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Review of Accounting Studies

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20-11-2019

The term structure of implied costs of equity capital

Authors: Jeffrey L. Callen, Matthew R. Lyle

Published in: Review of Accounting Studies | Issue 1/2020

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Abstract

We model and estimate the term structure of implied costs of equity capital (and implied risk premia) at the firm level for the years 1996–2015 from forward looking option contracts. Empirical tests reject the assumption that the term structure of implied firm-level costs of equity is constant over different time horizons. Instead, we find that the term structure is often upward sloping and concave. However, we also find that the term structure flattened during the 1998 and 2007–2008 crises and even sloped downward during part of 2008. Term structure estimates are shown to predict future stock returns and volatilities over multiple horizons. In contrast to static implied cost of capital models, the term structure estimates can capture ex ante the well-documented earnings announcement premium. Moreover, various firm-level characteristics related to firm performance and risk are shown to explain some of the cross-sectional variation in the shape of the term structure.

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1000/(1.025) + 1000/((1.025)(1.05)) = 1000/(1.033) + 1000/(1.033)² = 1,905.

These percentages are large in part because we assumed a 100% increase in the expected cost of equity from year 1 to year 2. Nevertheless, the example is informative if somewhat dramatic.

Of course, we are not the first to note the investment distortions that potentially arise from assuming static implied costs of equity capital. See Cready (2001), for example.

In an earlier version of this paper, we also compared the term structure estimates with traditional factor-based estimates. The latter were found severely wanting, which is hardly surprising given the well-known literature on factor-based models. These results are available from the authors.

From this point on until the formal hypotheses, we refer to the term structure of implied costs of equity, with the term structure of implied risk premia implicitly understood.

See the survey by Easton (2009), for example.

We use the term cash flows generically. In many of these models, cash flows are in fact replaced by a multiple of expected earnings.

Often only one-year-ahead and sometimes two-year analyst forecasts are the only forward looking information incorporated in these models. Longer term forecasts tend to be extrapolated from the latter.

Early work by Botosan (1997) employed expected dividend and terminal price forecasts to estimate the cost of equity. Claus and Thomas (2001), Gebhardt et al. (2001), and Morel (2003) use various implementations of the residual income and Ohlson (1995) models in their estimation procedures. Other studies (e.g., Gode and Mohanram 2003) focus on deriving cost of equity capital estimates, using common ratios and exploiting the Ohlson and Juettner Nauroth (2005) model. A more up-to-date literature incorporates risk into implied costs of equity, following Feltham and Ohlson (1999). In particular, see Nekrasov and Shroff (2009) and Lyle et al. (2013). Lyle et al. estimate dynamic costs of capital empirically based on extended Ohlson (1995) dynamics, but they do not explore the term structure issue.

See also Pastor et al. (2008) at the aggregate level.

See Campbell and Shiller (1988), Campbell (1991), Hodrick (1992), Jagannathan and Wang (1996), Fama and French (1997, 2002), Lamont (1998), Jagannathan et al. (2000), Lettau and Ludvigson (2001, 2002), Vuolteenaho (2002), Chen (2003), Campbell and Vuolteenaho (2004), Ang and Liu (2004), Callen and Segal (2004), Callen et al. (2005, 2006), Petkova and Zhang (2005), Cochrane (2011), and Lyle et al. (2013).

See Campbell and Cochrane (1999), Bansal and Yaron (2004), Barro (2006), van Binsbergen et al. (2012), Gabaix (2012), Lettau and Wachter (2007), Belo et al. (2015), and Ai et al. (2018), and many others. For a review of this fast growing literature, see van Binsbergen and Koijen (2017).

See Patell and Wolfson (1979, 1981) for an early attempt to extract forward looking earnings information from option prices.

Additionally, a number of papers have incorporated option information into CAPM type models, following a suggestion by French et al. (1983). For example, Buss and Vilkov (2012) and Chang et al. (2011) find that incorporating option information in the CAPM is useful in predicting betas and market returns.

One cannot simply generalize equity term structure findings based on aggregate data to the firm level. One need only be reminded of how very different are the implications to equity returns of shocks to discount rates, relative to earnings shocks at the aggregate level by comparison to the firm level. See Vuolteenaho (2002).

There are small exchanges, for example, the company One Chicago, www.onechicago.com, that provide a platform for trading individual futures contracts. However, these exchanges are in their infancy.

This issue is important because costs of capital are ultimately unobservable, and the only method available for confirming a cost of capital computation is by reference to future returns (adjusted perhaps for shocks) and future return volatilities.

We assume for simplicity that dividends (cash flows) are paid out at the end of the period.

The equation follows because the expected discounted difference between the time t futures price F_{t, t+T} and the future stock price S_t+T is necessarily zero (Duffie 2001; Back 2010); that is, \(E_{t}\left [ \frac {{\Lambda }_{t+T}}{{\Lambda } _{t}}(S_{t+T}-F_{t,t+T})\right ] =0\). Given that F_{t, t+T} is known at time t and \(E_{t}[\frac {{\Lambda }_{t+T}}{{\Lambda }_{t} }]=Rf_{t,t+T}^{-1},\) one obtains (6).

The stochastic discount factor can be interpreted economically as the marginal rate of substitution of consumption between t and t + T for a representative agent in the economy (e.g., Cochrane 2005).

Here we use the relation: \(Cov_{t}(R_{t,t+T}^{M},\frac {S_{t+T}}{S_{t}})=Corr_{t}(R_{t,t+T} ^{M},\frac {S_{t+T}}{S_{t}})\sigma _{t,t+T}^{M}\sigma _{t,t+T}^{\mu }\) for T ≥ 1. This formulation also assumes a representative investor risk aversion coefficient of one, consistent with empirical results described in the ??.

Because Pastor et al. (2008) ignore Jensen’s inequality, the linear relation is somewhat unclear.

The 360,000 months proxies for infinity.

See Section 5.5.5 below on empirical estimates of these biases.

Specifically, options that have an American-style exercise feature are priced by OptionMetrics, using a proprietary pricing algorithm that is based on the industry-standard Cox et al. (1979) binomial tree model. This model can accommodate underlying securities with either discrete dividend payments or a continuous dividend yield. We substitute these implied volatilities into the Black-Scholes formula to produce synthetic European option prices. The latter are then used to determine synthetic futures prices from Eq. 11. This procedure does not imply that European options are priced in a Black-Scholes economy (log-normal asset prices), only that the Black-Scholes model provides a simple one-to-one mapping from implied volatilities to option prices (see Figlewski 2010 on this issue), which then allows us to compute the price of the futures contract.

While studies in accounting research have mostly used the OptionMetrics standardized options data set (which itself is constructed from the volatility surface data), we can dramatically increase the sample size by using the volatility surface files directly.

OptionMetrics uses (nonparametric kernel) smoothing techniques on the raw data to generate their volatility surface to capture important patterns in the data while reducing noise. For additional information about the OptionMetrics data set, see the extensive details provided in their online manual, which is available on the WRDS website.

The delta of an option is the change in the price of option contract for a given change in the price of the underlying equity.

Interpolation is sometimes required to get precise estimates for at-the-money volatilities whenever the at-the-money strike price differs from the strike prices of traded options. For example, interpolation is necessary to recover the at-the-money implied volatility if the at-the-money strike price is 10.15 but the only available option data have strikes of 9, 10, 11 and 12. We use interpolation, closely following Figlewski (2010).

In addition, one goal of this study is to derive a term structure of implied costs of equity that is relatively easy to implement by academics and practitioners, and a backward looking measure of correlation simplifies the analysis considerably.

We choose to show the one-month-ahead estimates, in addition to the quarterly, because monthly returns are a common frequency for empirical asset pricing tests.

Contract specifics can be found on the CME website: http://www.cmegroup.com/trading/equity-index/us-index/sandp-500_contract_specifications.html

Note that the Ohlson and Juettner-Nauroth and Easton models have fewer observations because these models require positive (expected) earnings growth. Following Hou et al. (2012), the composite average is computed over all estimates that are non-missing.

A limitation of the term structure model is that it cannot generate as many cost of equity estimates as other models because of its dependence on option prices. Using the Lewellen estimates as the baseline, we find that the proportion of monthly estimates obtained, relative to Lewellen, is 43% for the term structure, 92% for Gebhardt, Lee and Swaminathan, 88% for Claus and Thomas, 74% for Ohlson and Juettner-Nauroth, 76% for Easton, and 71% for Gordon.

Johnson and So (2012) find that the OS variable helps to predict equity returns.

With the exception of the M2 box with days to expiry greater than 183 where the slope coefficient is significantly different from one. Although the intercept coefficient estimates are significant statistically, they are not significant economically.

Untabulated results compute term structure implied costs of equity stratified by the Fama and French (1997) 48-industry classification. Overall, implied costs of equity tend to be persistently upward sloping and concave along the term structure maturity, irrespective of the industry. Similar results (untabulated) obtain for implied risk premia.

If the function is concave, the coefficient on the quadratic term will be negative.

Test results using ranks of implied costs of equity yield qualitatively similar results (untabulated).

Given Eq. 9, it is natural to ask whether the term structure estimates capture risk primarily through the historical covariance with the market component or through the futures price F_{t, t+T} component. Because our term structure estimate combines both of these variables non-linearly over future horizons, this is a difficult question to answer in general. However, we can most easily separate these two components for the one month-ahead returns. Perhaps not surprisingly, we find that the futures component of the term structure estimate is a stronger predictor of future one month-ahead returns than the covariance component. Nevertheless, both add to each other in the prediction.

It could be argued that one should use the entire term structure of implied cost of capital rather than analyzing each date separately or evaluating the various slopes of the term structure. By incorporating all of the term structure estimates in the regressions, implied costs of equity might help to provide additional information about expected returns beyond the individual date. Unfortunately, term structure implied cost of equity estimates for different horizons tend to be highly correlated, making the results from such a regression unreliable. Additionally, the goal of our paper is not return forecasting maximization but rather to validate our measures and to show that they are not merely manifestations of common firm characteristics.

We estimate historical volatility over different estimation windows since there is no consensus in the literature about the appropriate estimation window.

In untabulated tests, we adjust excess returns for each decile portfolio by the Fama and French (1992, 1993) standard three, four, and five factors and compute the high minus low alpha. The return results are robust to this adjustment, suggesting that our results are not driven by return factors that are commonly used in empirical asset pricing.

Lewellen (2015) examines three empirically inspired specifications. We estimate his Model 2, as he shows that additional predictors do not enhance out-of-sample stock return predictability.

This result is consistent with findings of Hasan et al. (2015), who correlate firms’ weighted average cost of capital with the Dickinson’s (2011) life cycle proxy for Australian equities.

More formally, we define the residual volatility ResVol_{t, t+T} as the difference between the volatility σ_{t, t+T} and the mean volatility \(\frac {1}{T}{\sum }_{\tau =1}^{T}\sigma _{t,t+\tau }\) over the period t to t + T.

Needless to say, no such residual volatility measure can be computed for static implied costs of equity.

On model-free expected volatility estimation, see, for example, Britten-Jones and Neuberger (2000), Bakshi et al. (2003), Jiang and Tian (2005), Bollerslev et al. (2009), Carr and Wu (2009), and Carr and Lee (2009).

Similar concerns apply to static implied costs of capital based on analyst forecasts in that analysts tend to follow larger firms.

These untabulated empirical results are available from the authors.

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Title: The term structure of implied costs of equity capital
Authors: Jeffrey L. Callen
Matthew R. Lyle
Publication date: 20-11-2019
Publisher: Springer US
Published in: Review of Accounting Studies / Issue 1/2020
Print ISSN: 1380-6653
Electronic ISSN: 1573-7136
DOI: https://doi.org/10.1007/s11142-019-09513-z

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