Predicted and unpredicted returns across styles

The empirical results in the previous section suggest that the relative style returns based on firm characteristics PC, DY, MTBV and MV may be caused by the business cycle risks or investors’ underreaction to specific asset classes. This section explores the relative importance of the predicted and unpredicted component from the business cycle model in explaining the style return premiums.

Recall that Equation (7) predicts the one-month-ahead single stock returns. The predicted return of stock i for a given point of time t is:

̂ ̂ ̂ ̂ ̂ (8)

Where ̂ is the vector of estimated coefficients obtained from a time-series recursive regression based on the 60-month rolling window that contains stocks with at least 24 months return data.

Equation (8) stands for exact pricing specification and the unpredicted return portion of Equation (7) is ̂ , representing stock returns adjusted for the business cycle risk. The estimated intercept of Equation (7) is excluded from the explained portion of Equation (7). Chordia and Shivakumar (2002) argue that this time-varying intercept may capture some of the return patterns in the formation periods and therefore could lead to control for the cross-sectional variations in average returns that are unrelated with the business cycles.

To better understand the dynamics of predicted and unpredicted stock returns around the portfolio formation point, Figure 3-3 plots the median predicted and unpredicted returns for stocks within quintiles 1, 3 and 5. The quintiles are formed the same as in Table 3-3. For brevity only styles based on formation and testing period (12, 6) are presented. For a given stock i in each month t, the model parameters are estimated using equation (7) based on the observations from months t19 to t1. Using the estimated coefficients, the predicted

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returns for that stock from time period t18to t5 _{are recorded and}

the above procedures are repeated until all the stocks in that quintile are covered. If economic exogenous forces are the key factor affecting equity style returns over time, one would expect to see that the business cycle model predicts stock returns in a consistent and systematic way.

Figure 3-3 suggests that the predicted and unpredicted stock returns from the business cycle model seem to vary systematically across different quintiles. For quintiles sorted on characteristics PC and MTBV, the predicted portions are systematically lower for value stocks (Q1) than for growth stocks (Q5) around the formation period, and the unpredicted returns of value stocks appear to be systematically larger than growth stocks before and after the formation point. Such systematic patterns are strongest for size quintiles. This suggests that the macroeconomic variables are unable to capture the divergent return patters of stocks across quintiles sorted on PC, MTBV and MV. Instead, the pricing errors, namely the business cycle risk-adjusted returns, point to the right sign of observed size and value premiums.

However, stocks sorted on equity characteristics DY seem to tell a different story. The predicted returns of value stocks in DY quintiles are always larger than growth stocks before and in the formation period, and the unpredicted returns of value stocks are smaller than growth stocks. Although the business cycle model predicts that small size value stocks of high dividend yield do not outperform in the testing period, larger size value stocks could comfortably outperform growth stocks. Moreover, consistent with the evidence of strong value premium based on realized returns of DY quintiles, business cycle risk adjusted value premiums in the testing periods are negative, indicating that the business cycle model could indeed capture the dynamics of relative stock returns across DY quintiles.

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In summary, given the evidence of significant size and value premiums based on the realised stock returns, it is tempting to conclude that the relative returns for stocks in quintiles sorted on firm characteristic of PC, MTBV and MV are mainly determined by the unpredicted portions of the business cycle model, while the divergent style return for stocks sorted by characteristics of DY are captured by Equation (7). Hence value premiums based on characteristics PC and MTBV, and the size premium in the U.K. stock market are likely due to the mispricing of stock prices relative to common risk factors. But the outperformance of value stocks characterised by high DY values is likely to be driven by business cycle conditions, and therefore such value premium may be interpreted as the compensation for bearing business cycle risks.

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Figure 3-3 Median predicted and unpredicted returns around formation period

In each month t, all U.K. non-financial stocks are classified into 5 quintiles in ascending order based on the average previous J-month characteristics PC, DY, MTBV and MV. Each stock must have at least 24-month observations and the expected return of individual stock is estimated by Equation (7) using a set of economic pervasive variables relating to the business cycles. This Figure depicts the median predicted and unpredicted returns of quintile portfolios Q1, Q3 and Q5 for the 6-month holding period around the 12-month formation period (i.e. from t-18 to t+5 month, J = 12, K = 6). It is suggested that the unpredicted return components from the business cycle model vary systematically across quintiles based on PC, MTBV and MV, while the business cycle model captures the variations on average returns in DY quintiles. -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

PC Quintiles BS Model Predicted Returns

Q1_EW (Value) Q3_EW Q5_EW (Growth) -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

PC Quintiles BS Model Unpredicted Returns

Q1_EW (Value) Q3_EW Q5_EW (Growth) -0.05 -0.04 -0.04 -0.03 -0.03 -0.02 -0.02 -0.01 -0.01 0.00 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

DY Quintiles BS Model Predicted Returns

Q5_EW (Value) Q3_EW Q1_EW (Growth) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

DY Quintiles BS Model Unpredicted Returns

Q5_EW (Value) Q3_EW -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MTBV Quintiles BS Model Predicted Returns

Q1_EW (Value) Q3_EW Q5_EW (Growth) -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MTBV BS Model Unpredicted Returns

Q1_EW (Value) Q3_EW

95 Figure 3-3 (continued) -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MV Quintiles BS Model Predicted Returns

Q1_EW (Small) Q3_EW Q5_EW (Large) -0.05 0.00 0.05 0.10 0.15 0.20 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MV Quintiles BS Model Unpredicted Returns

Q1_EW (Small) Q3_EW 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

PC Quintiles BS Model Predicted Returns

Q1_VW (Value) Q3_VW -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

PC Quintiles BS Model Unpredicted Returns

Q1_VW (Value) Q3_VW 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

DY Quintiles BS Model Predicted Returns

Q5_VW (Value) Q3_VW Q1_VW (Growth) -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

DY Quintiles BS Model Unpredicted Returns

Q5_VW (Value) Q3_VW 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MTBV Quintiles BS Model Predicted Returns

Q1_VW (Value) Q3_VW -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MTBV BS Model Unpredicted Returns

Q1_VW (Value) Q3_VW Q5_VW (Growth) -0.20 -0.15 -0.10 -0.05 0.00 0.05 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MV Quintiles BS Model Predicted Returns

Q1_VW (Small) Q3_VW Q5_VW (Large) -0.05 0.00 0.05 0.10 0.15 0.20 0.25 -19 -17 -15 -13 -11 -9 -7 -5 -3 -1 1 3 5

MV Quintiles BS Model Unpredicted Returns

Q1_VW (Small) Q3_VW Q5_VW (Large)

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3.4.4 Style premiums after adjusting for the predicted returns