Question #4

This question consists of three parts, Parts 4.1, 4.2, and 4.3. All three parts are to be answered. For this purpose, the file ‘6923.Q4(20-2).xls’ contains 324 time-series observations of the following (by column):

• year and month (January 1974 to December 2000) (month 1 is January) (columns A & B)

• monthly returns to ten size-ranked portfolios of Australian stocks (columns H – Q) (note – as labeled, Portfolio 1 is comprised of the smallest stocks while Portfolio 10 is comprised of the largest stocks)

• returns to the equally-weighted and value-weighted market indices (columns R & S)

• the risk-free rate of return (column T)

• for columns D, E, F & G, the formulas reveal the calculations

These data should be used to answer each of the three parts of this question (Parts 4.1, 4.2, and 4.3)

Part 4.1

It is well documented in the Finance literature that average equity returns in January exceed those of other months. Further, the evidence suggests that this so-called January effect differs between portfolios of small stocks and large stocks.

Required:

(a) Using the regression model presented immediately below, separately estimate the January effect for the portfolio of the smallest stocks (Portfolio 1) and the portfolio of the largest stocks (Portfolio 10). For the regression, use the equal-weighted market return.

Rp,t – Rf,t = a1 + a2 Dt + bp ( RM,t – Rf,t ) + et (Q4.1)

where Rp,t  = the return to portfolio p in month t;

Rf,t   = the risk-free rate of return for month t;

RM,t  = the return to the market portfolio in month t; and

Dt = 1 for January and 0 otherwise.

Note, within the context of this model as specified:

Bp  = the beta of the particular size portfolio, p;

a1 = the average incremental abnormal return in non-January; and  

a2 = the incremental abnormal return in January.

(b) Present and estimate a single regression model that allows for a formal statistical test of whether there is a small firm effect in the data, and also whether the January effect differs between portfolios of small stocks and large stocks. Again, use the equal-weighted market return.

(i) Based upon your regression output, is there a small firm effect? 

(ii) Based upon your regression output, does the January effect differ between the small stock and large stock portfolios?

Hints: The model you develop should be a variant of the model above, modified to incorporate a dummy variable to capture size.

Your analysis should be based only on the portfolio of the smallest stocks (Portfolio 1) and the portfolio of the largest stocks (Portfolio 10).

To run the regression, the data for these two portfolios should be pooled. 

Part 4.2

In U.S. studies, it is well documented that average equity returns in January exceed those in other months. Tax-loss selling around the U.S. year-end is offered as an explanation. Interestingly, Australia also has a well-documented January effect, despite having a 30 June financial year-end.

Required:

(a) Present a dummy-variable regression model that can be used to determine whether Australia exhibits a July effect (to coincide with the Australian tax year) as well as a January effect. In presenting the model, clearly explain all terms and variables.

(b) Using this model, estimate the average abnormal return in (i) January, (ii) July, and (iii) the other ten months combined for both the portfolio of small stocks (Portfolio 1) and the portfolio of large stocks (Portfolio 10). Again, use the equal-weighted market return for RM,t in both regressions.

Part 4.3

You have a theory that the magnitude of the January effect has changed since the October 1987 stock market crash. You have decided to test this theory using raw returns for the value-weighted market index. In order to remove any potential confounding effects of the crash, you have decided to delete the data for the year 1987 from your sample. Thus, you are interested in two periods: (1) January 1974 through December 1986 (Period 1) and (2) January 1988 through December 2000 (Period 2).

Required:

(a) Use a single regression model incorporating dummy variables to estimate:

(i) the average January return in Period 1

(ii) the average January return in Period 2

(iii) the average return in non-January months during Period 1

(iv) the average return in non-January months during Period 2

(b) Conduct a statistical test on the following hypotheses:

(i) there is no difference in January returns between the two sub- periods

(ii) there is no difference in non-January returns between the two sub-periods.

Note: There are a number of different ways to set up the regression model, all of which will estimate the four parameters. However, the most appropriate form, which will allow for a direct test of the hypotheses may take some thought and planning.

In answering the question, write out the regression model used and briefly explain what each estimate measures. Also include a printout of the regression results. Show all workings involved in conducting the hypothesis tests. Provide a clear interpretation of your tests.