2. Size Premium

**Figure A2.** Positive relationship between expected return and size premium.

Size premium is the di fference between small companies' average return and big companies' average return in a diversified portfolio or in the market. Normally, small companies have higher returns than the bigger ones. Because at the same period of time, small companies' profit is easy to grow faster than big companies and the growth rate of their dividend also is higher.

According to the DDM model, the price of security depends on the discounted present value of future dividend. In this formula, 'P' is the expected present price of one stock. ' *D*' is the dividend of this year and '*g*' is the constant growth rate of the dividend (it also is the growth rate of profit). For the last, '*i*' is the dividend interest rate, which may refer to risk-free rate.

$$\begin{array}{lcl} \mathrm{P} &= \lim\_{\mathbf{n} \to \infty} [\frac{D(1+\emptyset)}{1+i} + \frac{D(1+\emptyset)^2}{(1+i)^2} + \frac{D(1+\emptyset)^3}{(1+i)^3} + \dots + \frac{D(1+\emptyset)^n}{(1+i)^n}] \\ &= \lim\_{\mathbf{n} \to \infty} [\frac{D(1+\emptyset)}{1+i} \* \frac{\left(\frac{1+\emptyset}{1+i}\right)^n - 1}{\frac{1+\emptyset}{1+i} - 1}] \end{array}$$

Given by *i* > *g*,

$$\left(\frac{1+g}{1+i}\right)^m \to 0$$

So, P = *<sup>D</sup>*(<sup>1</sup>+*g*) *i*−*g*

$$\text{f.P.}\,P\_1 = \frac{D\_1(1+\mathcal{g}\_1)}{i-\mathcal{g}\_1},\,\,P\_2 = \frac{D\_2(1+\mathcal{g}\_1)}{i-\mathcal{g}\_1}$$

If we assume *D*1 = *D*2

Hence, the return rate of the stock can be represented by

$$R\_i = \frac{P\_2 - P\_1}{P\_1} = \frac{(\text{g2} - \text{g1})(i+1)}{(1 + \text{g1})(i - \text{g2})}$$

In terms of small companies and big companies, *g*1(*the growth rate o f first term*) *and i* (*risk* − *f ree rate*) *are the same*.

However, in the second term, with the control of other effects, growth rate (*g*2) of small companies is higher than the big one, so the expected return (*Ri*) of small companies is generally higher than the large one.

In other words, high returns of small companies indicate they also carry higher risk. If investors can suffer the risk brought by small companies, they can gain the risk premium which is called 'size premium'.

3. Book-to-Market Premium

**Figure A3.** Positive relationship between expected return and book-to-market premium.

Book-to-market premium is the difference between high B/M ratio companies' average return and low B/M ratio companies' average return in a diversified portfolio or in the market.

There is an effect named B/M effect which indicates that higher B/M ratio companies has higher excess return. It can be illustrated by prospect theory easily. However, in this case, the *x*-axis is MV and the *y*-axis is the value.

**Figure A4.** Relationship between true and cognitive value of stocks with the growth of MV/B.

For the higher B/M ratio companies, the MV/B is relative lower and people always overprice the true value of a stock, it leads to the demand for those companies is higher. With the growth of demand and stock price, the return of those stocks also is higher.

Vice versa, higher B/M ratio companies has lower expected return and excess return (expected return minus risk of free rate).

Therefore, if the investor prefers higher B/M ratio companies, they take the risk of B/M effect on one hand, they gain the B/M premium on the other hand.

4. Profitability Premium

**Figure A5.** Positive relationship between expected return and profitability premium.

Profitability premium is the difference between robust profitability (higher ROE) companies' average return and weak profitability (lower ROE) companies' average return in a diversified portfolio or in the market.

With the control of other effects, companies with robust profitability—measured by the level of ROE (return of equity)—outperform in their expected rate of return and take greater variance. This is because companies with high profits also distribute high dividends.

$$\mathbb{P} = \frac{D(1+\mathbb{g})}{i-\mathbb{g}}$$

According DDM (dividend discount model) formula, higher dividend means higher price and demand which will enhance the level of expected return. If an investor purchases companies with robust profit, they may ge<sup>t</sup> higher excess return and fluctuation at the same time. Vice versa, weak profitability companies bring people low excess return and risk.

### 5. Investment Growth Premium

**Figure A6.** Positive relationship between expected return and investment growth premium.

Investment growth premium is the difference between conservative (lower growth rate of investment or lower growth rate of assets) companies' average return and aggressive (higher growth rate of investment or higher growth rate of assets) companies average return in a diversified portfolio or in the market.

The reason why aggressive companies may have low excess return and risk is that these kinds of firms allocate more profit into reinvestment rather than dividends, thus it decreases the expected price and return, which leads to low risk. Vice versa, conservative companies bring higher excess return and risk because of larger amount of dividend rather investment.

6. Momentum Premium

**Figure A7.** Positive relationship between expected return and momentum premium.

Momentum premium is the difference between higher momentum (higher accumulated return) companies' average return and lower momentum (lower accumulated return) companies' average return in a diversified portfolio or in the market.

Momentum is the accumulated return in one quarter. The higher one means the stock is popular with high return and risk. Vice versa, people invest in low momentum companies with low premium and risk.

### 7. Asset Turnover Premium

**Figure A8.** Positive relationship between expected return and asset turnover premium.

Asset turnover premium is the difference between higher asset turnover companies' average return and lower asset turnover companies' average return in a diversified portfolio in the market.

Asset turnover is the total revenue divided by total asset. The higher one means the stock is popular with high return and risk. Vice versa, people invest in low momentum companies with low premium and risk. The regression analysis was conducted with a opposite direction.

### **Appendix B. Chi-Square Test of Industry in Di**ff**erent Factors**

Table A1 recorded the total chi-square value of 28 industries for seven factors. By conducting the chi-square test, we can find out the different effect of factors to various industries. When the total chi-square value is larger than the critical value (when df = 27), it means that each industry is responding differently to the specific factors. According to the result, six factors passed the test, while Rm-Rf is insignificant. Therefore, we need to discuss the effect of factors based on different industry.

**Table A1.** Chi-square test of industry in different factors.


### **Appendix C. Significance Level and Correlation E**ff**ect**

**Table A2.** Significance level of factors in different industries.



**Table A2.** *Cont.*

In this table, 0 represents that the factor is insignificant to the industry. The absolute value represents the percentage of the significant coefficients. A positive number represents that the factor has positive effect on the industry, while negative number represents that the factor has a negative effect on the industry.

### **Appendix D. Forecasting the Direction of Factors**

**Figure A9.** Use SMB to forecast the direction of SMB.

**Figure A11.** Use CMA to forecast the direction of CMA.

**Figure A13.** Use HML and CRMHL to forecast the direction of CRMHL.

**Figure A14.** Use RMW and AMLH to forecast the direction of AMLH.


**Figure A15.** Use RMW, AMLH, and Rm-Rf to forecast the direction of Rm-Rf.

### **Appendix E. Result of Stability Test**


**Figure A16.** Significance level of variables.

The figures in the red frame are the significance level of time, timeˆ2 and season.
