_{1}

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We propose a Total Cash Flow present value model to estimate the implied risk premium equity indices typically used in financial models as proxies for the aggregate stock market. Following a bottom - up approach, we use individual companies’ financial information as well as analysts’ earnings estimates in order to reconstruct the index total cash flow and expected future cash flows. Using the prices for these indices, we then extract the implied Equity Risk Premia and analyze their evolution through time. To illustrate our approach, we run our model over a period of 10 years surrounding the 2008 financial crisis on five stock indices, namely the S & P 500, the Euro Stoxx 50, the FTSE 100, the Hang Seng Index and the TOPIX 100. We find that our model is a conceptually robust approach to modeling long-term market Equity Risk Premia and provides a normative view of expected returns for practitioners.

Equity Risk Premia are playing a fundamental role in modeling risk and returns characteristics of traded equities. Equity Risk Premia are widely used by practitioners and by researchers for valuation purposes as well as for the estimation of firms’ Cost of Equity Capital. Despite the central position of the Equity Risk Premium in many financial models, a consensus on the best way to estimate the Equity Risk Premium has not emerged. The main current methods used by researchers and practitioners follow very different logics, which in turn lead to large discrepancies of results. These discrepancies are far from being neutral as they lead in turn to substantial estimation errors when included in models such as in the capital asset pricing model. Ferson and Locke [

In this paper, we propose a model based on cash flows in order to extract the implied equity risk premium from various stock indices. We start by laying down the theoretical framework for Equity Risk Premia and their relationship with investors’ perceptions and attitudes towards risk. We follow with a review of the main approaches pursued to measure Equity Risk Premia, and lay down the theoretical foundations behind cash flow based methods.

We then introduce our forward-looking multiple-stage cash flow model. We discuss the logic of each stage and the way they relate to our general theoretical frame. Next, we apply this model to recent financial data the obtained premia. We finally conclude with a discussion on the meaning of the resulting Equity Risk Premia and their potential ability to predict realized returns.

The main questions one should ask, when thinking about Equity Risk Premia, are why Equity Risk Premia matter and how the current main methods for estimating Equity Risk Premiago about to reflect the structure of the underlying economic reality.

Let us start with defining the equity risk premium: the Equity Risk Premium is the average extra return demanded by investors, on top of a risk free rate, as a compensation for investing in equity securities with average risk. We use the term “risk” in the classical sense of variance of returns around the mean expected return. Under this definition, a risk free return is a return with no possible variations. Damodaran [

The Equity Risk Premium is therefore added to the risk free rate in order to obtain the total expected return on an equity investment with average risk. Average risk here is to be understood as the risk for well-diversified investors and is typically measured as the market risk. The resulting market expected returns are then used for valuation purposes to discount expected cash flows with average risk. Various risk-return models, such as the Capital Asset Pricing Model, use the Market Equity Risk Premium to assess the returns that investors should expect from individual securities. By extension, such expected returns are used by practitioners as proxies for companies Cost of Equity Capital which is then incorporated to the discount rates used in capital budgeting processes.

Obviously, changes in Equity Risk Premia will have a direct influence on the pricing of equities as well as on the valuation of projects considered by companies. In a sense, Equity Risk Premia can be viewed as the price of risk. When investors perceive higher risk linked to future cash flows of an investment, they attribute a lower value to the cash flows attached to such risky investment. This lower value reflects investors’ anticipation that the cash flows of the investment considered will exhibit a higher volatility, making the final outcome more uncertain.

Unlike for bonds, where required yields can easily be inferred from their current market prices and their future cash flows, bond cash flows being established contractually, Equities do not allow for such straightforward inferences. Indeed, the future cash flows of equity shares remain uncertain and provide no contractual guarantee. The consequence of such uncertainty is the lack of a general agreement on the best method to estimate the future expected returns implied by current prices, and by extension the Equity Risk Premium.

The academic literature has explored various methods, and an intense debate, on which approach to use and which one leads to the best estimates, continues within the finance community. While Campbel and Shiller suggested long run returns were “highly predictable” more recent research casted serious doubt on such claim. Ang and Bekaert [

We can sort the methods for estimating Equity Risk Premium into several categories:

The use of historical data for estimating Equity Risk Premia is an easy and widely spread approach. Essentially, long-term average realized index returns are used with average realized returns on a governmental security (serving as proxy for risk free return). One big limitation of using historical data is that it only reflects realized returns, failing to capture expectations ex ante. Conceptually, such approach does not provide much explanation why past equity risk premia would predict future ones. Yet, despite this lack of theoretical justification, it seems that the historical track record for ERP exerts some influence on investors’ expectations in a feedback mechanism. In particular, Goyal & Welch [

A second possible approach is to rely on surveys. One could survey analysts, professional and retail investors, corporate officers or even academics. One of the most extensive surveys by Fernandez et al. [

Independently of any predictive (in) ability, such survey serves the purpose of measuring some aggregate sentiment, with no particular link to any objective criteria.

For the past century, factor models have been used to try to predict future performance of equities or realized Equity Risk Premia. Typical variables used include a variety of financial ratios such as earnings to price, book to market, dividend payout, as well as other financial measures such interest rates, default spreads, inflation rates, etc.

The vast choice of possible variables, time frame as well as regressions methods, such as time-series regressions or cross-sectional regressions, has led to the development of an abundant literature with often contradicting results. Many papers, let us point to some of the most influential ones, Campbel and Shiller [

A specific category of models aiming at assessing Equity Risk Premia is that of implied premium models. These models essentially rely on cash flow projections (for a given index) coupled with the underlying current prices of the index in order to extract the implied Equity Risk Premium.

Such models avoid looking backwards to estimate future returns as they are market driven and do not require per se to rely on historical data. What we like in such models is that they are quite flexible, can be used in any market, even with limited historical track record. Such models tend to be very explicit about their modeling of the future, unlike surveys where the logic and parameters used by respondents are generally unknown. In addition to these attractive features, cash flow models offer a conceptually robust description of the fundamental elements that are key to investment decisions: size of cash flows, growth and risk of those cash flows.

Very concretely, given a price and a series of cash flow one can infer the risk imbedded in the pricing. As market prices are observable, the primary difficulty is to properly model the expected future cash flows, which requires the ability to estimate potential growth of those cash flows far in the future. This is most likely the main difficulty faced by investors and academic alike when using such models: long-term growth rates are notoriously difficult to assess precisely. Yet robust estimates are particularly important for Discounted Cash Flow models: the sensitivity of the obtained premia to estimates of the average long-term growth of cash flows implies that small estimation errors in those long-term growth rates lead to large variations of implied Equity Risk Premia. The potential subjectivity of such estimates and the unlikeliness of homogeneous expectations among investors lead Pablo Fernandez [

In order to capture the Market Equity Risk Premium implied by a given pricing we developed a multiple stage discounted cash flow model. Our methodology aims at offering a forward-looking assessment of the long-term risk premium implied by the most current pricing of an equity index. We therefore take the view that long-term economic perspectives dominate the pricing of equity markets, especially at the aggregate market level and in major stock indices.

Let us recall that the Equity Risk Premium corresponds to the excess rate of return over the risk-free rate that is required by investors as compensation for the higher risk inherent to equities.

Equity Risk Premium = Expected Return on Equities – Risk Free Rate.

Our model takes its roots in classical dividend discount models; While classical approach, relying on Miller and Modigliani (1961)’s irrelevance Theorem, assumed prices of equity shares should not be influenced by the actual payout, we took into account the observations of Mauboussin [

Furthermore, Damodaran [

Taking into account those observations, we developed a multiple stage growth model of cash distributions for equity indices in which we add aggregate share buybacks to the aggregate dividends paid, allowing us to get a better measure of total cash flow to equity (CFE) for the index. Therefore, the present value of the index (reflected in its price) is equal to the sum of discounted total cash flows to equity. In its most simple form the corresponding formula is as follows:

Value of Equity Index = ∑ t = 1 N E ( C F E t ) ( 1 + k e ) t + E ( C F E N + 1 ) ( k e − g N ) ( 1 + k e ) N

In this equation, E(CFE_{t}) is the expected aggregated cash flow to equity (potential dividends + share buybacks of the index) in year t, k_{e} is the rate of return required by equity investors and g_{N} is the stable growth rate (after year N).

g_{N} corresponds to a long term sustainable growth rate.

Our model is however expanded in order to allow for different growth parameters.

More specifically, we model these expected cash flows through the following steps:

We start by computing the most recent CFE of the index. In order to do so, we pursue a bottom up approach. We retrieve the latest data of 12-month trailing dividends as well as 12-month trailing share buybacks for each individual company composing the chosen index. We aggregate our results to obtain the past 12-month CFE of the index.

In order to estimate future CFE we then estimate the potential growth of such CFE. We assume that the growth of cash flow distributed to investors is ultimately tied to the growth of earnings of those corporations. While individual and temporary deviations may arise and such short term swings in earnings may only be partly translated in changes of distributions, our combination of aggregated data from many companies and the long term perspective we take, allow us to affirm that, at the index level, the growth of distributions should not be able to deviate durably from earnings growth. Therefore, we can use earnings growth estimates to infer the growth of CFE.

We have defined three stages, with different earnings growth regimes. Those three stages are described below:

In stage 1 earnings growth, and therefore CFE growth, is based on analyst consensus expectations. Analyst pays a particular attention to these earnings estimates, more so than for dividend and share buybacks estimates as discussed by Wilcox [

One special case concerns the growth estimate of the first period. In times of great financial distress, it is possible to observe some breakdown in the relationship between actual cash flows distributed to investors and earnings reported. A common practice in publicly traded companies is to increase dividend payout in order to continue to provide investors with cash payment despite a low or negative earnings situation.

When such occurrence at the index level is identified, we use the one-year dividend consensus forecast to estimate the total cash flow in that period instead of relying on earnings growth estimates which are distorted by the excessive swing in earnings.

In our second stage, starting in year t4 until year t10, we assume CFE will continue to rise but that their growth will linearly converge towards the perpetual growth used from year t11 on. While it may be tempting to provide with specific estimates for each of the years in our second stage, the lack of specific data provides little ground to support, especially in the absence of analysts’ estimates, the accuracy of such projections. We hence follow the practice described by financial textbooks [

Our third stage starts on the eleventh year. Since we cannot estimate forever our CFE, we introduce a terminal value of the index, modeled as a growing perpetuity.

T V = E ( C F E N + 1 ) ( k e − g N )

We assume that the growth of the index CFE rise from the eleventh year on at a constant rate. This rate should reflect long-term economic growth and should therefore be tied to long-term earnings growth. In our model, this rate g_{N} is proxied by the long term yields of treasuries (30-year rate) giving us a conservative perpetual growth parameter. Indeed the average yield of 30-year US treasury over the past 30 years on monthly data (6.28% pa) is very close with long term average growth of the S & P 500 earnings (from 5.98% to 6.17% for 30-year and 50-year average earnings’ growth)

We provide below the formal formula for our 3-stage model:

Value of Equity Index 3stage model = ∑ t = 1 3 ( C F E t ) ( 1 + k e ) t + ∑ t = 4 10 ( C F E t − 1 × ( 1 + ( g t − 1 × ( g 3 − g T V ) 8 ) ) ) ( 1 + k e ) t + ( C F E 10 ) × ( 1 + g T V ) ( k e − g T V ) ( 1 + k e ) 10

Once we have estimated the aggregated cash flows to the index for all three stages of the model, we implement a bottom up reconstruction of the index in order to obtain the total capitalization of that index. This bottom up approach gives us great flexibility to adapt our analysis framework to any existing index or any potential market proxy, including industry or sector subgroups. Once we have obtained the total market capitalization for our chosen index, we then use our multistage discounted cash flow model to infer the missing element: the discount rate. This discount rate acts as an internal rate of return (IRR) giving a unique solution. It can be interpreted as the implied required rate of return for that index and corresponds to the average investors’ expectations for the index long-term rate of return. This is a very a similar logic to that of the Yield to Maturity for a bond. This Yield to Maturity represents the implied discount rate that allows the present value of all cash flows provided by the bond to match its current price.

In order to compute an expected implied Equity Risk Premium from that implied rate of return we must also chose an appropriate risk free rate. Indeed, Equity Premia correspond to the extra return obtained above a chosen risk free rate. We step away from the common practice, found many other empirical studies, of using monthly returns of treasury bills. As we take a long-term perspective on Equity Premia, we instead match our Premia with risk free rates linked to long-term maturity debt instruments. In our model we follow the duration matching solution^{1}; concretely we specify a single risk free rate by choosing default-free securities (long term treasuries in our case) whose durations match most closely the durations of the indices analyzed. For those equity indices, the cash flows are modeled until infinity. To approximate the duration we can

therefore use the duration formula for growing perpetuity: 1 + k e k e − g

Using the average expected returns given by our model we obtain durations equal or above 20 years. In our search for best fit, we however do not pursue “perfect matching” through off-the-run securities in order to avoid liquidity issues and linked pricing distortions. In our analysis the best duration fit comes from the 30 year treasury bonds whose durations hover around 20 years, hence our choice of using 30 year Treasury bonds rate, when available for the markets analyzed, as a proxy for the risk free rate required to measure our Equity Risk Premium.

In order to provide some perspective of such model over a period including major swings in markets, including one of the most acute financial crisis since Second World War, we run an analysis over five major indices from 2003 until 2013. We use monthly data from January 2003 to December 2013 on five stock markets indices: S & P500, Euro Stoxx 50, FTSE 100, Hang Seng Index and the TOPIX 100. All data have been extracted from Bloomberg.

In all indices, the period surrounding the financial crisis, from September 2009 until March 2009, shows significant increases in ERPs as prices collapse with most spiking around March 2009. It is most visible on the Hang Seng and TOPIX 100 where declines in stock market prices have been particularly sharp in the aftermath of the Lehman Brothers bankruptcy in October 2008. However, the ERP for the Japanese index continues to increase very strongly following March 2009 as prices of the TOPIX index continue to tumble until 2010.

Looking at the Euro Stoxx 50 and the FTSE 100, we see swings in the ERP during these months of crisis, reflecting the combination of the downward adjustments of analysts’ future earnings expectations and the great volatility of prices.

^{1}See Damodaran, A. [

The US S&P 500 estimates show a peculiar trend. The pattern of the US ERP shows a rather steady increase of the ERP in the five years leading to the crisis. What could be taken for a smooth transition in fact highlights a double trend: on the one hand a gradual increase in future expected earnings until 2008, on the other hand from the summer of 2007 until March 2009 a decline in stock prices. Once the prices start to recover from april 2009 on, the ERP initiates its descent towards the pre-crisis levels by the end of our sample.

We conclude with a discussion on the meaning of the resulting Equity Risk Premia and their potential ability to predict realized returns.

We find that Cash flow based models are conceptually sounder than historical or survey based methods. Based on the sample analyzed, our results confirm that our total cash flow model is able to provide a robust description of an index’ Equity Risk Premium. Major financial events are reflected in a coherent manner and it will be interesting to analyze this model’s predictive ability on long-term

Dates | S&P 500 | EURO STOXX 50 | HANG SENG | FTSE 100 | TOPIX 100 |
---|---|---|---|---|---|

7/31/2007 | 9.42% | 9.60% | 6.21% | 6.69% | 4.82% |

8/31/2007 | 8.64% | 10.36% | 6.34% | 7.33% | 5.14% |

9/28/2007 | 7.93% | 10.35% | 6.19% | 7.40% | 5.52% |

10/31/2007 | 8.08% | 9.80% | 5.47% | 7.27% | 5.10% |

11/30/2007 | 8.59% | 10.39% | 5.73% | 7.54% | 5.44% |

12/31/2007 | 8.02% | 10.41% | 5.82% | 7.53% | 5.66% |

1/31/2008 | 8.41% | 10.74% | 6.99% | 7.31% | 5.76% |

2/29/2008 | 9.13% | 12.11% | 7.37% | 7.89% | 5.81% |

3/31/2008 | 9.06% | 11.87% | 4.00% | 6.52% | 6.21% |

4/30/2008 | 8.98% | 11.25% | 5.92% | 7.30% | 6.00% |

5/30/2008 | 8.98% | 10.86% | 5.95% | 6.97% | 5.02% |

6/30/2008 | 10.84% | 11.35% | 6.55% | 7.14% | 5.42% |

7/31/2008 | 10.44% | 12.70% | 7.24% | 7.85% | 5.99% |

8/29/2008 | 10.93% | 12.15% | 7.22% | 7.90% | 6.09% |

9/30/2008 | 9.91% | 12.86% | 7.80% | 8.22% | 6.36% |

10/31/2008 | 13.62% | 12.77% | 8.74% | 8.64% | 6.94% |

11/28/2008 | 13.17% | 13.70% | 9.55% | 8.33% | 6.46% |

12/31/2008 | 12.87% | 15.69% | 8.77% | 8.91% | 7.81% |

1/30/2009 | 11.52% | 9.35% | 6.73% | 6.02% | 7.22% |

2/27/2009 | 11.03% | 13.18% | 12.67% | 8.09% | 9.16% |

3/31/2009 | 10.55% | 15.33% | 10.81% | 10.37% | 13.66% |

4/30/2009 | 10.78% | 12.33% | 6.85% | 7.89% | 10.62% |

5/29/2009 | 10.83% | 11.64% | 6.54% | 7.65% | 10.18% |

6/30/2009 | 10.90% | 11.86% | 6.20% | 7.84% | 10.69% |

7/31/2009 | 10.65% | 12.65% | 6.50% | 8.26% | 11.20% |

data set. One should remember that the obtained Premia have a rather long-term perspective. In a similar manner that one would not assume that the Yield to Maturity of a long-term bond would offer accurate predictions of short-term price dynamics, one should not expect to use our model for short-term estimations of Stock markets behavior. While the chosen data set could not offer sufficient historical depth to provide some answers about longer-term dynamics, our position is that it should be unlikely that our model outperforms other measures proposed in the literature. Various models in the literature have attempted to produce Risk Premia that offer some predictability in and out of sample, yet the empirical performance of such predictions seems to remain statistically very weak, see Welch & Goyal [

Yet, looking at our model, one must remember that our primary goal was for it to reflect properly the underlying structure of an index cash flow and their evolution. One of the biggest advantages of cash flow models is to provide explicit assumptions about cash flows and growth parameters, and to link these assumptions with a particular pricing. Despite this orientation, several components of the model rely on the quality of the pricing mechanism of stock and bond markets. The recent interventions from Central Banks over the past few years are potentially affecting the pricing mechanism in substantial ways, creating distortions in its ability to link properly certain factors, such as long-term rates, to long-term estimates of economic growth. The implied ERPs do not only reflect what the model predicts but what the market itself predicts through the pricing mechanism. It is therefore very dependent on the efficiency of the market pricing mechanism. In order for such pricing and the resulting implied discount rate to predict future returns, one still needs to assume that the market can assess the future. This assumption remains unsupported by evidence, so far.

The author declares no conflicts of interest regarding the publication of this paper.

Moscato, G. (2019) Forward Looking Equity Risk Premium: A Normative Long-Term View. Theoretical Economics Letters, 9, 3021-3033. https://doi.org/10.4236/tel.2019.98186