Estimates of Tobin’s Q for the United States using publicly available

data present an apparent puzzle: it is systematically less than unity. This

paper sets out a simple model consistent with rational stock market valuation under conditions of asymmetric information that provides a possible

explanation of this puzzle

Robertson, D.

Wright, Stephen

English

Birkbeck Institutional Research Online

BIROn - Birkbeck Institutional Research OnlineWright, Stephen and Robertson, D. (2005) Tobin’s Q, asymmetric Informationand aggregate stock market valuations. Working Paper. Birkbeck, Universityof London, London, UK.Downloaded from: http://eprints.bbk.ac.uk/31972/Usage Guidelines:Please refer to usage guidelines at http://eprints.bbk.ac.uk/policies.html or alternativelycontact lib-eprints@bbk.ac.uk.Tobin’s Q, Asymmetric Information andAggregate Stock Market ValuationsDonald RobertsonFaculty of Economics and PoliticsUniversity of CambridgeAustin Robinson BuildingSidgwick AvenueCambridge CB3 9DDdonald.robertson@econ.cam.ac.ukStephen WrightDepartment of EconomicsBirkbeck CollegeUniversity of LondonMalet StreetLondon W1E 7HXs.wright@bbk.ac.ukFebruary 16, 2005AbstractEstimates of Tobin’s Q for the United States using publicly availabledata present an apparent puzzle: it is systematically less than unity. Thispaper sets out a simple model consistent with rational stock market valu-ation under conditions of asymmetric information that provides a possibleexplanation of this puzzleJEL Classifications: C32, C53, E44, G10, G14.Estimates of Tobin’s Q for the United States using publicly available datapresent an apparent puzzle. It is usually assumed that the ratio of the stockmarket’s valuation of US corporations to their underlying recorded assets at re-placement cost should be close to unity, at least on average. Indeed, if there aresignificant monopoly rents, or if (as has been claimed by a number of authors inrecent years - eg, Hall, 2000; Laitner & Stolyarov, 2003) statisticians typicallyunder-record intangible assets, then it would be expected that Tobin’s Q wouldtypically be above unity. In fact the data show that it has typically been wellbelow unity.One possible response to this apparent puzzle is to conclude that the datamust be wrong. This would require that statisticians habitually overestimatedthe replacement cost of capital for most of period over which Q can be calculated,but (at least according to the authors cited above) switched to underestimating itin the immediate past. Understandably, statisticians, who spend a lot more timeand effort on data construction than economists do, would dispute both theseclaims. It therefore seems worth pursuing the possibility that the statisticians,rather than the economists, are getting it right.There may indeed be an alternative explanation. Since Grossman and Stiglitz(1980) it has been established that a perfectly efficient stock market, in whichprice always equals underlying value, must be a logical impossibility, due to thecosts of information gathering. This paper explores the link between rationalstock market valuations and the corporate sector’s underlying aggregate value, inthe presence of asymmetric information. It asks whether systematic differencesbetween price and underlying value are a possible alternative explanation of theproperties of recorded q.1. The Puzzle of Tobin’s QTable 1, using data from Wright (2004,2005) shows that mean values of Tobin’sQ, constructed from a range of publicly available data, have been well below unityin the postwar era both for the US business sector as a whole,1 and for the betterdocumented non-farm non-financial corporate sector. The same feature holds indata for the nonfinancial corporate sector over the course of the entire twentieth1Note that Laitner & Stolyarov (op cit) derive Tobin’s Q estimates for the business sectorthat have an average value greater than unity in the period 1953-2000, and use this as supportingevidence for claims that there are significant quantities of intangible assets. Wright (2005) showshowever that this feature of their data is due to major errors in data construction.2century; and for quoted companies (defining Q on an equity basis) on data since1871.Table 1. Alternative Estimates of Tobin’s QGeometric Averages1953-2000 1900-2002 1871-2002All business 0.859∗ n/a n/aNonfarm, Nonfinancial Corporations 0.670∗ 0.648∗∗ n/aS&P 500 Companies2 0.590-0.695∗∗ 0.525-0.707∗∗ 0.545-0.862∗∗Sources: ∗ Wright (2004); ∗∗Wright (2005)2. Stock Market Valuation with Asymmetric Information2.1. FirmsConsider the value maximisation problem of a representative unleveraged firm.Our analysis is standard, and closely related to that of Hayashi (1982); Abel andBlanchard (1986) and many others. Let Ωt represent information available to thefirm at the end of period t. Assume the firm maximises the value of the firm:Vt = Et [Θt+1 (Dt+1 + Vt+1) |Ωt] (2.1)where Dt is the total net cashflow to equity holders.3 The firm makes investmentdecisions, and chooses factors of production, to maximise its current value, andsimultaneously prices its resulting maximised value with reference to Θt+1, theone-period ahead stochastic discount factor. Because the firm’s information set isgreater than the public information set, due to the costs of information-gathering,the firm is in a unique position to assess its own underlying value. It does so,however, using a market-based measure of (stochastic) opportunity cost.2.2. Stock MarketsAssume that stock markets price the same income flow, using the same stochasticdiscount factor, but on the basis of public information, St:Pt = Et [Θt+1 (Dt+1 + Pt+1) |St] (2.2)2Ratio of stock price to net worth per share, derived by cumulating retentions. Estimatesrequire starting values for Q: range shown based on initial values of 0.5 and 2 in 1871. Forfurther details of calculations see Wright (2004).3(For referees: See Endnote A)3where Pt = stock market value. Rearranging (2.1) and (2.2), we have the noarbitrage pricing conditionEt [Θt+1 (1 + It+1) |Ωt] = Et [Θt+1 (1 +Rt+1) |St] = 1 (2.3)where It+1 =Dt+1+∆Vt+1Vtis the internal return, and Rt+1 =Dt+1+∆Pt+1Ptis the stockreturn. With symmetric information, this condition would be automatically sat-isfied by the equality of Pt with Vt for all t and hence of the stock return with thefirm’s internal return. More generally, however, with costly information (whichhere can be viewed as providing the rationale for the existence of firms), the infor-mation sets, and hence the two returns must differ. Since the underlying incomeflow being valued is identical, this must in turn imply that Pt must differ fromVt.4 The pricing condition will nonetheless provide a crucial link between the stockmarket’s valuation and underlying value.2.3. Tobin’s q and “Noise”For simplicity we assume constant returns to scale and no investment adjustmentcosts,5 and hence that firms’ investment ensures aggregate Vt is always equal tothe replacement cost of capital, which we shall assume that statisticians measureperfectly in aggregate. These are very strong assumptions, but we make themsolely for the sake of isolating the potential role of asymmetric information. Thereis no necessary conflict between these assumptions and that of asymmetric infor-mation, so long as market participants cannot infer value at the individual firmlevel from the aggregate.6 Tobin’s Q will then be given byQt =PtVt(2.4)In contrast to the usual interpretation of Q, as reflecting investment adjust-ment costs, in this framework it simply captures the deviation of market pricesfrom their underlying value if there were symmetric information. We are inter-ested in whether in this restrictive framework Q may differ systematically fromunity.4While this feature is a logical implication of rational pricing under asymmetric informationit may of course also arise under other conditions.5(For referees: See Endnote B)6(For referees: see Endnote C)4We first make the pricing condition in (2.3) more transparent by log-linearising.Letting rt = ln(1 + Rt), qt = lnQt, etc, we can use the Campbell-Shiller (1988)approximation to writert+1 ' ϕ+∆pt+1 + (1− ρ)(dt+1 − pt+1) (2.5)where ρ = 11+exp(d−p) ; ϕ = ln¡1 + exp(d− p)¢−(1−ρ)(d− p) and d− p = E(d−p).For values of q sufficiently close to zero, we can apply the same log-linearisationcoefficients to it = ln (1 + It) (replacing p with v) and relate the two log returnsby writing:rt+1 ' it+1 + εt+1 (2.6)where, using (2.4) and (2.5)εt+1 = ρqt+1 − qt (2.7)Following standard practice, we refer to εt+1 as “noise” even though in our frame-work we assume it does not arise from any irrationality.Assuming log normality, taking unconditional expectations of the pricing con-dition (2.3), using (2.6) and ruling out a deterministic trend in q impliesq ' 11− ρE∙covt(εt+1, θt+1) + covt(εt+1, it+1) +vart(εt+1)2¸(2.8)Covariances and variances may in principle be time-varying. But, so long asthey are themselves stationary, qt will also be stationary (hence pt and vt must becointegrated). This general result (which places no other restriction on the processfor qt than that it be stationary) arises simply from the minimal assumption thatthe market and the representative firm use the same stochastic discount factor.73. Can “Noise” Explain Apparent Under-Pricing?From (2.8) it is evident that q may in principle be less than zero. This then offersat least a possible explanation for the observed less-than-unit geometric meanof Tobin’s Q. A relatively simple calibration exercise, exploiting some furtheridentifying assumptions, can be used to assess its plausibility.To simplify, we assume, first that conditional covariances are time-invariant;8and second that covt(ε, i) = 0. The second assumption is a stronger one, but is7(For referees: see Endnote D)8This is an innocuous assumption as long as they are themselves stationary.5consistent with efficient information-gathering by market participants.9 We canthen re-write (2.8) as:q ' vart(εt+1)1− ρ∙βθε +12¸(3.1)where βθε = covt(θt+1, εt+1)/vart (εt+1) is the response of the stochastic discountfactor to a unit innovation in εt+1. Since εt+1 will by assumption be observationallyindistinguishable from it+1 using the market information set, St+1 we can on apriori grounds set βθε = βθi = βθr. This latter coefficient can be treated as knownin our framework, since, using standard assumptions10 the log of the mean equitypremium must equal, with opposite sign, the mean covariance of the stock returnwith the unobservable stochastic discount factor. That is, if Rst+1 is the safe return,and we defineη ≡ logEµ1 +Rt+11 +Rst+1¶= −covt(θt+1, rt+1) = βθrvart (rt+1) (3.2)then, given that we can observe both η and vart (rt+1) , and hence βθr, the onlyunknown element in (3.1) is the conditional variance of εt+1. So we can can inferthe required degree of volatility of “noise” if this is to be the sole explanatoryfactor of the observed negative mean of q.It turns out that, given available emprical estimates, the above expressionsimplifies yet further. A mid-range estimate of η (see, eg, Campbell et al, opcit) is of the order of 0.04. Conveniently, this is also a reasonable estimate ofboth var(rt+1)11 and the linearisation parameter 1− ρ, so the expression in (3.1)simplifies further toq ' −12vart(εt+1)vart(rt+1)The figures for Tobin’s Q for the nonfinancial corporate sector shown in Table 1imply q ' −13. Thus if we assumed that “noise” was the sole explanatory factor,we would need the conditional variance of the noise process to be around twothirds of the conditional variance of the log stock return.9(For referees: see Endnote E)10For a standard treatment,see eg, Campbell, Lo and Mackinlay, 1997, Ch 8.11(For referees: See Endnote F)64. ConclusionsThis paper has set out a simple model consistent with rational stock market valua-tion under asymmetric information that appears to provide a possible explanationfor the empirical puzzle that Tobin’s Q is systematically less than unity. The in-tuition for this result is that, viewed as an investment asset, corporate capitalis relatively safe compared to the stocks and shares that provide notional titleto that capital. It is relatively safe because it is relatively uncorrelated with thestochastic discount factor, for the simple reason that its return is only imperfectlyobservable.We make no claim that this is actually the sole explanation of Tobin’s Q beingpredominantly below unity. Indeed, if it were, this would require “noise” to be thedominant element in the variability of aggregate stock returns. But our analysisdoes suggest that the empirical implications of less than perfect market efficiencyfor aggregate stock market valuations are potentially significant.References[1] Abel, A and Blanchard, O (1986), “The Present Value of Profits and CyclicalMovements in Investment” Econometrica 54(2) pp 249-273[2] Campbell, J and Shiller, R (1988) “The Dividend-Price Ratio and Expecta-tions of Future Dividends and Discount Factors”, Review of Financial Studies,1. 195-227[3] Campbell, J, Lo, A J and MacKinlay, A C (1997), The Econometrics ofFinancial Markets, Princeton University Press, Princeton, NJ[4] Grossman, S, Stiglitz, J (1980), “On the Impossibility of InformationallyEfficient Markets”, American Economic Review, 70(3), June 1980, pages 393-408.[5] Hall, R (2001) “The Stock Market and Capital Accumulation,” AmericanEconomic Review 91:1185-1202[6] Hayashi, F(1982), “Tobin’s Marginal q and Average q: A Neoclassical Inter-pretation”, Econometrica, 50(1), Jan. 1982, pages 213-24.7[7] Laitner, J and Stolyarov, D (2003), “Technological Change and the StockMarket”, American Economic Review, vol. 93, no. 4 1240-67[8] Wright, S (2004) “Measures of Stock Market Value and Returns for the USNonfinancial Corporate Sector, 1900-2002”, Review of Income and Wealth 50(4) pp 561-584[9] Wright, S (2005) “Technological Change and the Stock Market, A Comment",Under revision for American Economic Review8WorkingsDerivation of (2.6) and (2.7)If we apply the same linearisation as in (2.5) to the log internal return (whichis acceptable as long as q ' 0→ d− v ' d− p)it+1 ≡ ln (1 + It+1) ' ϕ+∆vt+1 + (1− ρ)(dt+1 − vt+1) (4.1)and subtract from (2.5) we have, using (2.4) and (2.7)rt+1 − it+1 ' ∆pt+1 −∆vt+1 + (1− ρ) [dt+1 − pt+1 − (dt+1 − vt+1)]' ∆qt+1 − (1− ρ)qt+1' ρqt+1 − qt' εt+1Derivation of (2.8)Substituting from (2.6) into (2.3), and using θt = logΘt givesEt exp(θt+1 + it+1)|Ωt = Et exp(θt+1 + it+1 + εt+1)|Sttaking logs, exploiting lognormality, and letting σ2x = vart(xt+1); σxy = covt(xt+1, yt+1),Et∙(θt+1 + it+1) +12¡σ2θ + σ2i + 2σiθ¢¸ |Ωt= Et∙(θt+1 + it+1 + εt+1) +12¡σ2θ + σ2i + σ2ε + 2σiθ + 2σεθ + 2σεi¢¸ |StIf we apply the law of iterated expectations repeatedly to derive unconditionalexpectations, and assume homoscedasticity, the differences in information setsdisappear, and we getE∙(θt+1 + it+1) +12¡σ2θ + σ2i + 2σiθ¢¸= E∙(θt+1 + it+1 + εt+1) +12¡σ2θ + σ2i + σ2ε + 2σiθ + 2σεθ + 2σεi¢¸9which allows us to cancel terms,0 = E (εt+1) +E∙σ2ε2+ σεθ + σεi¸Hence, using (2.7),−E (εt+1) ' −∆q + (1− ρ)q ' E∙σ2ε2+ σεθ + σεi¸ruling out a deterministic trend in q implies ∆q = 0, hence we can solve to give(2.8).Derivation of (3.1)To get (3.1) we set σεi = 0 and apply the homoscedasticity assumption (which,it should be noted, simply implies that conditional variances and covariances areconstant over time - it does not imply that they equal their unconditional values).Endnotes for RefereesA.We could equivalently consider the problem of a leveraged firm in a Miller-Modigliani setting, where leverage does not affect firm value. In this caseDt wouldbe the net cashflow to equity and bondholders. This would be more consistentwith the data on q, which include the market value of debt in the numerator; butwe simplify here for the purposes of exposition.B. Since we are only concerned with long-run features we could equally wellassume that adjustment costs have mean zero, which is explicitly or implicitlyassumed in most q theories of investment; and that any variance and covarianceterms associated with adjustment costs have a minimal impact on mean q.C. Here we respond to a number of points raised by past referees relating to theissue of measurement of aggregate vs firm level capital. 1) Aggregate capital couldin principle be measured perfectly if aggregate investment figures and depreciationfactors were measured perfectly, which does not require firm level information. 2)Typically there will be considerable time lags in data collection, and even if weaccept that statisticians get it right in the end they typically do so through aprocess of regular revisions. Thus it is highly unlikely that Qt will be part of St,the market information set. This in itself precludes any possible arbitrage based10on Qt. But even if aggregate Qt were part of St, and thus investors knew that themarket was over- or under-valued in aggregate, their informational disadvantageat the firm level would not allow them to exploit this information. 3) Some firmlevel data on assets are of course available from balance sheets, but only at bookvalue; and in the era of Enron it seems reasonable to assume that this gives atbest an imperfect picture of firm-level replacement cost assets.D. If q 6= 0, then, to the extent that the linearisation coefficents are sufficientlydifferent, there will be an additional term in d− p, the mean “cashflow” yield, sothe result will still hold as long as the cashflow yield is stationary for which thereis strong evidence. See Robertson, D and Wright, S (2004a), “Dividends, TotalCashflows to Shareholders and Predictive Return Regressions (under revision forReview of Economics and Statistics)E. It is perhaps worth stressing that this does not require εt itself to be whitenoise. So far all that we have established is the stationarity of q. Hence wecan write q = q + a(L)ωt where ωt is an innovation, hence, from (2.7) εt 'ε + (ρ − L)a(L)ωt. Similarly we can assume some process for it = b(L)ut whereut may be some structural innovation like a technology shock. Efficient use of Stwould imply E (ωtut) = 0, and hence cov(εt, it) = 0, but this makes no restrictionon the time series properties of εt.F. Assuming an annual volatility of 20%: ie, we use the unconditional variance;but the answer would barely differ if there were some predictability of rt+1. (Theapproximate equality of η and vart (rt+1) implies βθr ' −1, which we wouldexpect)11

Tobin's Q, asymmetric Information and aggregate stock market valuations

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Tobin's Q, asymmetric Information and aggregate stock market valuations

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