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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 $m$?

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  • $\begingroup$ Interesting question. Knowing the (co)variances of the factors spanning the SDF allows you to estimate the volatility of the SDF. The beta of an asset return will be bounded by the ratio of asset return volatility by SDF volatility. However, the volatility of the test assets can be arbitrarily large? You can lever up every return and achieve every level of volatility? $\endgroup$
    – Kevin
    Dec 18, 2022 at 16:30
  • $\begingroup$ It's very confusing for me. $\lambda \in \mathbb{R}^n$ and $\lambda'$ is the transpose of $\lambda$? If yes, $\beta_i$ must be a vector in $\mathbb{R}^n$? And so you want a bound for all element of $\beta_i$ ? But then, the formula $$\beta_i = \frac{cov(R_i ^e, \lambda')}{var(\lambda')}$$ is meaningless unless $\beta_i \in \mathbb{R}$ (a scalar, not a vector) $\endgroup$
    – NN2
    Dec 18, 2022 at 22:43
  • $\begingroup$ @NN2 i am using the notation of Cochrane (Asset pricing), p. 106-109 and p. 16-19. $\endgroup$ Dec 23, 2022 at 9:54

1 Answer 1

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I do not think such an upper bound exists for beta in terms of moments of the SDF alone. The upper bound of beta depends on the volatility of the SDF and the volatility of the asset returns. The former is given from its factor mimicking portfolio, the latter can be arbitrarily large.

Sharpe ratios

To start with, you write

Based on Hansen/Jagannathan, the set of means and variances of returns is limited.

This is not true because the Hansen and Jagannathan (1991) bound only limits assets' Sharpe ratios by moments of the SDF. There is no bound on the individual means and variances of asset returns.

The Hansen and Jagannathan (1991) bound essentially says: high Sharpe ratios only exist if the SDF is very volatile [if there's much aggregated risk]. Empirical asset pricing often applies the theorem to test whether a candidate SDF is volatile enough to match the observable Sharpe ratios in the data.

Trivial bound

A regression coefficient is bounded by assuming perfect correlation, \begin{align} \beta_i=\frac{Cov(R_i,M)}{Var[M]}\leq \frac{\sigma_i}{\sigma_M}. \end{align} Beta is bounded if $\sigma_i$ is bounded and if $\sigma_M$ isn't to close to zero.

Knowing the factors mimicking the SDF means we know the mean and variance of the SDF. The Hansen and Jagannathan (1991) bound tells that the true SDF needs to be sufficiently volatile. Otherwise, we can't justify the high Sharpe ratios in the data. So, the second concern about a low $\sigma_M$ is less of a concern.

However, note that some candidate SDFs are known for their low volatility. The CCAPM uses aggregate consumption growth as SDF, $M_{t,t+1}=\delta\left(\frac{C_{t+1}}{C_t}\right)^{-\gamma}$. However, in the data, consumption growth isn't very volatile. It's this low volatility which causes the sound rejection of the model. More modern macro-finance models include habits, long run risks or disaster risks to make their SDFs more volatile and better able to match the data.

Leverage

Returning to $\beta_i$, if we can rule out $\sigma_M$ running to zero, we can be assured that $\beta_i$ is bounded if $\sigma_i$ is. However, this is a problem. In your example, you use the famous 25 size/book-market portfolios as test assets. Other popular examples include 125 portfolios sorted on size/book-market/momentum or industry portfolios. However, we can choose our test assets fairly flexibly. Why not take any arbitrary portfolio and lever it up? This is still a tradable portfolio which the model should be able to price. However, by levering up a portfolio, we can arbitrarily increase its volatility (and beta). Crucially, note that levering doesn't impact the Hansen and Jagannathan (1991) bound which is concerned with Sharpe ratios, means divided by volatilities, and is insensitive to leverage.

Take your R code, replace the 25 portfolios by levered versions and see how your betas change. To bound the betas, you'd need to assume that investors can't arbitrarily lever up their portfolios. You'd need to restrict the set of valid test assets. There are surely models incorporating such frictions, but they're absent from the standard introduction of the SDF.

Easy illustration

To see the points more easily, let $R_i$ be an excess return and $\lambda>0$. Then, $R_\lambda=\lambda R_i$ is also a valid return and \begin{align} \mu_{\lambda} &= \lambda \mu_i, \\ \sigma_\lambda &= \lambda \sigma_i,\\ \beta_\lambda &= \lambda \beta_i, \\ \frac{\mu_\lambda}{\sigma_\lambda} &= \frac{\mu_i}{\sigma_i}. \end{align}

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  • $\begingroup$ My statement "Based on Hansen/Jagannathan, the set of means and variances of returns is limited." is a direct quote by Cochrane, Asset Pricing, p. 17 :-) I agree that there is no definite bound because of leverage (a similar statement i am quite often reading is "stock prices have no upper limit"). However, from an empirical point of view, what would you suggest to do to come up with an upper bound? Using 125 test portfolios and observe the maximum of the 125 betas? $\endgroup$ Dec 23, 2022 at 10:00
  • $\begingroup$ @skoestlmeier You know I love Cochrane, his research and his teaching material, and I know you do too. So I'd never criticise his book, but we both agree that he means the ratio of means and volatilities is limited. So the space of means and variances is indeed limited. You probability can't have an expected return of a million per cent with only 1 per cent volatility. Nonetheless, ultimately only Sharpe ratios are bounded, so you can have huge expected returns, if you're sufficiently risky $\endgroup$
    – Kevin
    Dec 23, 2022 at 10:16
  • $\begingroup$ @skoestlmeier Empirically, it depends on the research question and what you're trying to do. As we said, with leverage, we can achieve any beta. However, if we think about usual applications, I'd take the largest set of test assets possible. The 25 portfolios aren't nearly as broad as necessary here (particularly because book/market in particular gives rise to low volatility strategies). The 125 portfolios better. But why stop here? I'd include the entire data bases from Hou, Xue and Zhang (2020), Chen and Zimmermann (2022) and Jensen, Kelly and Pedersen (2022). $\endgroup$
    – Kevin
    Dec 23, 2022 at 10:20
  • $\begingroup$ This will cover hundreds of portfolios sorted along various dimensions of the cross-section. You might also want to play around with frequent rebalancing (monthly), all-sample breakpoints (not only NYSE), and equally weighted portfolios. This way, you'll end up with a huge number of test assets and a huge number of betas whose maximum you can determine. $\endgroup$
    – Kevin
    Dec 23, 2022 at 10:21
  • $\begingroup$ Of course, there's always a possibility to increase their volatility further, include market timing etc, But you'll get a decent idea where "usual" betas reach their maximum, for common portfolio construction. But I'd always keep in mind that it's (easily) possible to increase the beta. Another point to think about are conditional betas. There's time variation, and you may want to consider rolling windows or subsample analysis? But all of this would very much depend on your actual research question. $\endgroup$
    – Kevin
    Dec 23, 2022 at 10:24

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