1

An Objective Approach to Relative Valuation1,2

Ruben D. Cohen3

SSB Citi Asset Management GroupCottons Centre, London SE1 2QT, United Kingdom

Abstract

A fundamental approach to describing the behaviour of the equity price index ispresented. The method centres on the contention that, under a constant discount rateand in a market that is efficient and in equilibrium, the forward-looking risk premium isequivalent to the expected dividend yield, and both are equal to zero. Extending thisspecial-case scenario to one that involves time-wise variations in the discount rateleads to a special co-ordinate transformation [or mapping], which addresses how theindex should behave correspondingly.

Applying the same principle to both, corporate earnings and the nominal grossdomestic product [GDP], leads to a similar transformation. This, consequently, makesway for objective comparisons between the equity index, corporate earnings and theGDP, thereby raising the notion of relative valuation in this context. A practicaldemonstration of this is ultimately provided for the US and UK economies and equitymarkets.

1 Working paper.2 April/May 2000.3 E-mail: ruben.cohen@citicorp.com. Phone: +44 (020) 7500 2717.

1. IntroductionRelative valuation is an important

concept in investment and finance. Itssignificance is observed in day-to-dayinvestment activities, where gaps or spreads inyields, interest rates and other rates of growth,in general, are exploited (Fabozzi, 1999).

The main advantage of relative valuationover ad-hoc approaches is that it paves theway for unbiased comparisons – i.e. givencertain stocks, which one(s) should one investin, or is the equity index over/undervaluedrelative to what the earnings, GDP, etc.indicate. In this context, therefore, relativevaluation eliminates the need for an absolutemeasure, which is, arguably, an impossiblefeat to achieve.

Relative-valuation measures typicallyinvolve multiples, and these have beenapplied time after time to comparing, amongothers, growth stocks (Peters, 1991), bonds(Benari, 1988), funds (De Long and Shliefer,1992), as well as indices across different

countries (Arnott and Henricksson, 1989). Inaddition, such measures have even been usedto provide a process by which financial healthcould be assessed (Barth et al, 1998). Giventhat the above represent only a small fractionof the literature on how and why relativevaluation is put into use, it is, therefore,inevitable that this notion has a lot to offerwhen it comes to practical investment.

Here, as well, we intend to apply thenotion of relative valuation for comparisonpurposes. Our aim, as it turns out, is topropose and develop an objective way forcomparing three basic elements with eachother - namely, the state of the economy, asrepresented by the nominal GDP, corporateearnings and, finally, the equity price index.Multiples representing these are most likelyavailable and used commonly by both,economists and investment/financial analysts,to tie together the economy and the stockmarket. Such a measure of relative valuationcould, thus, enable one to assess whether the

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stock market is over, under or fairly valued incomparison with earnings and/or GDP.

Our primary concern here is to derive afundamental method for describing the time-dependent behaviour of the stock market,earnings and the GDP in relation to changes inthe rates of discount. What these discountrates encompass – i.e. short or long-term bondor other types of yields and interest rates –and how implementing different ones shouldalter the final results, are important issues thatwill be discussed as we go along.

In the process, we shall note twoattributes that might raise doubts on ourmethods and conclusions. Firstly, therelationships derived here will look differentfrom those typically found in the literature.This is because ours are extracted from anapproach that follows a different routealtogether. Secondly, the issue of spuriousrelations could be raised. To overcome this,we will try our best to supply proofs andcharts at every step of the way. In addition,we shall keep the paper as objective aspossible by limiting it to factual observationsand leaving out any hypothetical andspeculative explanations.

2. The ApproachWe plan to develop the method in two

ways – one focusing on equity [Section 2a]and the other on GDP and corporate earnings[Section 2b]. The latter two occupy the samesection because their underlying principleshappen to be the same. The final results willthen be combined together to establish therelative valuation measures. For sake ofbrevity, all derivations that already exist in theliterature cited shall be omitted.

2a. The Equity ModelIt is useful to begin with our previous

contention that, in a constant-discount rateenvironment and where the market is efficientand at steady-state equilibrium, the equity riskpremium is zero (Cohen, 2000). This clearlyrepresents a highly idealised scenario, butnonetheless it can and will be generalised hereto account for time-wise changes.

Let us begin by recalling that in theperfect state defined above, all rates of growth– that is, in price, dividends and earnings –will be equal to each other, as well as to aconstant. This, obviously, leads to the zero-dividend-yield criterion.

It can further be shown that upon lettingthe discount rate, R, be equal to this presumedconstant, R*, we obtain the followingexpression:

*ln

*R

tS

constantRR=

∂∂

==

���

� (1)

where S is the price and the quantity on theleft-hand side represents the percent rate ofgrowth in price [i.e. capital gains], conditionalon the discount rate being constant at R* - thatis R = R* = constant. Note that Equation 1 issimply the market’s rate of return,incorporating the zero-dividend-yieldcondition discussed earlier. It thus followsfrom 1 that the [logarithm of] price, ln S, couldbe written as a function of time, t, as well as R*

- that is:

),*(lnln tRSS = (2)

Holding the discount rate constant in theabove clearly imposes a daunting constraint onS. This, however, may easily be relaxed withthe help of a simple mathematical procedure,which entails the notion of the exactdifferential. The details of this procedure shallbe left out from here simply because they areavailable in almost any intermediate-level texton differential calculus.

Very briefly, the approach is as follows.In place of writing ln S(R*,t) as we have donein 2, we express it as

),(lnln tRSS = (3)

which generalises S to account for a time-variable discount rate, R = R(t), instead.

The rationale behind Equation 3 is thatthe effects of the market, and the economy ingeneral, on S are presumed to enter separatelythrough two fundamental elements, one whichis R and the other, which comprises everythingelse that falls outside the reign of R. As thesecond variable appears as time, t, it rendersEquation 3 general and, hence, along withR(t), it should capture all the economic andmarket effects on the price, S. In other words,expressing S in the form of Equation 3effectively removes all the restrictionsimposed on it earlier.

In light of the above, we take the totaltime differential of Equation 3 and obtain:

3

tR

RS

tS

ttRS

tR ∆∆

∂∂

+∂

∂=

∆∆ �

��

����

���

� lnln),(ln (4)

While the first partial differential – i.e.

RtS )/ln( ∂∂ - has been shown to be equal toR [see Equation 1], the second, tRS )/ln( ∂∂ ,

is simply the stock duration, which is thesensitivity of the price to changes in thediscount or interest rate at some point in time.The second term in full, which also includesvariations in the discount rate, embodies therisk premium as well.

Being a differential of an exact function,therefore, the two components in Equation 4are coupled to each other via:

1lnln

=∂

∂

∂

∂=

∂

∂

∂

∂ ����

���

��

����

��

�

����

���

�

RtR

S

ttRt

S

R(5)

which incorporates also Equation 1. Thiscould be integrated twice to yield the generalsolution to Equation 4, whereby

)(~ln 10 RRtRS Ψ+++= αα (6)

with α0 and α1 being integration constants and)(~ RΨ a yet unknown function of only R.Alternatively, we may recast 6 as:

)(ln 0 RtRS Ψ+=− α (7)

where )(RΨ is another function of R. Thelatter representation conveniently absorbsboth )(~ RΨ and α1R into a single function,

)(RΨ .It thus follows from 7 that plotting

tRS −ln against R should, in theory, producea single curve, depending only on R. Thistransformation, as a result, brings in all theeffects of time on tRS −ln through R. Aschematic illustration is presented in Figure 1,where a mapping of S versus R into tRS −lnversus R is shown to introduce some type ofregularity to a relatively disordered graph.

As per our derivation so far, we find itnecessary to mention two points. First, eventhough Equation 7 is extracted from whatappears to be formidable and perhaps too

theoretical an approach, it is indeed very easyto apply and, also, as it shall be demonstratedshortly, it does possess real and practical uses.Second, questions relating to the discount ratehave undoubtedly been raised by now. Forinstance, what is the discount rate, how shouldit be defined and, more importantly, howshould one deal with it? The answer to these,as it will turn out in Section 3, happens to besurprisingly straightforward. Beforehand,though, let us go ahead and apply the samelogic to both, the nominal GDP and earnings.

2b. Applications to GDP and EarningsIt is well accepted that movements in the

equity price index are closely tied to corporateearnings and, even more generally, to theeconomy. Common sense further dictates thata bull market typically comes with a strongeconomy and a bear market with a weakeconomy. One probable explanation for this issimply that the market comprises a subset ofthe economy – i.e. corporate earningsconstitute a [small] fraction of the GDP. This,therefore, should enable one to derive a GDPrelationship that is analogous to the one forequity, as well as for corporate earnings.

Before we begin, however, we need tointroduce a couple of analogies to the equityprice index. This is possible with the help ofthe earnings discount model4. For this, let usdefine VG and VE, respectively, as the “values”associated with the nominal GDP andcorporate earnings. Based on the above,therefore, VG could be represented by

R

tGtV f

G

)()( ≡ (8a)

and VE by

R

tetV f

E

)()( ≡ (8b)

4 Conditions under which the earnings discountmodel becomes as valid as the dividend discountmodel have been discussed earlier (Cohen, 2000).This entails an efficient market in which the firm-generated valuation equals that of the investor’s.Accordingly, therefore, the approach introducedin the present paper will not be suitable to pre1950, where this type of behaviour was notfollowed, at least by the S&P index.

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where Gf(t) and ef(t), respectively, are thetime-t expectation of the nominal GDP andcorporate earnings one year ahead, at t+1.Thus, with R being the discount rate, anearnings-discount-type valuation model isbeing imposed on the economy as well.5 Itshould further be stressed that the one-year-ahead nominal GDP, i.e. G(t+1), will fromnow on be implemented instead of theexpected purely for convenience, as we shallassume that the two are equal in aninformation-efficient economy. For corporateearnings, on the other hand, Datastream’saggregate I/B/E/S forecasts will be presumedsufficient for our purposes.

Now, with the above analogy in place, itis not difficult to demonstrate that the samerules that dominate the price index shouldapply to VG and VE as well, yieldingexpressions similar to Equation 7, but with VGand VE substituted for S. This, consequently,leads to:

)(ln 0 RtRVG Φ+=− β (9a)

and

)(ln 0 RtRVE Ξ+=− γ (9b) (20)

where βο and γο are integration constants andΦ(R) and Ξ(R) are functions of R only.

Aside from noting that the sametransformation that presides over the equitymodel applies to here as well, Φ(R) and Ξ(R)may not necessarily be the same as Ψ(R). Acomparison of these shall be undertaken laterin Section 6, however we need to addresscertain issues beforehand, namely of thediscount rate, “reversibility” and “structural orregime shifts.”

3. The Discount RateAs mentioned at the end of Section 2a,

the issue of the discount rate is an importantone. Putting it more precisely, what shouldone use for R in Equations 7 and 9 in order tobe able to test their validity?

Obviously, several choices exist. Theseinclude some measure of risk premium addedto some short/long-term risk-free interest rate, 5 Note that if we let R in Equation 8b be the USgovernment 10-year nominal bond yield, weeffectively recover the Federal Reserve’svaluation model (Greenspan, 1997).

bond yield, etc. Clearly, therefore, this makesroom for a lot of subjectivity. But, never theless, we will attempt to settle this for ourpurposes here.

Let us begin by recalling that, in a perfectmarket and conditional on R = R* = constant,all discount rates and rates of return will beconstant and equal to each other. Moreover,because of the risk premium being equal tozero, these will all converge to the rate ofinterest, which is also constant.

The zero-risk-premium constraint, whicheliminates all uncertainties on futureprojections, will additionally render all yieldand term structure curves flat by removing thespreads between the long and short-terminterest rates and yields. We, therefore, willhave remaining only one constant rate ofinterest, which we shall denote by r*. All this,of course, sets up the stage for a highlyidealised, as well as unrealistic, base-casescenario, which clearly extends the Goldenrule of economics to finance.

Bearing this in mind, it will be useful tohypothesise that any real-life scenario could beconsidered as merely a perturbation away fromthe base case. This line of reasoning willprove to be important here, as it will enable usto modify the highly superficial situation to areal one.

To deal with this, let us refer to eitherEquation 7 or 9. For convenience, we shallwork with Equation 7, although the logic thatfollows could equally as well apply to both, 9aand 9b.

Let us begin with the assumption that thisidealised, base-case scenario, where theinterest rate is equal to the constant discountrate, is in place. In addition, recall that theleft-hand side of Equation 7, which is

tRS −ln , is a function of the singleparameter, R, i.e. Ψ(R), the implication ofwhich is schematically presented in Figure 1.Putting these together gives

)*(*ln 0 RtRS Ψ+=− α (10)

and

)*(*ln 0 rtrS Ψ+=− α (11)

Moreover, the condition R* = r* makes wayfor the equality

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trStRS *ln*ln −=− (12)

simply because

)*()*( rR Ψ=Ψ (13)

We now move away from this idealisedstate by introducing time-variable ratesinstead. This can be accomplished byperturbing R* by some increment ∆R and r* by∆r, whereby both, ∆R and ∆r are free to varyin time, t. This, consequently, modifiesEquations 10 and 11 to:

)*(]*[ln 0 RRtRRS ∆+Ψ+=∆+− α (14)

and

)*(]*[ln 0 rrtrrS ∆+Ψ+=∆+− α (15)

respectively. It is important to recognise that,because the function Ψ is dependent only onthe single parameter - be it R or r - the effectof time on Ψ enters indirectly through either,R or r - i.e. R(t) and r(t).

Generalising this scenario even further bychoosing two different points in time - i.e. t1

and t2 - such that ∆R(t1) equals ∆r(t2), enablesus to equate 14 and 15 and get:

21 ]*[ln]*[ln trrStRRS ∆+−=∆+− (16)

simply because R*+∆R is chosen to be equalto r*+∆r. The significance of this is that itmakes no difference, whatsoever, as to whatone incorporates for R in Equations 7 and 9.For this matter, R could be either the discountrate, if known, or any of the several availablechoices for the interest rate, be it short term orlong.

The above could be observed inmappings that incorporate yields ofgovernment bonds with different maturities.Data convergence can be detected in virtuallyall cases, although, for sake of brevity, wehave decided not to include any exampleshere.6 Having said this, we now go on todiscuss the concepts of reversibility andstructural shifts, and how they enter this work. 6 The interested reader could obtain these directlyfrom the author.

4. Reversibility and Structural ShiftsThe equity price, GDP and earnings

representations provided in Equations 7 and9a-b lead to some important conclusionsregarding “reversibility” and “structuralshifts.” Realising that structural shifts tend toalter the behaviour of the economy and themarkets, an important objective here, as in anyeconomic and financial analysis, wouldthereby consist of defining ways for detectingand, possibly, classifying them.

To carry this out here, we start with theobservation that RtS −ln , RtVG −ln and

RtVE −ln must depend solely on R via thefunctions Ψ(R), Φ(R), and Ξ(R), respectively.The effects of time, as mentioned earlier, enterindirectly through R. Whether or not thisfunctional dependence of R on t is the same inall situations is not of concern now, but, in anycase, it shall be dealt with shortly.

An important outcome of suchdependence is the notion of “reversibility,”which may be explained as follows. Take, forinstance, Figure 2, where we focus on thetime-dependent behaviour of one of UKbenchmark government bond yields, namelythe 10-year, as obtained from Datastream. Wecould have very well selected other yieldsinstead, as the choice makes no difference towhat follows hereafter. The time frame herecovers the period Q1-80, to Q4-99.

Let us now, for the sake of example,identify and highlight 7 points where the yieldcrossed the value of 10%, all occurring ondifferent dates between 1986 and 1992. Veryclearly, even though the yields were identicalon these dates, which fall some years apart,one does not expect the corresponding valuesfor S, GDP and corporate earnings to remain inany way the same. Strictly speaking,therefore, S, GDP and corporate earnings, bythemselves, are “irreversible” functions of R,as they obviously tend to move in waysdifferent than R.

The concept of reversibility, never theless, comes into play when we consider,instead, the mapping prescribed by Equations7 and 9. Here we expect the transformedrelations, RtS −ln , RtVG −ln and

RtVE −ln to reposition with R, as theequations suggest. Therefore, if R were, say,equal to 10% in 1986, varies randomly overtime and in 1991, about five years later,

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reverts back to 10%, the transformed relationsshould also revert back to their original 1986values. This concept, which, in a sense,forces us to move away from the notion oftime series, is also schematically presented inFigure 1. The consequence of such behaviourshall be referred to as reversibility and willplay a dominant role in our work.

Alternatively, a structural or regime shiftimplies the contrary. If, for instance, a plot ofthe above-mentioned points fall on notablydisparate lines, then a structural shift,separating these points, might have occurredin between. Schematically, a structural shiftis exemplified in Figure 3, where mapping Sversus R into tRS −ln versus R over a giventime frame leads to distinctive characteristicpatterns. Empirical evidence of bothphenomena, namely reversibility and regimeshift, will be provided next.

4a. Evidence of Reversibility andStructural Shifts in the UK

We have argued so far that one couldincorporate any of the available interest ratesin Equations 7 and 9a-b. If our hypothesiswere correct, then on plotting RtX −lnagainst R, where X could equally represent S,VG or VE, we should expect to obtain a singlecurve, or, more generally, a series of curves,each pertaining to some particular structuralregime in the market and/or the economy.Although this mapping technique shall beapplied here only to the US and the UK, weshould emphasise that, in addition, a numberof other economies and markets tend todisplay similar behaviours (Cohen andChibumba, 1999).

First, let us go to Figures 4-6, whichillustrate RtS −ln , RtVG −ln and

RtVE −ln , respectively, plotted against R forthe FTSE 100 index and the UK economy.We have implemented here for R somenominal benchmark government bond yield.This is being utilised purely for consistency,because Datastream provides similar series fora number of other markets as well. It shouldalso be mentioned that all data are quarterly,beginning in 1980, which is when Datastreamstarted to provide them, to the present.Moreover, we note that frequency is not anissue since it does not alter any of the resultsthat follow next.

We refer to Figures 4a and 4b, whichdepict S and RtS −ln , respectively, plottedagainst R. Again, for R we have incorporatedthe above-mentioned 10-year yield, eventhough similar conclusions apply as well to theother yields.

On comparing Figure 4a to 4b, each alsocontaining a best-fit cubic-polynomial curvefor reference, we do observe a convergence ofdata where the proposed transformation hasbeen applied. From a statistical perspective,we implement the standard deviation relativeto the best fit as a measure of the scatter. Thisyields 0.247 and 0.088, respectively,belonging to Figures 4a and 4b. Clearly,therefore, the mapping does introduce asignificant amount of data convergence,consistent with the prediction.

Although in this case the mappingreduces the scatter [in terms of standarddeviation by a factor of ca. 3], we observe thatstill not all the scatter has been eliminated.This, we believe, is owed to the fact that theindex comprises various types ofinvestors/firms, each providing a valuationbased on a different discount rate. Therefore,aggregating all these together into a singleequity price, S, should naturally create adispersion in the mapped plane.

We have, in addition, highlighted in bothfigures the data for the 3 quarters immediatelypreceding the October 1987 stock marketcrash. It is remarkable that while there isvirtually no indication of “over pricing” inFigure 4a, the phenomenon becomes clearlyvisible as ouliers once the mapping isimposed. This is clearly portrayed in Figure4b. Interestingly also, a similar observation,although less conspicuous but equallyprominent, applies to the S&P 500 market aswell [see Figure 10]. In view of this,therefore, the methodology could potentiallybe useful as a means for detecting over/underpricing. However, being highly speculative atthis point, there is need for more rigoroustesting before this claim could be confirmed.

Returning now to the analysis, we notethat the scatter in Figure 4b is even morediminished if we were to graph RtVG −lnversus R instead, with R again representing abond yield. This is shown in Figure 5, wheredata convergence is much more noticeablethan in the previous case. This, we presume,is because VG takes into account thedifferences in investors – that is, in arriving at

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VG, an investor who, let us suppose, discountsat 5% will compute a VG different from onethat discounts at, let us say, 10%.

Last, but not least, the same logic appliesto earnings as well, which is graphed againstthe nominal 10-year yield in Figures 6a and6b, both alone and transformed, respectively.Note once again the marked data convergencein Figure 6b, where the mapping RtVE −ln isimplemented instead.

As for reversibility and structural shifts,they are evident in all the graphs, more so inFigures 5 and 6b, where streak-like patternsemerge. Each of these streaks corresponds towhat we believe to be a distinct structuralregime. Samples of these are circled inFigures 5 and 6b, which, for convenience, arealso shown expanded in Figures 7 and 8. Wenote here that data convergence is indeedremarkable under the proposedtransformation. This, therefore, stronglysupports the underlying hypothesis regardingreversibility and structural shifts.

It thus follows that reversibility occursalong any of the streaks in Figures 5 and 6b,where movements in interest rates, in somecases over many years, do not appear to throwany of the data points out of its course. This,presumably, happens because all these pointsbelong to a distinct structural regime. A moredetailed assessment of this will follow oncewe examine the situation for the US.

4b. Evidence of Reversibility andStructural Shifts in the US

Figures 10-12 depict the threeparameters, namely, RtS −ln , RtVG −ln and

RtVE −ln , all plotted against R. We observethat, as in the UK’s case, the relevantconclusions are identical. These are basically(1) all graphs exhibit data convergence, and(2) reversibility and regime shifts are againevident, particularly in Figures 11 and 12where the scatter is markedly less.

Apart from these similarities with theUK, an important feature further manifestsitself in Figures 10 and 12, where prices andearnings are concerned. In both, there appearsto be a branching of data, especially at yieldrates lower than 7%. One of the branches,which is highlighted and whose time frame isindicated, certainly belongs to a distinctstructural regime. We shall return to this

when we discuss both, the role of relativevaluation in, and its relationship to, this work.

5. Relative ValuationWithin the framework of our analysis,

relative-valuation measures could be arrived atby simply superimposing the data in Figures 4-12. The outcome of this shall be demonstratedhere for both the UK and US. We shouldwarn, however, that this is not an exercise in“forecasting.” It is merely a methodology bywhich intrinsic values could be comparedobjectively against one another.

It is also useful to mention that while theconclusions that follow are based more onvisual comparisons, an in-depth statisticalanalysis is necessary to put these findings onmore solid grounds. This, however, will comein later in Section 6 after the initial concept islaid out.

5a. Relative Valuation in the UKThe relative-valuation measures for the

UK are examined here by overlaying Figures5-7 on top of one another. For example,superimposing Figure 5 on 4b depicts how theFTSE 100 price index matches against theeconomy, both historically and currently.Similarly, laying Figure 6b on 4b and 6b on 5,respectively, illustrates how the economycompares against the equity market andcorporate earnings.

Let us begin with Figure 13, whichsuperimposes the price index on the nominalGDP, both in their special co-ordinatetransformations. We note here that, over thelong run, the two parameters appear to movetogether. This provides some justification tothe principle that, in the long term, equityprices and the GDP are related. Moreover,Figures 14 and 15, respectively, which overlaythe price index on the earnings and theearnings on the GDP, tell a similar story.Here, as well, the three elements, at least in theUK, appear to be in balance, moving verymuch together in the long run.

5b. Relative Valuation in the USSection 5a summarised the situation for

the UK. The US, as we shall see now, leads tosome different conclusions.

Figure 16 displays an overlay of the S&P500 price index over the GDP. We note herethat from 1980, which is the start of the data,to 1990, the points fall, more or less, on top of

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each other. This basically indicates thatduring this period, the two parameters weremoving in tandem with one another, aseconomic theory dictates.

In contrast to the above, however, weobserve that a breakdown in the relationshipoccurs somewhere in the middle, when thetwo begin to go in separate directions. Thedeviation begins sometime in 1995, when theS&P 500 index and the GDP appear to movealong different paths altogether. Whether thismeans that the index is overpriced relative tothe GDP and/or the components of the indexhave, on aggregate, been outperforming theaverage economy is perhaps speculation. But,whatever the reason, the insinuation is thatfrom 1995 onwards, the S&P 500 market hasfailed to represent the economy, bothadequately and fairly.

Figure 17 illustrates the S&P 500superimposed over corporate earnings. Thesituation in this case is somewhat clearer thanin the previous figure. Here, for instance, weobserve that the post-1994 increases in equityprices have followed the sudden jump in thecorporate earnings seen in Figure 12. Evenhere, there seems to be a post-1994 out-performance of the price index relative toearnings.

Finally, we superimpose corporateearnings over the GDP in Figure 18. Onceagain, avoiding all hypothetical explanationsand relying strictly on visual comparison7, wenote that until around the beginning of 1995,corporate earnings moved together withnominal GDP. Past that period, however, ashift in corporate earnings caused it tooutperform the GDP, with the latter stillcontinuing in its pre-1995 course. This isconsistent with our observation in Figure 16,where we concluded that, post 1994, the S&P500 equity market has not adequatelyrepresented the economy. Very clearly,therefore, the same applies to corporateearnings as well.

6. Some Remarks and Preliminary StatsIt is important to mention once again that

our conclusions so far have been based onless-than-elaborate statistical scrutiny. This,up till now, has simply entailed either visualcontrasts or computing standard deviations 7 The differences in this case are so obvious thatthey could be compared visually, but, never theless, qualitatively.

relative to some best-fit curve to prove thatdata do indeed converge when mappedaccording to our proposed methodology. Anexample of this has been included in Figures4a and 4b.

We now find it necessary now to conductsome preliminary statistics on how themapping functions, Ψ(R), Φ(R) and Ξ(R), asdescribed in Sections 2a and 2b, relate to oneanother. Respectively, these correspond to theprice, the GDP and aggregated I/B/E/Scorporate earnings forecasts.

The testing strategy is basic and goes asfollows. First, we assume that there is a linearrelationship between each of the functions –i.e.

Ψ(R) = a0=+=a1Φ(R), (17a)

Ψ(R) = b0=+=b1Ξ (R) (17b)

and

Φ(R) = c0=+=c1Ξ (R) (17c)

where the coefficients a0,1, b0,1 and c0,1 areregression constants. Second, in accordancewith our initial conjecture, if the hypothesised“golden rule”8 were to hold to within anyreason, then

Ψ(R) = Φ(R), (18a)

Ψ(R) = Ξ (R) (18b)

and

Φ(R) = Ξ (R) (18c)

all of which are obtained from

a0, b0 and c0 = 0 (19a)

and

a1, b1 and c1 = 1 (19b)

Effectively, therefore, this so-called goldenrule attempts to tie in together the financial

8 The golden rule here will signify the situationwhere the three functions, Ψ(R), Φ(R) and Ξ(R),are, at any point in time, exactly equal to oneanother.

9

markets with the economy on a strict, one-to-one basis [i.e. Ψ(R) = Φ(R) = Ξ(R)]. Theoutcome of this test, which involves runningthe regressions in 17a-c and assessing thesignificance of the null hypotheses in 19a-b, ispresented in Table 1.

Before we proceed with any discussions,it is necessary to note that the US has beenexcluded from here. The reason again refersto Footnote 7 – very simply, a visualexamination of the overlays in Figures 16-18is sufficient to strongly reject either or bothnulls, 19a and/or b. It is very clear in all thesefigures the absence of any strong one-to-onecorrespondence between Ψ(R), Φ(R) andΞ(R). This is perhaps because the threeappear to react differently to shocks.

In contrast to the US, the UK leads to anoutcome that is apparently unlike. Here, forexample, a quick visual assessment of theoverlays portrayed in Figures 13-15 doesindicate some amount of balance between thethree elements.

The test statistics, notwithstanding,which are outlined in Table 1, provide a moredetailed story. For instance, while the p-values indicate that the price index does notcorrespond that strongly with either, theeconomy [i.e. UK GDP] or corporateearnings, there does seem to be a strong 1:1relationship between corporate earnings andthe economy. A plausible explanation for thisis that, on aggregate, the FTSE earningsforecasts do indeed represent a fair sample ofthe UK’s economy. As for why the othersdon’t match so well, it might be that (1) thereis insufficient data, as I/B/E/S earningsforecasts for the FTSE 100 index becameavailable only after 1987, and/or (2) ourhypothesis of the golden rule may simply notbe valid within the time frame tested.

To further help clarify the matter, wehave included Figures 19a-b as well. Figure19a is just a plot of the FTSE 100 aggregated-earnings forecast against the correspondingprice index. Note the absence of any firmand/or conclusive relationship. Figure 19b, onthe other had, portrays 19a as Ξ versus Ψ,which is just a re-plot of Figure 14. The solid,diagonal line displays our theorised conjecturein Equation 18c, whereas the dashed line is alinear best fit through the data. Simply stated,therefore, the corresponding p-value of 11%-15% in Table 1 is the level of confidence withwhich one could claim that this diagonal line

is instead the best fit through the data in Figure19b. As noted, the confidence level is not thathigh for this particular case.

Lastly, we find it necessary to mention acouple of difficulties that we encountered intesting our overall results. First, as detectionand identification of structural tests comprisean integral part of this paper, we are not awareof any tests that check for the structural breakswithin the context of our work. Recall that thetraditional time-series-based methods fordetecting structural breaks are not valid herepurely because the mapping we incorporated isnot time series. Secondly, as of yet, we haveno theory to describe the shape of any of thefunctions, Ψ(R), Φ(R) and Ξ(R). The cubic-polynomial, best-fit curve shown in Figure 4bwas used here for convenience to help uscompute the standard deviation. With theabove in mind, therefore, it goes withoutsaying that (1) a more elaborate theory isneeded to describe the factors that underlie theshapes of the above functions, and (2) a morerobust and solid stronghold could be gainedonly after the results of our approach are fullysubjected to more rigorous statistical scrutinyand testing. These, by themselves, providescope and direction for further research alongthese lines.

7. Summary and ConclusionsAn objective, and [hopefully] practical,

approach to relative valuation has beenproposed. The method, which entails asimple mapping, enables one to objectivelycompare the nominal GDP, corporate earningsand equity index relative to one another.

On applying this to the UK and USmarkets and economies, we reach certainconclusions, some of which are listed below.

(a) On the basis of this work, acomparison of the UK equity index,aggregated corporate earnings and thenominal GDP, as shown in Figures 13-15,suggests that the three have been, more orless, in balance with one another over thepast 20 years. In other words, with theexception of a few quarters prior to theOctober, 1987, stock market crash, the UKshows no evidence of gross over/underpricing of any of these in relation to oneanother.

(b) A basic statistical test, focusing on therelationships between the price index, the

10

aggregated earnings forecast and theeconomy, has led to some inconclusiveresults regarding the golden rule. While itwas shown that the FTSE 100 corporateearnings compare well against theeconomy, any strong evidence of a one-to-one relationship between the [mapped]FTSE 100 price index with either, thecorporate earnings and the UK GDP,appears to be lacking.

(c) The situation in the US is markedlydifferent. Notwithstanding the lack ofstatistical tests here and relying more onvisual comparisons, our work indicatesthat from 1980 to 1994, the three elementswere roughly, and in terms of order ofmagnitude only, in balance with oneanother, as Figures 16-18 indicate. Post1994, however, according to Figure 18 anupward shift in corporate earnings pushedthe valuations above those implied by thenominal GDP. Moreover, during thesame period, the equity price index hassurpassed even the corporate earnings interms of relative valuation [see Figure17]. This peculiar behaviour providesclear evidence of what has been describedas the “new economy,” which presumablybegan in the US soon after 1994. Thereare certainly severe inconsistencies in thevaluations, all which have been happeningthroughout this last regime.

(d) It is ironic that this so-called “neweconomy” in the US seems to apply to theequity market only, which consists ofboth, prices and corporate earnings, butnot to the whole economy, as reflectedhere by the GDP. The US GDP, whenmapped accordingly [see Figures 11, 16and 18], appears to be immune from theimpacts of this structural shift. This,perhaps, is owed to the GDP being toolarge to be measurably influenced by theeffects of this new economy.

We finally conclude here by stressingonce again that our approach here has nothing

to do with forecasting. It simply manifests along-term measure of relative valuationbetween an index, its aggregated corporateearnings and the GDP.

9. References

Arnott, R.D. and Henriksson, R.D. (1989) “ADisciplined Approach to Global AssetAllocation,” Financial Analysts Journal(Mar/Apr), pp 17-29.

Barth, M.E., Beaver, W.H. and Landsman,W.R. (1998) “Relative Valuation Roles ofEquity Book Value and Net Income as aFunction of Financial Health,” Journal ofAccounting and Economics 25, pp. 1-34.

Benari, Y. (1988) “A Bond Market TimingModel,” Journal of Portfolio Management15, pp 45-49.

Cohen, R.D. (2000) “The Long-run Behaviourof the S&P Composite Price Index and itsRisk Premium," Internal Circulation, SSBCiti Asset Management Group, London.

Cohen, R.D. and Chibumba, A. (1999) “AnEquity Model,” Working paper, SSB CitiAsset Management Group, London.

De Long, J.B. and Shliefer, A. (1992)“Closed-end Fund Discounts,” Journal ofPortfolio Management 18, pp 46-64.

Fabozzi, F.J. Investment Management, 2nd Ed.,Prentice Hall, NJ, 1999.

Greenspan, A. (July 22, 1997) MonetaryPolicy Report to the Congress.

Peters, K.J. (1991) “Valuing a Growth Stock,”Journal of Portfolio Management 17, pp49-66.

11

Relationship Implication Available data range &no. of observations

Significance level9,10

[p-value]

Ψ(R) = Φ(R)1:1 relationship betweenequity price index and

economy80:1 – 98:3

(75)26%, 37%

Ψ(R) = Ξ (R)1:1 relationship betweenequity price index and

corporate earnings87:3 – 99:4

(50)11%, 15%

Φ(R) = Ξ (R)1:1 relationship betweencorporate earnings and

economy87:3 – 98:3

(45)88%, 99%

Table 1 – Significance levels, in terms of p-values, for one-to-one relationships between theFTSE 100 price index, economy and the aggregated corporate earnings. These are representedby the symbols Ψ(R), Φ(R) and Ξ (R), respectively. The stats for the US have been excluded

owing to reasons stated in Section 6.

Figure 1 – Schematic diagram of convergence of data points under the proposed mapping.

9 There are two p-values. The first refers to the null in Equation 19a and the second to 19b.10 The p-values are based on two-tailed tests around the Student-t distribution. The degrees of freedom arethe number of observations [in column 3] minus 2.

S

R

xx x

x x

xx

xx x

xx

x x

xx

x x

xxx x

x

l n (S

) - R

t

R

Ψ(R)

maps into

x

x

x

xx

xx

xx x x

x

xx

xx xx

12

Figure 2 – Behaviour of the 10-year UK benchmark government bond yield between Q1-80 to Q1-00. The circled regions, which are numbered, are the locations where the yield crossed the 10%

line during the past 20 years.

Figure 3 – Schematic diagram of how a regime shift manifests itself under the suggestedmapping. A mapping of S versus R into RtS −ln versus R leads to distinctive characteristic

functions, depicted here by Ψ1(R) and Ψ2(R), each belonging to a separate regime.

S

R

x

xx

x x

x

x

x

x x

x

x

x x

x

x

x x

x

xx x

x

ln (S

) - R

t

R

Ψ2(R)

maps into

x

x

xx

x

xx

x

xxx

x xx

xx x

Ψ1(R)

xx

4

6

8

10

12

14

16Q

1 80

Q1

81

Q1

82

Q1

83

Q1

84

Q1

85

Q1

86

Q1

87

Q1

88

Q1

89

Q1

90

Q1

91

Q1

92

Q1

93

Q1

94

Q1

95

Q1

96

Q1

97

Q1

98

Q1

99

Q1

00

UK 10y

1 5 6

732

4

13

Figure 4a – The logarithm of the FTSE 100 price index, ln S, plotted against the nominal UK 10-year benchmark government bond yield. Note that this is not a time-series plot. Data rangefrom Q1-80 to Q1-00. The triangles correspond to the 3 quarters immediately preceding the

October 1987 crash. The line is a best-fit cubic polynomial and σ is the standard deviation ofthe data relative to it.

Figure 4b – The FTSE 100 price index in transformed co-ordinates, ln S – Rt, plotted against thenominal UK 10-year benchmark government bond yield. Data range from Q1-80 to Q1-00.

Note tightness of data relative to Figure 4a. The triangles correspond to the 3 quartersimmediately preceding the October 1987 crash. The line is a best-fit cubic polynomial and σ is

the standard deviation of the data relative to it.

6

6.5

7

7.5

8

8.5

9

4 6 8 10 12 14 16

σ===0.247

6

6.5

7

7.5

8

4 6 8 10 12 14 16

σ===0.088

14

Figure 5 – The UK nominal GDP in transformed co-ordinates plotted against the UK 10-yearbenchmark government bond yield. Data range from Q1-80 to Q1-00. The circled region

presumably belongs to a distinctive structural regime.

Figure 6a – The FTSE 100 corporate earnings, as is, plotted against the nominal UK 10-yearbenchmark government bond yield. Data range from Q1-80 to Q1-00.

160

180

200

220

240

260

280

300

320

340

4 5 6 7 8 9 10 11 12 13

6

6.5

7

7.5

8

4 6 8 10 12 14 16

Post Q1-94

15

Figure 6b – The FTSE 100 corporate earnings in Figure 6a plotted in transformed co-ordinatesagainst the nominal UK 10-year benchmark government bond yield. Data range from Q1-80 to

Q1-00. The circled region presumably belongs to a distinctive structural regime.

Figure 7 – Expanded view of the structural regime highlighted in Figure 5.

6.5

7

7.5

8

5 5.5 6 6.5 7 7.5 8 8.5 9

post Q1-94

6

6.5

7

7.5

8

8.5

4 5 6 7 8 9 10 11 12 13

Containing data Q4-94 to Q4-98

16

Figure 8 – Expanded view of the structural regime highlighted in Figure 6.

Figure 9 – The same structural regime shown in Figure 8, but plotted as is – i.e. ef versus thenominal 10-year yield. This is a subset of Figure 6a. Note the loss of data convergence in

comparison with its counterpart in Figure 8.

6.5

7

7.5

8

4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Q4-94 to Q4-98

260

280

300

320

4 5 6 7 8 9

Q4-94 to Q4-98

17

Figure 10 – The S&P 500 price index in transformed co-ordinates plotted against the nominalUS 10-year benchmark government bond yield. Data range from Q1-80 to Q1-00. The trianglebelongs to the quarter immediately preceding the October 1987 crash, while the circled region

highlights a distinct structural regime covering the more recent time frame.

Figure 11 - The US nominal GDP in transformed co-ordinates plotted against the nominal US10-year benchmark government bond yield. Data range from Q1-80 to Q1-00.

4

4.5

5

5.5

6

6.5

4 6 8 10 12 14

post Q2-95

4

4.5

5

5.5

6

4 6 8 10 12 14

18

Figure 12 - The S&P 500 corporate earnings in transformed co-ordinates plotted against thenominal US 10-year benchmark government bond yield. Data range from Q1-80 to Q1-00. The

circled region presumably belongs to a distinctive structural regime.

Figure 13 – Overlay of Figure 4 on 5, showing the FTSE 100 equity price index in comparisonwith the UK nominal GDP. Both parameters are in transformed co-ordinates. The most recent

data point is highlighted.

6

6.5

7

7.5

8

4 6 8 10 12 14 16

GDPFTSE 100Most recent

4.5

5

5.5

6

6.5

4 5 6 7 8 9 10 11 12

post Q2-95

19

Figure 14 – Overlay of Figure 4 on 6, showing the FTSE 100 equity price index in comparisonwith the corporate earnings. Both parameters are in transformed co-ordinates. The most recent

data points are highlighted.

Figure 15 – Overlay of Figure 5 on 6, comparing the UK nominal GDP with the FTSE 100earnings. Both parameters are transformed. The most recent data point is highlighted.

6

6.5

7

7.5

8

4 6 8 10 12 14 16

EarningsFTSE 100Most recent

6

6.5

7

7.5

8

4 6 8 10 12 14 16

EarningsGDPMost recent

20

Figure 16 – Overlay of Figure 10 on 11, showing the S&P 500 equity price index in comparisonwith the US nominal GDP. Both parameters are in transformed co-ordinates. The most recent

data point is highlighted.

Figure 17 – Overlay of Figure 10 on 12, showing the S&P 500 equity price index in comparisonwith the corporate earnings. Both parameters are in transformed co-ordinates. The most recent

data point is highlighted.

4

4.5

5

5.5

6

6.5

4 5 6 7 8 9 10 11 12 13

Earnings

S&P 500Most recent

4

4.5

5

5.5

6

6.5

4 6 8 10 12 14

GDPS&P 500

Most recent

21

Figure 18 - Overlay of Figure 11 on 12, showing the US nominal GDP in comparison with theS&P 500 corporate earnings. Both parameters are transformed. The structural regime circled in

Figure 12 is again highlighted here.

Figure 19a – The I/B/E/S aggregatedearnings forecast for the FTSE 100 versusthe corresponding price index. Note theabsence of any firm and conclusiverelationship.

Figure 19b – Earnings, Ξ, versus the priceindex, Ψ, both in transformed co-ordinates.This is simply a re-plot of Figure 14. Thedashed line is a linear best fit, whereas thesolid diagonal satisfies one of our theoreticalconjectures, namely Ξ==Ψ [see Equation 18b].

4.2

4.6

5

5.4

5.8

6.2

4 5 6 7 8 9 10 11 12 13

EarningsGDP

post Q2-95

100

150

200

250

300

350

1000 3000 5000 7000

FTSE 100 Price IndexI/B/E

/S A

ggre

gate

d E

arni

ngs

Fore

cast

6

6.5

7

7.5

8

6 6.5 7 7.5 8

Ψ(R)

Ξ(R

)

(b)

Linear best fit

Line satisfyingΞ===Ψ

(a)