Historical Methods of estimating ERP

Freeman and Davidson (1999) estimate ex-ante ERP by employing standard C-CAPM model to show that ex-ante ERP is not an unbiased estimate of ex post ERP. They estimate following model using the UK data for the period of 1974-1987:

𝐸(𝑅_𝑖) − 𝑅_𝑓 =𝛼. 𝐶𝑜𝑣 (𝑅𝑖, 𝑅𝑐) 1 − 𝛼. 𝐸(𝑅_𝑐)

where, E(Ri) –Rf = ERP (Difference between the return on risky asset i and risk-free

asset 𝑅𝑓) 𝛼 = risk aversion coefficient and Rc is the growth rate of aggregate

consumption and show that ERP estimated using standard economic model cannot be an unbiased estimate of ex-post ERP, a result similar to the US economy as studied by Mehra and Prescott (1985). On the other hand, O’Hanlon and Steele (2000) use accounting method to estimate ex-ante ERP in the UK by using accounting data of 172 UK companies between the period 1968-1995. The estimated ERP was in the range of

4%-6% using 3-month UK Gilts as risk free rate. Accounting methods was also implemented by Fitzgerald, Gray, Hall and Jeyaraj (2013) to estimate ERP in the US for the period 1999-2008. They employ firm specific data of 5144 firms in the US to show that average ERP was 5.3%.

Fama and French (2002) use the fundamental approach to estimate ex-ante ERP for the period 1872-2000 in the US by estimating average stock return using dividend growth model and earnings growth model.

𝐴(𝑅𝑑,𝑡) = 𝐴 (

𝐷_𝑡

𝑃𝑡−1) + 𝐴(𝐺𝑃𝑡)

Where A (Rd) is the average return on stocks using dividend growth model, whereas the

first term on the right hand side is the average dividend yield and the second term is the average capital gains. They argue that if dividend-price ratio is stationary over a long period, then the average capital gain approaches to average dividend growth rate. So they estimate the average stock return using following relation

𝐴(𝑅𝑑,𝑡) = 𝐴 (

𝐷_𝑡

𝑃𝑡−1) + 𝐴(𝐺𝐷𝑡) − − − −𝑑𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑚𝑜𝑑𝑒𝑙

where GDt is the growth rate of dividends. The earnings model they employed was:

𝐴(𝑅_𝑒,𝑡) = 𝐴 ( 𝐷𝑡

𝑃𝑡−1) + 𝐴(𝐺𝑒) − − − −𝑒𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝑔𝑟𝑜𝑤𝑡ℎ 𝑚𝑜𝑑𝑒𝑙

where, Re, t is the average stocks returns using the earnings model and Ge is the earnings

Table 2.2 Results of Fama and French, (2002)

Period ERP Estimates (%)

Dividend Model Earnings Model Actual

1872-2000 3.54 NA 5.57

1872-1950 4.17 NA 4.4

1950-2000 2.55 4.32 7.43

They also show that by using 1-month T-bills rate as a risk-free rate instead of using 6- month commercial paper rate (which they use for the period 1872-1926), the ERP increases by 1%. A similar approach was taken by Claus and Thomas (2001), however they use the abnormal earnings approach to estimate ERP in the US, UK, France, Germany, Canada and Japan. They show that the average ERP for the six developed economies is no more than 3%. However, when they use the dividend growth model 𝑘∗ ₌ 𝐷1

𝑃0 + 𝑔 they find that the average ERP for the US, Canada, the UK, Germany,

France and Japan is 7.34%, 5.89%, 7.91%, 6.58%, 7.90% and 5.83% respectively for the period 1985-1998.

Campbell (2008) also uses the fundamental valuation approach to estimate the ERP for the US, Canada and MSCI World index. He uses a slightly modified version of earnings growth model used by Fama and French (2002). He estimates implied ERP assuming constant Return on Equity (RoE) of around 50%. This ERP is 3.3% for the MSCI World Index, 3.2% for the US and 3.1% for Canada. He uses the return on inflation-indexed bond as risk-free rate for US and Canada. When he uses the 3-year moving average for dividend pay-out ratio and 3-year moving average for RoE, then the

estimates of ERP, using 0.75 weight on the long term estimate and 0.25 weight on the short term estimate, was 3.9% for the World Index, 4.1% for the US and 3.6% for Canada by the end of March 2007.

Advanced modelling techniques such Markov switching models, time series models and Bayesian techniques have also been employed to estimate ERP. Mayfield (2004) uses two state (low volatility and high volatility) Markov process with structural shifts in the volatility to estimate ERP. He estimates the following model:

𝑅𝑖,𝑡− 𝑅𝑓,𝑡 = 𝛾𝜎𝑡2− 𝜋𝑡ln(1 + 𝐽𝑡)(1 + 𝐾𝑡∗)−𝛾

where 𝑅_𝑖,𝑡− 𝑅_𝑓,𝑡 is the ERP, γ is coefficient of relative risk aversion, σ2 is the variance of returns which takes two sets of values in low and high volatility states, πt is the

instantaneous probability of change in the state, Jt is the change in wealth associated

with change in state and K* is change in optimal consumption level due to change in the wealth. The average ERP estimate in the low state was 12.4% and -17.9% in high volatility state. He also shows that ERP depends on volatility of returns and that about half of the estimated ERP is associated with future changes in volatility. On the other hand Pastor and Stambaugh (2001) use Bayesian technique to estimate ERP in stable and transition regimes. They show that ERP fluctuates between 3.9% and 6% in the US for the period 1834-1999. The inclusion of structural breaks improves the precision of the estimates and due to this ERP changes from 6.5% to 5.9% in the 1990s. They also show that across the sample, with the inclusion of structural breaks, ERP is related to volatility of returns and ERP has changed over time and is decreasing since 1930s with few jumps in 1970s.

Time series modelling technique with simulated method of moments requiring numerical solution, have been implemented by Donaldson , Kamstra and Kramer (2010) to estimate ERP in the US. The moments are simulated by AR (1), MA (1) and ARCH (1,1) technique. They show that, by simulating the dividend growth rates and interest rates, the estimated ERP for the period 1952-2004 broadly matches the US data and is around 3.5%. ±50bps.

An altogether different approach is adopted by Appelbaum and Basu (2010) to estimate ERP and consumption process. They estimate an empirically tractable ERP and consumption functions, independent of each other, and which were dependent only on the moments of the state variables. The consumption function involving the moments of the state variables, which they estimate is

𝐶𝑡≡ 𝑓(𝑊𝑡, 𝑅𝑡, 𝐼𝑡, 𝜎𝑅, 𝜎𝐼) + 𝜀𝑐,𝑡

and the actual observed ERP function is

𝐸𝑅𝑃𝑡 ≡ 𝑓(𝑊𝑡, 𝑅𝑡, 𝐼𝑡, 𝜎𝑅, 𝜎𝐼) + 𝜀𝑒,𝑡

where, 𝑊_𝑡is the household’s wealth, 𝑅_𝑡 is the return on equity investment, 𝐼_𝑡 is the household’s income and 𝜎𝑅, 𝜎𝐼 are respectively the standard deviations of 𝑅𝑡and 𝐼𝑡. By

estimating the parameters by non-parametric method in the above ERP function for the period of 1921-2001 they estimate the actual observed ERP in the US and show that the ERP is varies with time.

The above discussion highlights the inconsistent nature of ERP estimation techniques. The literature sometime suggests to use the returns on long term government bonds and at other, short term T-bills returns as a proxy of risk-free rate in estimation of ERP.

However, recent events like the European Sovereign Debt Crisis, the downgrade of US government debt in August 2011, the downgrade of UK and French sovereign bonds have shown that government bonds and bills cannot be considered entirely as risk-free. The issue is not just whether to consider government debt as risk-free or not, the issue is that there is no clear consensus in the literature as to what should be the maturity of the government debt to be qualified as a risk-free. In addition to this Mehra (2011) argued that it is not incontrovertible to argue that 3-months Government bills cannot proxy risk free rate based on the fact that households have little or no 3-months T-bills in their portfolio of savings which they can use to smooth the intertemporal consumption. “Hence, T-bills and short-term debt are not reasonable empirical counterparts to the risk-free asset” (Mehra 2011, p.150).

The foregoing literature shows that the literature produces different estimates of ERP by using the same variables such as dividend yields, consumption growth rates, etc. when using different modelling techniques. Interestingly the estimates of ERP from the survey method are not close to the one estimated by using either the fundamentals or time series models. And not only that, there is general consensus in the literature that ex-post ERP cannot be an unbiased estimate of ex-ante ERP and yet the profession continues to use historic ERP as an estimate of unconditional ERP. In the literature the most common way of estimating ex-post ERP is by arithmetic averaging of returns.