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Estimating risk premiums for regulated firms when accounting for reference-day variation and high-order moments of return volatility

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Abstract

In many jurisdictions, the determination of the acceptable rate of return for the assets of a regulated utility is based partially on the Capital Asset Pricing Model (CAPM) to determine risk premia. However, the traditional estimation of CAPM can be criticized for not including considerations of reference-day risk as well as the higher moments of the rates of return. In this paper, we attempt to account for both the potential variation induced by the definition of specific reference days and the higher moments of rates of return in the estimates of Beta. This paper provides a new methodology to account for reference-day variation. We construct a set of pseudo-monthly rates of return to identify the influence of reference-day choice. These pseudo-monthly asset returns are used to estimate measures of asset systematic risk for an international panel of regulated firms. To evaluate the influence of return rate volatility we examine the errors from the estimation of the CAPM with least squares, least absolute deviation and a partially adaptive maximum likelihood specification.

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Notes

  1. These models have been applied in the finance literature as well as the macroeconomics cases as demonstrated in Andreou (2016).

  2. The reference day could be the middle of the month such as the 15th as well.

  3. The algorithm used can be made more efficient with the appropriate modifications such as suggested in Gleason's (1988) BB3 method.

  4. Only major holidays such as Christmas and New Year’s Day were added to the weekends. Asset prices from the preceding days are used for cases when there are gaps in the series.

  5. The permutations of the month definitions are not repeated in any of the 5000 cases considered here. The distribution of dates of the simulated ending days is available from the authors on request.

  6. All the daily asset price data, government bond rates, market indices and dividend data were obtained from Bloomberg data services.

  7. Here, the bias we refer to can be classed as a form of a bootstrap estimate of bias as described in Efron and Tibshirani (1993).

  8. The boxplots are defined where the upper and lower limits of the boxes are defined by the 75 and 25 percentiles, the centre line is the median and the whiskers extend to the upper and lower bounds or 1.5 times the interquartile distance whichever is smaller.

  9. Several studies have noted that time-varying volatility is not a problem when using low-frequency data (e.g. McDonald and Xu 1995, Fong 1997, Mills and Markellos 2008). We also examine the potential influence of a GARCH process in these errors as was investigated by (Mills and Markellos, chapter 8, 2008). Although our resampling method preserves the sequence of the observations and we could include time-varying aspects in our models fit to the pseudo-months, we found no indication of the presence of these processes had any significant influence on our results.

  10. We define significance in this case to be 2 standard deviations where the standard deviation for the skewness is given by \(\sqrt {{\raise0.7ex\hbox{$6$} \!\mathord{\left/ {\vphantom {6 N}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$N$}}}\) and for the excess kurtosis \(\sqrt {{\raise0.7ex\hbox{${24}$} \!\mathord{\left/ {\vphantom {{24} N}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$N$}}}\), where N is the number of observations.

  11. We estimate the covariance of the parameter estimates to allow for heteroscedastic errors to account for the use of computed regressors in the model.

  12. Note that these results are strongly influenced by the use of a weighted OLS approach for this estimation, where the weights are based on the variance of the mean of the bootstrap estimates.

  13. The QMLE was obtained using the OLS estimates for α and β as starting values with restrictions on the values of .05 to 10.0 for p, -1 to 1 for λ and 0 to 1 for ϕ. The optimization was done with a Newton–Raphson method as implemented in the nlpnra routine in SAS Proc IML.

  14. Note that these results are strongly influenced by the use of a weighted OLS approach for this estimation, where the weights are based on the variance of the mean of the bootstrap estimates.

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Authors and Affiliations

  1. Department of Economics, University of Melbourne, Melbourne, 3010, Australia

    Joe Hirschberg & Jenny Lye

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  1. Joe Hirschberg
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  2. Jenny Lye
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Correspondence to Jenny Lye.

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Hirschberg, J., Lye, J. Estimating risk premiums for regulated firms when accounting for reference-day variation and high-order moments of return volatility. Environ Syst Decis 41, 455–467 (2021). https://doi.org/10.1007/s10669-021-09812-4

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