Question:
7-8, Page 275:
Year
1
2
3
4
5
6
7
8
9
10
11
Mean=
σ=
Historical Rates of Return
NYSE
4.0%
14.3
19.0
(14.7)
(26.5)
37.2
23.8
(7.2)
6.6
20.5
30.6
9.8%
19.6%
Stock Y
3.0%
18.2
9.1
(6.0)
(15.3)
33.1
6.1
3.2
14.8
24.1
18.0
9.8%
13.8%
a. Construct a scatter diagram showing the relationship between returns on
Stock Y and the market, and then draw a freehand approximation of the
regression line. What is the approximate value of the beta coefficient? If you
have a calculator with a linear regression function or a spreadsheet, check
the approximate value of beta obtained from the graph.
Approximate Value of Beta is 0.6 – please see the attached excel sheets for calculations
and scatter diagram
b. Give a verbal interpretation of what the regression line and the beta
coefficient show about Stock Y’s volatility and relative riskiness as compared
with those of other stocks.
The Beta Coefficient is a measure of risk, it shows how volatile stock Y is with respect to
the NYSE index. A Beta Value of 0.6 means that, when the NYSE index is up by 1%,
stock Y is up by 0.6%. From this we can see that Stock Y is less risky than other stocks.
c. Suppose the scatter of points had been more spread out, but the regression
line was exactly where your present graph shows it. How would this effect
(1) the firm’s risk if the stock is held in a one-asset portfolio and (2) the actual
risk premium on the stock if the CAPM holds exactly?
If the stock is held in a one-asset portfolio, we are concerned about both systematic and
unsystematic risks. When the points are scattered, this means that the Unique risk
(unsystematic risk) is high, hence a one-asset portfolio would be a risky portfolio.
When the stock is held in a big portfolio and follows CAPM, we are concerned with the
Systematic (market) risk, the Unsystematic risk is somewhat eliminated by
diversification. In this situation, there will not be any change in the risk premium, the
only thing of concern would be the slope. If the slope is the same, the risk premium will
be the same no matter how spread the data is.
d. Suppose the regression line had been downward sloping and the beta
coefficient had been negative. What would this imply about (1) Stock Y’s
relative riskiness, (2) its correlation with the market, and (3) its probable risk
premium?
A downward sloping regression line and a negative beta coefficient indicate that the stock
is negatively correlated with the NYSE index. Stocks with such a negative relationship
with NYSE reduce the overall risk of a portfolio. From this we can see that stock Y has a
low relative risk and a negative correlation with the market. With such stocks, there is
usually a negative risk premium (risk discount) since the stock reduces the overall risk of
the portfolio; the risk premium however will depend on the standard deviation of the
stock.
e. Construct an illustrative probability distribution graph of returns on
portfolios consisting of (1) only Stock Y, (2) 1% each of 100 stocks with beta
coefficients similar to that of Stock Y, and (3) all stocks (i.e. the distribution
of returns on the market). Use as the expected rate of return the arithmetic
mean as given previously for both Stock Y and the market and assume that
the distributions are normal. Are the expected returns “reasonable”; that is,
is it reasonable that % 8 . 9 ˆ ˆ = = M Y k k ?
Please see the attached excel sheet