Assignment 1.

In this exercise, we use monthly data to estimate the ‘betas’ of stocks (equities) traded on the New York Stock Exchange. We study equations of the form:

r_jt = α_j + β_jr_mt + u_jt,                                                                 (1)

where r_jt represents the actual return to holding company j’s stock in month t^∗, r_mt is the return      on the market portfolio in month t (i.e., the portfolio consisting of all stocks, held in the same proportions as the market as a whole) and u_jt represents other influences on returns of the stock j.

The strict form of the capital asset pricing model (CAPM) predicts that this equation fully ‘ex- plains’ stock returns. Specifically, this means that u_jt depends only on random effects particular to company j, and is not predictable by macroeconomic variables. According to the CAPM, when markets operate efficiently in response to complete information, market equilibrium implies that r_mt contains all such information, relevant to individual stock returns. This assumption can be tested econometrically.

The parameter β (the ‘beta’) is an indicator of the risk and return associated with the stock. When β_j = 1, the expected net return is the same as that of the market portfolio. When β_j > 1, the ex- pected return exceeds that of the market portfolio, but there is correspondingly greater risk. When β_j < 1 there is lower return, but also less risk. Thus, market equilibrium ensures the existence of a risk-return trade-off. The model also predicts that α_j = 0.

The Data

Copy the file QM Ex2.tsm from the web  page.   Double-click the file to open TSM and create        the data file beta.xls. This file contains series of monthly returns on the stocks of 21 American companies, as follows:

∗Consider the exercise of buying a unit of stock today,  and selling it again after one month.  The pay-out from     one month’s holding of a unit of stock is d + p1 p0, where p0 and p1 are this period’s and next period’s price, respectively, and d is the dividend per unit. The monthly rate of return is accordingly

r = (d + p1 − p0)/p0.

The variables p1 and d are uncertain ex ante, and r is treated as a random variable in the analysis of the agent’s decision.

(Note: Conoco was taken over by Dupont in September 1981). The data file also contains the following variables:

The Exercise

Choose some stocks to analyse. Look at as many cases as you wish, but you must write up your analysis for two stocks, from different industries. It is also interesting to analyse the return on gold. The measure of r_mt is MARKET. It is recommend that you divide the sample into two 5-year periods, and choose one of the periods for your analysis. Over a longer time span, a company’s beta might change significantly (you can test this).

1. Plot your data set. Check how the returns behave in interesting periods, such as the market crash of October 1987. The scatter plot of r_jt against r_mt may also be of
2. Use ordinary least squares to estimate α and β. Calculate the approximated 95% confidence intervals for the parameters, and test the null hypotheses, α = 0, and also β = 1 against the alternative hypothesis β > Interpret the result of the latter test.
3. The standard deviation of the OLS residuals measures the individual risk of the stock, and        the R² of the regression measures the proportion of the risk attributable to the market (as opposed to individual factors). Comment on the values you

• Use the Chow stability test to check whether the model is stable over the full 10-year period of the sample. In the Options / Forecasting dialog, make sure that the Forecast Type is set to Ex-Post 1 Step. Then select the first 5 years as the sample for estimation, and choose the maximum forecast The Chow statistic is computed automatically when you run the regression.

5. Test the strict CAPM against the alternative of the Arbitrage Pricing Model (APM) by  testing   the ability of macroeconomic variables to predict returns.  Use the rate of inflation (RINF),     the growth in industrial production (GIND), and changes in the real oil price (ROIL) in a variable addition test. If the test finds significant effects,  re-estimate the equation and report  how much the value of beta changes as a result of including these

The variables you require for this test can be generated from the data series CPI, FRBIND and POIL in the data set. Do the following:

• Go to Setup / Data Transformation and Editing, or press b.Click the Formula Type

RPOIL = POIL/CPI

in the text field, and click <<< GO>>>. (This creates the real oil price).

c.Now click Transform, and select the option %- Difference. Highlight CPI, FRBIND and RPOIL and click GO. (This creates the percentage growth rates).

d.(Optional) Click Edit, select Rename, highlight D%-CPI, D%-FRBIND, and D%- RPOIL, and click Go. Supply the names RINF, GIND and ROIL, respectively. (These are the names used in Berndt’s book – see ‘Further Reading’, below.)

e.Click “Save Modified Data”.

Please satisfy yourself that your new variables RINF, GIND and ROIL contain the percentage monthly changes in CPI, FRBIND and POIL/CPI respectively. Add these variables to the regression. Note their individual significance by considering the t statistics.

Also test their joint significance, using an F test. Do as follows:

1. In the linear regression dialog, check Wald Test of Constraints, and press the square button next to the checkbox to open the Constraints
2. Make sure that the radio button Zero Restrictions is selected and check the F – Stat. checkbox.
3. Press Values, and in the column headed Wald Test, check the boxes of each variable to be
4. Run the regression. The test statistic appears following the usual
5. The last suggested exercise is a little unorthodox, and is optional, but it could help to illumi- nate the important idea of “expectation”. Recall the CAPM equation:

β  = E(r_jt) − r_ft ,                                                             (2)

j        E(r    ) − r

f t

where r_ft denotes the risk-free rate. This equation describes the subjective expectations of market traders in equilibrium at date t, which are of course unobserved. However, we might expect that the average values of E(r_mt) and E(r_jt) over time might be approximated by the sample average realized returns over time, which we do observe.

While r_ft is not constant (although varying much  less than the equity returns) we  might also expect that the time average of this variable (measured by RKFREE) would measure the average risk-free rate. An informal check on the CAPM might in this case be provided by calculating the ratio on the right-hand side of equation (2) with sample means replacing theoretical values, and comparing this with the OLS estimate of β_j for the same time period. To get the sample means, go to Setup / Compute Summary Statistics, highlight the variables in the list, select the required sample period, and click Go.

Comment on what you find.