A simplified approach to benchmark, relative return, and asset pricing

ABSTRACT

This analysis provides a much simpler and more intuitive derivation of Bergeron’s (2021) benchmark model. Bergeron states that there are no assumptions regarding arbitrage and equilibrium, but this analysis shows that the same result obtained under no-arbitrage equilibrium conditions. Missing from Bergeron’s analysis is an extension to Black’s (1972) zero-beta CAPM so that result is presented here. The analysis concludes with a simple empirical example highlighting the importance of choosing a benchmark that is mean-variance efficient.

KEYWORDS:

• Benchmarking
• asset pricing
• efficient portfolios
• arbitrage
• CAPM

JEL CLASSIFICATION:

• G10
• G11

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

¹ Bergeron describes the traditional CAPM (Sharpe, Lintner, and Treynor) as a special case but does not advance the model to Black’s Zero-Beta CAPM.

² The notation is identical to Bergeron (2021) and Campbell (2018) for easy comparison.

³ Equation (5)(5) $E_t\left({\tilde{R}}_{\mathit{i},\mathit{t}+1}\right)=E_t\left({\tilde{R}}_{\mathit{l},\mathit{t}+1}\right)+{\beta }_{\mathit{Bit}}{\lambda }_t$(5) is identical to Bergeron’s (2021) equation (29) on page 1501 but its development is much simpler with significantly fewer substitutions and relationships. Moreover, it demonstrates that the Bergeron model implicitly assumes arbitrage and equilibrium conditions.

⁴ The zero-beta asset is differentiated from a riskless asset in that the riskless asset has a beta of zero and a standard deviation of zero. The standard deviation for ‘z’ may be greater than or equal to zero but its systematic risk is zero, usually to three decimal places. In Table 1 its standard deviation is slightly above zero.

Table 1. Basic statistics.

⁵ Difference of Means Between RUI and SPY.

For a simple difference of means test with different variances, compute $\mathit{t}$, where

$\mathit{t}=\frac{\left(x_1-x_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}}$

The value is $\mathit{t}=.365$ which is less than the critical value of $\mathit{t}=2.334$ at $\propto =.01$ with 478 degrees of freedom and a hypothesized difference of means of 0.

Difference of Variances for RUI and SPY

For a difference in variances compute an F statistic, where

$\mathit{F}=\frac{S_1^2}{S_2^2}$

The computed value is $\mathit{F}=1.011$ which is less than the critical value of $\mathit{F}=1.238$ at $\propto =.01$ with 239 degrees of freedom for both the numerator and denominator.

Both tests fail to reject the null hypotheses that RUI and SPY are statistically identical in terms of mean and variance.