capital theory and depreciation
TRANSCRIPT
British Accounting Review (1998) 30, 39–72
CAPITAL THEORY AND DEPRECIATION∗
KEVIN HOLLAND, HUW RHYS and MARK TIPPETTUniversity of Wales, Aberystwyth
Our concern in the present paper is with an empirical evaluation of the relevanceof published depreciation numbers. We report results of an attempt to assess thecorrespondence between the depreciation rates implied by published financial state-ments and the market-based rates implied by time series variations in corporate equityreturns. Our modelling procedures are based on the ‘Crusonia plant’ construct,developed, in the first instance, by Frank Knight. This leads to a simple capital theorymodel under which firms are regarded as a reservoir of unused (homogeneous) capitalservices. Empirical results based on this construct indicate that estimated market ratesof depreciation have a significant influence on corporate depreciation policies. Simplenon-parametric correlation tests conducted at an industry level show that there is asignificant association between book rates of depreciation and the market rates impliedby an accumulation model based on Knight’s ‘Crusonia plant’ construct.
1998 Academic Press Limited
1. INTRODUCTION
The methods of theoretical finance have provided accountants with a richset of tools with which to investigate theoretical and empirical issues ofimportance to the accounting discipline. Uppermost amongst these is theSharpe (1964) ‘pure exchange’ model of capital asset pricing which is atthe heart of so many significant advances in the discipline that even a partialsummary of them would occupy an article in itself. As a model of firmbehaviour, however, the capital asset pricing model is not without somesignificant limitations. Not least amongst these is that it has little, if anything,to say about supply side issues; that is, how asset returns are shapedby production technologies, physical endowments, socio-legal-economiccustoms and conventions, etc. Whilst attempts have been made to explainhow the capital asset pricing model’s ‘assumed’ return dynamics might be
∗This paper has benefited significantly from the constructive criticism and encouragement of KenPeasnell and an anonymous referee as well as useful discussions with and encouragement from JohnForker and Andy Stark. Since we have not always followed their counsel, however, none can be heldresponsible for what remains. We also gratefully acknowledge financial support of the Research Boardof The Institute of Chartered Accountants in England and Wales.
Correspondence should be addressed to: Mark Tippett, Department of Accounting and Finance,University of Wales, Aberystwyth, SY23 3DY, UK.
Received 30 November 1995; revised 13 August 1996; accepted 6 November 1996
0890–8389/98/010039+34 $25.00/0 ba960051 1998 Academic Press Limited
. .40
consistent with such factors (Merton, 1973, pp. 870–871) significant pro-gress has really only come with the formulation of more general modelswhich explicitly incorporate all relevant supply side considerations.
Of particular significance here is the seminal ‘pure capital’ model of Cox,Ingersoll & Ross (1985a) which uses a ‘representative’ economic agentdevice in conjunction with assumptions about the production technologyemployed by firms and random technological uncertainty to derive a moregeneral intertemporal equilibrium model of capital asset prices. A simpleinterpretation of the model has it that the ‘state variable’ describing tech-nological uncertainty evolves in terms of a continuous-time gamma dis-tribution (Cox, Ingersoll & Ross, 1985b, pp. 390–392), in which case theincremental return to ‘an investment in any production process can benegative, thus reflecting random physical depreciation . . .’ (Cox et al., 1985a,p. 365).1 This suggests the possibility of employing capital theory modelslike that of Cox et al. (1985a) in the empirical evaluation of accountingprocedures; in this instance, the accounting practices in relation to de-preciation (Ball & Brown, 1968; Kaplan & Roll, 1972). In the Cox et al.(1985a, 1985b) model, however, instantaneous changes in technologicaluncertainty, the capital stock, physical depreciation and the spot risk-freerate of interest are all summarized by a common state variable and aretherefore, perfectly correlated (Duffie, 1988, pp. 301–303). Whilst thismight create difficulties for the ‘realism’ of the model, of more seriousconcern are its implications for parameter estimation in an environmentwhere it is known that these variables are less than perfectly correlated.Fortunately, it is a relatively simple task to adapt the Cox et al. (1985b)model so as to avoid this outcome.
Hence, our purpose here is two-fold. First, we adapt the Cox et al.(1985b) model, so that there is a more realistic focus on technologicalshocks due to random physical depreciation. Since this involves what mightbe unfamiliar concepts and ideas from the capital theory literature, we spenda little time explaining why our model takes the form that it does. Much ofthe discussion focuses on why models of the accumulation process arenormally developed in terms of a numeraire commodity, which a rep-resentative economic agent must allocate either to consumption or toinvestment; the so-called ‘Crusonia plant’ construct developed in its mostrefined form in the early work of Frank Knight (1944). However, consistentwith prior empirical work and assumptions on returns generating processes(Samuelson, 1965; Merton, 1973; Cox et al., 1985a), we also developMarkovian models of random physical depreciation and other forms oftechnological uncertainty, demonstrating the principles involved throughthe use of numerical examples, where appropriate. Our principal concern,however, is to report results of an attempt to employ the model we developas a device for assessing the correspondence between the depreciationnumbers contained in published financial statements and the depreciationrates implied by time series variations in corporate equity returns.
41
The paper contains five further sections. The next section lays down theneoclassical model of capital accumulation, on which our analysis is based,whilst section 3 illustrates how this model is used to obtain market-basedestimates of firm depreciation rates. Section 4 describes the samplingprocedures used to identify the data used in our empirical work. This sectionalso briefly examines some time series properties of the Central StatisticalOffice’s ‘Price Indices for Current Cost Accounting’, since our estimatingtechniques require use of real returns and these indices were used to filterout the ‘holding’ gains in nominal equity returns. Section 5 outlines thestatistical procedures used in our study and summarizes the results obtainedfrom them. Section 6 contains our summary conclusions and suggests somepossible avenues for future research.
2. A NEOCLASSICAL MODEL OF CAPITALACCUMULATION2
Hirshleifer (1970, pp. 5, 158–159) notes that capital theory is widelyconsidered to be amongst the most difficult areas of economic theory andso it is usual to cut into the complexity of the general problem by constructinga particular simplified picture of real capital and its role in the productionprocess. An obvious example is the ‘perfect and complete’ markets/in-tertemporally stable and homogeneous expectations assumptions whichunderscores market-based accounting research founded on the Sharpe(1964) and Merton (1973) versions of the pure exchange capital assetpricing (market) model. Amongst other things, these models imply that firmvalue is merely the sum of its constituent assets (there is no ‘synergy’ andso ‘goodwill’ is identically zero) (Merton, 1973, pp. 870–871) and, if returnsare not drawn from an elliptical distribution, that risky assets are inferiorgoods (Ingersoll, 1987, pp. 104–113; Varian, 1984, pp. 158–162). However,such limitations have not hindered the model’s application in practice,probably because alternatives based on more ‘realistic’ assumptions normallydepend on individual preferences and/or endowments or are so complexand intractable that they have limited empirical potential. We are remindedof the oft-repeated dictum (Friedman, 1946, p. 631) that ‘A man who hasa burning interest . . . to learn how the economic system really works . . . isnot likely to stay within the bounds of a method of analysis that denies himthe knowledge he seeks . . .’
Indeed, these considerations probably explain why much of contemporarycapital theory originates from a simplifying tradition which can be tracedback to Ricardo (and perhaps even earlier) (Cohen, 1989, pp. 237–238),but whose popularization in the finance literature stems from the path-breaking works of Irving Fisher (1907, 1930) and Frank Knight (1936a,1936b, 1944).3 Knight (1944, pp. 28–30), for example, notes that:4
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‘The only procedure which seems to be feasible for the analysis of a highlycomplex situation is to simplify it by abstraction to the utmost possible degreeof generality. . . . This . . . is particularly called for in the study of the theoryof investment [where] . . . fundamental economic . . . relationships have es-sentially the same meaning in a “Crusoe”economy that they have in a socialeconomy . . .’
He goes on to suggest that we might, therefore, base our modelling pro-cedures on an:
‘. . . hypothetical economy ha[ving] only one “want” and that this is gratifiedby a single productive agent or resource. The resource must, of course, be ofthe nature of capital . . . We may think of our Crusonia as living on the naturalgrowth of some perennial which grows indefinitely . . . except as new tissue iscut away for consumption . . . wear[s] out or [is] used up . . .’
Hirshleifer (1970, p. 159) suggests that under this ‘Crusonia plant’ constructwe might think of capital as an undifferentiated mass of consumables(Knight’s fungus) that physically grows at some overall proportionate rate.It can, therefore, be regarded as an ‘inventory’ concept of capital—that is,real capital as a stock of finished (consumable) goods.5 The significance ofone-commodity models like this lies in their facility to generalize ‘pureexchange’ results to economies incorporating capital and production in arelatively simple way (Cohen, 1989, pp. 232–233), something that is ofconsiderable importance given the present context. Against this, it is wellknown that results from one-commodity models do not necessarily holdin more general settings (Merton, 1973, p. 874). However, pedagogicalconvenience, combined with the fact that ‘Such simple models . . . haveconsiderable heuristic value in giving insights into the fundamentals ofinterest theory in all its complexities . . . ’ (Samuelson, 1962, p. 193), hasbeen taken as sufficient justification for their application both here and inother well-known applications in the capital theory literature (Merton, 1973,p. 874); Cox et al., 1985a, p. 364).
If we abstract from consumption, the ‘Crusonia plant’ construct impliesthat variations in the capital stock arise from two sources. First, capital isaugmented through production (Knight, 1944, p. 31). Production, however,results in the depreciation or ‘using up’ of pre-existing capital (Knight,1936a, p. 443; 1944, p. 28). Following this line of reasoning, consider afirm which puts S(0) units of capital into service at time zero, and that thisis financed entirely by equity.6 At time h, x1(h)S(0) units of capital remainin service, where x1(h) is the proportion of the firm’s beginning (time zero)capital which remains in service at time h. It follows that the depreciatedvalue of the firm’s capital at time h is R(h)x1(h)S(0), where R(h) is theprice of a unit of capital at time h. Furthermore, suppose a cash flow ofC(h)h is also received, where C(h) is the cash flow on an annualized basis.It then follows that the market value of the firm’s capital at time h will beC(h)h+R(h)x1(h)S(0). We follow conventional practice in assuming that
43
the production process exhibits stochastic constant returns to scale in whichcase the cash flow is (stochastically) proportional to the market value of thefirm’s end of period capital (Cox et al., 1985a, p. 365). It then follows that
C(h)h=[exp {rh+W1(h)}−1]R(h)x1(h)S(0)
where W1(h) is a ‘white noise’7 process with variance parameter r2 and[exp {rh+W1(h)}−1] is the (stochastic) proportionality factor.8 Now sup-pose the firm invests its cash flows in additional capital, in which case itwill have
S(h)=C(h)h+R(h)x1(h)S(0)
R(h)=exp[rh+W1(h)]S(0)x1(h)
units of capital in service at time h.If we now define x2(h) as the proportion of capital which the firm puts
into service at time h which also remains in service at time 2h, then itfollows that the firm will have x2(h)S(h) units of its time h capital remainingin service at time 2h. The value of this capital will be R(2h)x2(h)S(h), whereR(2h) is the price of a unit of capital at time 2h. Furthermore, sincethe firm receives a cash flow during each period which is stochasticallyproportional to the value of the firm’s end of period capital, it follows thata cash flow amounting to
C(2h)h=[exp {rh+W2(h)}−1]R(2h)x2(h)S(h)
will also have been received, where C(2h) is the cash flow received at time2h (on an annualized basis), W2(h) is a white noise process with varianceparameter r2 and [exp {rh+W2(h)}−1] is the stochastic proportionalityfactor relating to the period which begins at time h and ends at time 2h.Now, since the firm invests its cash flows in additional capital, we have thatits time 2h capital will end up being
S(2h)=C(2h)h+R(2h)x2(h)S(h)
R(2h)
=exp[rh+W2(h)]x2(h)S(h)
From the previous paragraph, however, the capital in service at time h was
S(h)=exp[rh+W1(h)]x1(h)S(0)
. .44
and so the time 2h capital may be restated as
S(2h)=exp[rh+W2(h)]x2(h)S(h)
=exp[rh+W2(h)]x2(h) exp[rh+W1(h)]x1(h)S(0)
or
S(2h)=exp[2rh+W1(h)+W2(h)]x2(h)x1(h)S(0)
=expC2rh+]2
j=1
Wj(h)DS(0)\2
j=1
xj(h)
Continuing this process to times 3h, 4h, 5h and so on shows that after nperiods, which we denote as time t=nh, the firm will have
S(t)=S(nh)=expCrt+]n
j=1
Wj(h)DS(0)\n
j=1
xj(h)
units of capital in service, where Wj(h) is the white noise stochastic com-ponent of the cash flow proportionality factor in the j th period and xj(h)denotes the proportion of capital in service at time ( j−1)h which remainsin service at time jh.9 Since the Wj(h) are white noise terms with the samevariance parameter, it follows that
]n
j=1
Wj(h)=W(t)
is normally distributed with zero mean and variance r2t (Egginton et al.1989, p. 265).10 We thus have that the firm’s time t capital will be
S(t)=exp[rt+W(t)]S(0)\n
j=1
xj(h)
Taking logarithms across this expression shows that the ‘real’ (continuouslycompounded) return to the firm’s capital over the time period [0, t] will be:
logS(t)S(0)
=rt+W(t)+]n
j=1
log[xj(h)]
where Rnj=1 log[xj(h)] is the accumulated (real) continuously compounded
45
depreciation rate on the firm’s capital.11 Hence, further progress dependson more concrete assumptions about the nature of this variable.
Here, Pollard & Tippett (1994) and Rhys et al. (1994) have proposed adiscrete time depreciation model, under which a firm is endowed withS(0)=j units of capital at time zero. The probability of there being S(h)=k≤j units of this capital remaining in service at time h is given by:
pj;k=qC(k+q)C( j+1)
C(k+1)C( j+q+1)
where q>0 is a parameter and C(m)=(m−1)!=(m−1).(m−2).. . . .1 is thegamma function of mathematical statistics (Freund & Walpole, 1987, p.210). Rhys et al. (1994, p. 165) show that this transition mechanism impliesthat a firm which puts S(0) units of capital into service at time zero willhave, on average,
E0[S(h)]=A qq+1B
h
S(0)
units of capital remaining in service at time h, where E0( .) is the expectationsoperator taken at time zero. Hence, if we define the depreciation rate asthe units of capital which are taken out of service in proportionate termsover the period [0, h], or
d0;h=S(h)−S(0)
S(0)
it then follows that the expected depreciation rate is
E0[d0;h]=E0[S(h)]−S(0)
S(0)=A q
q+1Bh
−1
This is the formula for ‘reducing balance’ depreciation encountered in mostintroductory accounting texts (Baxter, 1981, pp. 27–28).12
We can illustrate this model for a firm which puts S(0)=j=3 units ofcapital into service at time zero with an expected annual depreciation rateof 20%. Since h=1, this implies
E0[d0;1]=q
q+1−1=
−1q+1
=−0·20
or q=4, and so capital depreciates in accordance with the following four-state transition matrix:
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P=A1 0 0 015
45 0 0
115
45
1015 0
135
435
1035
2035B
Here, the fourth row gives the probability of having k=0, k=1, k=2 ork=3 units of capital in service at time h, given that there were j=3 unitsof capital in service at time zero. Similarly, the second row gives theprobability of having k=0 or k=1 units of capital in service at time h, giventhat the firm had j=1 unit of capital in service at time zero. The first rowrecords the fact that if the firm starts a period with no units of capital inplace, it will also have no capital in place at the end of the period. Theother rows are similarly interpreted. Hence, if a firm puts j=3 units ofcapital into service at time zero, then the probability of all k=3 units ofcapital remaining in service at time h is
p3;3=4(3+4−1)!(3+1−1)!
(3+1−1)!(3+4+1−1)!=
47=
2035
Similarly, if the firm has j=3 units of capital in service at time zero, thenthe probability of k=1 unit of capital remaining in service at time h is
p3;1=4(1+4−1)!(3+1−1)!
(1+1−1)!(3+4+1−1)!=
435
If, on the other hand, the firm had j=2 units of capital in service at timezero, then the probability of having both units of capital in service at timeh is
p2;2=4(2+4−1)!(2+1−1)!
(2+1−1)!(2+4+1−1)!=
23=
1015
Similarly, the probability of the firm having j=2 units of capital in serviceat time zero and k=0 units of capital in service at time h is
p2;0=4(0+4−1)!(2+1−1)!
(0+1−1)!(2+4+1−1)!=
115
The remaining conditional probabilities are similarly computed. Theseprobabilities imply that the initial capital endowment will eventually betaken out of service entirely and that the probability of this occurringincreases with the passage of time. Amongst other things, this reflects the
47
increased probability of obsolescence implied by the passage of time (Baxter,1981, p. 13).13
Rhys et al. (1994) show that if the initial capital endowment, S(0), is‘large’, then the probability density for the pj,k collapses to:
f(x)=qxq−1[−q log(x)]h−1
C(h)
where 0<x(h)<1 is the proportion of capital in service at time zero whichremains in service at time h. This is the log-gamma distribution withparameters q and h, so called because the substitution z=log(x) transformsf(x) to the gamma distribution of mathematical statistics (Freund & Walpole,1987, pp. 211–212).14 Here, we should note that Ijiri & Kaplan (1969, pp.747, 755) report that this distribution or transformations of it, provides‘. . . an excellent approximation to the . . . life characteristics of 52 types ofphysical property . . .’ examined by Kurtz (1930) and so, it would appearthat this is a reasonable distribution from an empirical point of view as well.We can illustrate its distributional properties by letting the expected annualdepreciation rate be 8%= 2
25 so that, as noted in the previous paragraph, wethen have
E0[d0;1]=−1q+1
=−225
or q=11.5. We choose this rate of depreciation because the empirical workreported below shows this to be a reasonable estimate for the depreciationrate of the ‘average’ firm in a large sample of publicly listed UK corporations.Panels (a)–(f) of Figure 1 plot the probability distributions correspondingto q=11.5 when h=1, 5, 10, 15, 25 and 35 years, respectively. Note thatthe bulk of the probability density gradually moves from the extreme rightto the extreme left of the graph, reflecting the fact that a gradually decliningproportion of capital remains in service with the passage of time.
The analysis of Rhys et al. (1994, pp. 164–166) can also be used to showthat the log-gamma probability density is based on Bose–Einstein statisticsand, therefore, satisfies a crucial additivity requirement, something we nowbriefly develop (Ijiri & Simon, 1975).15 We have previously noted that−log[xj(h)] is distributed as a gamma variate with parameters q and h andso, if the xj(h) are independent, then Freund & Walpole (1987, pp. 214,266) show that
−]n
j=1
log[xj(h)]=−log[x(nh)]=−log[x(t)]
is also distributed as a gamma variate but with parameters q and nh=t.
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1
12
Proportion in use0.4
10
8
6
4
2
0.1 0.2 0.3 0.5 0.6 0.7 0.90
Rho = 11.5 Time = 1
Probability density
0.8
1
3.5
Proportion in use0.4
3
2.5
2
1.5
1
0.1 0.2 0.3 0.5 0.6 0.7 0.90
Rho = 11.5 Time = 5
0.8
0.5
Figure 1
49
1
3.5
Proportion in use0.4
3
2.5
2
1.5
1
0.1 0.2 0.3 0.5 0.6 0.7 0.90 0.8
1
4.5
Proportion in use0.4
4
3.5
3
2.5
2
0.1 0.2 0.3 0.5 0.6 0.7 0.90
Rho = 11.5 Time = 15
0.8
0.5
0.5
1
1.5
Rho = 11.5 Time = 10
Figure 1
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1
9
Proportion in use0.4
8
7
6
5
4
0.1 0.2 0.3 0.5 0.6 0.7 0.90
Rho = 11.5 Time = 25
0.8
1
18
Proportion in use0.4
16
14
12
10
8
0.1 0.2 0.3 0.5 0.6 0.7 0.90
Rho = 11.5 Time = 35
0.8
2
3
2
1
4
6
Figure 1. The log-gamma probability density with parameters rho and time.
This, in turn, implies that the real return to an investment in the firm maybe restated as:
logS(t)S(0)
=rt+W(t)+]n
j=1
log[xj(h)]
=rt+W(t)+log[x(t)]
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Now we show above that W(t) is normally distributed with zero meanand variance r2t. In the Appendix, we also show that log[x(t)] has mean(−t/q) and variance (t/q2).16 It thus follows that the mean real continuouslycompounded return to equity is given by
E0 ClogS(t)s(0)D=Ar−1
qB t
where E0( .) is the expectations operator taken at time zero. Hence, consistentwith the ‘Crusonia plant’ construct, the expected real return to equity isthe outcome of two competing forces—the growth in capital as captured bythe parameter r, and the depreciation of capital as captured by the parameterq. If we also assume that W(t) and log[x(t)] are uncorrelated, it followsthat the variance of the continuously compounded return is given by
Var0 ClogS(t)S(0)D=Var0[W(t)]+Var0[log{x(t)}]
=Ar2+1q2B t
where Var0(.) is the variance operator taken at time zero (Freeman, 1963,p. 44); Freund & Walpole, 1987, pp. 166–167). It then follows that thevariance of the return is also the sum of technological uncertainty causedby capital growth and technological uncertainty as reflected in the randomrate of depreciation.17
3. ESTIMATION PROCEDURES
The analysis to date is based on the assumption that the firm’s activitiesare financed entirely by equity. We can generalize the model to take accountof any debt financing by using the analysis of Modigliani & Miller (1958).Here, we recall that the discrete return to the equity of an unlevered firm,Yu, over the period [0, t] is
Yu=Pu(t)−Pu(0)
Pu(0)
where Pu(t)=R(t)S(t) is the value of equity at time t, R(t) is the price of aunit of the capital good at time t, S(t) is the number of units of capitalwhich the firm has in service at time t and Pu(0) is the value of equity at
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time zero. Now, Modigliani & Miller (1958, p. 272) show that the equilibriumrelationship between the return on the equity of a levered firm, YL, and thereturn on the equity of an ‘equivalent’ unlevered firm will be:
YL=Yu+(Yu−r)BL
PL
where BL is the value of the levered firm’s debt, PL is the value of the leveredfirm’s equity and r is the interest cost of the levered firm’s debt. Hence, ifwe know a levered firm’s debt to equity ratio, the interest cost of its debtand the return on its equity, we can determine the discrete return thatwould have been earned on an investment in the equity of an equivalentunlevered firm, namely:
Yu=YL+r .BL
PL
1+BL
PL
It then follows that the nominal continuously compounded return to aninvestment in the equity of an equivalent unlevered firm is given by:
logCPu(t)Pu(0)D=log(1+Yu)=logC1+
YL+r .BL
PL
1+BL
PL
DUsing the fact that
logCPu(t)Pu(0)D=logC R(t).S(t)
R(0).S(0)D=logCR(t)R(0)D+logCS(t)
S(0)Dand so
logCS(t)S(0)D=logCPu(t)
Pu(0)D−logCR(t)R(0)D
appropriate substitution then shows that the real continuously compoundedreturn to an investment in the equity of an equivalent unlevered firm is:
53
logC1+YL+r .BL
PL
1+BL
PLD−logCR(t)
R(0)D
=rt+W(t)+log[x(t)]=logC(1+Yu)R(0)R(t) D
Taking expectations through this expression implies that the theoreticalmean and variance of the real return to an equivalent all equity financedfirm are given by
E0Clog(1+Yu)R(0)
R(t) D=Ar−1qB t
and
Var0Clog(1+Yu)R(0)
R(t) D=Ar2+1q2B t
respectively (Rhys et al., 1994, p. 168). This result can be used in conjunctionwith the method of moments and certain other simplifying assumptions toobtain consistent estimates of the parameters r and q. Basically, the methodof moments estimates parameters by equating the empirically estimatedmoments with the theoretical moments (Freund & Walpole, 1987, pp.348–350).
Nominal periodic returns for publicly listed corporations are easily ob-tained from the Datastream system. Similarly, we can proxy the inflationrate for capital equipment by using the Central Statistical Office’s ‘PriceIndices for Current Cost Accounting’, which are also available on the Datastreamsystem. To illustrate, Table 1 contains share price, dividend, interest anddebt/equity information relating to British Aerospace PLC for the yearending 31 December 1990. Hence, the nominal (discrete) return to equityfor the month of March is
496·11+20·53−491·19491·19
=0·05181
or YL=5·18%. The annual interest expense (Datastream item #153 plusitem #148) expressed as a percentage of the average (of beginningand ending) book value of debt (Datastream item #391 plus item #389
. .54
TABLE 1
BRITISH AEROSPACE PLCPrice, dividend, inflation and return statistics for the year ending 31 December 1990
Share Nominal Unlevrd Plant Plant RealDate price Dividend return return index infl rate return
31–12–89 595·53 114·231–01–90 551·23 −0·07439 −0·01604 114·4 0·0017 −0·0177928–02–90 491·19 −0·10892 −0·02438 114·7 0·0026 −0·0270031–03–90 496·11 20·53 0·05181 0·01386 115·6 0·0078 0·0060530–04–90 509·89 0·02778 0·00824 115·2 −0·0035 0·0117031–05–90 541·39 0·06178 0·01619 114·9 −0·0026 0·0188030–06–90 520·72 −0·03818 −0·00737 114·3 −0·0052 −0·0021331–07–90 550·25 0·05671 0·01501 113·9 −0·0035 0·0185131–08–90 535·49 −0·02682 −0·00467 113·8 −0·0009 −0·0037930–09–90 554·19 12·51 0·05828 0·01537 114·1 0·0026 0·0127431–10–90 529·58 −0·04441 −0·00886 114·5 0·0035 −0·0123630–11–90 523·67 −0·01116 −0·00095 114·8 0·0026 −0·0035631–12–90 514·81 −0·01692 −0·00231 116·2 0·0121 −0·01443
Interest 0·00222 q 23·15841 Mean −0·00111
Debt/ 3·22441 Dep rate 0·04139 Variance 0·00022equity
minus item #305) is 0·02698. The monthly equivalent of this figure is r=(1·02698)1/12−1=0·00222 or about 0·22%. Furthermore, the average ofthe beginning and end of year (book) debt/equity (item #305) ratio is (BL/PL)=3·22441. These figures thus imply that during the month of Marchthe nominal continuously compounded return to an equivalent pure equityfinanced firm would have been:
logC1+YL+r .BL
PL
1+BL
PLD
=logC1+0·05181+0·00222∗3·224414·22441 D=0·01386
or about 1·39%. Over this period the Central Statistical Office’s Plant andMachinery Index for the Aerospace Industry (UKPBAEROF) increasedfrom 114·7 to 115·6, implying a continuously compounded inflation ratefor the firm’s capital equipment of [115·6/114·7]=0·00782. Hence, the real
55
return for the month ending 31 March (after adjusting for leverage) is
logC(1+Yu)R(0)R(t) D=logC1+
YL+r.BL
PL
1+BL
PL D−logCR(t)R(0)D
=0·01386−0·00782=0·00605
or about 0·605%. Further calculation (as contained in Table 1) shows thatthe empirical mean and variance of the unlevered real return for the yearare−0·0011058 and 0·0002232, respectively. Now, since the expected realreturn to an unlevered firm is
E0Clog(1+Yu)R(0)
R(t) D=Ar−1qB t
whilst its variance is
Var0Clog(1+Yu)R(0)
R(t) D=Ar2+1q2B t
the method of moments implies that
Ar−1qB
12≈−0·0011058 and
Ar2+1q2B
12≈0·0002232
or upon simultaneous solution, q≈23·15841. Hence, the market-basedestimate of the depreciation rate, which we proxy by the expected one-period rate, is
−11+q
≈−1
1+23·15841≈−0·04139
or about 4·14% per annum.18
. .56
It is interesting to compare this estimate with figures taken from BritishAerospace’s financial statements. According to the Datastream system,British Aerospace’s accounts record a depreciation charge of £280 millionfor the year ending 31 December 1990. The average (of the beginning andending) gross book value of British Aerospace’s fixed assets for the yearended 31 December 1990 amounts to £4,787 million. The book depreciationrate is therefore (280/4,787)=0·0585 or about 5·85%. Hence, for the periodconsidered, it would appear that there is a significant difference betweenthe market-based (4·14%) and book depreciation (5·85%) figures. However,a conclusion on whether this is generally the case across all firms awaits theempirical results to be reported below.19
Before concluding this section, however, we should note that there areobjections which might be raised to our estimating procedures. Corporatebalance sheets, for example, include both depreciable and non-depreciableassets (e.g. stocks and work in progress, trade debtors, etc.) and, in so faras modelling procedures do not make provision for this, it might be arguedthat they are based on a potentially distorted view of the capital accumulationprocess.20 Here, we should note, however, that our estimating proceduresmust of necessity be:21
‘. . . very incomplete. It is never possible to deal . . . with . . . the vast complexityof factors entering into a normal real situation . . . the [analytical] methoddepends on the fact that in . . . problem situations certain elements arecommon . . . The laws of these few elements, therefore, enable us to reach anapproximation to the law of the situation as a whole . . . When the number offactors taken into account . . . becomes too large, the process rapidly becomesunmanageable and errors creep in, while the results lose in generality ofapplication more significance than they gain by the closeness of the ap-proximation to the fact given case. It is better to stop dealing with elementsseparately before they get too numerous and deal with the final stages of theapproximation by applying corrections empirically determined’ (Knight, 1921,pp. 4–8).
The important point being made here is that our ‘Crusonia plant’ assumptiontakes it that capital is uniformly homogenous and in so far as there is doubtabout the ‘realism’ of this assumption we can either vary our model in thehope that we might improve its descriptive validity or we can adopt Knight’scounsel of ‘. . . dealing . . . with the final stages of the approximation byapplying corrections empirically determined’. Given the significant com-plications that would arise as a result of efforts to increase the ‘realism’ ofour model, it seems sensible, in the first instance, to work the simplifiedmodel out to its logical conclusions and, then, to attempt such correctionsas appear to be necessary, given the empirical results obtained from themodel.
57
4. DATA AND SAMPLE DETAILS
The sample used in our study consists of all companies with:
• a complete monthly history of equity prices and the required annualaccounting data on the Datastream system for the 4-year period endingwith the 1992 balance sheet reference date;
• a complete record of dividends, including ex-dividend dates, on theDatastream system for the 4-year period ending with the 1992 balancesheet reference date. When a dividend record did not exist on theDatastream system, the non-payment of dividends was confirmed byreference to the company’s accounts. This approach was used to avoidany biases which might arise from ignoring the payment of dividends;and
• a complete history of annual interest expense, book value of debt,depreciation expense and gross book value of fixed assets on theDatastream system for the 4-year period ending with the 1992 balancesheet reference date.
A total of 690 companies satisfied the selection criteria. Table 2 providesdetails of the sample’s composition by industry type, the number of com-panies in each industry, the average balance sheet value of total assets andaverage sales, both for the 4-year period ending with the 1992 balance sheetdate and the standard deviation for each industry of both the average balancesheet asset values and sales. Note that the sample provides good coverageby any of these measures. Having identified the basic sample, the Datastreamfiles were then accessed to extract equity prices (adjusted for rights, bonusand other capital issues) as well the basic accounting data required for thestudy. The depreciation rate was defined as the annual depreciation charge(Datastream item #136) divided by the average gross book value of thecorporation’s fixed assets (Datastream item #327+item #328).
The procedures used to obtain the market’s estimate of the depreciationrate were demonstrated using British Aerospace as an example in Table 1and its surrounding discussion. It will be recalled that the inflation rate onthe company’s capital equipment was a critical input to these calculationsand we now describe in more detail how this statistic was estimated. Firstly,each company’s FT Actuaries Industry classification was obtained. In thecase of British Aerospace, for example, this was the Engineering-AerospaceIndustry (Code AERSP). The FT Actuaries Industry classification was thencross-matched to one of the 39 Price Indices for Plant and MachineryBought as Fixed Assets prepared by the Central Statistical Office (CSO).Table 3 contains further details of these indices. For British Aerospace, theappropriate index was that for the Aerospace Industry (Code UKPBA-EROF). For most of the companies included in our study, the cross-matching procedure resulted in a relatively clear-cut choice of CSO index.However, there were a few instances where the choice of index was more
. .58
TABLE 2
Basic sample properties as at the end of (for) balance sheet date ending 1992
Industry No. Sales Std dev Tot ass Std devno. Description companies £ (thous) £ (thous) £ (thous) £ (thous)
1 Building materials 37 433,885 691,108 451,400 774,3202 Metals & metals 23 184,955 371,017 137,951 265,262
forming3 Textiles 59 119,374 378,846 111,641 379,3634 Contracting & 48 195,069 336,997 166,435 259,121
construction5 Miscellaneous 91 204,347 912,807 254,140 1,566,203
manufacturing6 Engineering 71 137,099 206,083 137,445 239,2377 Oil & gas 11 306,424 681,987 497,339 813,8028 Electronics 58 206,968 775,177 212,070 909,5699 Food 29 1,704,547 4,212,305 1,179,090 3,067,334
manufacturing10 Electricals 29 167,502 625,047 128,809 454,04311 Other industrials 24 636,897 1,460,772 695,987 1,560,75212 Hotels & leisure 27 258,888 772,344 319,059 889,59213 Motors 22 234,980 416,504 162,558 458,98614 Brewers & distillers 24 944,353 2,054,740 1,410,231 2,721,74715 Food retailing 20 1,429,706 2,070,532 719,301 1,131,48116 Engineering- 5 2,258,857 3,897,565 2,468,640 4,320,206
aerospace17 Health & 17 570,463 850,672 710,724 1,176,296
household18 Stores 44 712,682 1,162,813 578,279 974,25319 Conglomerates 8 1,312,324 2,194,276 3,576,911 4,275,89620 Media 1 18,245 — 19,455 —21 Packaging, paper & 4 71,790 89,693 57,635 80,960
print22 Business services 4 55,394 47,884 54,871 63,37623 Chemicals 17 1,052,550 2,961,392 1,314,632 3,722,56824 Transport 15 866,502 1,622,760 1,177,672 2,017,41425 Telephone network 2 7,437,063 5,032,188 16,339,879 11,784,119
Total/ 690 482,160 1,535,729 509,869 1,888,713averages
problematic. For example, companies classified into the FT ActuariesTransport Industry can be drawn from shipping, transport and freight and/or airlines. The only CSO index related to this area is that for Road Transport.However, by definition, this excludes shipping and airline transport. Hence,it is possible that the CSO indices might provide a poor measure of thecapital costs of companies operating in these industries. Fortunately, theempirical evidence indicates that the CSO indices are all highly correlated
59
TABLE 3
Central Statistical Office—price indices for plant & machinery bought as fixed assets
Index no. Mnemonic Index name
1 UKPBMANFF All manufacturing industry2 UKPBAGFFF Agriculture, forestry and fishing industry3 UKPBCOALF Coal extraction industry4 UKPBCOKEF Coke ovens industry5 UKPBMOIXF Extraction of mineral oil and natural gas6 UKPBELECF Production and distribution of electricity7 UKPBPGASF Public gas supply industry8 UKPBNFMTF Non-ferrous metals industry9 UKPBBLDMF Building materials and mineral extraction
10 UKPBPOTTF Pottery and glass industry11 UKPBCHMCF Chemicals industry12 UKPBMMFBF Production of man-made fibres industry13 UKPBMGNEF Metal goods not elsewhere specified14 UKPBMENGF Mechanical engineering industry15 UKPBDPRCF Data processing and office machines16 UKPBELENF Electrical and electronic engineering17 UKPBMVEHF Motor vehicles industry18 UKPBSHPBF Shipbuilding industry19 UKPBOVEHF Other vehicles industry20 UKPBAEROF Aerospace industry21 UKPBIENGF Instrument engineering industry22 UKPBFODIF Food industry23 UKPBSPRTF Spirit distilling and compounding industry24 UKPBODNKF Other drinks industry25 UKPBTOBCF Tobacco industry26 UKPBTXTLF Textiles industry27 UKPBLETHF Leather, footwear and clothing industry28 UKPBTIMBF Timber and wooden furniture industry29 UKPBPULPF Pulp, paper and board industry30 UKPBPDPBF Paper products, printing and publishing31 UKPBRUBBF Rubber and plastics industry32 UKPBOMANF Other manufacturing industry33 UKPBCONSF Construction industry34 UKBWHXOF Wholesaling excluding oil industry35 UKPBOILWF Oil wholesaling industry36 UKPBRETRF Retailing and repairs industry37 UKPBHOTLF Hotels and catering industry38 UKPBROADF Road transport industry39 UKPBPOSTF Postal services and telecommunications
and that real returns are relatively insensitive to which index is used incalculating them.22 In any event, Table 4 provides further details on howthe FT Actuaries Industry classifications were cross-matched with the CSOindices.
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TABLE 4
Cross-matching of FT actuaries industrial classification and Central Statistical Office price indicesfor plant & machinery
FT actuaries industry Central Statistical Office—price Indexclassification index no.
Brewers and distillers (BRDIS) Spirit distilling and compounding 23industry
Building materials (BLDNG) Building materials and mineral 9extraction
Business services (BSERV) Data processing and office machinery 15Chemicals (CHMCL) Chemicals industry 11Conglomerates (CONGL) All manufacturing industry 1Contracting and construction Construction industry 33(CONTR)Electricals (ELTCA) Electrical and electronic engineering 16Electronics (ELTNC) Electrical and electronic engineering 16Engineering aerospace (AERSP) Aerospace industry 20Engineering general (MEENG) Mechanical engineering industry 14Food manufacturing (FDMFG) Food industry 22Food retailing (FDRET) Food industry 22Health and household products Chemicals industry 11(HELTH)Hotels and leisure (LESUR) Hotels and catering industry 37Media (MEDIA) Paper products, printing and 30
publishingMetals and metal forming (METFM) Metal goods not elsewhere specified 13Miscellaneous manufacturing Other manufacturing industry 32(MISCS)Motors (MOTGP) Motor vehicles industry 17Oil and gas (OILCO) Extraction of mineral oil and natural 5
gasOther industrial (OINMT) Other manufacturing industry 32Packaging and paper (PKPAP) Paper products, printing and 30
publishingStores (STORE) Retailing and repairs industry 36Telephone network (PHONE) Postal services and 39
telecommunicationsTextiles (TEXTL) Textiles industry 26Transport (TRNSP) Road transport industry 38
5. EMPIRICAL RESULTS
The procedures illustrated in the previous section were then used to estimatethe market rate of depreciation for the 4-year period ending with the 1992balance sheet reference date for each of the 690 companies composing oursample. Eight companies returned complex-valued estimates of the marketdepreciation rate and could not be used further.23 For the 682 companies
61
TABLE 5
Estimated average market rate of depreciation and average book rate of depreciation by industry
Industry No. Book Std Market Stdno. Description companies rate dev rate dev
1 Building materials 37 0·07133 0·02885 0·07804 0·040622 Metals & metals 23 0·06969 0·03363 0·08850 0·04934
forming3 Textiles 59 0·07038 0·02487 0·11297 0·057434 Contracting & 47 0·08917 0·03478 0·10748 0·04368
construction5 Miscellaneous 90 0·07890 0·03311 0·09378 0·05805
manufacturing6 Engineering 71 0·06930 0·01747 0·08260 0·051147 Oil & gas 11 0·07424 0·05121 0·08365 0·072838 Electronics 56 0·11127 0·04233 0·10794 0·053649 Food 29 0·08079 0·03388 0·06073 0·04295
manufacturing10 Electricals 29 0·08203 0·02293 0·08646 0·0552511 Other industrials 24 0·08037 0·02406 0·08773 0·0526812 Hotels & leisure 27 0·08232 0·06918 0·08798 0·0581413 Motors 22 0·06415 0·02875 0·09407 0·0514314 Brewers & distillers 23 0·04481 0·01713 0·08151 0·0577015 Food retailing 20 0·04314 0·01772 0·05003 0·0414016 Engineering- 5 0·05448 0·03466 0·05592 0·02998
aerospace17 Health & 16 0·07560 0·05316 0·06140 0·06135
household prod18 Stores 43 0·06767 0·02650 0·07795 0·0574919 Conglomerates 8 0·07216 0·02951 0·07578 0·0651420 Media 1 0·21548 — 0·14150 —21 Packaging, paper & 4 0·08091 0·01336 0·08936 0·06453
print22 Business services 3 0·08372 0·02831 0·08417 0·1046623 Chemicals 17 0·08368 0·01577 0·07385 0·0479524 Transport 15 0·06479 0·03066 0·06384 0·0466025 Telephone network 2 0·06361 0·00038 0·05789 0·00848
Total/ 682 0·07896 0·02968 0·08341 0·05302averages
which remained, Table 5 compares the average estimated (continuouslycompounded) market rate with the average (continuously compounded)book rate of depreciation by industry. The book rate of depreciation foreach corporation is defined as the average of the four annual rates computedfrom the published financial statements ending with the 1992 balance
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0.25
0.16
Book rate of depreciation
0.1
0.14
0.12
0.1
0.08
0.06
0.05 0.150 0.2
0.02
0.04Mar
ket
rate
of
depr
ecia
tion
Figure 2. Book rate vs market rate of depreciation (by industry).
sheet reference date.24 Figure 2 provides a graphical representation of therelationship between the two statistics. The average of the 37 estimateddepreciation rates composing the Building Materials Industry, for example,is 0·07804 or about 7·80% (per annum). The standard deviation of these37 estimates is about 4·06%. The average book rate for the firms composingthe Building Materials Industry is 7·13% with a standard deviation of2·89%. For the 682 companies comprising the sample as a whole, theaverage market depreciation rate amounts to 8·34% with a standard deviationof 5·30%. The average book rate for the sample as a whole is 7·90% witha standard deviation of 2·97%. The other entries in this table are to besimilarly interpreted.
An important characteristic of Table 5 is that the standard deviation ofthe estimated market rates are about twice those of the book rates ofdepreciation. This suggests that market rates might be affected by thefamiliar ‘errors in variables’ problem (Johnston, 1984, pp. 428–435), andthis, in turn, has important implications for the statistical analyses whichcan legitimately be applied to them.25 We attempted to address this issue byemploying non-parametric statistical procedures. In general, non-parametrictests are based on ordering relationships; for example, the ranks of a set ofnumbers rather than the numbers themselves (Conover, 1980, pp. 64–67).Intuition suggests that such procedures ought to reduce the impact ofmeasurement errors, since it is conceivable that underlying ordering re-lationships will still be preserved, despite the presence of errors in the
63
underlying data. Finally, if measurement errors possess the usual ‘fair game’properties, it is likely that industry averages will be less affected by sucherrors when compared to the individual firm depreciation rates.26 Hence,in what follows, we apply non-parametric testing procedures to the industryaverages summarized in Table 5 to assess whether there might be a significantassociation between the book- and market-based depreciation measures.
A formal test of the hypothesis that there is no association between thetwo depreciation measures is provided by the Spearman rank correlationtest (Conover, 1980, pp. 252–256). Applying this test to the industryaverages contained in Table 5 returns a sample Spearman rank correlationtest statistic of r=0·53231. This is significantly different from zero at the1% level (on a one-tailed test) (Conover, 1980, p. 456). If we exclude theMedia Industry on the ground that it is based on a single observation whichhas all the hallmarks of an ‘outlier’, the Spearman rank correlation coefficientfalls to 0·47130, which only just fails to be significant at the 1% level andis extremely significant at the 2·5% level. It follows that, at an industrylevel, book rates do appear to be significantly associated with the estimatedmarket rates.27
It warrants emphasizing, however, that it is still possible for systematicbiases to exist, even though there is a significant association between thebook- and market-based depreciation measures. We attempted to assesswhether this might be the case by hypothesizing a regression equation ofthe form b=a+bm, with a=0 and b=1, as describing the relationshipbetween the book rate (b) and the estimated market rate (m). Conover(1980, pp. 265–268) shows that a non-parametric test of this hypothesisinvolves computing the residuals from the hypothesized regression andthen determining the Spearman rank correlation coefficient between theresiduals and the independent variable (market depreciation rate) fromwhich they were determined. This procedure returned a Spearman rankcorrelation test statistic of 0·37231 if all 25 industries are included and0·29044 if the Media Industry is excluded. The first of these statistics issignificant at the 5% level (on a one-tailed test), whilst the second issignificant at the 10% level. This provides broad support for the hypothesisthat a≠0 and/or b≠1. Furthermore, a simple sign test conducted ondifferences across the 25 industry averages shows that the average marketrate is significantly larger than the average book rate at the 5% level (on aone-tailed test). These results suggest that whilst book rates appear to beslightly lower than their market counterparts, book rates do, nevertheless,appear to be significantly influenced by underlying market rates of de-preciation.
It is important to note, however, that there are any number of factorswhich might affect a firm’s depreciation policies and/or the rate at whichits capital depreciates in the market and these might be responsible for thesystematic biases which appear to emerge from the simple statistical analysesreported above. It is possible to mount arguments, for example, which
. .64
suggest that a firm’s debt to equity ratio, its capital intensity, profitabilityand size, etc., could all have effects on either the book- or market-basedmeasures of depreciation. The length of the current paper means thatdetailed development of these arguments awaits another occasion, but, sinceTable 2 already reports information on firm size, we can sketch the sizehypothesis in a little more detail at relatively little cost. We begin by notingthat larger firms will probably claim economies of scale that are not availableto smaller firms and this could have the effect of reducing the rate at whichcapital depreciates. Varian (1984, p. 19) puts this familiar ‘returns to scale’argument in terms of the following simple example:
‘Consider . . . a firm that builds an oil pipeline between two points and usesas inputs labor, machines and steel to construct the pipeline. We may take therelevant measure of output for this firm to be the capacity of the resultingline. Then it is clear that if we double all inputs to the production process,the output may more than double since increasing the surface area by twowill increase the volume by a factor of four.’
In other words, doubling the pipeline’s construction cost quadruples itscubic capacity. Other factors remaining equal, this also has the effect ofhalving the depreciation expense (per unit of oil piped) and so it is clearthat there can be scale-related factors which do influence the rate at whicha firm’s capital depreciates. And, as the above example demonstrates, forlarge firms, these scale effects are more likely to reduce, rather than increase,the rate at which capital depreciates.
We attempted to test this hypothesis by computing the Spearman rankcorrelation coefficient between the average size of the firms in the 25industries on which our study is based and the average industry bookdepreciation rates in the first instance, and then the average estimatedmarket depreciation rates. Size was measured by average industry sales asreported in Table 2, whilst the average book and market depreciation ratesare contained in Table 5. These calculations returned a test statistic of r=-0·52308 based on the book rates, which is significantly different from zeroat the 1% level. The test statistic based on the market depreciation rate isr=−0·82077, which is also significant, but at much greater than the 1%level. This indicates that there is a significant association between both themarket and book rates of depreciation and firm size and, furthermore, thatthis relationship is in the hypothesized direction. However, since size andindustry classification are also obviously highly correlated, one needs to becareful about the proper interpretation of these results.
What is clear from our analysis is that there is a case for more refinedstatistical analyses in this area and, to increase the degrees of freedomavailable, it would probably be best undertaken at the level of the
65
individual firm. Here, though, the ‘errors in variables’ issue poses asignificant problem.
6. SUMMARY CONCLUSIONS
Our concern in the present paper is with an empirical evaluation of therelevance of published depreciation numbers. We report results of anattempt to assess the correspondence between the depreciation ratesimplied by published financial statements and the market-based ratesimplied by time series variations in corporate equity returns. Our modellingprocedures are based on the ‘Crusonia plant’ construct developed, inthe first instance, by Frank Knight. This leads to a simple model ofcapital theory under which firms are regarded as a reservoir of unused(homogeneous) capital services. Our results are consistent with thehypothesis that estimated market rates of depreciation have a significantinfluence on corporate depreciation policies. Simple non-parametriccorrelation tests conducted at an industry level show that there is asignificant association between book rates of depreciation and the marketrates implied by Knight’s ‘Crusonia’-based model of capital accumulation.There is also clear evidence that this relationship holds at the individualfirm level, despite the existence of the familiar ‘errors in variables’problem.
Here, it is important to note that our analysis is based upon acontentious methodology; namely, the supposition that the ‘behaviour of. . . market prices of ordinary shares’ might be seen as ‘a way of testingempirically the propriety of accounting rules . . .’ (Chambers, 1974, p.39). There are those who hold to the view that stock markets are too‘extraordinarily complex’ and influenced by ‘so many other facts, factorsand fancies, that what goes on at the market surface has little identifiablerelation to what accounts convey’ (Chambers, 1974, p. 54). Those whosubscribe to this view will, no doubt, reject our analysis or, in the veryleast, see it as yet further evidence for the contention of Keynes (1936,pp. 153–154) that ‘day to day fluctuations’ in stock market prices are‘obviously of an ephemeral and non-significant character . . .’ Added tothis, there is the issue of our modelling procedures, based as they are,on what Samuelson (1962, p. 200) disparagingly refers to as the ‘jelly’model. This model assumes that firms produce a single consumptiongood which, in turn, is the only factor of production available in theeconomy. Knight (1944, p. 28), who referred to such economies as‘Crusonia’, justified this approach on the ground that it is the ‘onlyprocedure which seems to be feasible for the analysis of a highly complexsituation . . .’ It may well be, however, that the naıvete of this modelbiases our estimation procedures and that more realistic models of the
. .66
accumulation process could lead to entirely different results (Samuelson,1962).
The above caveats aside, and in so far as stock market prices are theproduct of a rational pricing mechanism, our results indicate thattraditional depreciation policies may not represent the arbitrary andincorrigible procedures and/or the ‘mere conventions’ which some havesuggested. Nor is it clear that the ‘radical reform’ and jettison of theentire allocation process, as espoused by Thomas (1969, 1974, 1979)and others, is necessarily the most sensible way forward. At the veryleast, rational debate founded on more refined empirical testing proceduresneeds to be fostered and encouraged, and perhaps after this we mightbe in a better position to reach some concrete conclusions.
N
1. Probably the best way to describe the Cox et al. (1985a) model is that it can beinterpreted as a stochastic continuous time re-interpretation of the Leontief (1969)deterministic input–output model.
2. Hicks (1974) provides a good summary of the relationship between the neoclassicalaccumulation approach to capital theory adopted here and the more traditionalaccounting models of depreciation based on ‘economic’ concepts of income. Hirshleifer(1970, Chapter 6) contains a much more exhaustive treatment of the competingapproaches to capital theory than can be given here.
3. The capital theory briefly summarized here was laid down in an at times acrimoniousdebate between Frank Knight (1934, 1936a, 1936b, 1938, 1944) and NicholasKaldor (1937, 1938). Hirshleifer (1967) provides a particularly readable account ofthe principles behind the debate. Further developments in this area are to be foundin the works of Pigou (1935), Solow (1956), Phelps (1961, 1965), Hirshleifer (1970),Merton (1975), Friedman (1976), Cohen (1989) and particularly Sandelin (1989).Stigler (1973) and Wick (1973) provide interesting historical insights.
4. These are portmanteau quotations—we have run together some non-consecutivepages.
5. See Fisher (1930, p. 519) for the formulation of a similar approach to capital theory.6. The position of a firm which is financed by both debt and equity is considered
below.7. Broadly speaking, a stochastic process is said to constitute ‘white noise’ if its
increments are independent normal variates which are identically distributed withzero mean and constant variance. In the present context, this implies that W1(h) isnormally distributed with zero mean and variance r2h. See Hoel, Port & Stone(1972, pp. 122–124, 141 et seq.) for a good introduction to this area.
8. Here, exp(x)=ex is the exponential operator.9. Cox, Ross & Rubinstein (1979, pp. 254–255) derive an options pricing model which
assumes a continuous deterministic upward trend in the underlying equity price butthat this is punctuated by radical downward movements defined in terms of a Poisson(jump) process. If we assume cash flows are deterministically proportional to themarket value of the firm’s end of period capital, then it follows that our model issimilar to the Cox et al. (1979, p. 255) equity pricing model. However, we arepersuaded by our peers and some empirical work we conducted that the deterministiccash flow assumption leads to unrealistically high estimates of depreciation rates.Hence, we follow Cox et al. (1985b) in assuming a diffusion-based cash flow model
67
where the variance is a key determinant of the stochastic proportionality factor.Empirical results reported below show that this assumption leads to estimates ofdepreciation rates which are more consistent with the book rates implied by corporatefinancial statements.
10. This follows from the fact that the sum of n independent normal variates, each ofwhich is distributed with zero mean and variance r2h, will also be normally distributedwith zero mean but with variance n(r2h)=r2t, since nh=t by definition. For furtherdetails, see Freund & Walpole (1987, pp. 166–167, 273–278).
11. The return will be in real terms because S(t) is defined to be the number of(physical) units of capital in place at time t. Similarly, xj(h) is defined as theproportion of the firm’s capital put into service at time ( j−1)h which remains inservice at time jh. This, too, is measured in physical units and so log[xj(h)] will bethe continuously compounded depreciation rate over the time interval [( j−1)h,jh].
12. A Taylor series expansion shows
E0[d0;h]=A qq+1B
h
−1=h logA qq+1B+. . .
where we have omitted terms involving hj for j≥2 (Apostol, 1967, pp. 435–436).Dividing by h and taking limits thus shows
Limith ∩ 0
E0[d0;h]h=logA q
q+1Bin which case we say log(q/q+1) is the continuously compounded depreciation ratecorresponding to the parameter q. This defines the continuously compoundeddepreciation rate used in the empirical work reported in Section 5.
13. This is called the ‘one-hoss shay’ model of depreciation, which, next to Poissondecay, is the most popular depreciation assumption in the capital theory literature(Dixit & Pindyck, 1994, p. 205). We now show that by redefining the initial statespace, this model leads to a probabilistic depreciation model which is related to thePoisson, gamma and chi-square distributions. Butler, Rhys & Tippett (1994, pp.75–79) also demonstrate how the transition matrix given here may be interpreted ina ‘straight line’ depreciation context.
14. See Feller (1966, pp. 24–25) for further examples of the empirical application ofthe simplest version (q=1) of this model. It is also worth noting that the well-known Cox et al. (1985b, pp. 390–392) ‘square root process’ for modelling interestrates and technological uncertainty has the same ‘steady state’ distribution as log(x)(Rhys & Tippett, 1993, pp. 501–502).
15. Satyendra Nath Bose was a young Bengali physicist at the University of Dacca whenin June 1924 he sent a letter to Albert Einstein together with a paper that had justbeen rejected by the Philosophical Magazine, a British physics journal. The paper’sprincipal brief was with Planck’s radiation law, but Einstein soon realized it hadmuch wider implications. Based on the ideas of Bose, Einstein showed that theentropy measure of the second law of thermodynamics based on Maxwell–Boltzmannstatistics was an approximation which holds true only in an asymptotic sense. Hethen showed that entropy measures defined in terms of Bose–Einstein statistics didnot suffer from this problem. Interestingly, one of us has shown that many of thedeterministic depreciation formulae currently employed in practice can be derivedfrom a Markovian probability structure similar to that employed by Pollard & Tippett(1994), but based on Maxwell–Boltzmann statistics where the initial capital is ‘large’.For further details, see Pollard & Tippett (1994, pp. 71–72) and Rhys (1995).
. .68
16. In the discrete state space model demonstrated earlier, Rhys et al. (1994, p. 164)show that (1/q) is a kind of continuity index—higher values of q imply smallerprobabilities of radical downward movements in state occupation. In the continuousstate space model, higher values of q imply a lower variance in the underlyingdepreciation rate. Hence, it follows that our model implies there will, in general, bemore violent fluctuations in the equity price as q decreases. In other words, despiteits simplifying assumptions, our model implies equity price behaviour which, in thevery least, is both consistent with recent stock market experience and with priormodels of equity price behaviour (Cox et al., 1979, pp. 254–255). See the Appendixfor some additional properties of the log-gamma distribution and also for someproperties of the joint distribution of W(t) and log[x(t)].
17. We could summarize the analysis of this section by saying that we follow Cox et al.(1985a), in assuming a stochastic constant returns to scale production technologywith two sources of technological uncertainty; namely, that relating to physicaldepreciation (as summarized by a log-gamma variate, x(t)) and that relating toproduced output (as summarized by a Wiener process, W(t), designed to capture alltechnological uncertainty except that relating to depreciation). Using the modellingprocedures laid down in Cox et al. (1985a, pp. 364–366), however, it is a relativelyeasy matter to generalize the analysis to capture technological uncertainty relating toother ‘factors of production’. Basically this involves splitting the Wiener process,W(t), into multiplicative components representing the other sources of technologicaluncertainty. This would enable us to address some interesting accounting issues (e.g.the effects of human capital on equity returns). However, given the added complexitythis would introduce, we follow Cox et al. (1985a, p. 364) in considering only ‘apure capital growth model . . . This provides a more streamlined setting for theissues which we wish to stress.’
18. This is the annual discrete rate of depreciation. From footnote 12, however, itfollows that the continuously compounded annual depreciation rate amounts to
logA qq+1B=23·15841
24·15841=−0·04227
or about 4·23%.19. Carey (1994) also reports some anecdotal evidence on this issue.20. We are not alone in being subject to this criticism. Indeed, most of the well-known
models of capital theory would be ‘knocked out’ on this score. The Sharpe (1964)Capital Asset Pricing Model, for example, is a ‘pure exchange’ model and thereforehas nothing to say about supply-side considerations (Sharpe, 1964; Merton, 1973;Cox et al., 1985a, p. 364).
21. These are again portmanteau quotations—we have run together some non-consecutivepages.
22. We regressed the monthly inflation rates implied by each CSO index against the 19other CSO indices used in the cross-matching exercise. The sample R2 statistics arecomputed from the 59 monthly observations covering the period from January 1988to December 1992, this being the period covered by our study. The average t statisticacross these regressions is 6·07, whilst 81 and 70% of the t statistics are significantlydifferent from zero at the 5 and 1% levels, respectively. We also replicated ouranalyses using the CSO all Manufacturing Industry Index, which is a weightedaverage of the other 38 CSO indices. There were no significant differences betweenresults obtained using this index and the results reported in the text.
23. Two of these companies were from the Electonics Industry, whilst there was onecompany from Contracting & Construction, Miscellaneous, Brewers & Distillers,Health & Household Products, Stores and Packaging, Paper & Printing.
69
24. See footnotes 12 and 18 for details of how the continuously compounded marketrate of depreciation was determined. The continuously compounded book rate wastaken as the natural logarithm of one plus the average book rate.
25. Johnston (1984, pp. 428–435), for example, develops a standard scenario to showthat stochastic regressors bias both the regression coefficient and its associatedPearson product moment correlation coefficient towards zero.
26. See Fama & Macbeth (1973, pp. 614–615) for some further discussion of this issue.27. At the individual firm level, Textiles, Oil & Gas, Food Manufacturing, Hotels &
Liesure, Engineering Aerospace and Packaging, Paper & Printing all returnedSpearman Rank Correlation test statistics which were significantly different from zeroat at least the 5% level.
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APPENDIX
BRIEF PROPERTIES OF THE LOG-GAMMA DISTRIBUTION
1. Moments of the Log-Gamma Distribution
For integral k we have:
E[(−q log x)k]=P1
0
qx(q−1)[−q log(x)]t+k−1
C(t)dx
where E(.) is the expectations operator. Make the substitution
x=expA−zq B , dz=−q
dxx
to give:
. .72
E[(−q log x)k]=Px
0
ezzt+k−1
C(t)dz
E[(−q log x)k]=C(t+k)
C(t)=t(t+1)(t+2). .(t+k−1)
From this it follows:
E[(log x)k]=t(t+1)(t+2). .(t+k−1)
(−q)k
Letting k=1 shows E[log x(t)=(−t/q). Letting k=2 shows
E[{−log x(t)}2]=[t(t+1)/q2].
NowsinceVar[log x(t)]=E[{−log x(t)}2]−(E[log x(t)])2 whereVar(.) is thevariance operator, it follows that
Var[log x(t)]=t(t+1)
q2 −A−tq B
2
=tq2
2. Joint Distribution of W(t) and log[x(t)]
Rhys et al. (1994, p. 166) show that −2q log[x(t)] is distributed as a Chi-square variate with 2t degrees of freedom. Furthermore, by assumption,W(t) is distributed independently of log[x(t)] as a normal variate with zeromean and variance r2t. It follows that
u=
W(t)
rJt
J−2q log[x(t)].J2t=
W(t)
rJ−q log[x(t)]
is distributed as Student’s distribution with 2t degrees of freedom (Freund& Walpole, 1987, p. 289).