Theory:
Based on Hansen/Jagannathan, the set of means and variances of returns is limited. With $R^f$ as the risk-free rate, $R_i^e$ as the return of stock $i$ in excess of $R^f$ and a stochastic discount factor $m$, we know:
$$\frac{\sigma(m)}{\mathrm{E(m)}} \geq \frac{\lvert \mathrm{E}(R_i^e)-R^f \rvert}{\sigma(R_i^e)} $$
We further know, that Beta pricing models are equivalent to linear models for the discount factor $m$:
$$\mathrm{E}(R_i^e) = \gamma + \lambda'\beta_i \leftrightarrow m = a + b'f,$$
with $\lambda$ being the factor risk-premium of factor $f$, where $\lambda = \mathrm{E}(f)$ holds (consider $f$ to be a tradable factor, e.g. the market excess return in case of the CAPM).
In empirical research we estimate $\beta_i$ by time-series regressions,
$$R_i^e = a_0 + \lambda' \beta_i + \epsilon_i$$
which implies
$$\beta_i = \frac{cov(R_i ^e, \lambda')}{var(\lambda')}$$
Does there exist an upper bound for $\beta_i$, dependent on $m$ or information about $\lambda'$?
My question is related to this one, where Matthew Gunn states that
Infinity is rather non-sensical.
However, the question asks for the market beta and not for an upper limit of coefficients ("beta") of proposed risk-factors (i.e. non-market factors).
Example:
Consider the Fama/French (1992, 1993) three factor model with a size factor SMB, the value factor HML and the market factor MKTRF and lets replicate Table 6 of the paper. We use 25 value-weighted portfolio returns, sorted on size and book-to-market ratio, and regress each of them onto our three risk factors:
$$R_i^e = a_0 + b_1 MKTRF+ b_2 SMB + b_3 HML + \epsilon_i,$$
and we are interested on the estimated coefficients $\hat{b}_2$ for the size-factor SMB.
R-Code:
library(FFdownload)
tempf <- tempfile(fileext = ".RData")
inputlist <- c("F-F_Research_Data_Factors", "25_Portfolios_5x5")
# download factors and 5x5 portfolios sorted on size and book-to-market
FFdownload(output_file = tempf, inputlist=inputlist, exclude_daily = TRUE, download = TRUE, download_only=FALSE)
load(tempf)
factors <- FFdata$`x_F-F_Research_Data_Factors`$monthly$Temp2[,c("Mkt.RF","SMB","HML")]
portfolios <- FFdata$x_25_Portfolios_5x5$monthly$average_value_weighted_returns
# mean and volatility of factor returns
apply(factors, 2, mean)
> Mkt.RF SMB HML
> 0.6704671 0.1931055 0.3590398
apply(factors, 2, sd)
> Mkt.RF SMB HML
> 5.352773 3.170227 3.565333
SIZE_BETAS <- vector(mode = "numeric", length = 25)
# regress portfolio returns on the three factors
for(i in 1:25){
ret <- portfolios[,i]
reg <- lm(ret ~ factors)
SIZE_BETAS[i] <- reg$coefficients[3] # extract estimated coef. for size factor
}
# results as in Fama/French (1993), Table 6, coefficient "s"
# deviations from the original Table are because data on French Data Lib. is updated
t(matrix(SIZE_BETAS, nrow = 5, ncol = 5))
[,1] [,2] [,3] [,4] [,5]
[1,] 1.4617511 1.5366500 1.2439367 1.2224793 1.3077786
[2,] 1.1348994 0.9895360 0.8200150 0.8118625 0.9160859
[3,] 0.8070132 0.5416144 0.4409870 0.4683932 0.5762129
[4,] 0.3306691 0.2294190 0.2040363 0.2017700 0.3108749
[5,] -0.1537455 -0.1927042 -0.2378840 -0.1890345 -0.1753790
The replication of size-factor coefficients using the 25 test portfolios gives estimates within the interval $[-0.24;1.54]$, so i would take the maximum value 1.54 as an upper bound in that case. However, using another set of test portfolios would yield other estimates. What can we say about an upper limit of $\hat{\beta}_i$ having information about risk-factors (here MKTRF, SMB and HML) spanning the discount factor $m$?