Question

Consider the daily percent changes of McDonald’s stock price and those of the Dow

Jones Industrial Average for trading days in the months of July and August 1987.

Data can be found in the Excel file S4.XLSX (Mc-Dow) in the Excel directory.

a. Draw a scatterplot of McDonald’s daily percent changes against the Dow Jones

percent changes.

b. Describe the relationship you see in this scatterplot.

c. Find the correlation between these percent changes. Does this agree with your

impression of the scatterplot?

d. Find the coefficient of determination (you may just square the correlation).

Interpret this number as “variation explained”. In financial terms, it represents

the proportion of non-diversifiable risk in McDonald’s. For example, if it were

100%, McDonald’s stock would track the market perfectly, and diversification

would introduce nothing new.

e. Find the proportion of diversifiable risk. This is just 1 - R2 (or 100% minus the

percentage of non-diversifiable risk). This indicates the extent to which you

can diversify away the risk of McDonald's stock by investing part of your

portfolio in the Dow Jones Industrial stocks.

f. Find the regression equation to predict the percent change in McDonald's stock

from the percent change in the Dow Jones Index. Identify the stock's so-called

beta, a measure used by market analysts, which is equal to the slope of this

line. According to the capital asset pricing model, stocks with large beta values

tend to give larger expected returns (on average, over time) than stocks with

smaller betas.

g. Find the 95% confidence interval for the slope coefficient.

h. Test at the 5% level to see whether or not the daily percent changes of

McDonald's and of the Dow Jones Index are significantly associated.

i. Test at the 5% level to see whether the beta of McDonald's is significantly

different from 1, which represents the beta of a highly diversified portfolio

McDonald’s stock vs Dow Jones
Dow Jones	McDonald’s
0.47	1.12
1.41	-0.29
0.70	0.83
0.69	0.58
-0.69	-0.52
-1.38	0.20
-2.34	-0.12
1.44	1.16
-0.24	0.10
0.47	0.53
-1.18	0.52
-1.19	-0.89
0.72	-0.80
0.72	0.09
0.95	0.07
0.94	0.54
0.93	0.35
0.92	1.04
0.92	0.78
0.45	1.10
0.23	0.18
-1.35	-0.58
-1.14	-0.41
2.08	0.78
3.17	1.07
-0.66	-0.09
1.99	1.69
3.03	1.69
-0.84	-0.42
1.48	0.83
-0.63	-0.23
2.31	0.56
-0.21	-1.70
-2.06	0.42
0.84	1.54
0.00	0.10
-2.08	-0.46
2.34	0.94
-0.21	0.76
-1.67	-0.99
-2.33	-1.33
1.08	0.89

Answer 1

a.

b. As we see a increasing trend there is positive correlation between x and y.

Further data are not that close to the line which means there is moderate correlation between x and y

c.

X - Mx	Y - My	(X - Mx)2	(Y - My)2	(X - Mx)(Y - My)
0.23	0.843	0.053	0.711	0.194
1.17	-0.567	1.369	0.321	-0.663
0.46	0.553	0.212	0.306	0.254
0.45	0.303	0.202	0.092	0.136
-0.93	-0.797	0.865	0.635	0.741
-1.62	-0.077	2.624	0.006	0.125
-2.58	-0.397	6.656	0.158	1.024
1.2	0.883	1.44	0.78	1.06
-0.48	-0.177	0.23	0.031	0.085
0.23	0.253	0.053	0.064	0.058
-1.42	0.243	2.016	0.059	-0.345
-1.43	-1.167	2.045	1.362	1.669
0.48	-1.077	0.23	1.16	-0.517
0.48	-0.187	0.23	0.035	-0.09
0.71	-0.207	0.504	0.043	-0.147
0.7	0.263	0.49	0.069	0.184
0.69	0.073	0.476	0.005	0.05
0.68	0.763	0.462	0.582	0.519
0.68	0.503	0.462	0.253	0.342
0.21	0.823	0.044	0.677	0.173
-0.01	-0.097	0	0.009	0.001
-1.59	-0.857	2.528	0.734	1.362
-1.38	-0.687	1.904	0.472	0.948
1.84	0.503	3.386	0.253	0.926
2.93	0.793	8.585	0.629	2.324
-0.9	-0.367	0.81	0.135	0.33
1.75	1.413	3.062	1.997	2.473
2.79	1.413	7.784	1.997	3.943
-1.08	-0.697	1.166	0.486	0.753
1.24	0.553	1.538	0.306	0.686
-0.87	-0.507	0.757	0.257	0.441
2.07	0.283	4.285	0.08	0.586
-0.45	-1.977	0.202	3.908	0.89
-2.3	0.143	5.29	0.02	-0.329
0.6	1.263	0.36	1.595	0.758
-0.24	-0.177	0.058	0.031	0.042
-2.32	-0.737	5.382	0.543	1.71
2.1	0.663	4.41	0.44	1.393
-0.45	0.483	0.202	0.233	-0.217
-1.91	-1.267	3.648	1.605	2.42
-2.57	-1.607	6.605	2.582	4.13
0.84	0.613	0.706	0.376	0.515
Mx: 0.240	My: 0.277	Sum: 83.335	Sum: 26.038	Sum: 30.934

X Values
∑ = 10.08
Mean = 0.24
∑(X - M_x)² = SS_x = 83.335

Y Values
∑ = 11.63
Mean = 0.277
∑(Y - M_y)² = SS_y = 26.038

X and Y Combined
N = 42
∑(X - M_x)(Y - M_y) = 30.934

R Calculation
r = ∑((X - M_y)(Y - M_x)) / √((SS_x)(SS_y))

r = 30.934 / √((83.335)(26.038)) = 0.6641

Yes i agree with the b.

d. Here r=0.6641, so r^2=0.4410

So 44.10% of variation in y is explained by y.