FINANCE , STATISTICS, FORECASTING. ACCURACY TESTING AND ERROR MATRICES

International Journal of Forecasting 33 (2017) 936–957

Contents lists available at ScienceDirect

International Journal of Forecasting

journal homepage: www.elsevier.com/locate/ijforecast

Beta forecasting at long horizons Tolga Cenesizoglu a,b, Fabio de Oliveira Ferrazoli Ribeiro c, Jonathan J. Reeves c,* a Alliance Manchester Business School, United Kingdom b HEC Montreal, Canada c UNSW Business School, University of New South Wales, Australia

a r t i c l e i n f o

Keywords: Evaluating forecasts Long term forecasting NPV analysis Realized beta Systematic risk

a b s t r a c t

Systematic (CAPM beta) risk forecasting for long horizons, such as one year, plays an important role in financial management. This paper evaluates a variety of beta forecasting procedures for long forecast horizons. The widely utilized Fama-MacBeth constant beta approach based on five years of monthly returns is found to be unreliable in terms of the mean absolute (and squared) forecast error and statistical bias. The most accurate forecasts are found to be those generated from an autoregressive model of the realized beta. In addition to analyzing the statistical properties of these forecasts, this paper demonstrates the economic significance of the different approaches through an evaluation of investment projects. © 2017 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction

Accurate forecasts of the Capital Asset Pricing Model1 (CAPM) beta at long horizons, such as one year, play an important role in financial management, including cost of capital estimations and performance measurement. Beta forecasts are usually generated by estimating the slope coefficient from a linear regression of individual stock re- turns on a constant and market index returns, typically using five years of monthly data as per Fama and MacBeth (1973). This method of estimating beta forecasts is the baseline for many empirical studies using the CAPM, as well as for numerous professional advisers on beta such as Bloomberg, Reuters, Standard & Poor’s and Value Line.

Motivated by advances in the financial econometrics of volatility measurement, namely realized volatility mea- surement (see Andersen, Bollerslev, Diebold, and Ebens, 2001a; Andersen, Bollerslev, Diebold, and Labys, 2001b, 2003), CAPM realized betas were developed in the work of

* Correspondence to: Banking and Finance, UNSW Business School, University of New South Wales, Sydney, NSW 2052, Australia.

E-mail address: [email protected] (J.J. Reeves). 1 See Sharpe (1964), Lintner (1965) and Mossin (1966).

Barndorff-Nielsen and Shephard (2004), Andersen, Boller- slev, Diebold, and Wu (2005) and Andersen, Bollerslev, Diebold, and Wu (2006). These betas are computed over a given period from a sufficiently high number of intra- period returns, and are consistent econometrically over a fixed interval. In recent studies, CAPM realized betas have served as a benchmark for evaluating the accuracy of beta forecasting approaches; see for example Hooper, Ng, and Reeves (2008), Chang, Christoffersen, Jacobs, and Vainberg (2012), Reeves and Wu (2013), Papageorgiou, Reeves, and Xie (2016) and Cenesizoglu, Liu, Reeves, and Wu (2016). This paper conducts a forecast evaluation study with realized betas computed over periods of six months or one year, and evaluates the standard (Fama & Mac- Beth, 1973) forecasting approach using proposed forecast- ing procedures based on realized beta estimation. Chang et al. (2012) also study the same long forecast horizons for beta as this paper. However, their forecasting approach is restricted to settings in which accurate stock option data are available, whereas our study’s main requirement is the availability of accurate daily stock return data, which gives it a far greater general applicability.

The primary aim of this paper is to evaluate (both statis- tically and economically) the performances of a variety of

http://dx.doi.org/10.1016/j.ijforecast.2017.06.004 0169-2070/© 2017 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 937

models for forecasting long horizon betas. We accomplish this by implementing two major classes of realized beta models (constant and autoregressive) and comparing their forecasting accuracies with that of the industry standard Fama-MacBeth constant five-year beta. A key feature of these forecasting approaches is their relative simplicity. This follows the fundamental principle of forecasting prac- tice of favoring models that are not unnecessarily com- plicated. Even though less simple forecasting models may perform better in-sample, their performances are often less robust out-of-sample, see Green and Armstrong (2015). For this reason, this paper does not study the forecasting performances of more complicated models for beta, such as multivariate GARCH models (see Braun, Nelson, and Sunier, 1995 and Brooks and Henry, 2002) or the Bayesian model of Jostova and Philipov (2005).

Our study uses daily data from 1st January 1952 to 31st December 2011 for 15 stocks from the DJIA index, and with the DJIA index as the market portfolio. By measuring the forecast performance using the mean absolute forecast er- ror (MAE) and the mean squared forecast error (MSE), and testing for bias in the forecasts, we find that the industry standard Fama-MacBeth constant five-year beta not only has much larger forecast errors than our other time series model, but is also downward biased, under-estimating the future value of beta.

We also test the forecast abilities of realized betas con- structed from daily data of varying lengths, from six to 60 months, as well as implementing five specifications of an autoregressive realized beta model, with lags of one to five. In general, we find that both constant realized betas and autoregressive models outperform the Fama-MacBeth con- stant five-year beta in terms of both MAE and MSE. The best constant method, realized beta with 18 months of daily data, reduces the mean absolute error by 26.5% (25.7%) for six-month (one-year) forecasts, while the best autore- gressive method, the AR(1), reduces it by 30.3% (29.8%) for six-month (one-year) forecasts relative to the standard Fama-MacBeth constant five-year beta. In addition, we also perform Mincer–Zarnowitz regressions (see Mincer and Zarnowitz, 1969) to test whether the predictions from the various forecasting models are biased. We find that the constant realized beta models and autoregressive realized beta models not only reduce the forecasting error, but also are less biased than Fama-MacBeth constant five-year betas, which are downward biased. Overall, we find an autoregressive model of the realized beta with one lag to be statistically unbiased, leading this model to be our preferred approach.

Our study focuses on Dow stocks due to their very high liquidity, which permits the use of daily historical return data going back to 1952. However, our approaches are not restricted to Dow stocks, but can be applied to any stocks that are sufficiently liquid that allow accurate daily return measurement. This set of stocks has been increasing con- stantly with the overall improvements in market liquidity, and typically includes sets of stocks such as those currently trading in the S&P 500 index, where daily returns over a number of years can be relied upon for most stocks.

In addition to the statistical results, this paper also demonstrates strong economic significance in an applica- tion to the cost of capital measurement and the evaluation

of investment projects. In these applications, the Fama- MacBeth constant five-year betas again result in both biases and a greater variability in the cost of capital mea- surement, which distorts the net present value calcula- tions. In particular, the convexity of the present value in beta means that the greater variability in the Fama- MacBeth constant five-year beta leads to the expected value of the risky asset being overstated, whereas the realized beta and autoregressive models display more favourable performances.

The rest of the paper is organized as follows. Section 2 reviews the realized beta estimator and discusses different approaches to forecasting beta at long horizons. Section 3 describes the data sources and our sample of US stocks. Section 4 investigates the empirical forecast performances of the various approaches for both the six-month and one- year forecast horizons. Section 5 demonstrates the eco- nomic significance of the different forecasting approaches by means of an evaluation of investment projects. The final section concludes the paper.

2. Realized betas and forecasting approaches

This section begins by discussing the estimation and theoretical justification of realized beta estimators, then discusses both the popular constant five-year beta fore- casting approach of Fama and MacBeth (1973) and new approaches to forecasting long horizon betas, utilizing re- alized beta estimates.

2.1. Realized beta measurement

We begin with a brief review of the realized beta es- timator and its theoretical justification, which was devel- oped and discussed by Barndorff-Nielsen and Shephard (2004) and Andersen et al. (2006). Suppose that prices follow a multivariate continuous-time stochastic volatility diffusion, with the N × 1 logarithmic vector price process pt:

dpt = µtdt + θtdWt , (1)

where µt is the vector of instantaneous drift, Ωt = θt θ ′t is the diffusion (variance–covariance) matrix, and Wt rep- resents a vector of standard Brownian motion innovations. The variance–covariance matrix and the drift vectors are not correlated with the Brownian motion process and are strictly stationary. To facilitate the interpretation, we can think of N as the number of stocks plus the market index, with the Nth element containing information on the index and each ith element containing information on stocks. By defining a time interval (for example, a day or a month) and denoting it h, we define the continuously compounded return in this period as rt+h,h ≡ pt+h − pt .

The realized beta of a period can be defined as the realized covariance between a security and the market index divided by the realized variance of the market. If ∆ is the sampling frequency, or the amount by which we divide the period h, the realized covariance during a time interval h, at time t + h, of a security i and the market index M, is defined as:

ν̂iM,t,t+h = ∑

j=1,...,[h/∆]

ri,t+j.∆,∆ · rN,t+j.∆,∆, (2)

938 T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957

while the realized market variance is defined as:

ν̂ 2 M,t,t+h =

∑ j=1,...,[h/∆]

r2N,t+j.∆,∆. (3)

The realized beta as a ratio of the realized covariance to the realized market variance is then:

β̂i,t,t+h = ν̂iM,t,t+h

ν̂2M,t,t+h =

∑ j=1,...,[h/∆] ri,t+j.∆,∆ · rN,t+j.∆,∆∑

j=1,...,[h/∆] r 2 N,t+j.∆,∆

p →

∫ h 0 ΩiN,t+τ dτ∫ h 0 ΩNN,t+τ dτ

= βi,t,t+h. (4)

This equation shows that, with finely sampled returns, the realized beta converges in probability to the true latent beta. This CAPM realized beta is equivalent to a linear regression of stock intra-period returns on market intra- period returns with the suppression of the constant term.

This paper uses Eq. (4) for the computation of realized betas which serve as benchmarks in the evaluation of fore- casts, where h is defined as either six months or one year, and ∆ is defined as daily. This follows the construction of the realized beta in the forecast evaluation study of Chang et al. (2012) at the six-month and one-year horizons. A number of other beta forecast evaluation studies at shorter forecast horizons have also utilized realized betas; see for example Hooper et al. (2008), Reeves and Wu (2013), Papageorgiou et al. (2016) and Cenesizoglu et al. (2016).

There are various important considerations that need to be taken into account when measuring the realized beta. Most importantly, the ∆ sampling frequency should not be too high relative to the liquidity of the asset, in order to avoid poor return measurement due to complications such as microstructure noise and non-trading effects, which lead to bias (typically downward) in the beta measure- ment.2 In addition, there is often some consideration given to capturing the time-varying nature of beta when choos- ing the period h. The period h must also be long enough (relative to the ∆ sampling frequency) to ensure that a sufficient number of return observations will be available to control the variability of the beta estimates.

2.2. Forecasting approaches

The standard approach to long horizon beta forecast- ing follows that of Fama and MacBeth (1973), where the beta forecast is the regression slope coefficient from five years of monthly stock returns regressed on a constant and the market returns. The popularity of this approach can be attributed to its simplicity and the fact that Fama and MacBeth’s paper was a seminal paper in the financial economics literature. However, there is a growing concern among the finance profession regarding the accuracy of this approach; see for example Acker and Duck (2007).

In addition to this standard approach, this paper also studies forecasting approaches that are based on realized

2 Dimson (1979) betas are sometimes utilized in settings in which the return measurement is too high relative to the liquidity of the asset. Some of the bias in these estimations with leads and lags of betas can be corrected; however, this typically comes at the cost of a greater variability in the beta estimates.

beta estimates. These realized beta approaches are based on estimations over various prior periods and include au- toregressive modeling.3 The first set of realized beta mod- els that we study are constant realized beta models. In this group of models, the forecast of beta for the next period is generated using the prior realized beta, RBetai,t,n, where i indexes each firm, t indexes the time at which the forecast is made and n represents the number of months of prior daily data that are used to compute the realized beta. We calculate realized betas with n equal to 6, 12, 18, 24, 48 and 60 months, denoted by 6M4 , 12M, 18M, 24M, 48M and 60M.

We also apply several specifications of the AR(p) model, computed on half-yearly and yearly realized betas formed from daily returns, with the general form:

βi,t = φi,0 +

p∑ j=1

φi,jβi,t−j + ϵi,t ,

ϵi,t ∼ iid(0, σ 2 i ), t = 1, 2, . . . , n,

(5)

where p denotes the number of lags of the realized betas used in the forecast. In this paper we use p = 1, . . . , 5, with three different in-sample estimation sizes. We estimate AR(p) models with 40 and 60 observations for the six- month forecast, and 30 observations for the yearly forecast. These autoregressive models are then utilized for the gen- eration of one-step-ahead forecasts.

3. Data

We calculate realized betas from daily returns for 15 US companies, with the Dow Jones Industrials Average (DJIA) index as the market portfolio. Equity data are sourced from CRSP through Wharton Research Data Services (WRDS) and the DJIA index through Datastream. We use the return series from CRSP, which adjusts for both capital structure events and cash dividends at the ex-dividend date. In order to be included in the sample, a company has to be listed in the DJIA index as of 31st December 2011, and to have complete daily return information going back to the 1950s. With these criteria we were able to collect daily data from 15 US companies from 1st January 1952 to 31st December 2011, which gives us a time series of 60 years (or 120 half- years) of realized betas. These sample selection criteria permit a long time span, though with the limitation that all of the stocks are largely capitalized companies. In future research it would be useful to study sets of stocks that have other characteristics. For example, if sufficient data existed, it would be interesting to also study stocks with smaller market capitalizations that have accurate daily returns.

4. Empirical forecast performance

This section begins by discussing the methodology of our evaluation of competing beta forecasting approaches, then presents and discusses the empirical results.

3 Andersen et al. (2005, 2006) provided the first in-depth studies of autoregressive models for CAPM realized betas, which were done at the monthly and quarterly frequencies.

4 6M is only used to forecast six-month-ahead betas.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 939

Fig. 1. Time series of six-month realized betas. The figure shows the time series of realized betas, calculated every half-year from daily returns data. For each stock, the betas are calculated at the end of the last trading day of the month of June and the last trading day of the year. Each individual time series comprises 120 data points from 1952:1 to 2011:2.

4.1. Methodology

Our study uses Eq. (4) to calculate the realized betas once every half-year (or year for the one-year-ahead fore- casting sample), following Barndorff-Nielsen and Shep- hard (2004) and Andersen et al. (2006), in order to have a sample of realized betas with non-overlapping windows of daily returns. The realized beta is calculated on the last trading days of June and December for the six-month real- ized beta series and the last trading day of December for the

yearly series. The out-of-sample forecast evaluation period for the half-year forecast is the 55 half-years starting from the second half of 1984. For the yearly forecast, the forecast evaluation period starts in 1987 and runs for 25 years.

The one-period-ahead forecast of beta, βn+1, generated by each model for each company, is compared with the benchmark beta, which we define as the six-month (or one-year) realized beta, calculated using daily returns at that given date. As betas are used by both executives of companies and external analysts, it is hard to define a

940 T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957

Fig. 2. Time series of one-year realized betas. The figure shows the time series of realized betas, calculated every year from daily returns data. The betas are calculated at the end of the last trading day of the year for each stock. Each individual time series comprises 60 data points, running from 1952 to 2011.

preferred direction of forecast errors (over- or underesti- mation of the true value).5 Hence, our baseline measures of the models’ forecast abilities — the mean absolute error

5 Errors in beta forecasting are not particularly worse if positive or negative because they lead to incorrect inference on the cost of capital calculation in either case, which can be bad or good depending on the use. If, for example, beta is used as an input when defining a hurdle rate for investment projects, on the one hand, an overestimation of beta could lead to a higher capital cost and thus a lower (or negative) present value of projects, which might be discarded erroneously. On the other hand, and following the same logic, an underestimation could lead to the approval of projects with a true negative present value.

(MAE) and mean squared error (MSE) — are based on sym- metric, quadratic and absolute loss functions as follows:

MAEi = 1 m

m∑ j=1

|β̂i,j − βi,j| (6)

MSEi = 1 m

m∑ j=1

(β̂i,j − βi,j) 2 , (7)

where m is the total number of forecasting periods, i in- dexes each company, β̂j is the forecast for the jth period

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 941

Fig. 3. ACFs and PACFs of selected companies. The figure shows the autocorrelation and partial autocorrelation functions for six-month and one-year realised betas for Coca Cola and IBM. The graphs show the functions from lags 0 to 24 for six months and lags 0 to 15 for one year. The shaded bands are 95% two-tailed confidence intervals.

beta and βj is the six-month (yearly) realized beta calcu- lated from daily returns in the jth period.

In addition, we evaluate the presence of systematic bias in the forecast of each model by following the seminal work of Mincer and Zarnowitz (1969) and estimating a regression of the next period’s realized beta on the current forecast of beta:

βt+h = α + γ βt+h|t + et+h|h, (8)

where t represents the current time period and the forecast is made for h steps ahead (in this paper, h is always 1). Once the parameters of this regression have been estimated, we test the joint hypothesis

H0 : α = 0, γ = 1

of no systematic bias in the forecast. We also test the null hypotheses H0 : α = 0 and H0 : γ = 1 individually, to investigate the origin of the bias. In the individual test, if

942 T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957

α = 0, we can conclude that there is no systematic bias, and if γ = 1, the forecast can be considered efficient.

4.2. Results

This section presents and discusses the main results of both the six-month-ahead and one-year-ahead forecasts of beta. First, we look at the time series of realized betas in Figs. 1 and 2, which display features of stationarity. In particular, betas over the stocks have not risen since the 1950s, indicating that the choice of a daily return measure- ment was still appropriate for these stocks in the early part of the sample, as it did not generate a downward bias in the beta measurements. It is also important to consider the dynamic structure of realized betas when modelling their behaviours, and consequently, estimating next pe- riod’s value. With that in mind, we calculated the autocor- relation (ACF) and partial autocorrelation (PACF) functions of realized betas for our entire sample. However, due to space limitations these are not displayed here, except for selected representative stocks in Fig. 3. For most stocks, these functions suggest that the structures of realized betas are modelled better by low order autoregressive processes, as the PACFs usually cut-off after lag 1 and the ACFs decay slowly.

Furthermore, we have time series of φ0,i and φ1,i esti- mates from the estimation of Eq. (5) with p = 1, i.e., AR(1) models, based on rolling windows of observations in our forecast evaluations. There is a simple interpretation of these coefficients from the AR(1) models in terms of the mean reversion speed (κi = 1 − φ1,i) and long-run mean (β̄i = φ0,i/(1 − φ1,i)) of realized betas. Table 1 presents the mean reversion speed and long-run mean of realized betas implied by the AR(1) models for each stock averaged over the corresponding out-of-sample periods. The average κ is greater than zero for all stocks, demonstrating mean reversion. Averaging over all stocks, the long-run mean of realized betas is 0.97.

We now move to the results regarding the forecasting abilities of the models that are ranked as contenders in this paper. Table 2 presents the values of the mean absolute error for the six-month-ahead forecast. We present first the numbers for each model for each company, then the average of each model. The first column is the standard Fama-MacBeth constant five-year (FM) beta that is used as a forecast for the next period. Columns 2 to 7 then present different specifications of the constant realized beta model, while columns 8 to 12 (columns 13 to 17) present five specifications of the AR(p) model with in- sample parameters estimated over 40 (60) periods (half- years). The best-performing model within each category (constant and autoregressive) is highlighted in each row.

For the half-year forecast, we obtain consistent results across all models regardless of whether we measure the forecast precision by MAE or MSE, though we only display the tables for MAE, due to space limitations. The FM beta is the worst performer of all models in both the overall av- erage and the individual companies, where it is the model with the highest error in 13 (12) of the 15 companies in terms of MAE (MSE). The realized beta with 18 months of daily data is the constant model with the smallest error in

the overall averages of MAE and MSE, although it is not the model with the smallest errors in the largest number of companies. Using this model to forecast beta six months ahead yields a reduction of 43.55% in MSE (24.87% in root mean squared error6) and 26.54% in MAE compared to the FM beta. For both sample sizes of the AR(p) models, 40 and 60 in-sample estimation periods, the models with one lag have the smallest errors. The AR(1) model with 60 in-sample estimation periods is the best model out of all categories in terms of lowest MSEs, with an overall square error of 0.0537 that represents a reduction of 48% (or 27.89% in root mean squared error) compared to the FM beta, while AR(1) with 40 in-sample estimation periods has an overall MSE of 0.0540, giving a reduction of 47.71% (or 27.69% in root mean squared error) relative to the FM beta. In terms of MAE, the model with the smallest error of all is the AR(1) with 40 in-sample estimation periods, with an absolute error of 0.1763, or a reduction of 30.28% relative to the FM beta, while AR(1) with 60 in-sample estimation periods has an overall MAE of 0.1764, a reduction of 30.24% compared to the FM beta.

Next, we repeat the six-month-ahead forecast eval- uation by splitting the evaluation period into two half- samples, 1984:2 to 1997:2 and 1998:1 to 2011:2. The results are displayed in Tables 3 and 4 respectively, and are similar to the full-sample results, though there are some minor differences over these half-samples (as we would expect, given that the evaluation periods are smaller, at 27 and 28 half-years respectively). For example, the overall average MAE in Table 4 is slightly lower with the AR(3) than with the AR(1), although the AR(1) still typically has the lowest forecast error when considering individual compa- nies.

Table 5 presents the values of the mean absolute error for the one-year-ahead forecast. The first column is the standard FM beta that is used as a forecast for the next period. Columns 2 to 6 present different specifications of the constant realized beta model, while columns 7 to 11 present five specifications of the AR model with in-sample parameters estimated over 30 periods (years). The best- performing model within each category (constant and au- toregressive) is highlighted in each row.

Firstly, though, the FM beta performs worst in all situa- tions in terms of MSE, having the highest MSE for 10 of the 15 firms and the largest overall MSE. Amongst the constant models, the realized beta calculated with 24 months of daily data has the lowest average MSE and also the lowest MSE in 5 of the 15 companies. Compared to the FM beta, using the 24M realized beta provides a reduction of 41.44% in MSE (23.48% in root mean squared error). Moving to the autoregressive specifications, the model with one lag is the best amongst the AR(p) models for 10 of the 15 companies, and has the smallest overall MSE, with a reduction in MSE of 46.18% (26.64% in root mean squared error) relative to the FM betas used for forecasting purposes.

The MAE results are mostly consistent with those using the MSE. Again, the standard FM beta is the worst per- former overall. Amongst the AR(p) models, the model with

6 The root mean squared error is calculated by taking the square root of the average MSE for each model.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 943

Table 1 Speed of mean reversion (κ) and long-run beta implied by AR(1) models.

Company Six-month-ahead Six-month-ahead One-year-ahead AR(1): 40 periods AR(1): 60 periods AR(1): 30 periods

κ β̄ κ β̄ κ β̄

Coca Cola 0.3621 0.9041 0.3539 0.8709 0.4340 0.8857 Du Pont 0.9225 1.0945 0.8028 1.0947 0.7357 1.0890 Exxon 0.5729 0.7827 0.5965 0.8043 0.7255 0.8008 GE 0.5010 1.1189 0.5733 1.1202 0.7031 1.1219 IBM 0.7647 1.0051 0.7887 1.0034 0.7494 1.0061 Chevron 0.4140 0.8577 0.3831 0.8954 0.4377 0.8732 United Tech 0.6881 1.0679 0.6466 1.0970 0.5748 1.0857 P&G 0.5811 0.8339 0.5530 0.8020 0.6390 0.8174 Caterpillar 0.5663 1.0436 0.6660 0.9983 0.6171 1.0096 Boeing 0.4768 1.0952 0.4196 1.1601 0.5568 1.1485 Pfizer 0.6247 0.9284 0.6285 0.9380 0.5873 0.9368 J&J 0.6212 0.8683 0.6627 0.8585 0.7786 0.8692 3M 0.8229 0.9407 0.7435 0.9567 0.8618 0.9564 Merck 0.7310 0.8937 0.7225 0.8783 0.7477 0.8790 Alcoa 0.5291 1.1185 0.5803 1.1293 0.7291 1.1280

Average 0.6119 0.9702 0.6081 0.9738 0.6585 0.9738

We estimate AR(1) models (βi,t = φ0,i + φ1,iβi,t−1 + ϵi,t , where ϵi,t ∼ iid(0, σ 2i )) based on rolling windows of either 40 or 60 semiannual periods to obtain six-month-ahead forecasts over the out-of-sample period between 1984 and 2011, and a rolling window of 30 years to obtain one-year-ahead forecasts over the out-of-sample period between 1987 and 2011. The table presents the speeds of mean reversion (κi = 1 − φ1,i) and the long-run betas (β̄i = φ0,i/(1 − φ1,i)) implied by different AR(1) models averaged over their corresponding out-of-sample periods.

one lag again performs the best, with an average MAE of 0.1670, which represents a reduction in absolute forecast error of 29.82% relative to the FM beta. The results from the constant models differ slightly in terms of MAE, with the model with realized betas calculated over 18 months of daily data (18M beta) now being the model with the smallest overall error, a MAE of 0.1768, which is a reduction of 25.71% compared with the FM beta. However, this model is the best performer of all constant models in only 3 of the 15 companies, and its overall value is very similar to that of 24M beta.

Tables 6 and 7 compare the summary statistics of the best forecasting AR(1) models (in terms of MAE and MSE) with those of the standard Fama-MacBeth constant five- year beta model. For each company we report the number of observations or out-of-sample forecasting evaluation periods, along with the mean, standard deviation, and min- imum and maximum forecast errors. The forecast error is the difference between the estimated beta at time t and the realized beta at time t+1, where the time may be measured in either half-years or years.

For the six-month forecast (Table 6), we find that the dispersion of forecast errors is much larger when the con- stant FM beta is used to forecast the next period’s stock beta. Here, the standard deviation of forecast errors using the FM model is larger than either of the AR(1) specifica- tions in 14 of the 15 stocks. In addition, the FM betas that are calculated as the standard in the literature again have the largest range of errors, with values ranging from −0.93 to 1.20, while the two autoregressive specifications have values from −0.87 to 0.95 and −0.84 to 0.91 (with 40 and 60 in-sample estimation periods, respectively).

Similar results are found for the one-year beta forecasts, see Table 7. The FM beta model has a larger standard deviation than the AR(1) for 11 of the 15 US stocks in our sample. Moreover, it also has the widest range of errors, with its global minimum and maximum being −0.70 and

1.14, compared with −0.74 and 0.66 for the autoregressive model. These results support the initial findings that the use of an autoregressive specification for the realized beta improves the forecasting ability for estimating stocks betas one year in advance relative to the standard practice in industry.

Having analysed the error characteristics of several classes of models, leading to the preliminary conclusion that the AR(1) model is the best in terms of having the smallest errors (though the constant model with 18 months also performs well in terms of both the MAE and MSE), we now analyse additional statistical properties of constant and autoregressive models, relative to the stan- dard FM beta, by implementing the methodology intro- duced by Mincer and Zarnowitz (1969).

When implementing the Mincer–Zarnowitz (MZ) re- gression, we first analyze the potential presence of bias in the forecast. We achieve this by testing the joint null hypothesis that α equals zero and γ equals one simulta- neously. This is done by conducting an F-test where the unrestricted model is the regression of the actual values on the forecasts and the restricted model is built by applying the restrictions α = 0 and γ = 1 to the MZ equation.7 We also investigate the characteristics of each model’s forecasts further by investigating whether the bias, if any, comes from the intercept (α) or the slope (γ ).

We begin by analysing the forecast biases for the six- month scenario in Tables 8–10. We have three columns for each model. The first two columns present the regression coefficients α and γ with individual tests of significance, while the third shows the p-values of the joint hypothesis of systematic bias; in each case, bold values indicate rejec- tion of the null at the 5% significance level, and in the third

7 The errors in the restricted model are essentially the forecasting errors of the MZ equation when restricted: yt+h = 0 + 1yt+h|t + ut , with y being the stock betas in our study.

944 T.Cenesizoglu

etal./InternationalJournalofForecasting 33

(2017) 936–957

Table 2 MAEs of six-month-ahead forecasts of betas.

Company FM 6M 12M 18M 24M 48M 60M 40 periods 60 periods

AR(1) AR(2) AR(3) AR(4) AR(5) AR(1) AR(2) AR(3) AR(4) AR(5)

Coca Cola 0.3350 0.1812 0.1621 0.1598 0.1737 0.1994 0.2084 0.1797 0.1767 0.1779 0.1812 0.1808 0.1819 0.1778 0.1757 0.1762 0.1760 Du Pont 0.1483 0.1805 0.1484 0.1420 0.1383 0.1364 0.1388 0.1449 0.1336 0.1373 0.1372 0.1417 0.1459 0.1348 0.1360 0.1365 0.1421 Exxon 0.2682 0.2257 0.2205 0.1960 0.1948 0.2221 0.2285 0.1988 0.2009 0.2009 0.2052 0.2133 0.2001 0.2032 0.1994 0.2003 0.2081 GE 0.2222 0.1449 0.1465 0.1543 0.1679 0.1955 0.1958 0.1323 0.1353 0.1386 0.1378 0.1417 0.1304 0.1323 0.1345 0.1327 0.1339 IBM 0.2711 0.2071 0.2079 0.1760 0.1756 0.1740 0.1805 0.1810 0.1862 0.1719 0.1746 0.1767 0.1755 0.1820 0.1686 0.1669 0.1691 Chevron 0.2617 0.2000 0.2116 0.1969 0.2015 0.2295 0.2349 0.2031 0.2069 0.1997 0.2055 0.2195 0.1981 0.1986 0.1930 0.1956 0.1988 United Tech 0.2696 0.1570 0.1635 0.1468 0.1439 0.1511 0.1535 0.1405 0.1441 0.1313 0.1329 0.1352 0.1401 0.1412 0.1334 0.1406 0.1396 P&G 0.2345 0.1639 0.1795 0.1905 0.1884 0.1928 0.2001 0.1807 0.1802 0.1834 0.1829 0.1792 0.1749 0.1706 0.1707 0.1687 0.1665 Caterpillar 0.2495 0.1955 0.2030 0.2144 0.2250 0.2483 0.2450 0.1964 0.2004 0.2027 0.2049 0.2083 0.2076 0.2092 0.2107 0.2117 0.2115 Boeing 0.3280 0.2289 0.2546 0.2770 0.2764 0.2879 0.2643 0.2193 0.2215 0.2269 0.2383 0.2513 0.2206 0.2234 0.2324 0.2407 0.2496 Pfizer 0.3326 0.1925 0.1812 0.1904 0.2068 0.2214 0.2119 0.1891 0.1910 0.2013 0.2064 0.1987 0.1750 0.1768 0.1847 0.1834 0.1918 J&J 0.2211 0.1790 0.1922 0.1946 0.1865 0.1804 0.1779 0.1749 0.1777 0.1883 0.1867 0.1901 0.1887 0.1900 0.1911 0.1864 0.1895 3M 0.1461 0.1471 0.1582 0.1408 0.1369 0.1460 0.1382 0.1313 0.1368 0.1341 0.1336 0.1361 0.1281 0.1264 0.1273 0.1311 0.1326 Merck 0.2280 0.1837 0.1818 0.1857 0.1814 0.1977 0.1960 0.1598 0.1616 0.1654 0.1719 0.1766 0.1598 0.1629 0.1685 0.1684 0.1714 Alcoa 0.2770 0.2104 0.2140 0.2212 0.2201 0.2164 0.2173 0.2126 0.2149 0.2113 0.2160 0.2312 0.2193 0.2224 0.2229 0.2278 0.2306 Average 0.2529 0.1865 0.1883 0.1858 0.1878 0.1999 0.1994 0.1763 0.1779 0.1781 0.1810 0.1854 0.1764 0.1768 0.1766 0.1778 0.1807

The table shows the mean absolute errors of six-month-ahead beta forecasts for each company and the average for the entire sample for each model: the Fama-MacBeth constant five-year; constant models of the realized beta with varying estimation sizes (6, 12, 18, 24, 48 and 60 months of daily data); and autoregressive models with 1–5 lags and estimation sizes of 40 and 60 periods (half-years). The out-of-sample evaluation runs from 1984:2 to 2011:2, with a total of 55 forecasting periods. Values in bold indicate the model with the smallest MAE within each model class.

T.Cenesizoglu etal./InternationalJournalofForecasting

33 (2017)

936–957 945

Table 3 MAEs of six-month-ahead forecasts of betas between 1984 and 1997.

Company FM 6M 12M 18M 24M 48M 60M 40 periods 60 periods

AR(1) AR(2) AR(3) AR(4) AR(5) AR(1) AR(2) AR(3) AR(4) AR(5)

Coca Cola 0.4225 0.1979 0.1882 0.1859 0.2058 0.2350 0.2432 0.2040 0.2112 0.2190 0.2250 0.2266 0.2116 0.2128 0.2132 0.2146 0.2173 Du Pont 0.1628 0.2063 0.1644 0.1473 0.1430 0.1416 0.1451 0.1420 0.1361 0.1369 0.1380 0.1451 0.1346 0.1315 0.1325 0.1322 0.1405 Exxon 0.2035 0.1782 0.1664 0.1479 0.1482 0.1993 0.2171 0.1619 0.1640 0.1663 0.1691 0.1706 0.1696 0.1736 0.1665 0.1631 0.1637 GE 0.1793 0.1395 0.1483 0.1463 0.1522 0.1755 0.1714 0.1311 0.1339 0.1343 0.1368 0.1405 0.1300 0.1303 0.1302 0.1295 0.1302 IBM 0.1993 0.1945 0.1938 0.1720 0.1608 0.1643 0.1711 0.1730 0.1796 0.1814 0.1884 0.1921 0.1666 0.1692 0.1669 0.1661 0.1688 Chevron 0.2264 0.1670 0.1934 0.1849 0.1846 0.2142 0.2277 0.1601 0.1632 0.1631 0.1662 0.1717 0.1606 0.1625 0.1598 0.1643 0.1607 United Tech 0.3448 0.1450 0.1608 0.1470 0.1364 0.1512 0.1570 0.1511 0.1510 0.1260 0.1283 0.1286 0.1565 0.1545 0.1381 0.1480 0.1465 P&G 0.1801 0.1574 0.1766 0.1927 0.1864 0.1645 0.1743 0.1724 0.1790 0.1825 0.1775 0.1751 0.1760 0.1759 0.1768 0.1719 0.1677 Caterpillar 0.3250 0.1998 0.2189 0.2247 0.2461 0.2603 0.2643 0.2277 0.2332 0.2346 0.2370 0.2383 0.2415 0.2427 0.2438 0.2467 0.2454 Boeing 0.4405 0.2539 0.2750 0.3073 0.3249 0.3546 0.3250 0.2365 0.2404 0.2447 0.2628 0.2762 0.2378 0.2424 0.2591 0.2717 0.2897 Pfizer 0.3480 0.2043 0.2052 0.2160 0.2352 0.2567 0.2418 0.2253 0.2299 0.2355 0.2569 0.2017 0.2050 0.2092 0.2242 0.2234 0.1881 J&J 0.2288 0.2000 0.2227 0.2100 0.1962 0.1886 0.1820 0.1863 0.1876 0.2090 0.2129 0.2021 0.2072 0.2084 0.2086 0.2042 0.2041 3M 0.1272 0.1536 0.1598 0.1414 0.1251 0.1442 0.1386 0.1260 0.1257 0.1222 0.1164 0.1161 0.1234 0.1204 0.1223 0.1214 0.1222 Merck 0.2348 0.1863 0.1989 0.2039 0.1916 0.2271 0.2175 0.1793 0.1770 0.1774 0.1822 0.1837 0.1770 0.1806 0.1877 0.1837 0.1861 Alcoa 0.2104 0.1800 0.1841 0.2013 0.2139 0.2219 0.2192 0.1741 0.1779 0.1773 0.1788 0.2034 0.1860 0.1899 0.1926 0.1885 0.1875 Average 0.2556 0.1842 0.1904 0.1886 0.1900 0.2066 0.2064 0.1767 0.1793 0.1807 0.1851 0.1848 0.1789 0.1803 0.1815 0.1819 0.1812

The table shows the mean absolute errors of six-month-ahead beta forecasts for each company and the average for the subsample period between 1984 and 1997 for each model: the Fama-MacBeth constant five-year; constant models of the realized beta with varying estimation sizes (6, 12, 18, 24, 48 and 60 months of daily data); and autoregressive models with 1–5 lags and estimation sizes of 40 and 60 periods (half-years). The out-of-sample evaluation runs from 1984:2 to 1997:2, with a total of 27 forecasting periods. Values in bold indicate the model with the smallest MAE within each model class.

946 T.Cenesizoglu

etal./InternationalJournalofForecasting 33

(2017) 936–957

Table 4 MAEs of six-month-ahead forecasts of betas between 1998 and 2011.

Company FM 6M 12M 18M 24M 48M 60M 40 periods 60 periods

AR(1) AR(2) AR(3) AR(4) AR(5) AR(1) AR(2) AR(3) AR(4) AR(5)

Coca Cola 0.2506 0.1650 0.1370 0.1347 0.1426 0.1651 0.1747 0.1563 0.1434 0.1382 0.1390 0.1366 0.1533 0.1440 0.1396 0.1391 0.1361 Du Pont 0.1343 0.1557 0.1329 0.1369 0.1337 0.1314 0.1327 0.1478 0.1312 0.1377 0.1365 0.1383 0.1569 0.1379 0.1393 0.1405 0.1437 Exxon 0.3305 0.2715 0.2728 0.2424 0.2398 0.2440 0.2395 0.2343 0.2364 0.2343 0.2400 0.2545 0.2295 0.2317 0.2311 0.2360 0.2508 GE 0.2636 0.1500 0.1447 0.1620 0.1831 0.2147 0.2192 0.1334 0.1366 0.1426 0.1389 0.1429 0.1308 0.1343 0.1386 0.1357 0.1373 IBM 0.3403 0.2192 0.2215 0.1799 0.1898 0.1833 0.1895 0.1888 0.1926 0.1627 0.1612 0.1619 0.1841 0.1943 0.1703 0.1676 0.1695 Chevron 0.2958 0.2318 0.2292 0.2084 0.2178 0.2443 0.2419 0.2446 0.2490 0.2349 0.2434 0.2656 0.2342 0.2335 0.2251 0.2258 0.2355 United Tech 0.1972 0.1686 0.1660 0.1466 0.1512 0.1510 0.1502 0.1302 0.1375 0.1364 0.1374 0.1416 0.1242 0.1283 0.1288 0.1334 0.1330 P&G 0.2869 0.1702 0.1822 0.1883 0.1902 0.2201 0.2250 0.1887 0.1814 0.1843 0.1880 0.1832 0.1737 0.1655 0.1647 0.1656 0.1654 Caterpillar 0.1768 0.1913 0.1876 0.2044 0.2046 0.2367 0.2264 0.1661 0.1688 0.1720 0.1739 0.1793 0.1748 0.1768 0.1787 0.1778 0.1788 Boeing 0.2196 0.2048 0.2348 0.2477 0.2295 0.2237 0.2057 0.2028 0.2031 0.2098 0.2146 0.2273 0.2040 0.2051 0.2067 0.2107 0.2108 Pfizer 0.3178 0.1812 0.1581 0.1657 0.1794 0.1874 0.1831 0.1542 0.1535 0.1683 0.1577 0.1958 0.1460 0.1456 0.1467 0.1449 0.1954 J&J 0.2136 0.1588 0.1628 0.1798 0.1771 0.1726 0.1738 0.1640 0.1680 0.1682 0.1614 0.1786 0.1709 0.1721 0.1743 0.1692 0.1754 3M 0.1644 0.1408 0.1566 0.1402 0.1482 0.1476 0.1378 0.1363 0.1476 0.1457 0.1502 0.1553 0.1326 0.1322 0.1321 0.1405 0.1427 Merck 0.2215 0.1812 0.1652 0.1681 0.1716 0.1693 0.1753 0.1411 0.1467 0.1539 0.1619 0.1697 0.1432 0.1459 0.1499 0.1537 0.1572 Alcoa 0.3413 0.2398 0.2428 0.2403 0.2262 0.2111 0.2155 0.2496 0.2506 0.2440 0.2520 0.2579 0.2513 0.2539 0.2522 0.2657 0.2721 Average 0.2503 0.1887 0.1863 0.1830 0.1857 0.1935 0.1927 0.1759 0.1764 0.1755 0.1771 0.1859 0.1740 0.1734 0.1719 0.1738 0.1802

The table shows the mean absolute errors of six-month-ahead beta forecasts for each company and the average for the subsample period between 1998 and 2011 for each model: the Fama-MacBeth constant five-year; constant models of the realized beta with varying estimation sizes (6, 12, 18, 24, 48 and 60 months of daily data); and autoregressive models with 1–5 lags and estimation sizes of 40 and 60 periods (half-years). The out-of-sample evaluation runs from 1998:1 to 2011:2, with a total of 28 forecasting periods. Values in bold indicate the model with the smallest MAE within each model class.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 947

Table 5 MAEs of one-year-ahead forecasts of betas.

Company FM 12M 18M 24M 48M 60M AR(1) AR(2) AR(3) AR(4) AR(5)

Coca Cola 0.3225 0.1366 0.1598 0.1640 0.1911 0.1889 0.1564 0.1685 0.1733 0.1717 0.1741 Du Pont 0.1306 0.1276 0.1171 0.1198 0.1086 0.1149 0.1074 0.1056 0.1030 0.1034 0.0992 Exxon 0.2661 0.2265 0.1876 0.1904 0.2257 0.2355 0.2016 0.2050 0.2123 0.2146 0.2244 GE 0.2290 0.1548 0.1674 0.1804 0.1860 0.1785 0.1191 0.1170 0.1121 0.1082 0.1180 IBM 0.2659 0.1399 0.1150 0.1094 0.1197 0.1278 0.1267 0.1274 0.1329 0.1353 0.1404 Chevron 0.2278 0.1975 0.1920 0.1971 0.2119 0.2189 0.1966 0.2055 0.2115 0.1974 0.2024 United Tech 0.2520 0.1334 0.1362 0.1315 0.1323 0.1400 0.1151 0.1142 0.1108 0.1277 0.1276 P&G 0.2248 0.1732 0.1819 0.1782 0.1588 0.1705 0.1686 0.1628 0.1832 0.1888 0.1957 Caterpillar 0.2417 0.2207 0.2438 0.2368 0.2466 0.2339 0.2177 0.2198 0.2243 0.2276 0.2342 Boeing 0.2895 0.2638 0.2754 0.2575 0.2989 0.2622 0.2532 0.2798 0.2883 0.2687 0.2640 Pfizer 0.2983 0.1732 0.1819 0.1922 0.1942 0.1811 0.1553 0.1584 0.1558 0.1568 0.1573 J&J 0.1917 0.1674 0.1708 0.1745 0.1663 0.1569 0.1824 0.1941 0.1701 0.1590 0.1690 3M 0.1485 0.1579 0.1362 0.1464 0.1475 0.1310 0.1188 0.1219 0.1221 0.1286 0.1296 Merck 0.2142 0.1741 0.1874 0.1731 0.1853 0.1799 0.1437 0.1484 0.1571 0.1678 0.1759 Alcoa 0.2663 0.2306 0.1992 0.2149 0.2122 0.2165 0.2422 0.2547 0.2612 0.2713 0.2781 Average 0.2379 0.1785 0.1768 0.1778 0.1857 0.1824 0.1670 0.1722 0.1745 0.1751 0.1793

The table shows the mean absolute error of one-year-ahead beta forecasts for each company and the average for the entire sample for each model: the Fama-MacBeth constant five-year; constant models of the realized beta with varying estimation sizes (12, 18, 24, 48 and 60 months of daily data); and autoregressive models with 1–5 lags and an estimation size of 30 periods (years). The out-of-sample evaluation runs from 1987 to 2011, with a total of 25 forecasting periods. Values in bold indicate the model with the smallest MAE within each model class.

column this represents bias in the forecast. The problems with using the industry standard FM beta to forecast the next period’s beta are evident here, with all companies showing bias in the forecasts through very strong rejec- tions of the joint null hypothesis (the rejection is significant at the 1% level or better in 14 of the 15 companies). The positive significant values of α indicate that the FM betas understate the value of the realized beta one period ahead. The constant realized beta models are mostly biased too, though the bias decreases almost monotonically as the number of months increases. The best model in terms of the MAE and MSE, the 18M beta, is biased in 10 of the 15 companies; however, it is still considered the best constant model, due to its MAE and MSE performances.

Next, we analyse the bias in autoregressive models, where we find substantial improvements. Starting with AR(p) models with 40 periods of in-sample estimation, we can see that the models with two and three lags are the best, with only one biased result among the 15 companies. Although the AR(1) model is not the best in terms of bias, it still does a good job, with only three biased companies. This model also presents strong support of the null when it is unbiased, with all p-values being higher than 20%. The final results for the bias are given in Table 10 for the autore- gressive models with 60 in-sample estimation periods. The AR(1) model is the best overall model for the six-month forecasts, with bias in only one company, Caterpillar, and strong support of the null of no bias, with p-values as high as 0.9824 in Merck, for example. The other specifications of the AR(p) model with 60 in-sample estimation periods also perform well in terms of the bias, but the AR(1) is still the best performing forecasting model when we consider the previous results in terms of MAE and MSE.

Tables 11 and 12 report the results for the main models (FM, constant realized beta and autoregressive realized beta) for one-year forecasts. Here, we can see again that the FM betas are mostly biased. The forecasts from this model are biased for 13 of the 15 companies in the sample, with the two companies for which the null is not rejected at the 5% level (Chevron and Caterpillar) still having p-values

under 10%. Most of the time the bias comes from both the intercept and the slope of the regression with the positive α values, as in the six-month scenario, which suggests an underestimation of the forecast. Moving to the constant realized beta models, we can see a general improvement in the bias as we increase the length of the measure (from 12 to 60 months of daily returns). Both the 18M and the 24M models, which have the lowest errors according to MAE and MSE, perform better than the FM betas, with only six of the 15 companies presenting systematic biases. The constant model which presents the lowest bias is the realized beta with 60 months of daily data, with only five biased results. However, as this model had a relatively weak performance according to MAE and MSE, the com- plete picture still indicates the 18M and 24M as the two most reliable constant models for forecasting beta.

Moving to Table 12, the improvement in bias reduction from the use of autoregressive models to forecast the next period’s (one-year-ahead) betas is clear. The AR(p) models are the least biased of all, with the bias increasing with the number of lags used in the specification, which again supports the idea that betas are at most autoregressive at one period. The AR(1) model outperforms all others, with only one incidence of bias in a company’s beta forecast, that for Boeing. This again provides strong support for the choice of the AR(1) as the most accurate beta forecasting model.

In summary, the analysis of the forecasting bias us- ing the Mincer–Zarnowitz regression provides additional support for the AR(1) being the most suitable forecasting model for both six-month and yearly beta forecasts. The results also show a substantial bias in the standard FM beta. Furthermore, our out-of-sample results favoring the AR(1) are consistent with our initial in-sample autocorrelation and partial autocorrelation statistics for realized betas.

5. An economic example: evaluating investments

One of the most important applications of long range beta forecasting in finance is in calculations of the cost

948 T.Cenesizoglu

etal./InternationalJournalofForecasting 33

(2017) 936–957

Table 6 Summary statistics of six-month-ahead forecast errors.

Company FM AR(1): 40 periods AR(1): 60 periods

No. obs Mean Std. Dev. Minimum Maximum No. obs Mean Std. Dev. Minimum Maximum No. obs Mean Std. Dev. Minimum Maximum

Coca Cola 55 −0.13 0.37 −0.93 0.63 55 0.02 0.23 −0.60 0.55 55 0.01 0.23 −0.60 0.52 Du Pont 55 0.08 0.16 −0.25 0.40 55 −0.01 0.20 −0.72 0.38 55 −0.01 0.19 −0.57 0.37 Exxon 55 −0.19 0.27 −0.88 0.50 55 −0.02 0.25 −0.72 0.52 55 0.00 0.25 −0.71 0.53 GE 55 −0.05 0.28 −0.65 0.69 55 −0.03 0.17 −0.56 0.32 55 −0.02 0.18 −0.56 0.39 IBM 55 0.07 0.33 −0.54 0.82 55 0.02 0.22 −0.60 0.42 55 0.01 0.22 −0.60 0.37 Chevron 55 −0.10 0.28 −0.54 0.57 55 0.01 0.26 −0.76 0.61 55 0.03 0.26 −0.78 0.66 United Tech 55 0.20 0.26 −0.58 0.76 55 0.02 0.20 −0.87 0.53 55 0.03 0.20 −0.84 0.51 P&G 55 −0.12 0.29 −0.83 0.52 55 0.02 0.24 −0.56 0.95 55 0.00 0.23 −0.59 0.91 Caterpillar 55 0.10 0.29 −0.50 0.68 55 −0.09 0.25 −0.64 0.45 55 −0.12 0.25 −0.68 0.35 Boeing 55 0.11 0.42 −0.71 1.20 55 0.02 0.27 −0.71 0.63 55 0.05 0.28 −0.66 0.65 Pfizer 55 −0.09 0.41 −0.93 0.93 55 0.04 0.25 −0.58 0.76 55 0.03 0.24 −0.59 0.77 J&J 55 0.00 0.29 −0.47 0.76 55 0.04 0.24 −0.34 0.81 55 0.03 0.25 −0.39 0.79 3M 55 −0.05 0.18 −0.41 0.40 55 0.01 0.16 −0.26 0.39 55 0.02 0.15 −0.25 0.37 Merck 55 −0.04 0.28 −0.46 0.87 55 0.00 0.23 −0.41 0.91 55 0.00 0.23 −0.36 0.90 Alcoa 55 0.20 0.29 −0.54 1.03 55 −0.04 0.26 −0.71 0.53 55 −0.03 0.27 −0.71 0.49

The table presents summary statistics of each company’s sample of 55 forecasting errors from the Fama-MacBeth constant five-year model and the two best autoregressive models (the AR(1) model with a 40- or 60-period in-sample estimation time). The forecasting error is measured as error = βt+h|t − βt+h. The out-of-sample evaluation runs from 1984:2 to 2011:2, with a total of 55 forecasting periods.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 949

Table 7 Summary statistics of one-year-ahead forecast errors.

Company FM AR(1): 30 periods

No. obs Mean Std. Dev. Minimum Maximum No. obs Mean Std. Dev. Minimum Maximum

Coca Cola 25 −0.10 0.37 −0.78 0.60 25 0.03 0.20 −0.50 0.32 Du Pont 25 0.08 0.13 −0.26 0.26 25 −0.01 0.15 −0.48 0.21 Exxon 25 −0.22 0.24 −0.79 0.32 25 −0.01 0.25 −0.47 0.46 GE 25 −0.05 0.29 −0.56 0.58 25 −0.02 0.18 −0.52 0.39 IBM 25 0.12 0.31 −0.30 0.78 25 0.03 0.16 −0.33 0.33 Chevron 25 −0.13 0.24 −0.51 0.41 25 0.02 0.24 −0.45 0.53 United Tech 25 0.19 0.25 −0.23 0.76 25 0.03 0.17 −0.51 0.47 P&G 25 −0.12 0.28 −0.77 0.46 25 0.01 0.21 −0.37 0.49 Caterpillar 25 0.08 0.29 −0.44 0.56 25 −0.13 0.26 −0.72 0.27 Boeing 25 0.05 0.38 −0.45 1.14 25 0.06 0.31 −0.49 0.66 Pfizer 25 −0.06 0.37 −0.72 0.88 25 0.04 0.21 −0.37 0.48 J&J 25 0.01 0.26 −0.41 0.54 25 0.07 0.23 −0.37 0.65 3M 25 −0.07 0.16 −0.40 0.22 25 0.02 0.15 −0.26 0.28 Merck 25 −0.01 0.26 −0.42 0.61 25 0.01 0.20 −0.43 0.52 Alcoa 25 0.20 0.24 −0.40 0.59 25 −0.07 0.29 −0.74 0.48

The table presents summary statistics of each company’s sample of 25 forecasting errors from the Fama-MacBeth constant five-year model and the best autoregressive model (AR(1) model with a 30-period in-sample estimation time). The forecasting error is measured as error = βt+h|t − βt+h. The out-of- sample evaluation runs from 1987 to 2011 with a total of 25 forecasting periods.

of capital, see Pratt and Grabowski (2014). In a survey regarding the practice of corporate finance, Graham and Harvey (2001) deals with the estimation of the cost of capital at length. The study surveys 392 US company ex- ecutives about their practice regarding the cost of capital, capital budgeting and capital structure, and document the widespread use of CAPM as a capital budgeting tool in the US. The results show that 73.5% of their respondents always or almost always used the CAPM, and the authors also show and discuss the fact that it had increased in popularity since previous studies.

We illustrate the impact of our results on economic applications by analysing four investment projects. The first three investment projects all have the same initial investment of 8000, a time frame of five years and a cash flow of 15,000, and differ only in the timing of the cash flow. In project A, cash is returned in equal amounts over the course of the project, at 3,000 per year; project B has a decreasing cash generation over the five years, of 5000, 4000, 3000, 2000 and 1000; and project C has an increasing cash generation over the five years, of 1000, 2000, 3000, 4000 and 5000. The fourth project is a perpetual cash flow (which is more similar to a going concern company), with an initial investment of 10,000 and a cash flow of 650 per year over an indefinite period.

We assess the economic effects of using different beta forecasts by calculating the net present value (NPV) of all four projects for all companies across all time periods of the out-of-sample evaluation period. These NPV calculations use a discount rate (r) from CAPM, assuming a 2% risk free rate and a 5.5% risk premium of the market over the risk free rate, i.e., r = 2% + β5.5% and

NPV = −C0 + ∞∑ i=1

Ci (1 + r)i

, (9)

where C0 is the initial investment and Ci is the cash flow in year i.

We have 375 NPV values for each project/forecast method combination for the one-year sample, and 825 for

the six-month beta. The results are presented in Figs. 4 and 5, which display the histograms of NPVs for each project for the six-month and one-year betas respectively, and in Table 13 with the summary statistics. Figs. 4 and 5 reveal that, for all four projects, the NPVs calculated from the AR(1) forecast method have less extreme values than those calculated with FM betas, and are also more concentrated around the mean. The greater variability in the NPVs calculated using FM betas is also evident in Ta- ble 13. In panel (a), the one-year beta sample, we can see that, for all four projects, the standard deviation of the NPVs calculated using FM betas is more than double that of the NPVs calculated using the AR(1) betas. In addition, the NPVs calculated with FM betas have more extreme maximums and minimums than the AR(1) forecasts. This is particularly striking for the perpetuity project, where the maximum NPV for the FM beta is 19,642, compared to only 5200 for AR(1). Furthermore, the interquartile ranges for FM are more than double those of AR(1). Similar results appear in panel (b) for the six-month beta sample, with the FM beta again displaying substantially more variability than the AR(1) betas.

The excessive variability in the FM beta relative to the AR(1) also results in an upward bias in the mean NPV calculated with FM betas. This is particularly strong for the perpetuity project. For the one-year forecast, the perpetu- ity mean NPV is −358 with the FM beta forecast, while only −961 with the AR(1) forecast (see panel (a) of Table 13). These findings are consistent with those of Armstrong, Banerjee, and Corona (2013), who show that, due to the convexity of the present value in the CAPM beta, a higher variability in beta results in the expected value of the risky asset being overstated.

Overall, these examples demonstrate that there are in- deed significant economic differences between the various forecasting methods. The results provide us with sufficient evidence that using the more accurate beta forecasting model from our econometric tests (the AR(1) model of realized betas) can change the investment, or project eval- uation, decision significantly relative to what is still the

950 T.Cenesizoglu

etal./InternationalJournalofForecasting 33

(2017) 936–957

Table 8 Forecasting bias (M–Z test): six-month-ahead forecasts (Fama-MacBeth and constant models).

Company FM 6M 12M 18M 24M 48M 60M

α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val

Coca Cola 0.88 −0.05 0.0000 0.24 0.71 0.0163 0.15 0.81 0.1810 0.11 0.86 0.3459 0.14 0.82 0.2456 0.20 0.74 0.1254 0.21 0.72 0.1114 Du Pont 0.27 0.70 0.0007 0.84 0.23 0.0000 0.63 0.43 0.0061 0.57 0.49 0.0326 0.51 0.55 0.0806 0.35 0.70 0.2941 0.47 0.59 0.2067 Exxon 0.48 0.53 0.0000 0.46 0.43 0.0001 0.40 0.51 0.0044 0.35 0.57 0.0222 0.32 0.61 0.0493 0.44 0.45 0.0272 0.52 0.35 0.0194 GE 0.94 0.20 0.0000 0.45 0.61 0.0033 0.41 0.65 0.0332 0.44 0.62 0.0493 0.54 0.54 0.0373 1.20 −0.03 0.0029 1.64 −0.41 0.0004 IBM 0.83 0.15 0.0000 0.78 0.21 0.0000 0.89 0.10 0.0000 0.48 0.52 0.0659 0.51 0.48 0.0566 0.44 0.54 0.1334 0.52 0.46 0.0719 Chevron 0.44 0.51 0.0125 0.39 0.51 0.0008 0.36 0.55 0.0068 0.29 0.63 0.0401 0.28 0.65 0.0732 0.39 0.50 0.0240 0.44 0.43 0.0100 United Tech 1.06 −0.01 0.0000 0.77 0.26 0.0000 0.75 0.28 0.0002 0.59 0.44 0.0081 0.53 0.50 0.0216 0.63 0.40 0.0046 0.68 0.35 0.0022 P&G 0.50 0.44 0.0000 0.36 0.55 0.0014 0.34 0.57 0.0094 0.30 0.61 0.0274 0.28 0.63 0.0405 0.29 0.61 0.0423 0.31 0.59 0.0347 Caterpillar 0.29 0.70 0.0069 0.38 0.69 0.0125 0.37 0.70 0.0619 0.38 0.71 0.1106 0.37 0.72 0.1455 0.27 0.85 0.0572 0.19 0.94 0.0222 Boeing 1.04 −0.02 0.0000 0.53 0.49 0.0004 0.60 0.41 0.0009 0.76 0.27 0.0004 0.87 0.16 0.0004 1.43 −0.41 0.0000 1.02 0.00 0.0043 Pfizer 0.99 −0.14 0.0000 0.43 0.51 0.0005 0.34 0.61 0.0260 0.36 0.58 0.0495 0.43 0.51 0.0435 0.98 −0.11 0.0056 1.04 −0.18 0.0265 J&J 0.37 0.52 0.0033 0.29 0.62 0.0039 0.27 0.64 0.0253 0.22 0.70 0.1037 0.16 0.78 0.2811 0.10 0.83 0.3093 0.08 0.84 0.2564 3M 0.99 −0.06 0.0000 0.67 0.28 0.0000 0.85 0.09 0.0000 0.79 0.15 0.0011 0.71 0.25 0.0070 0.92 0.01 0.0015 0.83 0.11 0.0058 Merck 0.64 0.28 0.0001 0.52 0.40 0.0001 0.52 0.40 0.0021 0.59 0.32 0.0031 0.61 0.30 0.0078 1.03 −0.18 0.0011 1.30 −0.49 0.0002 Alcoa 0.29 0.65 0.0000 0.36 0.71 0.0233 0.33 0.74 0.0812 0.27 0.79 0.2099 0.24 0.82 0.3157 0.18 0.89 0.3531 0.15 0.93 0.2331

The table presents the parameter estimates and p-values of the Mincer–Zarnowitz forecasting bias test for each company for the Fama-MacBeth constant five-year and constant realized beta models. The α and γ columns show the coefficients from a M–Z regression, with bold values indicating rejection of the null of individual tests α = 0 and γ = 1 at the 5% two-tailed significance level. The p-val columns show the p-values of the joint hypothesis test of systematic bias in the forecast H0 : α = 0, γ = 1, with bold values highlighting rejection of the null at the 5% two-tailed significance level. The out-of-sample evaluation runs from 1984:2 to 2011:2, with a total of 55 forecasting periods.

T.Cenesizoglu etal./InternationalJournalofForecasting

33 (2017)

936–957 951

Table 9 Forecasting bias (M–Z test): six-month-ahead forecasts (autoregressive models, 40 in-sample periods).

Company AR(1) AR(2) AR(3) AR(4) AR(5)

α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val

Coca Cola −0.01 0.99 0.7543 0.04 0.95 0.9180 0.06 0.92 0.8424 0.06 0.91 0.7028 0.08 0.89 0.7105 Du Pont 1.46 −0.33 0.0218 0.68 0.38 0.2168 0.88 0.20 0.1382 0.84 0.23 0.1184 0.98 0.11 0.0619 Exxon 0.22 0.74 0.5312 0.28 0.67 0.4069 0.30 0.64 0.2981 0.38 0.54 0.1416 0.45 0.45 0.0674 GE −0.01 1.03 0.5465 0.05 0.98 0.5409 0.06 0.97 0.5557 0.04 0.99 0.5449 0.10 0.94 0.4751 IBM 0.67 0.32 0.2208 0.86 0.12 0.0700 0.36 0.62 0.2902 0.43 0.56 0.1803 0.50 0.48 0.0962 Chevron 0.22 0.71 0.3212 0.30 0.61 0.1365 0.27 0.65 0.1530 0.32 0.59 0.0733 0.38 0.50 0.0221 United Tech 0.89 0.15 0.0472 1.00 0.04 0.0190 0.51 0.51 0.1419 0.57 0.45 0.0845 0.61 0.42 0.0576 P&G 0.03 0.94 0.8681 0.18 0.77 0.6546 0.25 0.68 0.4665 0.31 0.60 0.2695 0.38 0.52 0.1223 Caterpillar 0.17 0.93 0.0415 0.21 0.88 0.0651 0.26 0.84 0.0340 0.28 0.83 0.0355 0.31 0.81 0.0195 Boeing 0.32 0.68 0.2964 0.35 0.66 0.2714 0.40 0.60 0.1358 0.51 0.48 0.0193 0.61 0.39 0.0052 Pfizer 0.17 0.78 0.4378 0.20 0.75 0.4079 0.31 0.63 0.2146 0.39 0.55 0.0884 0.36 0.56 0.0119 J&J −0.13 1.10 0.3682 −0.08 1.05 0.4145 0.04 0.90 0.5502 0.12 0.83 0.5791 0.14 0.77 0.2494 3M 0.88 0.05 0.2150 0.91 0.02 0.0529 0.70 0.24 0.0988 0.73 0.21 0.0543 0.67 0.27 0.0213 Merck 0.24 0.73 0.7475 0.44 0.50 0.3600 0.51 0.41 0.2545 0.61 0.30 0.0957 0.64 0.26 0.0483 Alcoa −0.11 1.13 0.3698 −0.06 1.09 0.4374 −0.02 1.05 0.5798 0.03 1.01 0.5989 0.21 0.85 0.4891

The table presents the parameter estimates and p-values of the Mincer–Zarnowitz test of forecasting bias for each company for autoregressive models with an in-sample estimation size of 40 periods (half-years). The α and γ columns show the coefficients from a M–Z regression, with bold values indicating rejection of the null of individual tests α = 0 and γ = 1 at the 5% two-tailed significance level. The p-val columns show the p-values of the joint hypothesis test of systematic bias in the forecast H0 : α = 0, γ = 1, with bold values highlighting rejection of the null at the 5% two-tailed significance level. The out-of-sample evaluation runs from 1984:2 to 2011:2, with a total of 55 forecasting periods.

952 T.Cenesizoglu

etal./InternationalJournalofForecasting 33

(2017) 936–957

Table 10 Forecasting bias (M–Z test): six-month-ahead forecasts (autoregressive models, 60 in-sample periods).

Company AR(1) AR(2) AR(3) AR(4) AR(5)

α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val

Coca Cola −0.04 1.04 0.9369 −0.04 1.05 0.9426 −0.03 1.03 0.9607 −0.05 1.05 0.8751 −0.02 1.01 0.9712 Du Pont 1.28 −0.17 0.2706 0.27 0.76 0.8417 0.34 0.69 0.7600 0.43 0.61 0.6582 0.65 0.41 0.3419 Exxon 0.17 0.80 0.7717 0.23 0.72 0.6339 0.25 0.70 0.5097 0.32 0.61 0.2963 0.39 0.52 0.1616 GE −0.14 1.15 0.4723 −0.06 1.08 0.5296 −0.08 1.09 0.4908 −0.08 1.09 0.4612 −0.06 1.07 0.5311 IBM 0.67 0.31 0.4833 0.95 0.04 0.1694 0.21 0.77 0.7129 0.25 0.74 0.6892 0.37 0.61 0.4664 Chevron 0.21 0.71 0.2061 0.21 0.71 0.2231 0.16 0.77 0.3512 0.18 0.75 0.2719 0.20 0.72 0.1692 United Tech 0.64 0.38 0.1746 0.71 0.31 0.1622 0.43 0.58 0.3563 0.53 0.48 0.1850 0.56 0.45 0.1606 P&G −0.10 1.12 0.9154 −0.03 1.04 0.9793 0.01 0.99 0.9982 0.05 0.93 0.9576 0.07 0.91 0.9295 Caterpillar 0.17 0.95 0.0043 0.24 0.88 0.0156 0.25 0.87 0.0088 0.28 0.85 0.0092 0.30 0.83 0.0051 Boeing 0.33 0.65 0.1204 0.33 0.64 0.1153 0.38 0.59 0.0367 0.43 0.54 0.0114 0.48 0.49 0.0038 Pfizer −0.15 1.13 0.5674 −0.01 0.98 0.7738 0.14 0.82 0.5246 0.15 0.80 0.5273 0.27 0.67 0.0994 J&J −0.33 1.37 0.2874 −0.29 1.33 0.3631 −0.28 1.31 0.3601 −0.20 1.21 0.5254 −0.13 1.12 0.5485 3M 0.25 0.72 0.6146 0.18 0.79 0.6320 0.24 0.73 0.6393 0.40 0.56 0.3760 0.49 0.47 0.2047 Merck 0.07 0.92 0.9824 0.37 0.58 0.6260 0.66 0.25 0.2607 0.63 0.28 0.2671 0.67 0.24 0.2055 Alcoa −0.26 1.24 0.2944 −0.24 1.23 0.3212 −0.18 1.17 0.4461 −0.18 1.18 0.4087 −0.03 1.04 0.7865

The table presents the parameter estimates and p-values of the Mincer–Zarnowitz test of forecasting bias for each company for autoregressive models with an in-sample estimation size of 60 periods (half-years). The α and γ columns show the coefficients from a M–Z regression, with bold values indicating rejection of the null of individual tests α = 0 and γ = 1 at the 5% two-tailed significance level. The p-val columns show the p-values of the joint hypothesis test of systematic bias in the forecast H0 : α = 0, γ = 1, with bold values highlighting rejection of the null at the 5% two-tailed significance level. The out-of-sample evaluation runs from 1984:2 to 2011:2, with a total of 55 forecasting periods.

T.Cenesizoglu etal./InternationalJournalofForecasting

33 (2017)

936–957 953

Table 11 Forecasting bias (M–Z test): one-year-ahead forecasts (Fama-MacBeth and constant models).

Company FM 12M 18M 24M 48M 60M

α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val

Coca Cola 0.92 −0.13 0.0011 0.19 0.75 0.1223 0.19 0.74 0.1421 0.22 0.71 0.1426 0.28 0.62 0.1100 0.26 0.64 0.1216 Du Pont 0.37 0.62 0.0026 0.67 0.39 0.0205 0.53 0.53 0.0819 0.59 0.47 0.1004 0.49 0.57 0.2807 0.52 0.54 0.2466 Exxon 0.28 0.91 0.0007 0.48 0.42 0.0226 0.36 0.56 0.1156 0.40 0.51 0.1062 0.54 0.33 0.0742 0.63 0.23 0.0645 GE 1.03 0.11 0.0002 0.58 0.49 0.0355 0.67 0.42 0.0836 0.81 0.30 0.0277 1.50 −0.30 0.0090 1.82 −0.57 0.0072 IBM 0.82 0.14 0.0000 0.61 0.37 0.0117 0.51 0.47 0.0599 0.43 0.55 0.1576 0.42 0.55 0.1760 0.49 0.48 0.1162 Chevron 0.04 1.13 0.0561 0.35 0.57 0.0769 0.30 0.64 0.1779 0.30 0.64 0.2263 0.36 0.57 0.2281 0.40 0.51 0.2092 United Tech 1.15 −0.10 0.0000 0.69 0.33 0.0113 0.66 0.36 0.0175 0.68 0.34 0.0278 0.73 0.30 0.0173 0.79 0.24 0.0106 P&G 0.54 0.38 0.0000 0.33 0.58 0.0609 0.34 0.56 0.0341 0.29 0.61 0.1070 0.25 0.65 0.1344 0.28 0.61 0.1094 Caterpillar 0.49 0.57 0.0639 0.59 0.53 0.0454 0.60 0.53 0.0919 0.55 0.57 0.1507 0.52 0.63 0.1799 0.42 0.72 0.1405 Boeing 1.24 −0.21 0.0002 0.76 0.26 0.0055 0.95 0.07 0.0029 1.03 −0.01 0.0044 1.66 −0.67 0.0006 1.28 −0.28 0.0224 Pfizer 0.93 −0.09 0.0000 0.59 0.31 0.0073 0.66 0.23 0.0039 0.75 0.12 0.0055 1.44 −0.65 0.0004 1.59 −0.83 0.0019 J&J 0.40 0.46 0.0105 0.30 0.59 0.0555 0.29 0.60 0.0976 0.25 0.64 0.1567 0.18 0.72 0.2756 0.14 0.76 0.2454 3M 0.86 0.08 0.0056 0.90 0.03 0.0011 0.77 0.18 0.0287 0.85 0.09 0.0230 1.03 −0.11 0.0205 0.83 0.11 0.0694 Merck 0.73 0.17 0.0003 0.75 0.14 0.0011 0.78 0.10 0.0014 0.74 0.15 0.0135 1.18 −0.36 0.0024 1.36 −0.56 0.0018 Alcoa 0.18 0.73 0.0001 0.42 0.68 0.1437 0.28 0.81 0.3600 0.28 0.81 0.3378 0.15 0.94 0.3411 0.12 0.97 0.2364

The table presents the parameter estimates and p-values of the Mincer–Zarnowitz test of forecasting bias for each company for the Fama-MacBeth constant five-year and constant realized beta models. The α and γ columns show the coefficients from a M–Z regression, with bold values indicating rejection of the null of individual tests α = 0 and γ = 1 at the 5% two-tailed significance level. The p-val columns show the p-values of the joint hypothesis test of systematic bias in the forecast H0 : α = 0, γ = 1, with bold values highlighting rejection of the null at the 5% two-tailed significance level. The out-of-sample evaluation runs from 1987 to 2011, with a total of 25 forecasting periods.

954 T.Cenesizoglu

etal./InternationalJournalofForecasting 33

(2017) 936–957

Table 12 Forecasting bias (M–Z test): one-year-ahead forecasts (autoregressive models).

Company AR(1) AR(2) AR(3) AR(4) AR(5)

α γ p-val α γ p-val α γ p-val α γ p-val α γ p-val

Coca Cola −0.23 1.24 0.5151 −0.28 1.29 0.3982 −0.19 1.16 0.4040 −0.12 1.09 0.5200 −0.08 1.04 0.6356 Du Pont −0.19 1.18 0.9376 0.10 0.92 0.8790 0.12 0.90 0.8276 0.46 0.59 0.7551 0.32 0.72 0.8517 Exxon 0.14 0.83 0.9464 0.33 0.61 0.7143 0.50 0.38 0.4618 0.41 0.49 0.5206 0.61 0.25 0.2480 GE 0.05 0.97 0.9075 0.01 1.01 0.8369 −0.01 1.02 0.9438 0.06 0.96 0.9092 0.20 0.84 0.8349 IBM 0.19 0.78 0.5865 0.45 0.52 0.4598 0.82 0.16 0.1693 1.00 −0.03 0.0942 1.07 −0.10 0.0470 Chevron 0.15 0.79 0.6913 0.19 0.73 0.5092 0.23 0.68 0.3848 0.15 0.75 0.3176 0.21 0.70 0.3987 United Tech 0.83 0.19 0.3002 0.89 0.13 0.3317 0.90 0.13 0.3885 0.90 0.12 0.1697 0.93 0.10 0.1487 P&G −0.19 1.21 0.8601 0.03 0.94 0.8703 0.36 0.55 0.5036 0.54 0.32 0.1835 0.51 0.35 0.0932 Caterpillar 0.29 0.85 0.0660 0.36 0.80 0.0418 0.48 0.69 0.0357 0.49 0.69 0.0148 0.55 0.66 0.0029 Boeing 1.15 −0.13 0.0325 0.93 0.08 0.0116 0.98 0.04 0.0047 0.88 0.12 0.0303 0.84 0.17 0.0489 Pfizer 0.45 0.45 0.2102 0.31 0.59 0.1546 0.33 0.57 0.1198 0.32 0.58 0.0935 0.36 0.55 0.1115 J&J −0.08 1.01 0.3085 0.16 0.71 0.2233 0.07 0.84 0.4541 0.16 0.74 0.3943 0.24 0.64 0.2918 3M 0.76 0.18 0.5377 0.92 0.02 0.3806 0.90 0.04 0.4038 1.24 −0.33 0.1502 1.25 −0.33 0.0376 Merck 1.25 −0.43 0.0832 1.38 −0.58 0.0554 1.26 −0.44 0.0300 1.29 −0.47 0.0082 1.29 −0.47 0.0027 Alcoa −0.21 1.23 0.4332 −0.04 1.10 0.5364 0.13 0.94 0.6887 0.24 0.85 0.5602 0.39 0.73 0.3961

The table presents the parameter estimates and p-values of the Mincer–Zarnowitz forecasting bias test for each company for autoregressive models. The α and γ columns show the coefficients from a M–Z regression, with bold values indicating rejection of the null of individual tests α = 0 and γ = 1 at the 5% two-tailed significance level. The p-val columns show the p-values of the joint hypothesis test of systematic bias in the forecast H0 : α = 0, γ = 1, with bold values highlighting rejection of the null at the 5% two-tailed significance level. The out-of-sample evaluation runs from 1987 to 2011 with a total of 25 forecasting periods.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 955

Fig. 4. Net present value distributions of four sample investments: six-month-ahead forecasts. The figure shows the six-month beta frequency distributions of the net present values of four sample investment projects, calculated for all companies over all 55 out-of-sample forecasting periods. The net present values are calculated with a discount rate given by the CAPM, where beta is the forecast value of the AR(1) and FM betas, and with a risk free rate of 2% per annum and a market risk premium of 5.5%.

Fig. 5. Net present value distributions of four sample investments: one-year-ahead forecasts. This figure shows the one-year beta frequency distributions of the net present values of four sample investment projects, calculated for all companies over all 25 out-of-sample forecasting periods. The net present values are calculated with a discount rate given by the CAPM, where beta is the forecast value of the AR(1) and FM betas, and with a risk free rate of 2% per annum and a market risk premium of 5.5%.

956 T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957

Table 13 Summary statistics of net present value calculations.

(a) One-year betas

5-year project A 5-year project B 5-year project C Perpetuity

AR(1) FM AR(1) FM AR(1) FM AR(1) FM

Mean 4,205 4,237 4,776 4,799 3,634 3,674 −961 −358 Std 301 612 245 500 356 725 1,201 3,011 Max 5,252 6,062 5,622 6,265 4,883 5,858 5,200 19,642 Min 2,815 2,406 3,628 3,283 2,003 1,529 −4, 582 −5, 211 25% 4,004 3,842 4,612 4,480 3,396 3,205 −1, 792 −2, 294 75% 4,386 4,640 4,923 5,130 3,848 4,151 −361 850

(b) Six-month betas

5-year project A 5-year project B 5-year project C Perpetuity

AR(1)-40p AR(1)-60p FM AR(1)-40p AR(1)-60p FM AR(1)-40p AR(1)-60p FM AR(1)-40p AR(1)-60p FM

Mean 4,208 4,210 4,227 4,778 4,779 4,791 3,638 3,640 3,662 −917 −899 −423 Std 334 345 608 272 282 496 395 409 720 1,355 1,415 2,881 Max 5,375 5,382 6,062 5,720 5,726 6,265 5,030 5,039 5,858 6,462 6,547 19,642 Min 2,790 2,783 2,406 3,606 3,600 3,283 1,973 1,965 1,529 −4, 625 −4, 636 −5, 211 25% 3,999 4,015 3,850 4,609 4,621 4,486 3,390 3,409 3,214 −1, 806 −1, 755 −2, 271 75% 4,388 4,390 4,641 4,925 4,927 5,130 3,851 3,854 4,152 −350 −340 854

The table presents the summary statistics for net present values for four sample investment projects for all 25 (55) out-of-sample forecasting periods and all companies. The net present values are calculated with a discount rate given by the CAPM, with beta being the one-year (six-month) forecast value of the AR(1) and FM betas, a risk free rate of 2% per annum and a market risk premium of 5.5%.

industry standard (the Fama-MacBeth constant five-year beta forecast).

6. Conclusions

Beta forecasting for long horizons is important to many firms, particularly when they are calculating the cost of capital. This paper has demonstrated the inaccuracy of the standard forecasting approach that follows Fama and MacBeth (1973), from both the statistical and economic perspectives. We also propose a new approach to long range beta forecasting using an AR(1) model of the real- ized beta, which has a relatively low mean absolute (and squared) forecast error and is statistically unbiased, for use in settings where at least 20 years of daily return data are available.

In settings in which accurate daily return data are avail- able for only a limited historical period, we find that a constant realized beta model based on the previous 18 months of daily returns produces forecasts with the lowest mean absolute (and squared) forecast errors. However, these constant beta forecasts can often contain significant statistical biases, which highlights the fact that additional historical data are important in long range beta forecasting for horizons of six months or one year.

This paper has also found that the expected value of a project’s net present value is overstated when using the Fama-MacBeth constant five-year beta, due to the rela- tively high variability in this forecast. On the other hand, beta forecasts with an AR(1) model of the realized beta provide a higher level of accuracy in project net present value estimation.

It is hoped that the more accurate long run beta fore- casting approaches demonstrated in this paper will prove

useful not only in NPV analysis for capital budgeting deci- sions, but also in other applications in finance.

Acknowledgments

We would like to thank the handling editor, Profes- sor George Kapetanios, and two anonymous referees for very helpful comments that improved the paper. We have also benefited from discussions with Buly Cardak, Jae Kim, Michael Li, Kostas Nikolopoulos, participants at the 36th In- ternational Symposium on Forecasting in Santander, Spain, and seminar participants at La Trobe University, Australia.

References

Acker, D., & Duck, N. W. (2007). Reference-day risk and the use of monthly returns data. Journal of Accounting, Auditing and Finance, 22(4), 527–557.

Andersen, T. G., Bollerslev, T., Diebold, F. X., & Ebens, H. (2001). The distribution of realized stock return volatility. Journal of Financial Economics, 61(1), 43–76.

Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2001). The distribution of realized exchange rate volatility. J. Amer. Statist. Assoc., 96(453), 42–55.

Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71(2), 579–625.

Andersen, T. G., Bollerslev, T., Diebold, F. X., & Wu, G. (2005). A framework for exploring the macroeconomic determinants of systematic risk. The American Economic Review, 95(2), 398–404.

Andersen, T. G., Bollerslev, T., Diebold, F. X., & Wu, G. (2006). Realized beta: Persistence and predictability. In B. Fomby Thomas, & Dek Terrell (Eds.), Advances in econometrics: Econometric analysis of financial and economic time series: 20 part 2. (pp. 1–39).

Armstrong, C. S., Banerjee, S., & Corona, C. (2013). Factor-Loading uncer- tainty and expected returns. The Review of Financial Studies, 26(1), 158–207.

Barndorff-Nielsen, O. E., & Shephard, N. (2004). Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica, 72(3), 885–925.

T. Cenesizoglu et al. / International Journal of Forecasting 33 (2017) 936–957 957

Braun, P. A., Nelson, D. B., & Sunier, A. M. (1995). Good news, bad news, volatility, and betas. The Journal of Finance, 50(5), 1575–1603.

Brooks, C., & Henry, O. T. (2002). The impact of news on measures of un- diversifiable risk: Evidence from the UK stock market. Oxford Bulletin of Economics and Statistics, 64(5), 487–507.

Cenesizoglu, T., Liu, Q., Reeves, J. J., & Wu, H. (2016). Monthly beta forecast- ing with low-, medium- and high-frequency stock returns. Journal of Forecasting, 35(6), 528–541.

Chang, B.-Y., Christoffersen, P., Jacobs, K., & Vainberg, G. (2012). Option- implied measures of equity risk. Review of Finance, 16(2), 385–428.

Dimson, E. (1979). Risk measurement when shares are subject to infre- quent trading. Journal of Financial Economics, 7(2), 197–226.

Fama, E. F., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empiri- cal tests. Journal of Political Economy, 81(3), 607–636.

Graham, J. R., & Harvey, C. R. (2001). The theory and practice of corpo- rate finance: evidence from the field. Journal of Financial Economics, 60(2–3), 187–243.

Green, K. C., & Armstrong, J. S. (2015). Simple versus complex forecasting: The evidence. Journal of Business Research, 68(8), 1678–1685.

Hooper, V. J., Ng, K., & Reeves, J. J. (2008). Quarterly beta forecasting: An evaluation. Int. J. Forecast., 24(3), 480–489.

Jostova, G., & Philipov, A. (2005). Bayesian analysis of stochastic betas. Journal of Financial and Quantitative Analysis, 40(4), 747–778.

Lintner, J. (1965). Security prices, risk, and maximal gains from diversifi- cation. The Journal of Finance, 20(4), 587–615.

Mincer, J. A., & Zarnowitz, V. (1969). The evaluation of economic forecasts. In Economic forecasts and expectations: Analysis of forecasting behavior and performance (pp. 1–46). NBER.

Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica, 34(4), 768–783.

Papageorgiou, N., Reeves, J. J., & Xie, X. (2016). Betas and the myth of market neutrality. Int. J. Forecast., 32(2), 548–558.

Pratt, S. P., & Grabowski, R. J. (2014). Cost of capital: Applications and examples. (5th ed.). Wiley.

Reeves, J. J., & Wu, H. (2013). Constant versus time-varying beta models: further forecast evaluation. Journal of Forecasting, 32(3), 256–266.

Sharpe, W. F. (1964). Capital asset prices: a theory of market equilibrium under conditions of risk. The Journal of Finance, 19(3), 425–442.

Tolga Cenesizoglu is an associate professor of finance at HEC Montreal and a senior lecturer of finance at the Alliance Manchester Business School. He received his Ph.D. in economics from the University of Califor- nia, San Diego (UCSD). He also has a M.Sc. in statistics, M.A. in economics from UCSD and B.Sc. in industrial engineering from Bogazici University, Istanbul, Turkey. His current research interests are in the areas of empir- ical and theoretical asset pricing, financial econometrics, forecasting, and market microstructure. He has previously published in journals such as Management Science, Journal of Banking and Finance, International Journal of Forecasting, Journal of Forecasting, and Journal of Financial Research.

Fabio de Oliveira Ferrazoli Ribeiro received his Ph.M. in banking and finance from the University of New South Wales (UNSW). He also holds a Master of Applied Finance from Macquarie University. He currently works as a real estate investment specialist for a large financial institution in the US.

Jonathan J. Reeves Ph.D. (Economics) Queens University, Canada is Senior Lecturer of Finance, UNSW Business School, University of New South Wales, Sydney, Australia. His primary expertise is in forecasting of volatil- ity and correlation in financial markets, and more broadly in the fields of financial econometrics, asset pricing, financial risk management and forecasting. He has previously published in journals such as the Interna- tional Journal of Forecasting, Journal of Forecasting and Journal of Financial Econometrics. He has also consulted for financial services companies both locally and internationally.