1. Estimating the Beta for Nike, Inc.
Mia Attruia
According to Yahoo Finance on
Wednesday, November 18, 2015, Nike,
Inc. closing stock price was reported to
be $125.78. Yahoo Finance also reports
the beta, which they estimate using the
most recent 36 months of monthly
returns, of Nike’s stock to be 0.552343.
The task at hand is to estimate the
Capital Asset Pricing Model for
Goldman Sachs, Inc., confirm the beta
value and determine whether the stock is
fairly price.
1. The Capital Asset
Pricing Model
The Capital Asset Pricing Model
(CAPM) asserts that the expected excess
returns of any stock are proportional to
the expected excess returns of the
market. Let ri be the return on the equity
of any corporation “i”, rm be the market
return, and rf be the risk-free rate of
return. Then the CAPM equation is:
where E[ri-rf] is the expected excess
return (in excess of the risk-free rate of
return) for corporation “i” and E[rm-rf]
is the expected excess return of the
market. The parameter , beta, is the
proportionality factor and is regarded as
the market relevant measure of risk for
the stock of corporation “i”. If
beta > 1 (beta < 1) then firm “i” is
relatively more (less) risky thank the
overall market. The main point is that
the variance of the returns of firm “i” is
not the appropriate measure of risk, beta
is.
Since the expected excess returns
are not directly observable, we must
estimate the beta using a linear
regression over past observable returns:
Where is an error term with mean
zero and constant variance and is
independent of rm and rf.
CAPM predicts that a “fairly-priced”
stock would have an estimated value of
of zero. A value significantly
greater than zero would indicate that the
stock is getting higher returns than
predicted by the theory so investors
should purchase the stock to capture
these unexpectedly high returns. This
increase in demand would drive up the
price of the stock and thus lower its
expected return until the falls back to
zero. Similarly, if was significantly
lower than zero than investors should
short- sell the stock which would
eventually drive down the price of the
stock and the would rise to zero.
Investors refer to this process as
“chasing the alpha”.
2. Data
In order to be compatible with Yahoo
Finance, we use the more recent 36
months of returns to compute our beta.
November 2015 is not complete so we
will use the most recent 37 end of month
stock prices, 10/2012 through 10/2015,
for Nike to compute the percent monthly

2. returns for 11/2012 through 10/2015
using:
rNiket=100*diff(log(Niket))
Where Niket is the stock price of Nike at
date t.
The market price will be
approximated by the Standard & Poor
500 Index (trading symbol ^GSPC) with
monthly percent returns, rspt, computed
just as above.
For the risk-free rate we will use
the 13-week US Treasury Bill rate
(trading symbol ^IRX), which, since it is
an interest rate, is already reported in
percent.
Excess returns are then computed as
exrNiket=rgst-rft
for Nike and
exrSPt=rgst-rft
for the market
Key summary statistics are
reported in Table 1. The returns are
reported on a monthly base. To get
annualized values we would multiply the
mean by 12 and the standard deviation
by the square root of 12. The average
excess returns for Nike is almost twice
that of the market but the risk of Nike, as
measured by the standard deviation of
excess returns, is more than twice that of
the market.
Table 1: Sample means and standard deviations
for the excess returns of Nike (exrNike) and the
S&P500 Index (exrsp) from 11/2012 through
10/2015.
Mean St. Dev
exrNike 3.0025 4.9846
exrSP 1.0447 2.9870
The higher risk of Nike is associated
with higher returns on average, as
predicted by financial theory. A
commonly used measure of the risk
return trade-off is the Sharpe ratio,
which is computed as the expected
excess return of an asset, divided by its
standard deviation. The higher the
Sharpe ratio, the greater is the expected
return per unit of risk. The estimated
Sharpe ratio for Nike is
3.0025/4.9846=0.6024 and for the
market it is 1.0447/2.9870= 0.3497,
which suggests that the higher risk of the
market means that Nike is fully
compensated by the higher expected
return. Thus, Nike does not appear to be
a very risky stock.
The correlation between the
excess returns of Nike and the market is
and is significantly different from zero.
This positive correlation is evident in the
time series plot of the data shown if
Figure 1. The plot also visually confirms
that the excess returns for Nike are much
more volatile than the excess returns of
the market.
Figure 1: Excess returns of Nike and the S&P500
Index from 11/2012 to 10/2015.
3. Estimation Results
A summary of the estimation results for
the CAPM equation (1)
2.42041 + 0.557245
(0.843010) (0.269760)
= 36 = 0.111509 = 4.767076
F(1, 34)= 4.267149
The estimated value for beta is
0.557245, which is close to the beta of
0.552343 reported by Yahoo Finance.
Since the beta value is smaller than 1,

3. Nike returns are not as risky as the
overall market. If the market changes by
1% we would expect that the price of
Nike shares would change by 1.56%
A regression model is essentially
a variance decomposition model. If we
take the variance of both side of
equation 1 we get:
(4.9846)2= (0.557245)2(2.9870)2+ (4.767076)2
24.84623716=2.77052967+22.72501359
Thus, about 11%
(2.77052967/24.84623716) of the variance
of Nike excess returns is explained by
the CAPM. This proportion of explained
risk is referred to as “market risk” and
cannot be avoided by investors. It is this
portion of risk, determined by the market
risk and the stock’s beta, which
determines the fair market value of an
asset. About 89% of the variance
remains unexplained by the model.
Financial theory refers to this
unexplained risk as “idiosyncratic risk”
and the main point of portfolio
diversification is to eliminate such risk.
Since idiosyncratic risk can be
diversified away by the careful investor,
this portion of a stocks risk does not
contribute to a stocks market price.
The percent of explained
variation is precisely what the R2
=0.111509 statistic is designed to
measure and we confirm from the
Appendix table that the unadjusted R2 =
as we computed above. The F-test for
the R2 has a p-value of 0.0465 so the
CAPM is somewhat significant.
The 95% confidence intervals of
the estimated CAPM are shown in Table
2.
Table 2: 95% confidence intervals for the
estimated coefficient of the CAPM.
Variable Coefficient 95% Confidence
Interval
2.42041 0.707208 4.13361
0.557245 0.0090271 1.10546
The estimated constant term, , is not
significantly different from zero, as
predicted by the CAPM theory. We
accept the null for the test Ho: = 0
versus at the 5%
significance level since the 95%
confidence interval of does contain 0.
We conclude that Nike’s stock prices are
fairly (accurately) priced according to
the CAPM.
Similarly we could accept the
null for 1 versus
Since 1 is contained in the 95%
confidence interval for . Thus, we
cannot reject the null that the beta of
Nike equals the beta for the market,
which always equal 1 by definition. The
market is just as risky as the market.
4. Summary
We estimated the Capital Asset Pricing
Model for Nike using percent monthly
returns over the 36-month period
11/2012 through 10/2015. The Standard
& Poor 500 Index approximated the
market portfolio, which is a commonly
used broad market index. The risk-free
rate was approximated by the 13 week
US Treasury bill.
We find that Nike has higher
expected returns than the market but not
a higher risk. The Sharpe ratio of Nike is
0.6024 compared with the markets’
Sharpe ratio of 0.3497. This confirms
that Nike has a higher return per unit of
risk than does the market.
The estimated beta value for
Nike is 0.56. Because the beta value is
less than 1, it confirms that Nike is not

4. as risky than the market portfolio. This is
not surprising and illustrated the power
of diversification in financial
investments. The confidence interval for
beta is quite wide, however, and does
contain 1.
The estimated value of is not
significantly different from zero, but a
little higher than zero, which indicates
that, according to the CAPM, Nike’s
stock is not really fairly priced by the
market.
The not so fair price and
relatively low risk of Nike does not
mean that the stock is a poor investment.
The stock may be an important part of a
well-diversified investment portfolio.
Appendix: Complete estimation results.