concerns (see Appendix A).

The DataWe chose the Manufacturing Price Index, published by Statistics Canada,instead of the Consumer Price Index. This choice is based on the assump-tion that the assets included in constructing the Manufacturing PriceIndex are more representative of the assets held by business firms than theassets included in the Consumer Price Index.' The Canadian manufactur-ing price indexes have a two-level hierarchy: (1) an overall manufactur-ing price index (general price level index) and (2) 20 specific industryprice indexes.

In order to estimate the mean and covariance matrix of relative pricechanges, we used monthly data from January 1977 to December 1982.The use of data from this time period has three advantages. First, empir-ical data were collected in the United States during this time period, andthus, we can make a direct comparison between Canadian and U.S. data.Second, the relatively high rate of price changes during the 1977 to 1982time period (as opposed to the low rates of price changes being


experienced now) makes such data more relevant to understanding theeffects of high rates of price changes. Third, the data are from a periodwhen the interest in accounting for changing prices was at its peak inCanada and abroad. We can thus provide evidence of what would havehappened if the CICA had chosen to use price indexes instead of the morecomplex procedures specified in CICA Handbook Section 4510 (report-ing tihe effects of changing prices). Complete time series for this periodwere available only for 16 of the 20 price indexes. We used these 16indexes and renormalized the weights assigned to these 16 indexes to sumto 1 (see Table 1).

TABLE 1Components of industry selling pdce indexes: Monthly data 1977-1982 (72 observations)

12345678910111213141516

Code

D500001D511200D511500D513400D514500D516600D519100D523200D524200D527100D529400D532900D537300D541400D544000D54520O

Descdption

Food & BeverageTobacco ProductsRubber & PlasticsLeather IndustdesTextilesKnitting MillsWoodFurniturePaperPdmary MetalsMetal FabdcationMachinery IndustdesElectdcal ProductsNonmetallic IndustdesPetroleum & CoalChemical

Total

OdginalWeights

%

19.0161.0842.417

.8403.369

.8464.5151.5397.809-7.9707.1694.162

- 6.4703.0434.0446.270

80.563

NormalizedWeights

%

23.61.43.01.04.21.05.61.99.79.98.95.28.03.85.0

-_L8100.0

All Components of the indexes for which monthly data were available for the pedodfrom January 1977 to December 1982 are included in the study. The odginal weights ofthese goods and their adjusted weights are shown in the table. The weights are deter-mined by Statistics Canada based on surveys, questionnaires, and collection of data byother means at their disposal.

Empirical Estimates of the Accuracy EunctionIn our estimation of the accuracy function, we suppose that each firm inthe economy consists of randomly drawn bundles of assets (multinomialdistribution with parameter w) of n = 16 assets in varying proportions.Each of these 16 assets represents the bundle of goods included in calcu-


lating the corresponding price index by Statistics Canada. To estimate theaccuracy function H(k), we need to find out for each value of ^, which ofthe A -partitions have the smallest mean squared enor, and the magnitudeof that mean squared enor.

For example, consider k = 5.We need to consider the many ways ofpartitioning 16 assets into 5 nonempty subsets and find out which of thesepartitions yields the smallest mean squared enor. This is a combinatorialproblem, and we cannot apply the usual methods of calculus to find theminima. In addition, the number of possible partitions is very large, andexhaustive enumeration is not feasible. We therefore resort to usingheuristics based on the results of Shih and Sunder (1987) to search for aclose approximation of the accuracy function. Using this method, weidentified 15 partitions (and the conesponding valuation rules ) for eachvalue of k= 2, ... 15 (for both k = 1 and k = 16, the partition is unique).The findings are shown in Figures 1 and 2.

Figure 1 Mean and minimum MSE of 15 valuation rules for each value of ;t

1200

1000-

800-

600-

400-

200-

0 1 2 3 4 5 6 7 8 9 10 11 121314 15 16

Number of Price Indexes in the Valuation Rule (k)


Figure 2 Marginal gains in accuracy of valuation rules for each value of k (average andminimum of 15 valuation rules)

300

I

§

1

200-

1 2 3 4 5 6 7 8 91011 12131415

Number of Price Indexes in the Valuation Rule (k)

For each value of Jt = 1, 2 , . . . t6, the gray line in Figure 1 plots themean squared error associated with the ^-index valuation rules averagedover the 15 different valuation rules. The minimum MSE over the 15 rulesis shown by the dark line. The dark line can be thought of as an estimateof the valuation accuracy function for the Canadian economy. Becausethe accuracy function is defined as the global minimum and our estimateis the minimum of only the 15 valuation rules identified by our heuristic,we know that this estimated function is positively biased. Further experi-mentation with this algorithm (and more computer resources) may enableus to obtain an even better (lower) estimates of the accuracy function.Still, it is clear that the accuracy function is highly convex. Even the aver-age of MSE for the 15 valuation rules (the gray line) is highly convex.

Figure 2 plots the marginal reduction in accuracy from Figure 1 as ^increases. Convexity of the accuracy function requires the marginal curveto be monotonically decreasing. All except one marginal change is


decreasing in k. After the first four or five indexes, the marginal reductionin mean squared enor becomes quite small.

DiscussionThe empidcal evidence presented earlier suggests that the accuracy func-tion of the Canadian economy is highly convex. This implies that the mar-ginal gain in accuracy of valuation declines sharply as the number andspecificity of pdce indexes used for valuation increases. In Canada, as inthe United States, use of a small number of pdce indexes to adjust his-todcal costs will yield most of the benefits of current valuation in thesense of reducing the mean squared enor of valuation. Addition of moreand more indexes yield smaller and smaller additional benefits in theform of reduced mean squared enor.

Due to the high inflation rates expedenced in the 196O's and early197O's, standard setting bodies in Canada, the United Kingdom, and theUnited States issued standards for supplementary disclosure of informa-tion on the effects of changing pdces on financial statements. In Canada,the CICA issued Handbook Section 4510 (reporting the effects of chang-ing pdces) in December 1982, requidng companies to disclose the effectof changing prices on inventory, property plant and equipment, a financ-ing adjustment, and the gain or loss in general purchasing power result-ing from holding net monetary items. Section 4510 was introduced as anexperiment, and companies were encouraged to volunteer to comply withits requirements. In the United States, companies were allowed, but notrequired, to use pdce indexes to estimate the cunent value of their assets.

The experiment was unsuccessful. Compliance rates with Section4510 were very low. In 1986 only four percent of eligible companies vol-untadly complied with Section 4510 (Hanna 1988). The CICA, thus, ter-minated the expedment, and Section 4510 was withdrawn from the CICAHandbook. A sharp decrease in inflation rates dudng the expedment ped-od has been suggested as one explanation for the failure of the expedment(Hanna 1988). This explanation is consistent with the decline in interestin accounting for changing pdces when the rate of pdce changes is low.

A second explanation proposed by Price Waterhouse is that the stan-dard was very complex, and the costs of preparing the data exceeded thebenefits (Pdce Waterhouse, Review and Perspective on AccountingDevelopments 1992-93). This explanation suggests that devising effectiveaccounting standards for changing prices requires that the standardsshould be simple, vedfiable and should yield accurate valuation of assets.

The major result from the cunent study is that valuation mles basedon a few broad pdce indexes yield a simple, vedfiable, and relativelyaccurate valuation of assets. There appears to be relatively little gain fromobtaining more specific asset valuations. This is an important findingbecause it suggests that simple adjustments to histodcal cost data will


result in a relatively accurate valuation of assets, and thus, complicatedcomputations, such as those required by CICA Handbook Section 4510,may not be worth the additional effort required. Our results indicate thatthe structure of price changes in Canada is sufficiently similar to theUnited States and Australia for the key propositions and findings of thestatistical approach to valuation to hold for the Canadian economy.

The analysis conducted in this paper has focused on errors of valua-tion rules that arise from price changes over time. Lim and Sunder (1991)label this source of errors as "price movement errors." Lim and Sunder,however, have identified and modeled a second source of valuation errorsthat arise from errors of measurement in price data used to construct priceindexes (measurement error). Lim and Sunder (1991) then conducted asimulation to show that when both sources of error are taken into account,the number of price indexes needed to construct an efficient valuation ruledecreases even further. This suggests that the conclusions of this paperwould be strengthened if we could obtain data on measurement errors.Because there is no known source of Canadian data on price measurementerrors, such analysis has not been undertaken. However, the theory sug-gests that such additional empirical data, whenever it becomes available,will further strengthen our results.

The results suggest that, in the future, Canadian standard settersshould allow companies to adjust historical cost financial statementsusing highly aggregated price indexes. These adjustments will yield a rel-atively accurate, inexpensive, and verifiable method of approximating thecurrent value of a firms assets. More complicated, costly, or subjectiveadjustment methods should be mandated only if the standard setters candemonstrate that there are large measurement errors in constructing priceindexes for certain classes of assets in the Canadian economy.

Appendix AUnbiased EstimatorThe unbiased estimator of mean squared error constructed by Sunder andWaymire (1984,403) is:

where T represents the number of relative price change observations used

to estimate /J and 2 .

Search Algorithm(1) Collect time series of price data from Statistics Canada Database thathave complete monthly data over the six-year period from January 1977to December 1982. This resulted in selection of data from 16 industries.


(2) Compute time series of relative price changes, r=(P' - P'')/P°.A A

(3) Estimate mean (ft) and variance-covariance matrix (S) of each timeseries of relative price changes.(4) Obtain the relative weight vector w for the Statistics Canada priceindex data base and renormalize relative weights to the sum of weights forthe time series to 1.(5) Prepare an n x n matrix E, where

, A A - 2 2

e,j = (o,a>j(aii -f o}, - 2

(6) To construct a more accurate A;-index system, find a set 7 of (n - k -t-1)goods for which

S 2 e,j .,ju i*j ' IS small.

Shih and Sunder (1987) present theorems (1) to (4), which formalize theintuition that accurate valuation rules require placing assets with largevariances, very large or very small means, large weights, and smallcovariances each in a single good index. These four criteria were appliedheuristically to generate 15 relatively accurate valuation rules for eachvalue of K.

EndnotesI Even if we had used the Consumer Pdce Index data in this study, we doubt if our

results would have been significantly different. However, a definite answer to thisquestion can be provided only by an empidcal analysts of that data.

ReferencesCanadian Institute of Chartered Accountants. Reporting the Effects of Changing Pdces.

CICA Handbook Section 4510, 1982.Chambers, R.J. Accounting, Evaluation and Economic Behavior. Prentice-Hall, 1966.Edwards, E.O., and P. W. Bell. The Theory and Measurement of Business Income.

University of Califomia Press, 1961.Hall, T.W. An Empidcal Test of the Effect of Asset Aggregation on Valuation Accuracy.

Journal of Accounting Research (Spring 1982), 139-151., and K. Shdver. Econometric Properties of Asset Valuation Rules Under Pdce

Movement and Measurement Errors. The Accounting Review (July 1990),537-556.

Hanna, J. Accountitig for Changing Prices: Dead or Alive?. CA Magazine (May 1988),38-43.

Ijid, Y. The Foundations of Accounting Measurements: A Mathematical, Economic andBehavioral Inquiry. Prentice-Hall, 1967.

. The Linear Aggregation Coefficient as the Dual of the of the Linear CorrelationCoefficient. Econometrica (Apdl 1968), 252-259.

Lemke, K.W. and M.J. Page. Economic Determinants of Accounting Policy Choice: TheCase of Current Cost Accounting in the U.K.. Journal of Accounting andEconomics (1992), 87-114.


Lim, S.S. and S. Sunder. Accuracy of Linear Valuation Rules in Industry SegmentedEnvironments: Industry vs. Economy Weighted Indexes. Joumal of Accounting andEconomics (1990), 167-188.

. Efficiency of Asset Valuation Rules Under Price Movement and MeasurementErrors. The Accounting Review (December 1991), 669-693.

Mattessich, R. The Scientific Approach to Accounting. In Modem Accounting Research-History, Survey and Guide, ed. R. Mattessich. CGA Research Foundation, 1984.

Shih, S., and S. Sunder. Design and Test of an Efficient Search Algorithm for AccurateLinear Valuation Systems. Contemporary Accounting Research (Fall 1987), 16-31.

Shriver, K. Further Evidence on the Marginal Gains in Accuracy of Altemative Levels ofSpecificity of the Producer Price Indexes. Joumal of Accounting Research (1986),151-165.

Sterling, R.R. Theory of the Measurement of Enterprise Income. University of KansasPress, 1970.

Sunder, S. Accuracy of Exchange Valuation Rules. Journal of Accounting Research(Autumn 1978), 341-367.

, and G. Waymire. Marginal Gains in Accuracy of Valuation From IncreasinglySpecific Price Indexes: Empirical Evidence for the U.S. Economy. Joumal ofAccounting Research (Autumn 1983), 565-580.

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Convexity of Valuation Accuracy Function: Empirical Evidence for the Canadian Economy