The Implied Equity Duration - Empirical Evidence for Explaining the Value Premium

The Implied Equity Duration - Empirical Evidence for Explaining the Value Premium This version: April 16, 2010 (preliminary) Abstract In this empirical paper, we demonstrate that the observed value premium in stock markets might be a compensation for an underlying equity duration based risk factor. Similar to the bond duration concept, the equity duration captures the average timing of a company s cash flows to its shareholders. We find that low-duration stocks have both higher expected and realized returns than high-duration stocks. This difference in returns cannot be attributed to the shares risk exposure as measured by the market beta. Instead, our analysis presents evidence that the equity duration has very similar properties than the Fama-French factor B/M ratio, suggesting that this ratio acts as a proxy of the equity duration. JEL Classification: G12, G15 Keywords: equity duration, bond duration, value premium, analysts forecasts, stock returns, panel regressions, expected returns, implied cost of capital

1 1 Introduction The value premium is one of the most striking characteristics of stock returns. First detected by Graham and Dodd (1934), the value premium denotes the observation that stocks with a high book-to-market ratio of equity (value stocks) exhibit higher average returns than stocks with a low book-to-market ratio (growth stocks). More important, this additional return is not a compensation for investors to bear higher systematic risk as implied by the CAPM (Sharpe, 1964; Lintner, 1965), but seems to be an anomaly with respect to this well-known asset pricing model (Basu, 1983; Fama and French, 1992). These findings led Fama and French (1993, 1996) to incorporate the value effect as additional factor to explain the cross-sectional variation of stock returns. However, their so-called three factor model is only empirically motivated, and not substantiated by any comprehensive economic model. In the sequel, theory followed experiment, and researchers tried to find theoretical explanations for the value effect. Among others, Lettau and Wachter (2007) propose an equity duration based explanation of the value premium 1. In close analogy to bond duration, they define firms that pay out a large fraction of their cash flows in the distant future as high duration stocks, whereas companies that pay relatively more to their shareholders in the near future are called low duration stocks. The difference in the payment timing between both classes of stocks has considerable consequences on their price sensitivity to changes in the market environment, and thus its riskiness. In general, low duration stocks are more sensitive to changes in expected cash flows than longhorizon equity. On the other hand, high duration stocks are more sensitive to changes in the discount factor. Campbell and Voulteenaho (2004) show that for risk averse, long-term investors, only the latter risk should be priced. As a consequence, low duration stocks - which correspond in general to value stocks - are perceived to be riskier than low duration stocks. Essentially, the value premium is thus a compensation for a share s risk exposure to cash-flow surprises. Although a valuable concept in theory, empirical validations of the equity duration face the challenge of estimating the equity duration. In contrast to bond duration, future payments to shareholders are unknown and must be estimated. What is more, equity investments usu- 1 The literature proposed also other explanations for the value premium. On the one hand, Brennan et al. (2004) and Campbell et al. (2007) use macroeconomic factors to relate the value premium to stock returns. Another stream of the literature, see e.g. Bansal and Yaron (2004) and Cohen et al. (2008), examines dividend payments of companies that exhibit a high book-to-market ratio. A more comprehensive literature review of the various concepts to explain the value effect is contained in e.g. Lettau and Wachter (2007).

2 ally imply an infinite stream of cash-flows. Any equity duration estimation methodology has therefore to capture expected payments that occur in the very far, if not infinite future. The objective of this paper is twofold. First, we present a convenient estimation methodology of a share s equity duration. Second, we test whether our so-obtained duration measure can explain the value premium as observed in stock markets, as suggested by Lettau and Wachter (2007). To estimate the equity duration for the shares in our sample, we first propose a new representation of the equity duration. In analogy to bond duration, we define equity duration as a share s price sensitivity to changes in the discount rate, standardized by its market price. In the next step, we employ the so-called implied equity duration concept to estimate the duration of the shares in our sample. Derived from the closely related implied cost of capital concept, it relies on present value models as pricing function and information of analysts forecasts to estimate the duration of a share 2. More precisely, the implied equity duration is estimated as the slope coefficient of a share s present value formula with respect to its implied cost of capital estimate. The so-obtained equity duration estimates are then used to verify the duration based explanation of the value premium. In a first step we compare both measures for the value premium, i.e. equity duration estimate and traditional the book-to-market ratio with each other. Then we examine if the equity duration helps in explaining the cross-sectional variation of U.S. stock returns by carrying out regression test of stock returns on the duration estimate and firm risk variables to account for the risk-return relation implied by standard asset pricing models. We find that long-horizon equity has both lower average expected and future returns than short-horizon stocks. This difference in returns cannot be attributed to the shares market risk exposure as measured by the beta. Instead, our analysis presents evidence that the equity duration has very similar properties than the Fama-French factor B/M ratio, suggesting that this ratio acts as a noisy proxy of the equity duration. This paper is inspired by previous literature that relates the timing of a share s cash flows to its riskiness and returns. Campbell and Mei (1993) were among the first to note that the systematic riskiness of a share does not only originate from common shocks to cash flows, but also from variation in shocks to expected returns. Based on this observation, Cornell (1999b) demonstrates that the βs of growth stocks, i.e. companies with a high duration, are too high 2 The implied cost of capital (ICOC) is defined as the internal rate of return that equates current share price with discounted future cash flows to shareholders, where the expected cash flows are obtained from equity analysts forecasts. The ICOC represents a convenient transformation of information contained in equity analysts forecasts and the current market price into an expected return estimate.

3 to be solely explained by the cash flow beta as measure of systematic risk, and hence must originate from the timing of their cash flows. Brennan and Xia (2006) show that the total systematic risk of a share increases with its duration. Finally, Dechow et al. (2004) use the information implied in stock market prices to actually estimate an implied equity duration. By proposing a new representation of equity duration, our paper facilitates the estimation of the equity duration. Our new definition only requires a share s pricing function which is differentiable in its cost of capital. Hence, this definition can be universally applied to many different pricing functions and various ways to derive a company s cost of capital estimate. Compared to previous works, including Dechow et al. (2004), our model of implied equity duration is the first to make use of analysts forecasts to estimate the equity duration. This offers two advantages. First, the so-obtained equity duration is completely forward-looking, and not building on the doubtable premise that historical data conveys information about the future. Besides this conceptual advantage, the so-obtained duration estimate prove to be more reliable in empirical tests. This paper builds also on recent literature on the implied cost of capital (ICOC). Under the assumption that market prices reflect the fundamental value as predicted by analysts forecasts and that these forecasts are a good surrogate for the average investor s expectation, the ICOC is a good proxy for a stock s expected return. First proposed in the works by Cornell (1999a), Gebhardt et al. (2001) and Claus and Thomas (2001), it is used to estimate a forward looking equity risk premium by aggregating the ICOC over entire markets. Other studies rely on the ICOC to test asset pricing models (Lee et al., 2007), or the risk-return tradeoff of individual shares (Pástor et al., 2007). By relying on the ICOC to estimate the equity duration, our study extends the application of the ICOC to another relevant topic of capital markets. In line with previous papers based on the ICOC, this study underpins the benefits of using of analysts forecasts in the context of traditional asset pricing models. This paper proceeds as follows. The next section reviews the bond and equity duration concepts, describes the equity duration based explanation of the value premium in more detail, and presents the exact estimation methodology of the implied equity duration. Section 3 contains a brief description of the U.S. data sample. In section 4, we carry out first preliminary tests to better understand the relation of equity duration and other standard firm risk variables. In section 5, we validate the equity duration concept for the U.S. equity market by carrying out panel regression test on a broad set of stocks. Then, in section 6, we discuss our empirical

4 results and relate them to the existing literature. Section 7 offers some concluding remarks. 2 The Implied Equity Duration The equity duration is the adaption of the traditional bond duration measure to stocks. It denotes the average time at which a shareholder receives the cash flows from his investment. In this section, we first review the bond duration measure. Then we present how this duration measure can be transferred to equity investments, and demonstrate the equity duration s relation to the value premium. Finally the implied equity duration, the duration measure used in this paper is derived. 2.1 Bond Duration We start this section by presenting the standard definition of bond duration, as introduced by Macaulay (1938). Definition 1 (Bond Duration): Let P 0 denote the bond price at time 0, CF t the annual cash flows to bond holders at time t, and y the annualized bond s yield to maturity, as implied by the price P 0. Then the duration D 0 of a bond with a maturity of T years is defined as: D 0 = 1 P 0 T CF t t (1) (1 + y) t t=1 The yield to maturity of a bond is the internal rate of return that equates today s bond price with the coupon and principal payments to be received in the future. Note that this definition assumes a constant bond yield over the security s live span, as implied by a flat yield curve e.g. Quite evidently, y can also be conceived as the bond s risk-adjusted discount rate. Essentially, bond duration has two interpretations. First, it is the cash flow-weighted average term to payment of the bond s cash flows. More intuitively, it is the remaining average maturity of the cash flows. Thus, the duration of a zero coupon bond with a maturity of T years is t years, since the only cash flow will occur at time T. If there are coupon payments, the duration will be smaller. The second interpretation of duration is the bond s price sensitivity to interest rate movements, expressed in the number of years. To see this more clearly, consider the derivative of a bond s pricing function with respect to its yield:

5 P (y) y = P 0 D 0 1 + y = P 0D mod (2) where D/(1 + y) is often called modified duration D mod. Hence, it holds approximately 3 that P P D 1 + y y = D mod y (3) The reason for expressing this sensitivity in years is that the time that will elapse until a cash flow is received allows more interest to accumulate. Therefore the price of an asset with long term cash flows has more interest rate sensitivity than an asset with cash flows in the near future. For small interest rate changes, the duration is the approximate percentage by which the value of the bond will fall for a 1% per annum increase in market interest rate. So a 15-year bond with a duration of 7 would fall approximately 7% in value if the interest rate increased by 1% per annum. In other words, duration can be conceived as the elasticity of the bond s price with respect to interest rates. 2.2 Equity Duration Equity duration is the extension of the bond duration concept to shares. Instead of analyzing coupon payments to bond holders, the equity duration examines a company s cash flows to its share holders. Definition 2 (Equity duration): Let P 0 denote the share price at time 0, E 0 [CF t ] the expected annual cash flows to share holders at time t, and k the company s cost of capital. Then the equity duration D eq 0 is defined as: D eq 0 = 1 P 0 t=1 t E 0[CF t ] (1 + k) t (4) This formula demonstrates the three main differences to bond duration. First, cash flows from equities to shareholders are not fixed, but uncertain. Thus, the equity duration cannot be defined on fixed payments, but on expectations of future cash flows E 0 [CF t ]. Second, equity investments are in general a claim to an infinite stream of cash flows. Hence, the equity duration is defined over an infinite time horizon. Third, the price of a share is not a function of its yield to maturity, but its cost of capital k. These differences pose considerable challenges when 3 The relation is only approximate since duration itself is a function of the yield to maturity. Thus, it holds true only for small changes of y.

6 estimating the equity duration in practice. We will came back to this issue in section 2.4, where we present the estimation procedure of the equity duration. Similar to the bond duration, we can also derive an expression of the equity duration by taking the first derivative of the share s pricing function with respect to its cost of capital, likewise to equation (2). After rearranging, we obtain the following representation. Proposition 1 (Representation of equity duration): Let P (k) be the pricing function of a share, P 0 its price at time 0, and k the company s cost of capital. Then the equity duration D eq 0 can be represented as follows: D eq P (k) 1 + k 0 = (5) k P 0 Proposition 1 states that the equity duration can be represented as a share s price sensitivity to changes in the discount rate, standardized by the factor (1 + k)/p 0. Note that equation (5) is a very convenient definition of a share s duration. In essence, this representation demonstrates that all that is needed for estimating an equity duration is a share s pricing function which is differentiable in k. Hence, this definition can be universally used to estimate a share s equity duration, allowing for different pricing functions and various ways to derive cost of capital estimates. Similar to the case of a bond, equity duration duration has two interpretations. On the one hand, it is the cash-flow weighted average time at which shareholders receives the cash flows from their investment in a company s share. Consequently, stocks that pay out a large fraction of their cash flows in the far future are high duration stocks. Prominent examples of such stocks are those of rapidly growing technology companies, which might even not pay out any dividends in the first years after incorporation. In contrast, stocks of mature companies which exhibit high dividend-price ratios (such as utility companies) tend to be low duration stocks. On the other hand, equity duration is equally a measure of a share s price sensitivity to changes in the discount rate, or the company s cost of capital. Similar to bonds, shares with a high equity duration are more sensitive to changes in the discount rate than shares with a low duration. 2.3 Systematic Risk, Equity Duration, and Value Premium The price of a share is of course not only influenced by changes in the discount factor. Since future cash flows from equities are uncertain, the other major source of price adjustments are changes in a company s expected future cash flows. Hence, share prices are subject two

7 different kinds of risk: risk related to changes in the discount factor, and risk related to changes in expected future cash-flows. These two different components of a share s risk have important consequences when it comes to its valuation in equity markets. In their seminal paper, Campbell and Voulteenaho (2004) show that a share s systematic risk exposure (β) can be divided into two different components, its sensitivity to shocks in aggregate cash flows and to shocks to the discount rate: β i,m = Cov(ri, r m ) V ar(r m ) = Cov(ri, r m CF ) V ar(r m ) + Cov(ri, r m DR ) V ar(r m ) = β i,cf + β i,dr (6) where r CF and r DR are two independent components of the market return; the first due to surprises in expected cash flows and the second due to surprises in the discount rate, so that r m = r CF + r DR. 4 Using an intertemporal CAPM of the sort of Merton (1973), they show that for long-horizon risk-averse investors, a share s systematic risk related to changes in aggregate cash-flows (β i,cf ) should be priced significantly more than its discount rate component (β i,dr ). Intuitively, their argument is as follow: share prices can fall either because of negative surprises in expected cash-flows, or because investors demand a higher cost of capital, i.e. discount rate, for their equity investments. In the former situation, the value of equity decreases while leaving investment opportunities unchanged. In contrast, in the latter case, shares decrease in value while other investment opportunities improve, such as those offered by fixed income products. Since this second source of equity risk is hence (partially) hedged by improved investment conditions, it should be less priced in equity markets. Hence, the relative proportion of both sources of risk is important for a share s market price. Instead of relying on the return decomposition of Campbell and Voulteenaho (2004) to analyze a share s exposure to both cash flow and discount rate news, we resort to the equity duration concept to obtain a measure for a share s sensitivity both risk components. The second interpretation of equity duration, the sensitivity to changes in the discount factor, suggests that long-horizon equity is more affected by changes in the discount factor than by changes in expected cash flows. This implies in contrast, that - ceteris paribus - low duration stocks should be relatively more influenced by the second source of risk, i.e. changes in expected cash flows. Consequently, if a share s sensitivity to discount rate shocks is less priced than its risk exposure to cash flow surprises, low duration stocks should exhibit higher prices than high 4 Note that we present a simplified expression of their return decomposition framework only. For more details, please refer to Campbell and Voulteenaho (2004).

8 duration stocks 5. Lettau and Wachter (2007) build on the findings of Campbell and Voulteenaho (2004) to establish a relation between the timing of a share s cash flows (i.e. its duration) and the observed value premium. They push their idea further by assuming that the stochastic discount is completely uncorrelated to discount rate shocks. Hence, in their model, only cash-flow shocks are priced in equity markets. Since long-horizon equity is less affected by cash-flow shocks, they are less risky, and hence have lower average returns than short-horizon equity. What is more, Lettau and Wachter (2007) show that long-horizon firms exhibit lower B/M ration, thereby linking the value premium to equity duration. Hence, the value premium is essentially a compensation for a share s higher sensitivity to cash-flow surprises. Put differently, the model of Lettau and Wachter (2007) shows that the traditional value measure, the book-to-market ratio, might just be proxy for the underlying cash-flow based risk factor captured by the equity duration. Following the inverse relation between B/M ratio and equity duration, we hence hypothesize that the equity duration should be highly negatively correlated with the book-to-market ratio, and should be contribute to explaining the observed value premium in stock markets. Moreover, if the book-to-market ratio is only a proxy for an duration-based risk factor, joint regression test of stock returns on the B/M ratio and the equity duration should leave the latter without any statistical significance. 2.4 Implied Equity Duration When referring to the representation of equity duration in proposition 1, it is evident that three ingredients are needed to successfully estimate an equity duration: a pricing function for the share P (k), an estimate for the company s cost of capital k, and the share price P 0 6. A company s share price P 0 is readily available. As pricing function, we rely on standard discounted cash flow valuation models, which are all different variants of the present value statement: 5 There is however a subtle difference between both approaches. The equity duration is a measure of a share s price sensitivity to idiosyncratic shocks to the discount rate. However, Campbell and Voulteenaho (2004) suggest that not total price sensitivity should be priced, but only its systematic component, i.e. a share s covariance with market returns due to aggregate discount rate shocks. From an empirical point of view, fortunately, this differences does play a minor role only. We will discuss this relation more deeply in section 6. 6 The definition 4 of equity duration requires expected cash-flows to infinity instead of a pricing function. However, since any pricing function has to include predictions or assumptions on future cash flows, at least implicitly, this results in the same.

9 P 0 = t=1 E 0 [CF t ] (1 + k) t (7) These present value models state that the value of a share should equal its discounted future cash flows until infinity. Implementing such models in practice can however be somewhat cumbersome, since expectations about future cash flows until infinity are difficult to estimate. In this paper, we use a combination of equity analysts forecasts for the short-term horizon and assumptions of commonly used valuation formulas for the long-run to obtain sensible estimates for the equity duration 7. Below, we give an example of a simple dividend discount model as pricing function. The exact implementation used in this paper builds on a residual income model, and is presented in section 2.5. Finally, the equity duration requires an estimate for the share s expected cost of capital k, or equivalently, a share s rate of return. To achieve the best possible analogy to a bond s yield to maturity y as implied by its market price (see definition 1), we employ so-called the implied cost of capital as proxy for the shareholders expected return. The implied cost of capital (ICOC) is defined as the internal rate of return that equates current share price with discounted expected future cash flows to shareholders, where expected cash-flows are usually obtained from equity analysts. Accordingly, we rely on valuation functions that are based on the relation (7) to estimate the implied cost of capital k by solving the present value relation for its discount factor, using the analysts forecasts and the respective model s assumptions for expected cash flows, and the prevailing share price. Using the representation of equity duration in (5), we thus can derive the definition of implied equity duration as used in this paper. Definition 3 (Implied equity duration): Let P (k) be a pricing function of a share, P 0 current price at time 0, and k the company s implied cost of capital. Then the implied equity duration D i 0 is given as: its D i 0 = P (k) 1 + k (8) k P 0 Although similar in spirit, our concept of the implied equity duration is different from the definition used in Dechow et al. (2004). First, their model assumes an exogenous cost of capital of 12% across all companies at all times, whereas we estimate the cost of capital for each 7 The findings of Elton et al. (1981) suggest that analysts forecasts are a surrogate for the average investor s expectations. Note that expected cash flows in the very far future are rather irrelevant for the equity duration since their discounted present value is very small.

10 company. Second, they use historical data to forecast future cash-flows by setting up a meanreverting first-order autoregressive process for the roe. Dechow et al. (2004) themselves admit that their forecasting model to calculate the equity duration is rather crude. By relying on the combination of analysts forecasts and sensible valuation assumptions as presented in the next section, we believe to offer some considerable improvements over their model. Example. To get some intuition about the implied equity duration concept, it is helpful to examine the implied duration measure obtained from a simple pricing function. Consider the very basic Gordon (1962) growth model as pricing function. Definition 4 (Gordon growth model): Let g denote the constant growth rate of dividend payments to shareholders. Then the share price P 0 is given by: P 0 = E 0[CF 1 ] k g (9) The Gordon model assumes that future dividend payments to shareholders growth geometrically at a constant rate until infinity. In practice, this model does not capture the true dynamics of dividend payments declining usually over time, and is thus only a valid approximation for the share price of mature companies with stable dividend payments. Still, it can serve as a useful illustration of the implied equity duration. As shown in the appendix, the implied equity duration based on the Gordon model (9) is: D i 0 = P 0 (1 + g) + 1 (10) E 0 [CF 1 ] In this simple case, equity duration is hence approximately the inverse of the dividend yield, a point made first by Lintner (1971) and more recently by Bernstein (1995). This is quite intuitive, since the lower the dividend yield, the more time it takes for an investor to recoup his initial investment. Since dividends are the fraction of earnings that are paid to shareholders, equity duration can also be expressed in terms of the P/E ratio: D0 i = P 0 1 + g + 1 (11) E 0 [E 1 ] p where p denotes the (constant) payout ratio of the firm, and E 0 [E t ] the expected next year s earnings. Hence, the P/E is also a proxy for the equity duration, as pointed out by Dechow et al. (2004). The higher the P/E ratio, the higher the equity duration. Since earnings are less

11 subject to different payout policies, the P/E ratio might be more reliable for estimating the equity duration. Finally, we can derive in a similar way a duration approximation based on the P/B ratio, i.e. the inverse of the Fama-French risk factor book-to-market ratio. Since earnings can be expressed as the return on equity capital, see equation (13) below, we obtain following expression: D i 0 = P 0 B 0 1 + g p roe + 1 (12) Hence, using the simple Gordon growth model, we can see the inverse relationship between B/M ratio and equity duration. This observation matches the predictions of the theoretical model by Lettau and Wachter (2007), stating that equity duration is negatively related to the B/M ratio and can thus be conceived as noisy approximation thereof. 2.5 Empirical Implementation Given the well-known shortcomings of the Gordon (1962) valuation model, we resort to a more complex valuation model, the residual income model (RIM) as proposed by Gebhardt et al. (2001) to estimate the implied equity duration. Definition 5 (Residual income, residual income model): Let B t denote the book value of equity per share at the end of year t, E t the earnings per share in year t, roe t the company s return on equity, and k the company s cost of equity capital. Then the residual income R t per share is defined as: R t = E t k(b t 1 ) = (roe t k)b t 1 (13) If denotes E 0 [R t ] the expected residual income per share, then the price of a share P 0 is given as: P 0 = B 0 + t=1 E 0 [R t ] (1 + k) t = B 0 + t=1 E 0 [roe t ] k (1 + k) t B t 1 (14) This pricing function, first proposed by Preinreich (1938), states that the value of a company should equal its invested capital, plus the expected residual income from its future activities. Since exact predictions of future earnings cannot made to infinity, one has to make assumptions about expected cash-flows when implementing the model in practice. In this paper, we resort to the three-stage RIM approximation following Gebhardt et al. (2001).

12 Definition 6 (Three-stage residual income valuation): Let E 0 [iroe T ] denote the expected industry return on equity, and T the explicit forecast horizon. Then the price of a share is given by: 3 E 0 [roe t ] k T E 0 [roe t ] k P 0 = B 0 + B (1 + k) t t 1 + B (1 + k) t t 1 + E 0[iroe T ] k k(1 + k) B T 1 T 1 t=1 t=4 }{{}}{{}}{{} (15) Explicit forecasts Transition period Terminal value After the explicit forecast period of three years, this model assumes that the companies return on equity capital converges over a transition period of T 3 years to the long-term industry average. Similar to Gebhardt et al. (2001) we use explicit earnings forecasts by equity analysts as provided by IBES to calculate the expected return on equity for the next three years 8. The industry roe is calculated as the median of all (positive) realized roe across all companies of the respective sector over the preceding 60 months. This procedure aims to average out business cycle effects of the industry profitability. We use the industry sector codes of the GICS classification for sorting the companies 9. Future expected book values of equity are calculated using the so-called clean surplus relation: B t = B t 1 + E t (1 p t ). To that end, we have to make assumptions regarding future payout ratios. While the use of industry averages for the long-term return on equity has some appeal due to the findings of empirical analysis (Nissim and Penman, 2001; Soliman, 2008), the commonly used assumptions of constant payout ratios to construct future book values seem somewhat arbitrary to us. As a consequence, we consider a slightly modified version of the three-stage RIM that avoids relying on assumptions on future payout ratios. Following the literature on sustainable growth rates, we use the following identity between payout ratio p, return on equity roe and the long-term growth rate of the company g l : p = 1 g l roe (16) In order to estimate the future development of a company, one has to make assumptions for two out of the three parameters. Instead of assuming the long-run payout ratio p, we opt to fix 8 We use the median of expected earnings of all contributing sell-side equity analysts. In the case where the expected earnings estimate of year 3 were missing, we generated an earnings estimate for year 3 by applying the long-term consensus earnings growth rate to expected earnings in year 2. If the projected earnings in year two or three were negative, we dropped the observation from the sample. 9 In our standard implementation we fix T = 9. Instead of using the GICS classification, Gebhardt et al. (2001) rely on the 48 Fama and French (1997) industry classifications.

13 the long-term growth rate of the company g l. By setting g l equal to the expected GDP growth rate of the economy, we ensure that no company will persistently grow faster than the whole economy and eventually exceed it 10. Hence, we employ the RIM following Gebhardt et al. (2001) as presented in equation (15), but using different payout ratios for each industry. For each sector, we calculate the long-term industry payout ratio using the relation (16), given the expected GDP growth of the economy and the industry roe. In the transition period, we then fade both payout ratio and return on equity towards their long-term levels 11. 3 Data and Descriptive Statistics 3.1 Data and variable construction In this paper, we analyze the U.S. equity market from a time period from January 1990 to February 2006. We focus on all listed stocks covered by MSCI for which we obtain data of equity analysts to estimate the implied equity duration. The monthly data for prices, total returns, book values and dividends per share, market capitalization, and returns on equity are taken from MSCI. All market capitalization data are free float adjusted. Earnings estimates and the long-term earnings growth rates are taken from IBES. Time series data of national accounts to calculate the expected nominal GDP growth rate are obtained from Eurostat. Stock indices to derive market betas and deflate firm size data are taken from Datastream. To estimate the implied equity duration, we first calculate the industry roe for each of the twelve GICS industry classifications. Then, we use this industry mean to estimate the implied cost of capital (ICOC) for each share by solving equation (15) for the internal rate of return, given the cash flow forecasts and the prevailing share price. The solution is straightforward, since the pricing function is monotone in k, and can thus be solved iteratively. Because we require five years of company data to calculate the industry roe, we can only estimate the implied cost of capital from January 1995 on, leaving us with roughly 11 years of data. Then we estimate the slope coefficient of the pricing function at the ICOC estimate. The implied equity duration is then obtained by adjusting the slope coefficient by the factor (1 + k)/p 0. 12 10 In this study we use a simple moving average forecast model and calculate the expected GDP growth rate as the average geometric nominal GDP growth rate over the past 5 years. 11 In practice, our RIM3 specification does not diverge substantially from the model as proposed by Gebhardt et al. (2001). In the U.S., the correlation of the ICOC estimates obtained from both models is greater than 0.99. 12 Given the complexity of the model (15), an analytical derivation of the implied equity duration is not

14 Market beta is calculated as the standardized covariance of share price and market return over 60 months prior to the observation 13 Price momentum, finally, is calculated as the change in stock prices over the six months prior to the observation. We employ the last available information as required by the formulas at the end of each calendar month. Thereby, we make sure that the duration estimates are based only on publicly available information. Firms with an incomplete data set, i.e. one or more missing input variables where we could not resort to approximations as explained above, have been ignored 14. 3.2 Summary Statistics The U.S. data set contains 1,507 companies and over 122,000 monthly observations. Since the number of companies included in the study changes over time, we have an unbalanced panel data set. Before starting with the analysis, we remove apparent data errors(i.e. unreasonable data points) from the sample, thereby reducing the data set by less than 0.2%. Note that the examined sample size varies from analysis to analysis, since not all companies pay dividends or have positive earnings 15. Table 1 shows the summary statistics of the U.S. firm-level data. The first line indicates that the average equity duration was 17.7 years over the examined time period. In other words, an equity investor expected to wait on average about 18 years to get the money from his investment back. At first sight, this appears to be a reasonable average time span. For, example, when referring to the duration proxies presented at the end of section 2.4, we would obtain similar figures. Using e.g. the price-to-book approximation of equation (12), and abstracting from growth (g = 0), assuming a dividend payout ratio of 50% and a return on equity of 20% would imply an equity duration close to 24 years. The equity duration s standard deviation of 5.5 years indicates that most observations are in a reasonable range. The average implied cost of capital estimate is at around 8.2%, which seems to be a reasonable approximation of the average expected return of stocks. For instance, when assuming a nominal risk-free rate of 3%, this estimate implies an equally weighted market risk premium of roughly 5%. As was to be expected, the average beta estimate is very close to its theoretical value of possible. 13 If the share price is not available 60 months before the observation, we reduced the beta estimation period to 24 months. If the time was even shorter, we dropped the observation from the sample. 14 Note that we do not carry out any time adjustment procedures similar to other studies on the implied cost of capital. Since we use a monthly data set, such adjustments would require the exact dividend payout dates and book value adjustments for all companies since 1995. Such data is not easy to get hold of, nor is it reliable. 15 In the correlation analysis, we transform D/P and P/E ratio in their natural logarithms.

15 1. The average firm size of the sample is at around 18,000 mn USD. The B/M ratio is slightly below 0.5, implying that the average firm of the sample was valuated at about the double of its equity book value. Past 6-month price momentum is at round 4%. Finally, the average P/E ratio was about 22.2 and the dividend yield at 1.7%. These two figures are well above, respectively below, their long-term averages. This is very likely to be a consequence of the time frame of this study, coinciding with the years around the stock market bubble in 2000. Figure 1 displays the evolvement of the implied equity duration over time. The figure presents the median, the mean, and the market-cap weighted average of the equity duration from 1995 to 2006. Large companies exhibit a higher equity duration than small companies: the market-cap weighted average equity duration is always above the median estimate, moving with the stock market index reaching a high in the year 2000. In contrast, the median duration remained over the whole time in a fairly small range between 14 and 17 years 16. Finally, table 2 shows the correlation statistics. Since we use in the regression analysis below the logarithms of equity duration, B/M ratio, firm size, and the valuation multiples, the correlation matrix is calculated on the basis of their logarithms as well. One observation is striking: the highly negative correlation of the equity duration with the B/M ratio. Again, following the approximation of equation (12), this relation was to be expected. The other two valuation multiples, P/E ratio and D/P ratio, are considerably less related to the equity duration measure, as the last two lines shows. Besides, the implied equity duration is positively related to market beta, firm size and price momentum. The correlation structure of the different risk factors with each other exhibits the standard characteristics as documented in many empirical asset pricing tests. 4 Preliminary Analysis: Equity duration and firm characteristics Before examining the relation of equity duration to stock returns, we first present two preliminary analyzes to better understand the relation between the equity duration and other firm characteristics that are generally used as risk proxies. First, in section 4.1, we divide the whole data sample in two partitions according to their duration estimates and examine their average 16 Note that following the downturn of stock markets up to 2003 however, the market-cap weighted equity duration remained low, close to the median. Since the duration is positively related to prices, see again the priceto-book approximation, we can interpret this as an indicator that large cap stocks were no longer overvalued after 2003.

16 characteristics. Then, in section 4.2, we analyze the relationships in more detail by performing regression tests of the equity duration. 4.1 High and low duration stocks As a first assessment of the relation between equity duration and firm-risk characteristics, we divide all observations in two equal partitions according to their equity duration estimate, i.e. into a set of high-duration and low-duration stocks. The results are shown in table 3. We can draw several important conclusions from the table. First, high duration stocks are less risky then low duration stocks. The average cost of capital for long-term equity, as measured by the implied cost of capital, is more than 4% lower than for low duration stocks. Next, this difference in cost of equity capital cannot be explained by its total systematic risk exposure as measured by its market beta, since both sub samples have almost equal average beta estimates. What is more, low duration stocks have even a slightly lower average beta estimate compared to high duration stocks. However, this result is consistent with the idea that not total beta is decisive for the perceived riskiness of a share, but the relative proportion of cash-flow beta to total beta. Under the premise that low duration stocks exhibit relatively more discount cash-flow beta, they should be considered to more risky. Next, similar to the preceding correlation analysis, the table also confirms the alleged negative relation between B/M ratio and equity duration. High duration stocks are growth stocks, i.e. companies with a low B/M ratio. This finding is in line with the simple approximation provided in the last section, showing the inverse relationship of equity duration and B/M ratio. Further, these growth stocks are valued higher than value stocks, with their average market capitalization being twice as large as the value stocks (which might partially attributed to the tech bubble covered in the sample). Finally, high duration stocks are also past winners: on average, they have seen a price momentum of 8% over the six months prior to the duration estimation, which sharply contrasts with the low duration stocks, that did not change in value on average. This observation is rather intuitive: holding everything equal, such as expected future cash-flows, a rise in the share price implies that an investor has to wait longer for the amortization of the stock investment.

17 4.2 Regression tests of equity duration So far we have provided evidence that the relation of equity duration to simple valuation ratios is most pronounced for the B/M ratio. Compared to the B/M ratio, the other two proxies of equity duration are considerably less related to the duration measure. However, table 3 also indicates a rather strong relation of firm size with duration, although we do not have an explicit risk-based explanation for this effect. Most puzzling perhaps is the fact that duration is positive related to systematic risk as captured by the market beta, contrasting with negative relation to the expected cost of capital. Finally, we observe also a rather strong relation of price momentum with duration These observations might question the direct link between equity duration and the book yield. After all, the equity duration could just be a linear combination of several known equity risk factors, which would challenge the duration s ability of explaining the value premium. Hence, the question is whether the relation of B/M ratio to duration is independent of other firm risk effects. Since the correlation structure itself does not provide insights on the robustness of the relation between B/M ratio and equity duration, we perform regression tests of equity duration estimate on its proxies B/M ratio, E/P ratio and D/P ratio jointly with other risk variables, i.e. market beta, size, and price momentum. We adopt the panel regression approach, adjusting the t- statistics for serial correlation following Rogers (1993). To reduce the impact of outliers, we employ the natural logs of equity duration, the valuation ratios, and firm size in the regression tests. The results are displayed in table 4. The upper panel contains univariate regression results, the lower panel the joint regression tests. The upper panel shows that the B/M ratio is indeed most closely related to the equity duration estimate. The relation is inverse: the higher stocks are priced compared to their book value of equity capital, the more they tend to be long-horizon equity. This relation is also economical meaningful. Based on the log-log relation, a ten percent increase in B/M ratio implies a decrease of the equity duration of about 2.7%. The lower panel shows that this relation persists after controlling for common firm risk. Market beta and price momentum are also determinants of equity duration, with both variables being highly significant. This observations is again consistent with the notion that duration might be a proxy for a discount rate beta: the higher total beta, the higher on average also the discount rate sensitivity, and consequently the equity duration. The intuition for the impact of price momentum is similar to before: a rise in the share price implies that an investor has to wait

18 longer for the amortization of the investment. Firm size, in contrast, is no longer related to equity duration in joint regression tests, suggesting that the observed correlation is more likely to be induced by indirect risk relations than a sign of a true economic relationship between size and duration. 5 Empirical Results: Equity duration and stock returns In this section, we explore the relation of the equity duration to stock returns. More precisely, we use stock return regression tests in order to analyze the equity duration s ability to explain the cross-sectional variation in stock returns. After outlining the estimation methodology, we first examine the equity duration s explanatory power for stock returns in univariate regressions. Then, in section 5.3, we present joint regression tests, where we control the impact of equity duration on stock returns for other standard firm risk proxies. Under the premise that the timing of a share s cash flows is an independent source of risk, we should observe that high duration stocks have lower average returns. Next, when accepting the notion that the valuation ratios (B/M, P/E, D/P) are only crude proxies of this timing-based risk factor, equity duration should have a higher explanatory power for stock returns than these valuation ratios. 5.1 Methodology In the literature, there are two common econometric approaches to carry out regression tests on stock returns. Researchers either employ the Fama and MacBeth (1973) cross-sectional regressions, or they rely on the panel regression approach. In this study, we opt for panel regression tests, since they make a much better use of the information contained in the data. For a detailed discussion of the two regression approaches and further advantages of panel regressions, please refer to the appendix A.2. Our panel regression approach is the natural extension of the Fama and MacBeth (1973) cross-sectional regressions to multiple time periods. Instead of regressing each cross-section individually, we combine all monthly cross-sections to estimate the model (pooled time-series cross-section). In analogy to Fama and French (1992), we regress the firm s individual stock return r i,t on its duration estimate D i,t, the market beta β i,t (the firm s factor loading on the market excess return), and firm characteristics X i,t that have been identified as risk-factor in the literature:

19 r i,t = α + δd i,t + ϕd t β i,t + γ X i,t + u i,t (17) where the subscript i denotes the company (cross-section dimension) and t denotes the time period of the observation (time-series dimension) 17. The subsequent total stock return measured after having observed the risk-factors and firm characteristics is denoted by r i,t. Other firm characteristics include firm size (calculated as the log of the market capitalization divided by the level of the stock market index) and the historical six-month price momentum. The variable d t is a dummy for the ex post observed market risk premium (i.e. the difference between the market return and the risk-free rate), having a value of 1 when the market risk premium is positive during the holing period, and -1 if it is negative. This dummy is necessary to account for the fact that during periods when the realized market return is less than the risk-free rate, the relationship between predicted return and beta is reversed. More precisely, high-beta stocks should have lower returns when the ex post risk premium is negative (Pettengill et al., 1995). The general specification in (17) is known as pooled regression model. In many cases, however, the underlying assumption that an observation of a company at time t is independent of an observation of the same company at time s is not met. Hence, the regression equation is modified to allow for individual effects for each company. This model is known as the one-way individual effects model: r i,t = α i + δd i,t + ϕd t β i,t + γ X i,t + u i,t (18) One of the most common approaches to estimate such an individual effects model is to assume that the individual effect α i of each firm is constant over time. Relying on such a one-way fixed effect model hence implies that the returns of some companies are on average higher than the return of the market, whereas some other companies underperform the market on average. The rationale for employing individual fixed effects it to capture firm characteristics that are not included in the regression equation, such as the unobservable firm-specific factors (e.g. managerial skills, corporate culture). It also reduces the impact of outliers, and thus makes the coefficient estimates more reliable. In addition to the one-way individual effects model, we also consider a two-way individual effects model that allows not only for firm-specific effects, but 17 For similar regressions, see Brennan et al. (1998), or Lee et al. (1999, 2007). The use of the firm characteristics size and B/M ratio instead of the respective factor loadings is motivated by the work of Daniel and Titman (1997) who argue that it is rather the characteristics than the covariance structure that explains the variation in stock returns.