# The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties

## Abstract

**:**

## 1. Introduction

## 2. The Gibbons, Ross, and Shanken Test Statistic and Its Relevance

_{1}= α

_{2}= … = α

_{n}= 0. Ryan et al. [12], utilizing the GRS-test, look at the average value of absolute intercepts, α

_{i}= 0 to test whether the regression intercepts are jointly equal to zero, with the idea being that the intercept is indistinguishable from zero if an asset pricing model completely captures the expected returns (in which case the portfolio is efficient). Suarez and Alonso-Conde [13] looked at an entropy-based decomposition that captures the divergence between the factor-mimicking portfolio and the minimum-variance pricing kernel as distinct from quadratic test statistics, such as the GRS-test (determined as a function of pricing errors). Solórzano-Taborga et al. [14], utilize the GRS test for identifying restrictions (they termed ‘efficiency factor’) to test the null of asset pricing errors equaling zero. Barillas et al. [15] utilized the GRS test to accommodate the comparison of non-nested models as in a squared Sharpe-ratio (Sharpe [16]). Kamstra and Shi [17] rigorously generalized the Sharpe ratio-based interpretation of the GRS test to the multiple portfolio case but also suggested modifications to it when extended to multiple factors.

^{2}); $f\left(x\right)=\frac{1}{\mathsf{\sigma}\sqrt{2\mathsf{\pi}}}{e}^{-\frac{1}{2}{\left(\frac{x-\mathsf{\mu}}{\mathsf{\sigma}}\right)}^{2}}$, homoscedastic with the diagonal elements jj and kk of two var-cov matrices A and B being equal, ${\sum}_{A}jj={\sum}_{B}kk,\forall j=k$, and uncorrelated $\rho \left({\u03f5}_{j},{\u03f5}_{k}\right)=0$. Kamstra and Shi [17] asserted that the GRS statistic can lead to higher failure rates especially when the returned model has K-factors, K > 1 and N-assets, N < N*; “the bias to over-reject is non-negligible in small samples”. The short-selling constraint prevents the replication of an investible benchmark index, thus invoking Roll’s Critique [7] of whether the benchmark is representative of the test portfolio. However, tests of mean-variance efficiency with no short sales constraints have been proposed by Basak et al. [20]. Kim and Robinson [21] also point out that perfect efficiency cannot exist in practice and that it would be unrealistic that all intercept values were jointly and exactly zero, hence they introduce an interval-based hypothesis testing and get lower rejection rates with the GRS-test.

_{1}= α

_{2}= … = α

_{n}= 0. Ryan et al. [12], utilizing the GRS test, looked at the average value of absolute intercepts, α

_{i}= 0 to test whether the regression intercepts are jointly equal to zero, with the idea being that the intercept is indistinguishable from zero if an asset pricing model completely captures the expected returns (in which case the portfolio is efficient). It can be surmised that there is a substantive link between the GRS test, and portfolio performance evaluation and that it has been well-researched. This paper abstracts from those empirical tests and seeks to provide a trigonometric interpretation of the GRS test.

## 3. Suggested Recharacterization of the GRS-Statistic

^{*}be the angle between tangent OM and the X-axis (Figure 1), and ${\mathsf{\varphi}}_{\mathrm{p}}$ be the angle between the segment OP and the X-axis.

_{o}(which indicates that the test portfolio P is efficient), and ϕ

^{*}is the angle between the tangency portfolio and the X-axis:

_{p}is the angle between the test portfolio and the X-axis,

_{i}≡ x

_{i}generate a $\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}{x}_{1}.{x}_{2}\to \mathrm{v}\mathrm{a}\mathrm{r}\left(x\right).$

## 4. A Simulated Efficient Frontier and a Test of Portfolio Efficiency

^{2}) and having a p.d.f.:

_{i}was successively set at 0, 2, 4, 6, 8, 10 and σ

_{i}was set at 0, 5, 10, 15, 20, 25.

**x**$\ge 0$ is the short-selling constraint.

_{1}and x

_{2}respectively. Another approach, called the Risk parity based allocation approaches does not require an estimate of the return vectors. Lee [40] outlined a process that is elaborated in Appendix A.4).

^{2}) equals the sum of the values of the perpendicular (p

^{2})-squared and the base-squared (b

^{2}), i.e., h

^{2}= p

^{2}+ b

^{2}. For example, for the coordinates of the EW portfolio and the M portfolio (8, 4.67) and (12, 5.05); the values of the angles stretched by the two given coordinates are found as follows:

^{2}should be close to 1. Larger values of ψ

^{2}imply portfolio inefficiency arising out of the increased distance between the test portfolio and the global MV efficient portfolio on the frontier ($W={\mathsf{\Psi}}^{2}-1\to 0$ implies efficiency)

**.**In other words, for values of W close to zero, the test portfolio cannot be called inefficient (visual implementation in Appendix A.5).

_{0}: Portfolio is efficient.

_{0}is Rej. H

_{0}, iff. F(X

_{F}, N, T-N-1) < a threshold p-value. For the two portfolios with the given sample (r, r

_{f}, σ) in the table below, the various parameters required to determine the GRS statistic are displayed below:

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. The Kuhn-Tucker [41] Saddle Point Theorem

#### Appendix A.2. Polar Plot of the Table of Sharpe Ratios and the GRS−W Statistic

GRS~W Stat | θ = (r − r_{f})/s | ||||||||

r\s | 10 | 12 | 14 | 16 | 10 | 12 | 14 | 16 | s/r |

5 | 0.198 | 0.276 | 0.328 | 0.364 | 0.4 | 0.333 | 0.286 | 0.25 | 5 |

5.5 | 0.15 | 0.2374 | 0.297 | 0.339 | 0.45 | 0.375 | 0.321 | 0.281 | 5.5 |

6 | 0.101 | 0.198 | 0.265 | 0.313 | 0.5 | 0.417 | 0.357 | 0.313 | 6 |

6.5 | 0.053 | 0.158 | 0.232 | 0.285 | 0.55 | 0.458 | 0.393 | 0.344 | 6.5 |

7 | 0.005 | 0.117 | 0.198 | 0.257 | 0.6 | 0.5 | 0.429 | 0.375 | 7 |

7.5 | 0.077 | 0.163 | 0.228 | 0.542 | 0.464 | 0.406 | 7.5 | ||

8 | 0.037 | 0.129 | 0.198 | 0.583 | 0.5 | 0.438 | 8 | ||

8.5 | 0.094 | 0.168 | 0.536 | 0.469 | 8.5 | ||||

9 | 0.06 | 0.137 | 0.571 | 0.5 | 9 | ||||

9.5 | 0.025 | 0.107 | 0.607 | 0.531 | 9.5 | ||||

10 | 0.077 | 0.563 | 10 | ||||||

10.5 | 0.047 | 0.594 | 10.5 |

#### Appendix A.3. Values of the GRS-W Statistic for a Range of Angles

Test Port | |||||||||||||

ANGLE | 0 | 10 | 20 | 30 | 40 | 45 | 50 | 60 | 70 | 80 | 90 | ||

mkt tangency port | cosine (x^{o}) | 1.00 | 0.98 | 0.94 | 0.87 | 0.77 | 0.71 | 0.64 | 0.50 | 0.34 | 0.17 | 0.00 | |

0 | 1.00 | 0.000 | |||||||||||

10 | 0.98 | 0.031 | 0.000 | ||||||||||

20 | 0.94 | 0.132 | 0.098 | 0.000 | |||||||||

30 | 0.87 | 0.333 | 0.293 | 0.177 | 0.000 | ||||||||

40 | 0.77 | 0.704 | 0.653 | 0.505 | 0.278 | 0.000 | |||||||

45 | 0.71 | 1.000 | 0.940 | 0.766 | 0.500 | 0.174 | 0.000 | ||||||

50 | 0.64 | 1.420 | 1.347 | 1.137 | 0.815 | 0.420 | 0.210 | 0.000 | |||||

60 | 0.50 | 3.000 | 2.879 | 2.532 | 2.000 | 1.347 | 1.000 | 0.653 | 0.000 | ||||

70 | 0.34 | 7.549 | 7.291 | 6.549 | 5.411 | 4.017 | 3.274 | 2.532 | 1.137 | 0.000 | |||

80 | 0.17 | 32.163 | 31.163 | 28.284 | 23.873 | 18.461 | 15.582 | 12.702 | 7.291 | 2.879 | 0.000 | ||

90 | 0.00 | 2.6,10+32 | 2.58,10+32 | 2.35,10+32 | 1.99,10+32 | 1.56,10+32 | 1.33,10+32 | 1.10,10+32 | 6.66,10+31 | 3.12,10+31 | 8.04,10+30 | 0.000 |

#### Appendix A.4. Risk Parity: Avoiding the Problem of Error Maximization in a Mean-Variance Optimization Framework

_{12}is the correlation, σ

_{1}, σ

_{2}and σ

_{p}the standard deviations of the assets and the portfolio, and non-negative weights such that w

_{1}+ w

_{2}=1, the problem reduces to equating PCTR

_{i}= PCTR

_{j}$\forall i\ne j$:

_{i}, the weight of asset i is inversely proportional to its beta β

_{i}. Choueifaty and Coignard [55], also abstract from the classic mean-variance optimized portfolios and thereafter developed risk-parity based strategies.Yu [56], in a comprehensive study on the leading methodological issues surrounding CAPM modeling identified nine major mathematical issues that affect beta estimation, of which Agrrawal [46] and Fama and French [3] link to frequency, interval and dynamic weighting constructs that ultimately affect portfolio efficiency measures. In a recent paper, de Jong and diBartolomeo [57], discuss evolving implications of optimization that deal with new alpha sources emanating from multiple performance sources and portfolio efficiency measures. Stone et al. [58] find evidence that points to a fundamental revision in the theory of the relationship between cash levels in an financial system and central bank interest rates; this paper fundamentally recasts the GRS-W statistic based on its trigonometric properties.

#### Appendix A.5. Risk Parity: Efficient Portfolio Zone as a Floating Hyperplane, the Tangency Portfolio and the GRS-W Statisctic Gradient -3D Efficient Frontier with Actual ETFs

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**Figure 1.**The geometric basis of the GRS-W test statistic. The test coordinates of the EW portfolio (P) and the tangency portfolio (M) are plotted relative to the mean-variance efficient frontier. The further away P is from the tangency portfolio, the less efficient the portfolio P is.

**Figure 2.**Results of an actual simulation are plotted here, the MV portfolio (red square) and the test portfolio (triangular orange) are also utilized to derive the actual GRS−W statistic value (Table 2). The R

_{f}rate is 1%.

**Figure 3.**A plot of the left panel of Table 1, showing the spike in the GRS−W as variance approaches 16% with a low return of 5%, high “W stat” values imply inefficiency of the test portfolio.

GRS~W Stat | θ = (r − r_{f})/s | ||||||||
---|---|---|---|---|---|---|---|---|---|

r\s | 10.00 | 12.00 | 14.00 | 16.00 | 10.00 | 12.00 | 14.00 | 16.00 | s/r |

5.00 | 0.198 | 0.276 | 0.328 | 0.364 | 0.400 | 0.333 | 0.286 | 0.250 | 5.00 |

5.50 | 0.150 | 0.2374 | 0.297 | 0.339 | 0.450 | 0.375 | 0.321 | 0.281 | 5.50 |

6.00 | 0.101 | 0.198 | 0.265 | 0.313 | 0.500 | 0.417 | 0.357 | 0.313 | 6.00 |

6.50 | 0.053 | 0.158 | 0.232 | 0.285 | 0.550 | 0.458 | 0.393 | 0.344 | 6.50 |

7.00 | 0.005 | 0.117 | 0.198 | 0.257 | 0.600 | 0.500 | 0.429 | 0.375 | 7.00 |

7.50 | 0.077 | 0.163 | 0.228 | 0.542 | 0.464 | 0.406 | 7.50 | ||

8.00 | 0.037 | 0.129 | 0.198 | 0.583 | 0.500 | 0.438 | 8.00 | ||

8.50 | 0.094 | 0.168 | 0.536 | 0.469 | 8.50 | ||||

9.00 | 0.060 | 0.137 | 0.571 | 0.500 | 9.00 | ||||

9.50 | 0.025 | 0.107 | 0.607 | 0.531 | 9.50 | ||||

10.00 | 0.077 | 0.563 | 10.00 | ||||||

10.50 | 0.047 | 0.594 | 10.50 |

**Table 2.**These represent a range of actual GRS-W test statistic values and the associated Sharpe ratio values for a set of mean-variance points (as applied in Figure 2). This table has the p-values as well. N = 30 and a 10-year weekly period of T = 520. The top panel of the table illustrates the values arrived in the first two rows of the larger table, with the first row corresponding to the tangency portfolio (7.93 r, 9.83 σ), which is also applied in Figure 2, as the tangency portfolio on the efficient frontier. Cells in bold text are the efficient portfolios with low GRS-W statistics.

Tangency Portfolio (*) | Test Portfolio (p) | ||||||||
---|---|---|---|---|---|---|---|---|---|

mean, r | 7.93 | 10.50 | |||||||

risk free, r_{f} | 1.00 | 1.00 | |||||||

sigma, s | 9.83 | 8.00 | |||||||

θ = (r − r_{f})/s | 0.705 | 1.188 | |||||||

GRS~W Stat | 0.046 | ||||||||

N | No. of Assets | 30 | |||||||

T | No. of Weekly Intervals | 520 | |||||||

X_{F} | 0.759 | ||||||||

p-value | Rej. H_{0} (is an Efficient Port) iff p~0 | 0.8202 | |||||||

Mean, r | Sigma, s | θ = (r − r_{f})/s | GRS~W Stat | N | T | XF | p-Value | ||

Tangency Portfolio (*) | 7.93 | 9.83 | 0.705 | No. of Assets | No. of Weekly Intervals | Rej. H_{0} (Efficient Port) iff p~0 | |||

Test Portfolio (p) | 10.50 | 16.00 | 0.594 | 0.047 | 30 | 520 | 0.763 | 0.81564721900 | Eff. |

Test Portfolio (p) | 3.00 | 10.00 | 0.2 | 0.3737 | 30 | 520 | 6.115 | 0.00000000000 | Not Efficient |

Test Portfolio (p) | 5.00 | 10.00 | 0.400 | 0.198 | 30 | 520 | 3.238 | 0.00000004498 | Not Efficient |

Test Portfolio (p) | 5.50 | 10.00 | 0.450 | 0.150 | 30 | 520 | 2.448 | 0.00004374410 | Not Efficient |

6.00 | 10.00 | 0.500 | 0.101 | 30 | 520 | 1.652 | 0.01746340000 | Not Efficient | |

6.50 | 10.00 | 0.550 | 0.053 | 30 | 520 | 0.861 | 0.68131012700 | Eff. | |

7.00 | 10.00 | 0.600 | 0.005 | 30 | 520 | 0.081 | 1.00000000000 | Eff. | |

Test Portfolio (p) | 5.00 | 12.00 | 0.333 | 0.276 | 30 | 520 | 4.513 | 0.00000000000 | Not Efficient |

5.50 | 12.00 | 0.375 | 0.237 | 30 | 520 | 3.884 | 0.00000000012 | Not Efficient | |

6.00 | 12.00 | 0.417 | 0.198 | 30 | 520 | 3.238 | 0.00000004498 | Not Efficient | |

6.50 | 12.00 | 0.458 | 0.158 | 30 | 520 | 2.580 | 0.00001444100 | Not Efficient | |

7.00 | 12.00 | 0.500 | 0.117 | 30 | 520 | 1.917 | 0.00279042500 | Not Efficient | |

7.50 | 12.00 | 0.542 | 0.077 | 30 | 520 | 1.256 | 0.16816038300 | Eff. | |

8.00 | 12.00 | 0.583 | 0.037 | 30 | 520 | 0.599 | 0.95601098600 | Eff. | |

Test Portfolio (p) | 5.00 | 14.00 | 0.286 | 0.328 | 30 | 520 | 5.366 | 0.00000000000 | Not Efficient |

5.50 | 14.00 | 0.321 | 0.297 | 30 | 520 | 4.862 | 0.00000000000 | Not Efficient | |

6.00 | 14.00 | 0.357 | 0.265 | 30 | 520 | 4.336 | 0.00000000000 | Not Efficient | |

6.50 | 14.00 | 0.393 | 0.232 | 30 | 520 | 3.793 | 0.00000000028 | Not Efficient | |

7.00 | 14.00 | 0.429 | 0.198 | 30 | 520 | 3.238 | 0.00000004498 | Not Efficient | |

7.50 | 14.00 | 0.464 | 0.163 | 30 | 520 | 2.674 | 0.00000646460 | Not Efficient | |

8.00 | 14.00 | 0.500 | 0.129 | 30 | 520 | 2.107 | 0.00067275200 | Not Efficient | |

8.50 | 14.00 | 0.536 | 0.094 | 30 | 520 | 1.539 | 0.03571892400 | Not Efficient | |

9.00 | 14.00 | 0.571 | 0.060 | 30 | 520 | 0.973 | 0.50892160100 | Eff. | |

9.50 | 14.00 | 0.607 | 0.025 | 30 | 520 | 0.413 | 0.99780363100 | Eff. | |

Test Portfolio (p) | 5.00 | 16.00 | 0.250 | 0.364 | 30 | 520 | 5.958 | 0.00000000000 | Not Efficient |

5.50 | 16.00 | 0.281 | 0.339 | 30 | 520 | 5.549 | 0.00000000000 | Not Efficient | |

6.00 | 16.00 | 0.313 | 0.313 | 30 | 520 | 5.117 | 0.00000000000 | Not Efficient | |

6.50 | 16.00 | 0.344 | 0.285 | 30 | 520 | 4.667 | 0.00000000000 | Not Efficient | |

7.00 | 16.00 | 0.375 | 0.257 | 30 | 520 | 4.202 | 0.00000000001 | Not Efficient | |

7.50 | 16.00 | 0.406 | 0.228 | 30 | 520 | 3.724 | 0.00000000053 | Not Efficient | |

8.00 | 16.00 | 0.438 | 0.198 | 30 | 520 | 3.238 | 0.00000004498 | Not Efficient | |

8.50 | 16.00 | 0.469 | 0.168 | 30 | 520 | 2.745 | 0.00000351764 | Not Efficient | |

9.00 | 16.00 | 0.500 | 0.137 | 30 | 520 | 2.249 | 0.00022052900 | Not Efficient | |

9.50 | 16.00 | 0.531 | 0.107 | 30 | 520 | 1.752 | 0.00899892900 | Not Efficient | |

10.00 | 16.00 | 0.563 | 0.077 | 30 | 520 | 1.256 | 0.16816038300 | Eff. | |

10.50 | 16.00 | 0.594 | 0.047 | 30 | 520 | 0.763 | 0.81564721900 | Eff. |

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## Share and Cite

**MDPI and ACS Style**

Agrrawal, P.
The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties. *Mathematics* **2023**, *11*, 2198.
https://doi.org/10.3390/math11092198

**AMA Style**

Agrrawal P.
The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties. *Mathematics*. 2023; 11(9):2198.
https://doi.org/10.3390/math11092198

**Chicago/Turabian Style**

Agrrawal, Pankaj.
2023. "The Gibbons, Ross, and Shanken Test for Portfolio Efficiency: A Note Based on Its Trigonometric Properties" *Mathematics* 11, no. 9: 2198.
https://doi.org/10.3390/math11092198