# An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns

## Abstract

**:**

## 1. Introduction

## 2. Research Method

_{it}= γ

_{0t}+ Σ

_{m=1}

^{M}γ

_{mt}z

_{mt−1}+ u

_{it}

_{it}is the excess return on asset i at time t, z

_{mt−1}is the value of the mth stock characteristic of asset i at time t−1, and u

_{it}is a random error term of asset i at time t. The Fama and MacBeth cross-sectional regression assumes a linear functional form between the excess returns and stock characteristics. It is possible that nonlinearities are important in the releations between stock characteristics and excess returns. Kirby (2015) and Freyberger, Neuhier and Weber (Freyberger et al. 2017) provide different approaches to examine this issue. The expected excess returns are given by:

_{it}) = γ

_{0}+ Σ

_{m=1}

^{M}γ

_{m}z

_{mt−1}

_{0}and γ

_{m}are the time-series averages of the monthly γ

_{0t}and γ

_{mt}coefficients. Each month during the sample period, all stocks are ranked on the basis of their E(r

_{it}) and grouped into quintile portfolios as in Fama and French (2015). I then calculate the value weighted portfolio excess returns for each quintile portfolio. Where a security has missing return data during the month due to temporary suspension or death, I code the missing returns to zero as in Liu and Strong (2008). I correct for the delisting bias of Shumway (1997) by assigning a −100% return if the death is deemed valueless4 as in Dimson, Nagel and Quigley (Dimson et al. 2003).

_{0}and γ

_{m}implies that investors cannot implement these as portfolio strategies. As a result, my study focuses on in-sample performance rather than out-of-sample performance. Fama and French point out that using full sample slopes has much greater precision compared to using rolling window estimates. Likewise if the portfolios are formed using monthly γ

_{0t}and γ

_{mt}coefficients, then most of the spread in portfolio returns will be due to unexpected returns and not due to expected return patterns. Lewellen examines the predictive ability of expected excess returns using the Fama and MacBeth approach.

_{b}is the (2N, 1) vector of optimal weights in the benchmark investment universe, where the first N cells are zero and the remaining N cells are the optimal weights of the N risky assets in the benchmark investment universe.

_{b}

^{1/2},θ

_{b}= x

_{b}’u/(x

_{b}’Vx

_{b})

^{1/2}, u is a (2N, 1) vector of expected excess returns, and V is a (2N, 2N) covariance matrix. The DSharpe measure captures the increase in Sharpe performance in adding the quintile portfolios formed using expected excess returns from the extended model of stock characteristics to the benchmark investment universe. If the additional stock characteristics make no incremental contribution to the investment opportunity set, then DSharpe = 0. I estimate the DSharpe measures for the case where unrestricted short selling is allowed and for the case where no short selling is allowed in the risky assets. When the risk-free asset exists, all optimal portfolios (which are combinations of the risk-free asset and the tangency portfolio) have the same Sharpe performance. As a result, the DSharpe measure can be estimated using any optimal portfolio on the corresponding mean-variance frontiers of the benchmark and augmented investment universes. I estimate the optimal portfolios using a given value of risk aversion, which I set equal to 3 as in Tu and Zhou (2011).

_{s}and V

_{s}as the sample moments of the expected excess returns and covariance matrix, and r as the (T, 2N) matrix of excess returns of the risky assets. The posterior probability density function is given by:

_{s},T)•p(V|V

_{s}, T)

_{s}, T) is the conditional distribution of a multivariate normal (u

_{s}, (1/T)V) distribution and p(V|V

_{s}, T) is the marginal posterior distribution that has an inverse Wishart (TV, T − 1) distribution (Zellner 1971).

_{s}, T − 1) distribution. Second, a random u vector is drawn from a multivariate normal (u

_{s}, (1/T)V) distribution. Third, given the u and V from steps 1 and 2, the DSharpe measure is estimated from Equation (3)9. Fourth, steps 1 to 3 are repeated 1000 times as in Hodrick and Zhang (2014) to generate the approximate posterior distribution of the DSharpe measure.

## 3. Data

#### 3.1. Size

#### 3.2. Book-to-Market (BM) Ratio

#### 3.3. Momentum

#### 3.4. Stock Issues

#### 3.5. Accruals

#### 3.6. Profitability

#### 3.7. Asset Growth

## 4. Empirical Results

^{2}column is the time-series average of the adjusted R

^{2}from the monthly cross-sectional regressions.

^{2}column is the time-series average of the adjusted R

^{2}from the monthly cross-sectional regressions. Panel A of the table reports the cross-sectional regression results where each characteristic is included individually. Panels B and C report the cross-sectional regressions using the model 1 characteristics and the model 2 characteristics respectively.

^{2}column is the time-series average of the adjusted R

^{2}from the monthly cross-sectional regressions. Panel A of the table reports the cross-sectional regression results where each characteristic is included individually. Panels B and C report the cross-sectional regressions using the model 1 characteristics and the model 2 characteristics respectively.

## 5. Conclusions

## Conflicts of Interest

## References

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1 | |

2 | Almazan, Brown, Carlson, and Chapman (Almazan et al. 2004) find that only a tiny fraction of U.S. mutual funds engage in short selling. |

3 | Qing and Turner (2014) present a novel study which examines the impact of stock characteristics in the London market between 1825 and 1870. |

4 | Using the death event information on the London Share Price Database (LSPD) provided by London Business School. |

5 | A number of studies examine whether there is mean-variance spanning between two mean-variance frontiers when there is no risk-free asset. De Roon and Nijman (2001) and Kan and Zhou (2012) provide excellent reviews of alternative tests of mean-variance spanning when unrestricted short selling is allowed. De Roon et al. (2001) develop the corresponding tests of mean-variance spanning when there are no short selling constraints and transaction costs. |

6 | Recent applications of the Bayesian approach include Hodrick and Zhang (2014) and Liu (2016) in tests of the benefits of international diversification. |

7 | Basak et al. (2002) point out using a linear function may lead to a large approximation error when no short selling constraints are imposed. Basak et al find that the standard error of their mean-variance inefficiency measure increases when no short selling constraints are imposed, which is the opposite of Wang (1998) and Li et al. (2003). |

8 | We can view the normality assumption as a working approximation to monthly excess returns. A non-parametric test along the lines of Ledoit and Wolf (2008) could address this issue in future research. |

9 | If the optimal portfolios lie on the inefficient side of the mean-variance frontier, I set the corresponding Sharpe performance to zero. |

10 | Investment trusts are equivalent to U.S. closed-end funds. |

11 | Fama and French (2008) examine the same group of characteristics in their study in U.S. stock returns. |

12 | The predictive ability of the profitability characteristic is highly sensitive to the profitability measure used. Using the alternative profitability measures in Fama and French (2008, 2015) and Lewellen (2015), the average spreads can be tiny or even turn significantly negative. |

13 | Kirby (2015) uses the time-series of the monthly spreads as the set of payoffs to evaluate candidate stochastic discount factor models. This approach is used to examine whether the magnitude of the average spreads are consistent with asset pricing models. See also the related study by Back, Nishad and Ostdiek (Back et al. 2015). This approach can be used to examine whether the predictive ability of stock characteristics can be captured by risk factors. |

14 | Li et al. (2003) note a similar problem in the mean-variance spanning test of De Roon et al. (2001). |

15 | This result stems from the fact that no short selling constraints mitigate the impact of estimation risk in covariance matrix estimators with large sampling error such as the sample covariance matrix. |

16 | Gregory, Tharyan and Christidis (Gregory et al. 2013) use the largest 350 UK stocks to form the factors in the Fama and French (1993) and Carhart (1997) linear factor models. |

Characteristics | Mean | Standard Deviation | N |
---|---|---|---|

Excess return | 0.687 | 18.109 | 1647 |

Size | 10.508 | 2.075 | 1832 |

BM | −0.659 | 1.098 | 1292 |

Momentum | 0.130 | 0.507 | 1422 |

Stock issues | 0.253 | 0.511 | 1325 |

Accruals | 0.010 | 0.762 | 1158 |

Profitability | 0.354 | 0.282 | 1261 |

Asset growth | 0.031 | 0.408 | 1394 |

Panel A: Individual | Slope | t-Statistic | R^{2} |

Size | −0.185 | −4.09 ^{1} | 0.007 |

BM | 0.231 | 3.56 ^{1} | 0.007 |

Momentum | 0.888 | 4.28 ^{1} | 0.009 |

Stock issues | −0.462 | −3.72 ^{1} | 0.004 |

Accruals | −0.120 | −1.46 | 0.002 |

Profitability | 0.672 | 3.76 ^{1} | 0.003 |

Asset growth | −0.849 | −5.66 ^{1} | 0.004 |

Panel B: Model 1 | Slope | t-Statistic | R^{2} |

Size | −0.138 | −3.42 ^{1} | 0.024 |

BM | 0.324 | 5.84 ^{1} | |

Momentum | 1.312 | 7.06 ^{1} | |

Panel C: Model 2 | Slope | t-Statistic | R^{2} |

Size | −0.095 | −2.37 ^{1} | 0.036 |

BM | 0.394 | 5.46 ^{1} | |

Momentum | 1.306 | 7.30 ^{1} | |

Stock issues | −0.358 | −2.58 ^{1} | |

Accruals | −0.160 | −1.19 | |

Profitability | 0.807 | 5.01 ^{1} | |

Asset growth | −0.699 | −3.96 ^{1} |

^{1}Significant at 5%.

Panel A: Unconstrained | Mean | Standard Deviation | 5% | Median |

Momentum | 0.164 | 0.046 | 0.090 | 0.162 |

BM | 0.117 | 0.039 | 0.057 | 0.114 |

Size | 0.059 | 0.025 | 0.023 | 0.056 |

Panel B: Constrained | Mean | Standard Deviation | 5% | Median |

Momentum | 0.090 | 0.034 | 0.036 | 0.089 |

BM | 0.010 | 0.011 | 0 | 0.005 |

Size | 0.019 | 0.019 | 0 | 0.014 |

Panel C | Bench | Augment | ||

Momentum | −2.390 | −8.829 | ||

BM | −1.065 | −12.863 | ||

Size | −3.017 | −5.348 |

Panel A: Unconstrained | Mean | Standard Deviation | 5% | Median |

Stock issues | 0.073 | 0.030 | 0.029 | 0.071 |

Accruals | 0.030 | 0.018 | 0.007 | 0.026 |

Profitability | 0.092 | 0.034 | 0.041 | 0.089 |

Asset Growth | 0.063 | 0.028 | 0.023 | 0.061 |

All | 0.151 | 0.043 | 0.085 | 0.149 |

Panel B: Constrained | Mean | Standard Deviation | 5% | Median |

Stock issues | 0.012 | 0.011 | 0 | 0.009 |

Accruals | 0.005 | 0.007 | 0 | 0.002 |

Profitability | 0.031 | 0.015 | 0.006 | 0.030 |

Asset Growth | 0.012 | 0.010 | 0 | 0.010 |

All | 0.051 | 0.022 | 0.015 | 0.051 |

Panel C | Bench | Augment | ||

Stock issues | −2.730 | −10.119 | ||

Accruals | −2.722 | −4.528 | ||

Profitability | −2.711 | −6.753 | ||

Asset Growth | −2.728 | −10.565 | ||

All | −2.716 | −7.590 |

Panel A: Individual | Slope | t-Statistic | R^{2} |

Size | −0.013 | −0.28 | 0.014 |

BM | 0.088 | 1.11 | 0.019 |

Momentum | 1.381 | 5.02 ^{1} | 0.033 |

Stock issues | −0.829 | −4.56 ^{1} | 0.010 |

Accruals | 0.042 | 0.34 | 0.005 |

Profitability | 0.364 | 1.75 ^{2} | 0.009 |

Asset growth | −0.935 | −3.97 ^{1} | 0.012 |

Panel B: Model 1 | Slope | t-Statistic | R^{2} |

Size | 0.002 | 0.049 | 0.053 |

BM | 0.181 | 2.84 ^{1} | |

Momentum | 1.364 | 5.38 ^{1} | |

Panel C: Model 2 | Slope | t-Statistic | R^{2} |

Size | −0.025 | −0.52 | 0.074 |

BM | 0.288 | 3.53 ^{1} | |

Momentum | 1.599 | 6.14 ^{1} | |

Stock issues | −0.660 | −3.35 ^{1} | |

Accruals | −0.078 | −0.49 | |

Profitability | 0.510 | 2.62 ^{1} | |

Asset growth | −0.539 | −2.00^{1} |

^{1}Significant at 5%;

^{2}Significant at 10%.

Panel A: Unconstrained | Mean | Standard Deviation | 5% | Median |

Momentum | 0.176 | 0.048 | 0.100 | 0.176 |

BM | 0.071 | 0.032 | 0.026 | 0.067 |

Size | 0.050 | 0.024 | 0.017 | 0.047 |

Panel B: Constrained | Mean | Standard Deviation | 5% | Median |

Momentum | 0.030 | 0.019 | 0.001 | 0.029 |

BM | 0.005 | 0.007 | 0 | 0.002 |

Size | 0.002 | 0.005 | 0 | 0 |

Panel C | Bench | Augment | ||

Momentum | −0.455 | −4.918 | ||

BM | −1.918 | −7.886 | ||

Size | −2.142 | −5.100 |

**Table 7.**Posterior Distribution of the DSharpe Measure of the Additional Model 2 Characteristics: Large Stocks.

Panel A: Unconstrained | Mean | Standard Deviation | 5% | Median |

Stock issues | 0.046 | 0.024 | 0.013 | 0.042 |

Accruals | 0.039 | 0.021 | 0.011 | 0.036 |

Profitability | 0.057 | 0.028 | 0.016 | 0.055 |

Asset Growth | 0.028 | 0.017 | 0.007 | 0.025 |

All | 0.045 | 0.025 | 0.012 | 0.042 |

Panel B: Constrained | Mean | Standard Deviation | 5% | Median |

Stock issues | 0.013 | 0.010 | 0 | 0.012 |

Accruals | 0.006 | 0.007 | 0 | 0.003 |

Profitability | 0.016 | 0.012 | 0 | 0.013 |

Asset Growth | 0.001 | 0.003 | 0 | 0 |

All | 0.017 | 0.013 | 0 | 0.015 |

Panel C | Bench | Augment | ||

Stock issues | −2.344 | −5.705 | ||

Accruals | −2.336 | −5.378 | ||

Profitability | −2.340 | −3.792 | ||

Asset Growth | −2.340 | −5.608 | ||

All | −2.342 | −4.191 |

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**MDPI and ACS Style**

Fletcher, J.
An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns. *Int. J. Financial Stud.* **2017**, *5*, 21.
https://doi.org/10.3390/ijfs5040021

**AMA Style**

Fletcher J.
An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns. *International Journal of Financial Studies*. 2017; 5(4):21.
https://doi.org/10.3390/ijfs5040021

**Chicago/Turabian Style**

Fletcher, Jonathan.
2017. "An Empirical Examination of the Incremental Contribution of Stock Characteristics in UK Stock Returns" *International Journal of Financial Studies* 5, no. 4: 21.
https://doi.org/10.3390/ijfs5040021