PJ McCloskey

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Candidate number: 13141288

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Supervisor:

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Dr. Roald Versteeg

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September 2019

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Abstract

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Department of Economics, Mathematics and Statistics

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Birkbeck College, University of London

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Stock markets world-wide have rewarded patient investors, hence the common adviceto “buy-and-hold”. Yet, even with a large body of research over a prolonged period,proof remains an onerous exercise for academics. We use Tobin’s Q and the dividendyield to build an equilibrium relationship for the US aggregate stock market using 119years of data. The resulting VECM model supports practitioners making long-horizonpredictions and provides powerful forecasting ability. Our work is directly applicable toUK price controls

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Keywords: stock market; expected returns; predictability; valuation ratios; Tobin’s Q; VECM.

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* I thank Roald Versteeg, Donald Robertson and Simon Price for their generous support.

1. Introduction

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1.1. The Expected Market Return (EMR)

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The Expected Market Return (EMR) [1] is the real return that equity investors expect from a well-diversified portfolio. Its use in asset pricing models, investment allocation strategies, pension scheme valuations, and the cost of monopoly services such as water and energy utilities makes it a very important value for investors, consumers, and finance professionals.

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1.2. Practitioner approaches

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The EMR is not directly observable, so practitioners are faced with a common challenge: how should we estimate an expectation? Broadly, there are two options: forward-looking predictions or outturn averages.

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Most practitioners make forward-looking predictions. Examples include JP Morgan (2019), Schroders (2019), Vanguard (2019), Willis Towers Watson (2019), Aberdeen Standard Investments (2019), SEB Group (2019), UBS (2019), and Aon Hewitt (2018). However, this approach is inherently subjective and assumption intensive.

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In contrast, some practitioners focus on outturn averages. Examples include the Competition Commission (CC) (2014) and Ofgem (2019). Although this approach is less subjective, it assumes that the outturn average remains appropriate.

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1.3. Predictability

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We use the term “predictability” to refer to the accuracy of EMR forecasts over long holding periods, relative to realised returns. In this way, we need not contrast with the Efficient Market Hypothesis (EMH). We assume markets are very efficient in the short run but less efficient in the long run, and that the EMR varies through time.

2. Literature Review

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2.1. Predictability

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Some academics argue that the EMR is lower than historical averages imply. Mehra and Prescott (1985) argue that ex-post returns are inexplicable in the context of standard models of risk. Campbell and Shiller (1988) and Shiller (1990) focus on whether there is “excess volatility” relative to the present value of dividends, with Shiller arguing that markets can be irrationally exuberant. Fama and French (1988) assess the power of dividend yield regressions, showing that explanatory power grows with the investment horizon. They appeal to the simple logic that EMR changes are offset by immediate decreases in the current price, and thus mean reversion arises from time variation of expected returns. Blanchard (1993) argues that, relative to risk-free rates, ex-post equity returns are far in excess of what is justified by standard asset-pricing models with reasonable levels of risk aversion.

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A common thread in the literature is that, over long horizons, ex-post returns are less uncertain than expected, given the uncertainty of one-year returns. Dimson, Marsh and Staunton (DMS) (2001) show that worldwide stock returns have a degree of long-term consistency. Campbell and Viceira (2001) show how predictability is applied in portfolio allocation strategies. Robertson and Wright (2002b) present a Vector Error Correction Model (VECM), using dividends and asset values, Tobin’s Q, to explain reductions in volatility. In support of Smithers and Wright (2000), Harney and Tower (2003) demonstrate the predictive power of Tobin’s Q, disagreeing with Epstein (2000), who criticised Smithers and Wright. The most forceful advocate of predictability, Siegel (2008), argues that bonds are riskier than stocks in the long term. Cochrane (2008) attempts to address statistical concerns by explaining that the absence of dividend growth predictability supports the argument for return predictability. Tower (2011) develops an updated model for CAPE, Tobin’s Q and CAPER, and argues that Tobin’s Q plays the largest explanatory role.

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2.2. Non-predictability

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Kim, Nelson and Startz (1991) argue that mean reversion is entirely a pre-war phenomenon. Epstein (2000) dismisses the use of Tobin’s Q as a value indicator, arguing that value indication is not its intended purpose and that “new economy” and intangible factors are unduly ignored. Other academics challenge the econometric properties, the statistical significance, the power of the statistical tests, and the statistical bias in favour of predictability inference. Examples include Torous, Valkanov and Yan (2004), Campbell and Yogo (2006), Goyal and Welch (2003), and Campbell and Thompson (2008). Boudoukh, Richardson and Whitelaw (2005) argue that the econometric problem is one of overlapping observations and persistence of the predictive variable. Ang and Bekaert (2007) argue that, after accounting for small sample properties of standard tests, excess return predictability by the dividend yield is not statistically significant.

3. Our Contribution

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If we believe that the stock market can deviate from its fundamental value for a prolonged period, the current literature is light on detail with regards to Tobin’s Q or other long-run fundamentals, with CAPE found to be second best when both are used. Earnings ratios also lack consistent measurement over time given changes to accounting rules, and are difficult to consolidate at a macroeconomic level for long periods. Further, CAPE ratios can be strongly correlated with Tobin’s Q and may therefore lack added information value; hence, we do not include these. There is also a lack of evidence for low-frequency data, such as annual rather than monthly data, over long horizons of more than five years.

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Our approach to test predictability, using a long-horizon weak-form approach, therefore:

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• uses physical asset values, Tobin’s Q, and dividends as predictors
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• uses annual data, focusing on the US non-financial sector
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• tests return variances in 21 countries
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• tests stationarity and cointegration, building a VECM model for the US equilibrium
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• presents impulse responses
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• compares VECM predictions with outturn averages, in-sample and out-of-sample
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• applies this work to UK price controls

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In doing so, we follow closely Robertson and Wright, The Good News and the Bad News about Long-Run Stock Returns (2002b). We do not attempt to explain the risk premium of stock returns, as discussed by Favero and Gozluklu (2009), as this would, in our view, introduce more difficulty than is strictly necessary.

4. Methodology

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4.1. Tobin’s Q

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For the aggregate stock market, Tobin’s Q is the ratio of companies’ market value to the replacement value of companies’ assets. Theoretically, market value should equal asset replacement value – the ratio should never deviate from unity. By extension, a ratio more (less) than unity indicates poor (good) value. Empirically, measurement issues arise for assets – depreciation may be underestimated, intangible assets may not be captured, and/or markets for assets may be non-existent or illiquid. To avoid primary measurement issues, we focus on deviations from the average value of Tobin’s Q, rather than the absolute value. Thus, when the stock market is overvalued (undervalued), Tobin’s Q will be high (low), relative to its average value. A working paper from the Bank of England notes:

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“…the common perception that Q is interesting from a theoretical perspective, but of little empirical relevance, is not true. In contrast, it appears to be a rich source of information about real and financial quantities.” (Bank of England, 2006, p. 5)⁴

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In levels, equity Q can be defined as:

            Pₜ
        Qₜ = ───            (1)
            Kₜ

Where Pₜ represents stock market value and Kₜ represents capital replacement value. In logarithms,

        qₜ = pₜ − kₜ            (2)

We define log equity returns following Campbell & Shiller (1988) as follows,

        rₜ₊₁ ≡ ln(1 + Rₜ₊₁) = ln ⎛ Pₜ₊₁ + Dₜ₊₁ ⎞
                        ⎝  Pₜ  ⎠

          ≈ φ + Δpₜ + (1 − ρ)(dₜ₊₁ − pₜ₊₁)    (3)

Wherein ρ = 1 / (1 + exp(d̄ − p̄)) and φ = ln(1 + exp(d̄ − p̄)) − (1 − ρ)(d̄ − p̄) and d̄ − p̄ is the mean dividend yield. Substituting from (2) into (3) to eliminate pₜ, solving for qₜ, and iterating forwards, subject to the transversality condition limᵢ→∞ ρⁱqₜ₊ᵢ = 0, we obtain:

             φ
        qₜ ≅ ─────── + ∑ᵢ₌₁∞ ρⁱ⁻¹fₜ₊ᵢ      (4)
           (1 − ρ)

        where fₜ = Δkₜ + (1 − ρ)(dₜ − kₜ) − rₜ

In other words, q is asymptotically equal to a weighted sum of the future values of f, which in turn is

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In other words, q is asymptotically equal to a weighted sum of the future values of ƒ, which in turn is the sum of; changes in capital ∆k (i.e. total capital), future profitability, (d − k) and returns, 𝑟.However, this design assumes 100% equity financing; to extend this model further, we include theimpact of leverage, adding 𝐿𝑡to the numerator of (1), and updating (2) and (4) accordingly, to derive

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        qₜ ≅ ∑ᵢ₌₁∞ ρⁱ⁻¹eₜ₊ᵢ            (5)

    where fₜ = Δkₜ + (1 − ρ){(1 − ζ)(dₜ − kₜ) + ζ(lₜ − kₜ)} − ζΔlₜ − (1 − ζ)rₜ

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This model should be relatively uncontroversial given the following intuition.⁵ Simply, (5) shows that q reflects total capital Δkₜ, adjusted downwards for total liabilities ζΔlₜ, and further downwards for total returns to equity, (1 − ζ)rₜ, wherein ζ represents total leverage, such that equity returns decrease, in levels not proportions, as leverage increases. Further, any upward pressure on q, in fundamental rather than measured terms, must arise from increases in capital, increases in profitability as proxied by (dₜ − kₜ), reductions in liabilities, or reductions in returns.

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Thus, this formulation of q serves three very important purposes. First, it confirms an analytical grounding that links q to equity returns r. Second, we can naturally see that if q is mean-reverting, the components on the right-hand side, individually or collectively, must be mean-reverting also. Third, and most significantly, we have confirmed clear theoretical grounds that equity returns are bounded by underlying asset values.

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In this sense, “predictability” is an entirely natural occurrence, caused by rational economic forces. Plausibly, it could nonetheless be argued that, if mean-reversion of q is weak, r could still be random, or indistinguishable from random. However, Poterba & Summers (1983) demonstrate that q has little explanatory power for investment. Leaving three other parameters that mean-reversion of q could explain: (i) liabilities, l, which has considerably less uncertainty given natural bondage with capital and its fixed-income nature; (ii) dividends, d, which we address in the following section; and lastly, (iii) equity returns, r.

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4.2. Dividends and other distributions to equity

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The stock market should equal the discounted value of future dividends and other distributions to equity. With a constant discount rate and a constant growth rate, increases or decreases in dividends/distributions should be reflected in increases or decreases in Pₜ.⁶ If the dividend/price ratio is higher or lower than average, this may indicate the market is undervalued or overvalued. We therefore use the Dividend Yield (DY), or when including other cashflows to/from equity, the Adjusted Dividend Yield (ADY), as a fundamental indicator of value. The ADY captures all flows between equity investors and corporations, including dividends, stock repurchases, new issues, cash-financed mergers and acquisitions, and private equity issuance. In contrast, the DY captures only dividend payments. ADY is therefore a more comprehensive measure of value — we will demonstrate other differences below.

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4.3. Pre-conditions

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As is standard with all time-series analysis, we inhabit a log-linear world, following standard notation of using lower case letters to denote logs. To avoid making spurious inferences, q and ady need to be stationary, which we can refer loosely to as “mean-reversion”.

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Each indicator, but particularly q, is strongly linked to a time-series approach, motivating us to obtain a long time-series of 119 years, to then estimate how far, and for how long, stock prices can stray from underlying fundamentals. Westerlund & Narayan (2014), in contrast, argue that precision may be increased using panel data rather than a long time-series, effectively trading Ns for Ts. However, for our purposes, we are not persuaded. A larger number of Ns, which in our case could be international stock markets, instead of Ts, meaning longer time series, would come at a cost of understanding the maximum disequilibria and/or average equilibrium that a major market is content to bear. Our valuation theory is that equity values cannot stray unbounded from q or ady and that future returns reflect previous values of underlying fundamentals. Following standard practice, we use total real returns throughout.

5. Data

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5.1. US Nonfinancial Corporate Business

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We collect 119 years of annual data for the US non-financial sector, using data from Wright (2004b) for the years 1900 to 1945, and from FRED, US financial accounts, for the years 1945 to 2018 regarding annual balances⁷ and annual flows.⁸ Wright’s methodology (2004a) and data are published online, and we follow these carefully to build on the original research.⁹ Following Wright, and to avoid a discontinuity in the FRED data regarding land values, we re-create an alternative series that is more consistent over time. Further, we adjust corporate bonds and mortgages to market values on the basis that FRED data on these items are primarily recorded at book values. Without this adjustment, the use of book values causes two problems: first, q would be downward biased, as liabilities represent a significant value of corporate worth; second, a negative value for net liabilities makes it impossible to input log values to our econometric model. We convert from nominal using Shiller’s inflation series¹⁰, noting that it is simply a rebased version of Wright’s data. We publish our dataset including reconciliations to Wright’s original work.¹¹

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5.2. UK Fundamentals and Returns in 21 Countries

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To supplement this, we obtain an unrivalled dataset from DMS, on equities, bonds, inflation and exchange rates, for 21 countries for the same 119-year period, 1899 to 2018.¹² For the UK, and for the period 1995 to 2018, we derive q from ONS national balance sheet estimates for nonfinancial corporations and use S&P Capital IQ to collect FTSE All-share information for 631 firms’ ADY.¹³ We supplement this with a discontinued series from the Financial Times on FTSE All-share dividend yields from 1963 to 2016. We sought, but unfortunately did not obtain, data from the Bank of England on q.