The implied equity duration when discounting and forecasting parameters are industry-specific

Transcription

1 The implied equity duration when discounting and forecasting parameters are industry-specific Abstract: First draft: September 2012 This draft: December 2013 The procedures for estimating implied equity duration employed by Dechow et al. (2004) are, in their own words, parsimonious but relatively crude. Therefore, improvements in the procedures should lead to a more accurate and useful measure. Within this context, this paper aims to compute the implied equity duration thought forecasting and discount procedures that use implied industry characteristics with the objective of bridging the empirical issues with the theoretical formulation. The results show that firms IED changes 5.91 years on average and changes in absolute values up to years. From a qualitative point of view, when we rank firms by duration, the ranks change until 32 positions, and the average of absolute changes is rank positions. Then, we conclude that the cost of being parsimonious is high on average and also quite variable across firms, both quantitatively and qualitatively. Moreover, this cost is large enough to reverse the ranked ordering of firms based on duration. Keywords: Asset pricing; Financial analysis; Risk factors; Stock duration. JEL Classification: G12, G14, M41. 1

2 1. Introduction In contrast to the field of fixed income, the interest rate risk analysis for equities does not have a unique framework. We can find in this context analyses based on analytical stock valuation models, such as that by Leibowitz (1986), together with others built on empirical relations between stock prices and market interest rates, with a high variability in their results. Moreover, none of these methodologies has provided convincing enough results to become a reference methodology. In this context, Dechow et al. (2004) (DSS) developed a measure of implied equity duration (IED) based on the Macaulay bond duration, a risk measure commonly accepted and used within both academia and practice that effectively measures the sensitivity of a bond to changes in the discount rate in the fixed cash flow model, i.e., the yield to maturity. Their methodology also combines the two aforementioned approaches for the equity context, as they used an analytical model of stock valuation based on the discounted cash flows, but matching the market quote by adjusting the terminal cash flow scheme. DSS compute the IED for each of the companies using the available data from the NYSE, AMEX and NASDAQ between 1963 and 1998 and obtain an IED average of years, with a standard deviation of The results of their empirical tests show that the IED explains the risk characteristics of stock returns, resulting in a positive and significant relationship with the volatility of the stock returns and their betas and that the IED provides a greater prediction power of these variables than does its own lagged variables. Furthermore, the IED captures a strong common factor in the profitability of 2

3 the shares that encompass the common factor related to the book-to-market ratio (BtM) whose empirical properties were revealed by Fama and French (1993). As DSS note, implementation of their estimation procedure for IED requires as inputs four financial variables (book values, sales, earnings and market capitalisation) and four forecasting parameters (the autocorrelation coefficient for ROE, the autocorrelation coefficient for sales growth, the expected return on equity and the long-run growth rate). These four forecasting parameters, according to the IED former analytical building in DSS, are firm-specific; however, in the subsequent empirical analysis, they are firm (and time) constant. Only the use of a naïve forecast of the expected return on equity, assuming it to be a cross-sectional constant, was sought to be justified by the authors. Moreover, DSS emphasise that all of their forecasting procedures are relatively crude and that most of their forecasting parameters are likely to vary as a function of industry membership and other firm characteristics. Within this context, the objective of this paper is to compute the IED by alternatively using the DSS forecasting procedures and others forecasting procedures that use industry characteristics in order to compare the results both in quantitative and qualitative terms. Thereby, we will be able to know the effective ability of a relatively parsimonious empirical estimation procedure to produce an effective measure of implied equity duration, and also whether it is really implausible that expected return differences could be large enough to reverse the rank ordering of firms based on duration, as DSS claim. 3

4 We carry out this analysis for non-financial listed firms in the Spanish stock market with data available at the end of Fullana and Toscano (2013) compute the IED for eighty non-financial listed companies in the Spanish stock market at year-end 2011, using the available data from the SABI and COMPUSTAT Global Vantage databases. They obtain an IED average of years with a standard deviation of They find significant linear relations between the IED and the Earning-to-Price ratio, the Book-to-market ratio and the sales growth rate. However, to provide comparable results with those obtained by DSS for the US exchange market, these authors use the exact DSS methodology adapted to the Spanish context. As Santa-Clara (2004) has noted, the IED is an interesting new approach to measuring stock risk. This framework is linked within the empirical literature that relates the systematic risk of stocks with the premium value by breaking down the betas for assets in cash flow betas and discount-rate betas, and this fact increases the interest in deepening the IED analysis beyond its immediate practical applicability. In this context, Campbell and Mei (1993) find that the discount-rate betas represent the large part of the total beta of the companies, and based on that, Cornell (1999) suggests that the high betas for growth stocks are a consequence of a greater weight of the cash flows removed in time, or equivalently, of the longer duration of growth stocks. Campbell et al. (2009) show that the value premium is a consequence of the differences in the timed schedule of the expected cash flows by the shareholder represented by the duration. In addition, Da (2009) shows how cross-sectional differences in the duration of companies could explain an important part of the stock returns. 4

5 The rest of this paper is organised as follows. In the next section, we expose the DSS IED framework. In Section 3, we present our hypothesis and describe the implied industry-specific parameters estimation methodology used. Section 4 describes the data selection process and the implicit parameters estimated. Section 5 presents the results of computing the IED for each methodology and the marginal effect of adjusting any procedure. Finally, section 6 summarises the results and states the main conclusions. 2. The DSS implied equity duration framework. Macaulay (1938) first introduced the duration concept as the weighted average of the times until bondholders receive bond fixed cash flows, where the weights are equal to the present values of the payments normalised with respect to the price of the bond. Hicks (1939) further showed that the duration is essentially a measure of the elasticity of bonds to interest rates. From the bond price formula, it is straightforward to note that bond prices are inversely related to their yield to maturity as a unique risk factor. 1 However, in the equity valuation, there are many factors, including the interest rate, that affect cash flows beyond the discount rate used to compute their present values. 2 In the study by Boquist et al. (1975), the stock duration measure is built on the dividend discount model. Leibowitz (1986) used market quotes to develop an alternative 1 In the fixed-income field, the duration concept has been widely used and developed up to the concept of duration vector to a few key rates selected by regression analysis (Navarro and Nave, 1997 and 2001) or to unobservable factors extracted by principal component analysis or independent component analysis from the term structure of interest rates (Gonzalez and Nave, 2010). 2 Lintner (1971), Boquist et al. (1975) and Livingston (1978) initially proposed the concept of stock duration, and then, in the works of Leibowitz (Leibowitz, 1986 and Leibowitz et al., 1989), it was first computed for individual firms. More recent studies in this line are those by Cohen (2002), Hamelink et al. (2002) Lewin et al. (2007), Shaffer (2007) and Leibowitz et al. (2010). 5

6 measure. However, as Leibowitz and Kogelman (1993) noted, the values for both measures are significantly different, even when the compensation between price risk and reinvestment risk is taken into account, as proposed by Johnson (1989). 3 DDS proposed an alternative stock duration measure that we derive below, that is based on the Macaulay s duration for a bond. So, we start from the expression of Macaulay s duration, computed one time period before the first future cash flow: D 0 = T t=1 t CF t ( 1+ r) t P 0 (1) where: CF t r P 0 are the bond cash flows; is the bond s yield to maturity; and is the actual price of the bond. Differentiating the bond price expression with respect to the bond s yield to maturity, we obtain the relationship between changes in the price of a bond and changes in its yield to maturity as a function of the duration: P r = P D 1+ r (2) 3 These differences determine the so-called duration paradox and give rise to works, such as those by Leibowitz and Kogelman (1993) and Hurley and Johnson (1995), which attempt, though unsuccessfully, to conciliate the two measures. 6

7 We rewrite this relationship and express it in a discrete form to obtain the following relation between the relative bond price changes and the discrete changes in the bond s yield to maturity as a function of [D / (1 + r)], the so-called modified duration, MD: ΔP P D Δr = MD Δr (3) 1+ r As DSS note, extending the duration concept to equities introduces two drawbacks. The first one is related to the potentially infinite number of cash payments on equity securities in contrast to the finite number of cash payments on a plain vanilla bond. To address this problem, DDS assume that the cash flow stream for an equity security can be decomposed into a finite forecasting period and an infinite terminal expression with a value equal to the difference between the observed market capitalisation implicit in the stock price and the present value of the cash flows over the finite forecast period, described as follows. If we split the duration formula shown in equation (1) into two blocks, one with a finite horizon up to T and another with an infinite horizon from T, we obtain equation (4), which expresses the equity duration as a sum of the weighted values of the duration of the cash flows for the finite horizon and the duration of terminal cash flows. D = T t=1 T t=1 t CF t ( 1+ r) t CF t ( 1+ r) t T t=1 CF t ( 1+ r) t P + t=t+1 t=t+1 t CF t ( 1+ r) t CF t ( 1+ r) t t=t+1 CF t ( 1+ r) t P (4) where: 7

8 P CF t r is the actual market firm capitalisation; are the predicted firm payouts to the stockholders; and represents the expected return on firm equity. If we also assume that the terminal cash flows are a perpetual with an actual value equal to the difference between the market capitalisation and the present value of the predicted cash flows for the finite period, then: t=t+1 CF t ( 1+ r) t = P T t=1 CF t ( 1+ r) t (5) As the duration of the perpetual that begins in T periods is [T + (1 + r) / r], substituting (5) into (4), we obtain the following expression for the IED: D = T t=1 t CF t ( 1+ r) t P P # + % T + 1+ r & ( $ r ' T t=1 CF t ( 1+ r) t P (6) As a result, the second problem of extending the duration concept to equities arises. That is, while the amount and timing of bond cash flows are usually fixed in advance so that the bondholder has little uncertainty, the cash flows of equities are not specified in advance and may be subject to great uncertainty. To compute equation (6), we need predictions of firm cash flows for the finite period [0, T]. 8

9 In this sense, Dechow et al. (2004) used a cash flow prediction model based on previous results that relate accounting measures to future cash flows (Nissim and Penman, 2001). Thus, from the accounting identity of the flows distributed to shareholders from firm earnings and book values, we can obtain the following: CF t = E t (BV t BV t 1 ) (7) where: E t is the firm earnings at the end of period t; and BV t represents the firm s book value at the end of period t. Rewriting the right side of equation (7), we obtain: # E CF t = BV t 1 t BV BV & t t 1 % $ BV t 1 BV ( t 1 ' (8) Thus, a cash flow forecast for period t can be computed with: (i) the return on equity (ROE) in period t, i.e., E t /BV t-1 ; and (ii) the growth rate of equity in period t, i.e., (BV t BV t-1 )/BV t-1. DSS modelled ROE as a first-order autoregressive process with reversion to the historical market cost of equity and the growth rate of equity as a first-order autoregressive process of the sales growth rate with reversion to the long-term macroeconomic growth rate because the sales growth rate, according Nissim and Penman (2001), is the best proxy of the future growth rate of the equity. These naïve 9

10 and crude forecasting procedures permit a relative parsimonious empirical estimation procedure because only needs four market parameters to forecast the cash flows of any firm at any time. 3. Industry-specific discounting and forecasting parameters. As DSS argue, it is generally accepted that the ROE follows a mean reverting process (Stigler, 1963; and Penman, 1991) and that the economic intuition and the empirical evidence (Nissim and Penman, 2001) suggest that the mean that reverses the ROE approximates the cost of equity, but these results always refer to a firm-level context. Similarly, a mean reverting process of the sales growth rate that reverts to a long-term rate could be an appropriate parsimonious model for the growth rate in equity but also in a firm-level context. Assume that time- and firm-constant parameters may result in significant measure errors, and DSS did not formally check for these errors. Thus, we model ROE at industry-level as a mean reverting process that reverts to the industry long-term expected return on equity. We also model the industry-level growth rate in equity as a mean reverting process of the firm sales growth rate that reverts to the industry s long-term expected growth rate. Therefore, we need four industry forecasting parameters for firm cash flow forecasts. In line with the DSS framework for computing IED, we will use implicit expected industry rates of return and growth rates as long-term rates. Following Easton and Sommers (2007), we use a regression model to estimate these rates based on current accounting data: 4 10

11 eps jt bps jt 1 = δ 0 + δ 1 P jt bps jt bps jt 1 (9) where: δ 0 = r; δ 1 = (r g)/(1+g); and: eps jt is the j-firm earnings per share ratio at t; bps jt is the j-firm book value per share ratio at t; and P jt is the quote of a j-firm share at t. Then, the implicit expected rate of return by industry and the implicit growth rate by industry estimated are used as long-term rates in the two mean reverting processes with industry specific autocorrelation coefficients that, applied to firm-specific initial values, project the firm-specific future cash flows. As an output of the forecasting procedures described, we have an implicit expected rate of return by industry that allows us to address another problematic issue in the DSS empirical implementation of the IED. DSS assume an expected return on equity crosssectional (and time) constant that is also used as a discount rate and justify it in two footnotes with partial evidence and an incomplete robustness test. If we want to approximate a long IED of a firm to a short IED of another firm by changing the discount rate of return of one of the two firms, a large change in the discount rate is needed, of course. However, if we want to approximate two similar IED 4 Alternatively, this regression may be estimated at the firm-specific level by applying the model to time-series data, as done by O Hanlon and Steele (2000). However, our cross-section estimation by industry does not require more data than those used by DSS, and we prefer to preserve the comparability of results in as many dimensions as possible. 11

12 of two firms by changing the discount rate of one of the two firms, only a small change in the discount rate is needed, and consequently, it is plausible that the differences in expected equity returns used as discount rates could be large enough to reverse the rank ordering of firms based on duration, in contrast to what DSS claim. On the other hand, if we change the discount rate but keep it cross-sectional (and time) constant, we expect changes in the firms IED in the same direction and hence in their average, but the relative rankings of the durations across securities undergo little change. However, to check the influence on the results of the cross-sectional (and time) constant expected return on equity, this assumption should be relaxed, adopting a more plausible assumption, as we do in this study. We use industry-specific implicit expected rates of return as discount rates when computing the IED with industry forecasting parameters. Finally, we estimate the Easton and Sommers model in equation (9) with all of the firms in the sample to obtain implicit expected market rates of return and market growth rates that we use as constant long-term rates in the DSS framework. Thus, we effectively avoid the criticism that our empirical results are simply an artefact of the way in which we measure the expected returns. 4. Data Table 1 summarises the sample selection procedure. The initial sample we use includes all non-financial firms with available data listed in the Spanish Stock Market at the end of 2011, for a total of ninety firms. 5 Accounting data at the end of 2011 and 2010 are 12

13 obtained from financial statements included in SABI. 6 From these data, we compute firm-specific ROE and SGR. Figure 1 shows the pair of ratios for each one of the ninety firms. Then, we eliminate the influence of outside atypical values of these ratios using the minimum covariance determinant method for multivariate outlier detection, as done by Verardi and Dehon (2010). 7 The final sample includes sixty-two firms whose pairs of ROE and SGR are shown in Figure 2. From the Madrid Stock Exchange, we obtain each firm s capitalisation at the end of the year 2011 and the industry classification of firms in the final sample. Non-financial firms are grouped into five industries: Oil and Energy; Commodities, Industry and Construction; Consumer Goods; Consumer Services; and Technology and Communications. From the accounting data at the end of 2011 and 2010, we also estimate the market and industry-specific implied cost of equity and growth rate that we use as discount rates and as long-term rates in the mean reverting processes of the forecasting procedures. These rates are shown in Table 2. 5 The financial industry idiosyncrasy regarding interest rates makes the separation between financial and non-financial firms necessary, giving rise to a parallel line of research on the sensitivity of financial firms to interest rates movements. See Czaja et al. (2009) and the references therein. 6 SABI (Bureau Van Dijk Group) provides company financial statements, ratios, activities and information on managers and ownership structure for more than 950,000 Spanish businesses and around 300,000 Portuguese firms. 7 Verardi and Dehon (2010) show in a simulation example that the minimum covariance determinant estimator outperforms other methods such as the Hadi method. Verardi and Croux (1999 and 2010) programmed a fast algorithm of this estimator in Stata and made it available with the mcd command. We use the mcd command in Stata with values of the optional parameters to minimize the presence of outliers in the final sample. 13

14 We use SABI data from 1991 to 2011 to estimate both the ROE autocorrelation coefficient and the SGR autocorrelation coefficient of firms with at least 10 consecutive years of available data. Then, we compute the industry-specific parameter as an average of the available firm-specific ratios belonging to each sector. Similarly, we compute the market parameters as an average of all available firm-specific ratios. Table 2 shows these parameters as well. 4. Results. For the IED computation in equation (6), we use a finite prediction horizon of 10 years, as DSS do, after checking that the mean reversion both for the SGR and the ROE are mostly completed before this time. For the sample firms at the end of 2011, Table 3 shows the IED new (IEDn) computed with industry-level parameters and the IED computing a la DSS, i.e., using market parameters. This table also shows industry equally weighted averages and the other variables usually used as risk proxies (EPR, BtM, CAP, SGR) computed for the same date. In table 4, panel A, we summarise the main statistics for all of these variables. The IED average is years with a standard deviation of 6.35 years. The value that defines the lowest quartile is 16.32, and the one that defines the superior quartile is The minimum value of IED is 1.22 years, and the maximum is years. These results are slightly different from those obtained by Fullana and Toscano (2013) for the same market and the same date. The differences arise due to the most restricted outlier detection methodology, which reduces the standard deviation significantly, and the market implicit cost of equity and growth rate used here. 14

15 The IEDn are 2.65 years longer on average than IED are because the firm s distribution among industries, as we can see in Table 3. The IEDn are also more scattered than the IED, now owing to the different parameters used by industries. In table 4, panel B, we show the correlations between all the variables included in the analysis. The correlations between the IEDn and IED are approximately 0.50, which are positive and relatively high, as expected. Note, however, that the correlations between the IEDn and the other variables considered are generally smaller than the correlations with the IED, and in a meaningful way with the variables related to the ROE and GSR affected by the different forecast procedures used. Table 5 shows the most relevant results of this paper. We compute the difference between IEDn and IED for each firm in order to show the quantitative impact of the methodological changes carried out. The smallest absolute change is of 0.36 years and the largest of years, with a mean absolute change of 5.91 years. This result means that on average, the firms IED change upward or downward in 5.91 years, i.e., 27.31% of the average firm IED value of years. But we also conduct the same analysis qualitatively, computing changes in the ranks when we ordered firms with the IEDn with respect to the initial ordering by IED. In this way, the smallest absolute change of rank is of zero positions and the largest of 32 rank positions, and the mean absolute change is of rank positions. This mean absolute change in rank position accounts for 23.77% of the 62 total rank positions. Through the comprehensive view provided by Tables 2 and 5, we can detect the relative weight of the adjustments in the expected cost of equity and the expected equity growth. 15

16 Table 2 shows that for the Oil and Energy industry, the implied equity growth is almost equal to the implied equity growth of the Market, so that the differences of IED in Table 5 for firms in this industry, representing 16.00% of the IED average of this industry, are mainly due to the adjustment of the expected cost of equity. Similarly, the Consumer Services industry has an implied cost of equity very similar to that of the Market, and thus, the differences in this industry computed in Table 5, representing 13.78% of the IED industry average, are mostly due to the adjustment of the expected equity growth. Therefore, it seems that the relative weight of the adjustments in the expected cost of equity and the expected equity growth are very similar, perhaps slightly higher in the case of the adjustment of the expected cost of equity. However, to more precisely isolate the different effects that cause the results shown in Table 5, we next compute the IED but modifying one by one the three elements adapted to be industry-specific in calculating IEDn: (i) the discount rate; (ii) the parameters for ROE forecasting; and (iii) the parameters for SGR forecasting. The results of these partial analyses are shown in Table 6. We can observe that when we only adapt the discount rate to be industry-specific, the firms IED mean absolute change is of only 0.52 years, and absolute changes are between 0.00 and 3.47 years. As DSS note, the impact of adjusting the discount rate is quite small compared with the previous results: concretely, it represents the 8.80% of the IED total mean absolute change of 5.91 years. The qualitative analysis shows similar results: the rank position absolute changes are between 0 and 8 rank positions, and the mean absolute change is of 1.16 rank positions, 16

17 a small value if we compare it with the total mean absolute change of rank positions: it only represents 7.87% of the total effect. However, as we have already discussed above, this result does not necessarily imply that adjusting the expected cost of equity cannot reverse the rank ordering of firm based on IED. In fact, the results in Table 6 show that, when only the adjustment of the expected cost of equity is considered, 37 of the 62 firms, i.e., 59.68% of the sample, change their rank position, and 16 firms, 25.81% of the sample, change two or more positions. Now, when we only modify the parameters for ROE forecasting, adjusting them to be industry-specific, we can see in Table 6 that the firms IED mean absolute change is 9.79 years. The absolute changes are between 0.25 and years. These high and variable absolute changes significantly distort the rank order to the extent that the mean absolute change of rank positions is 17.15, and the absolute rank change values are between 0 and 57 rank positions. Finally, we modify only the parameters for SGR forecasting, adjusting them to be industry-specific, and the firms IED mean absolute change is of 4.19 years with values between 0.11 and years. These changes have a mean absolute effect in the rank ordering of 7.84 rank positions. The absolute rank changes vary from 1 to 42 positions. As we can also see in Table 6, the last two effects analysed mostly have opposite signs that offset when we adjust the parameters for ROE and SGR forecasting together, as would occur in practice. This is why the sum of these two partial effects is greater than the total effect shown in Table 5. The positive correlation between industry- 17

18 specific implied returns of equity and industry-specific implied equity growth rates, shown in Table 2, explains the opposite signs of these partial effects. In this sense, Table 6 also shows the more interesting effect of (iv) adjusting the parameters for ROE and SGR forecasting together to be industry-specific and remaining cross-sectional constant the expected equity return used as a discount rate. When we do so, the firms IED mean absolute change is 6.22 years with absolute change values between 0.03 and years, and the mean absolute change in rank position is with a minimum value of 0 and a maximum value of 49. Once more, the sum of this partial effect and the effect of adjusting the expected equity return used as a discount rate to be industry-specific is greater than the total effect shown in Table 5, which is also due to the negative correlation between these partial effects. 6. Conclusions In their seminal paper from 2004, Dechow, Sloan and Soliman develop the implicit equity duration in the image of Macaulay s bond duration in order to capture an important common factor in stock returns as Macaulay s duration does with respect to bond returns. However, in the same paper, the authors acknowledge that their procedure for estimating equity duration, while parsimonious, is relatively crude, and improvements in the procedure should lead to more accurate and useful measure of equity duration. Within this context, our paper aims to compute the IED thought forecasting and discount procedures that use implied industry characteristics as an alternative to the historic market parameters used in DSS forecasting and discount procedures with the 18

19 objective of bridging the empirical issues with the theoretical IED formulation. Then, we compare, both in quantitative and qualitative terms, the IED that arises with the results obtained by DSS procedures. The results show that when we use implied industry characteristics in forecasting and discount procedures, the firms absolute changes in IED are 5.91 years on average, with absolute values reaching years. From a qualitative point of view, the rank positions of firms by duration change up to a maximum of 32 positions, and the average of absolute changes is of rank positions. All firms except the one with the smallest IED change their rank position. When we isolate the effects corresponding to the distinct adjustments that we make in computing IED, we observe that a greater accuracy in the discount rate used involves relative small changes in IED as DSS suggest, approximately a half-year on average. However, these changes are neither insignificant nor always in the same direction. Moreover, as we suggest, these changes are large enough to reverse the rank ordering of firms based on IED, changing one of the firms in the sample up to eight rank positions. On the other hand, when adjusting only the forecasting parameters to be industryspecific, an effect slightly greater than the total effect in firms IED arises despite the logic offset between the effect of adjusting the forecasting parameters and the effect of adjusting the discount rate reflected in the results. Moreover, when only one forecasting model is adjusted, the results show even larger changes, being the firms IED more sensitive, approximately twice, to industry-specific adjustments in the ROE forecasting model than to industry-specific adjustments in the SGR forecasting parameters. 19

20 Thus, we can now conclude that the cost of being parsimonious in the DSS procedures is relative high on average and also quite variable across firms, both quantitatively and qualitatively. Moreover, this cost is large enough to reverse the rank ordering of firms based on duration. Therefore, more accurate empirical procedures, even those used in this paper that are limited by their industry-level nature, should be applied in order to proxy the theoretical IED that, like Macaulay s bond duration, is an asset-specific formulation. References: Boquist, J.A., Racette, G.C. and Schlarbaum, G.G., Duration and Risk Assessment for Bonds and Common Stocks. Journal of Finance, 30, Czaja, M. G., Scholz, H. and Wilkens, M., Interest rate risk of German financial institutions: the impact of level, slope, and curvature of the term structure. Review of Quantitative Finance and Accounting, 33(1), Campbell, J. and Mei, J., Where Do Beta Come From? Asset Price Dynamics and the Sources of Systematic Risk. The Review of Financial Studies, 6, Campbell, J.Y., Polk, C. and Voulteenaho, T., Growth or glamour? Fundamentals and systematic risk in stock returns. Review of Financial Studies, 23 (1), Chen, H.J., Firm life expectancy and the heterogeneity of the book-to-market effect. Journal of Financial Economics, 100 (2), Cohen, R.D., The relationship between the equity risk premium, duration and dividend yield. Wilmott Magazine, 6,

21 Cornell, B., Risk, Duration, and Capital Budgeting: New Evidence on Some Old Questions. Journal of Business, 72, Da, Z., Cash Flow, Consumption Risk, and the Cross-section of Stock Returns. Journal of Finance, 64, Dechow, P.M., Sloan, R.G. and Soliman, M.T., Implied Equity Duration: A New Measure of Equity Risk. Review of Accounting Studies, 9, Easton, P. D. and Sommers, G. A., Effect of analysts' optimism on estimates of the expected rate of return implied by earnings forecasts. Journal of Accounting Research, 45(5), Fama, E. F. and French, K.R., Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33, Fullana, O. and Toscano, D., The implied equity duration for the Spanish listed firms. Spanish Review of Financial Economics. DOI: /j.srfe González, M. and Nave, J.M., Portfolio immunization using independent component analysis. Spanish Review of Financial Economics, 21, Hamelink, F., MacGregor, B., Nanthakumaran, N. and Orr, A., A comparison of UK equity and property duration. Journal of Property Research, 19 (1), Hicks, J.R., Value and Capital. Oxford, UK: Oxford University Press. Hurley, W.J. and Johnson, L.D., A Note on the Measurement of Duration and Convexity. Financial Analysts Journal, May/June, Johnson, L. D. (1989). Equity duration: Another look. Financial Analysts Journal, 45(2), Leibowitz, M., Total Portfolio Duration: A New Perspective on Asset Allocation. Financial Analysts Journal, 42(5),

22 Leibowitz, M. L., Bova, A. and Hammond, P. B., Real Return Tents and Equity Durations. The Endowment Model of Investing: Return, Risk, and Diversification, Hoboken, NJ: John Wiley and Sons. Leibowitz, M. and Kogelman, S., Resolving the Equity Duration Paradox. Financial Analysts Journal, 49:1, Leibowitz, M.L., Sorensen, E.H., Arnott R.D. and Hanson H.N., A Total Differential Approach to Equity Duration. Financial Analysts Journal, September/October, Lewin, R.A., Sardy, M.J. and Satchell, S., UK measures of firm-lived equity duration. International Finance Review, 7, Lintner, J., Corporate Growth Under Uncertainty. The Corporate Economy. London, UK: MacMillan. Livingston, M., Duration and risk assessment for bonds and common stocks: a note. Journal of Finance, 33 (1), Macaulay, F.R., Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States since Cambridge, UK: NEBER books. Navarro, E. and Nave, J.M., A two-factor duration model for interest rate risk management. Investigaciones económicas, 21(1), Navarro, E. and Nave, J.M., The structure of spot rates and immunization: Some further results. Spanish Economic Review, 3(4): Nissim, D. and Penman, S.H., Ratio Analysis and Equity Valuation: From Research to Practice. Review of Accounting Studies, 6, O Hanlon, J. and Steele, A., Estimating the equity risk premium using accounting fundamentals. Journal of Business Finance and Accounting 27:

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24 Table 1. Summary of firm sample selection, financial variables and parameters used in the estimation of implied equity duration. Panel A. Firm sample selection SIBE listed firms on 31/12/ Financial sector companies 29 Accounting data not available 4 Outliers removed 28 Number of firms in the final sample 62 Panel B. Financial variables and database Book value of equity Earnings before extraordinary items Firm sales Market capitalisation SABI SABI SABI SABI Panel C. Forecasting parameters for market Autocorrelation coefficient for ROE 49.77% Estimated market equity return 8.81% Autocorrelation coefficient for growth sales 83.11% Estimated market growth rate 7.63% Panel D. Summary of final sample firms by size index and industry Industries # IBEX35 IBEX MC IBEX SC Oil & Energy Commodities Consumer Goods Consumer Services Technology Totals Accounting data have been obtained from the database for Spanish and Portuguese companies SABI (Bureau Van Dijk Group). In Panel D, industries and selective indices by size appear according to the Madrid Stock Exchange classifications. SABI provides company financial statements, ratios, activities and information on managers and ownership structure for more than 950,000 Spanish businesses. In Panel D, industries and size indices appear according to the Madrid Stock Exchange classifications. Firms listed in IBEX35 are the largest capitalisations. IBEX MC includes medium capitalisations firms. IBEX SC includes smallest capitalisations firms. 24

25 Table 2. Industry-specific forecasting parameters Autocorrelation Coefficient for ROE Autocorrelation Coefficient for Growth Sales r g Oil & Energy Commodities Consumer Goods Consumer Services Technology Market For each industry, this table shows: the Autocorrelation Coefficient both for Return over equity and for Growth Sales; long-run equity return (r) and the long-run equity Growth Rate (g) estimate by crosssection applying: eps jt bps jt 1 = δ 0 +δ 1 P jt bps jt bps jt 1, where: δ0 = r; δ1 = (r g)/(1+g); and eps jt is the j-firm earnings per share ratio at t, bps jt is the j-firm book value per share ratio at t; and P jt is the quote of a j-firm share at t. 25

26 Table 3. Implied equity duration new, implied equity duration, earnings-toprice ratio, book-to-market, market capitalisation and sales growth rate TICKER IEDn IED EPR BtM CAP SGR ENG , ELE , EGPW , IBE , REE , REP , Oil & Energy , # Obs , ABG , ANA , ACX , CAF , AZK CIE MDF ENO ECR FCC , FDR GAM GSJ LGT OHL , URA Commodities , # Obs ,

27 Table 3 (Cont.) TICKER IEDn IED EPR BtM CAP SGR ADZ ALM BDL RIO CFG CUN OLE EBRO , ENC FAE GRF , TVX IBG ROVI MCM NAT PVA PRM RDM SNC UPL VID VIS , ZEL Consumer Goods # Obs ABE , A3TV CBAV CDR FUN TL , MEL PRS PSG , VOC VLG Consumer Services , # Obs ,

28 Table 3 (Cont.) TICKER IEDn IED EPR BtM CAP SGR AMS , IDR , JAZ TEC TEF , Technology , # Obs , For each firm, this table shows: IEDn: the implied equity duration with industry-specific parameters at year-end 2011; IED: the implied equity duration a la DSS at year-end 2011; the earning-to-price ratio as annual earnings over market capitalisation; BtM: the book-to-market ratio calculated as year-end 2011 book value of equity over market capitalisation; the market capitalisation at year-end 2011 in millions of euros; and the sales growth rate as the annual sales difference over the previous-period sales [(Sales t -Sales t-1 )/Sales t-1 ]. 28

29 Table 4. Descriptive statistics for IEDn, IED and other risk related measures Panel A. Statistics Med SE Min Max 1Q 3Q Median IEDn IED EPR BtM CAP 3, , , , SGR Panel B. Correlations (Pearson/Spearman) IEDn IED EPR BtM CAP SGR IEDn IED EPR BtM CAP SGR IEDn: the implied equity duration with industry-specific parameters at year-end 2011; IED: the implied equity duration a la DSS at year-end 2011; EPR: the 2011 year-end earning-to-price ratio as annual earnings over market capitalisation; BtM: the book-tomarket ratio calculated as the year-end 2011 book value of equity over market capitalisation; CAP: the market capitalisation at year-end 2011 in millions of euros; and SGR: the sales growth rate as the annual sales difference over the previous-period sales [(Salest-Salest-1)/Salest-1]. Pearson linear correlation coefficients appear in Panel B below the diagonal, and Spearman linear correlation coefficients are above the diagonal. 29

30 Table 5. Total quantitative and qualitative IED changes TICKER IED IEDn Dif. Rank Rank IED IEDn Dif. ENG ELE EGPW IBE REE REP ABG ANA ACX CAF AZK CIE MDF ENO ECR FCC FDR GAM GSJ LGT OHL URA ADZ ALM BDL RIO CFG CUN OLE EBRO ENC FAE GRF TVX IBG ROVI MCM NAT PVA PRM RDM SNC UPL VID VIS ZEL

31 Table 5 (Cont.) TICKER IED IEDn Dif. Rank IED Rank IEDn Dif. ABE A3TV CBAV CDR FUN TL MEL PRS PSG VOC VLG AMS IDR JAZ TEC TEF For each firm, this table shows the difference between IEDn and IED in a quantitative way and in a qualitative way thought changes in the rank when we ordered firms with the IEDn with respect to the initial ordering by IED. 31

32 Table 6. Partial IED changes Panel A: Quantitative IED changes TICKER (i) Dif. (ii) Dif. (iii) Dif. (iv) Dif. ENG ELE EGPW IBE REE REP ABG ANA ACX CAF AZK CIE MDF ENO ECR FCC FDR GAM GSJ LGT OHL URA ADZ ALM BDL RIO CFG CUN OLE EBRO ENC FAE GRF TVX IBG ROVI MCM NAT PVA PRM RDM SNC UPL VID VIS ZEL

33 Table 6. Panel A (Cont.) TICKER (i) Dif. (ii) Dif. (iii) Dif. (iv) Dif. ABE A3TV CBAV CDR FUN TL MEL PRS PSG VOC VLG AMS IDR JAZ TEC TEF Panel B: Qualitative IED changes TICKER (i) Dif. (ii) Dif. (iii) Dif. (iv) Dif. ENG ELE EGPW IBE REE REP ABG ANA ACX CAF AZK CIE MDF ENO ECR FCC FDR GAM GSJ LGT OHL URA

34 Table 6. Panel B (Cont.) TICKER (i) Dif. (ii) Dif. (iii) Dif. (iv) Dif. ADZ ALM BDL RIO CFG CUN OLE EBRO ENC FAE GRF TVX IBG ROVI MCM NAT PVA PRM RDM SNC UPL VID VIS ZEL ABE A3TV CBAV CDR FUN TL MEL PRS PSG VOC VLG AMS IDR JAZ TEC TEF For each firm, this table shows the computed IED and rank but modifying one by one the three elements adapted to be industry-specific in calculating IEDn: (i) the discount rate; (ii) the parameters for ROE forecasting; (iii) the parameters for SGR forecasting; and (iv) modifying together the parameters for ROE forecasting and for SGR forecasting. 34