Beta Measures Risk

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6.3 Beta Measures Risk

A Market index reflects the state of the economy. When we regress a time series of an individual stock’s rates of return against a Market index, the simple linear regression slope β indicates the expected change in a stock’s rate of return in response to a unit change in the Market rate of return. We estimate β1 with b1 using a sample of stock prices:

6.3 Beta Measures Risk

t Market t

stocki b bRR

R

Rˆ , = 0+ 1 ,

Where

t stocki

RR , is the estimated rate of return of a stock i in month t, and

t Market

RR , is the rate of return of a Market index in month t.

In this specific case, the simple linear regression slope estimate b is called beta. If, in response to a unit change in the Market rate of return, the expected change in a stock’s rate of return b is greater than one, the stock is more volatile, and exaggerates Market movements. A one percent increase in the Market value is associated with an expected change in the stock’s price of more than one percent change. Conversely, if the expected change in a stock’s rate of return b is less than one, the stock dampens Market fluc- tuations and is less risky. A one percent change in the Market’s value is associated with an expected change in the stock’s price of less than one percent. Beta reflects the amount of risk a stock contributes to a well-diversified portfolio.

We know from Chapter 4 that the sample correlation coefficient between two variables rXY and their sample covariance covXY are closely related to the simple regression slope estimate b1:

2

1 cov /

Y XY X

Y

XY s

s r s

b = =

In a Leading Indicator model of an individual stock’s rate of return against a Market index, our estimate of beta is directly related to the sample correlation and sample covariance between the individual stock’s rate of return and the Market rate of return:

Market stock Market stock stock

stock s

s r

b

beta ⁱ

i i

i = = , ²

, /

covstock Market sMarket

i

=

Our estimate of beta is a direct function of the sample covariance between an individual stock’s rate of return and the Market rate of return, as well as Market sample variance.

Stocks with rates of return that are more strongly correlated with the Market rate of return and those with larger standard deviations have larger betas.

Example 6.2 Four diverse stocks.

To illustrate the relationship between individual stocks’ covariances and correlations with the Market and their betas, monthly rates of return for Lockheed Martin, General Electric, Apple and IBM are plotted in Figure 6.3 with monthly S&P500 rates of return over the two year period from September 2003 through October 2005.

153

Monthly Stock Rates of Return

-0.17 -0.09 0.00 0.09 0.17 0.26 0.34

9/26/03 3/26/04 9/24/04 3/25/05 9/23/05

Month

Rate of Return

SP500 RR Lockheed Martin RR GE RR IBM RR Apple RR

Figure 6.3 Monthly rates of return of four diverse stocks and S&P500 November 2000 – October 2005

Lockheed Martin and General Electric (fainter) in have smaller variances than the computer stocks (thicker). Lockheed Martin and General Electric are less risky investments. We also see that Lockheed Martin (fainter light) moves independently of the Market (black), while the other three tend to move with the Market.

We would expect Lockheed Martin to be relatively immune to economic swings, since much of their business consists of government contracts. We would also expect the two computer stocks to be riskier than General Electric, since the computers (MP3 players, software) are relatively expensive, luxury items. In boom cycles, the computer companies do more business. General Electric sells many necessities, including appliances and light bulbs. The demand for these products is affected less by economic swings, making GE stock relatively less correlated with Market swings, and, hence, less risky.

Only Lockheed Martin returns move opposite the Market and are negative in about a third of the months when the Market is gaining. Market returns never exceed ten percent, while individual stocks sometimes gain as much as thirty-four percent. Market losses are never greater than ten percent, while individual losses are as great as seventeen percent.

Table 6.2 contains sample correlation coefficients, covariances, and betas for each of the four stocks using five years of monthly data (December 2000 through October 2005).

6.3 Beta Measures Risk

correlation

with the Market rstock,Market

standard deviation

covariance with the Market covstock,Market

beta bstock

SP500 RR 0.047

Lockheed Martin RR -0.13 0.064 -0.00038 -0.18

GE RR 0.407^a 0.064 0.00119 0.55^a,b Apple RR 0.416^a 0.138 0.00265 1.22^a IBM RR 0.681^a 0.100 0.00313 1.45^a,c

aSignificant at .01.

bSignificantly less than 1.0 at a 95% confidence level.

cSignificantly greater than 1.0 at a 95% confidence level.

Table 6.2 Correlations, Standard Deviations, Covariances and Betas for Four Stocks November 2000 to October 2005

The correlation between Lockheed Martin’s monthly rate of return and the Market monthly rate of return does not differ from zero, confirming that Lockheed Martin’s returns move independently of the Market. Correlations between each of the other three stocks’ returns and the Market are significantly greater than zero, indicating that they do move with the Market. IBM, with its large correlation of .68, magnifies Market movement.

General Electric’s and Apple’s returns are both moderately correlated with the Market

index returns ( .4

,

, ≅ ≅

Market Apple Market ctric

GeneralEle r

r ). However, Apple returns are considerably

more volatile ( =.14< =.06

ctric GeneralEle

Apple s

s ). GE returns dampen market returns more,

as we see in a comparison of the covariances between the two stocks’ returns and Market returns (cov_GE,Market =.0012<cov_Apple,Market =.0027). Because Apple rates of return are more volatile than General Electric, Apple will also have a larger beta than General Electric.

Betas bstocki are shown in the last column of Table 6.3. A percent increase in the Market produces

• a zero expected change in Lockheed Martin’s price,

• less than one percent expected increase in General Electric’s price,

• a one percent expected increase in Apple’s price, and

• more than one percent expected increase in IBM’s price.

Beta estimates are shown in Table 6.3 and Figure 6.4.

155

SP500 RR Line Fit Plot

-0.30 -0.15 0.00 0.15 0.30 0.45

-0.20 -0.10 0.00 0.10

SP500 RR

Lockheed Martin RR

SP500 RR Line Fit Plot

-0.30 -0.15 0.00 0.15 0.30 0.45

-0.20 -0.10 0.00 0.10

SP500 RR

GE RR

Regression Statistics: Lockheed Martin Multiple R 0.130

R Square 0.017 Standard

Error 0.064

ANOVA df SS MS F Significance F

Regression 1 0.004 0.004 0.97 0.3280

Residual 57 0.234 0.004

Total 58 0.238

Coefficients

Standard

Error t Stat p value Lower 95%

Upper 95%

Intercept 0.011 0.008 1.3 0.1839 -0.005 0.028

SP500 RR -0.177 0.179 -1.0 0.3280 -0.536 0.182

Table 6.3 Estimates of betas for four diverse stocks

011RˆR_LMt =. - .177SP500t RRˆ_GEt =−.003+.55^aSP500t

(.008)(.179) (.008)(.16)

RSquare: .02 RSquare: .17^a

aSignificant at .01

6.3 Beta Measures Risk

SP500 RR Line Fit Plot

-0.30 -0.15 0.00 0.15 0.30 0.45

-0.20 -0.10 0.00 0.10

SP500 RR

Apple RR

SP500 RR Line Fit Plot

-0.30 -0.15 0.00 0.15 0.30 0.45

-0.20 -0.10 0.00 0.10

SP500 RR

IBM RR

a Apple_t

R

Rˆ =.046 +1.22^aSP500t RRˆ_IBMt =.007+1.45^aSP500t

RSquare: .17^a RSquare: .46^a

aSignificant at .01 ^aSignificant at .01

Figure 6.4 Response of four diverse stocks to The Market

A potential investor would conclude:

“Lockheed Martin, with an estimated beta of zero, is the least risky stock of the four. LM returns are relatively invulnerable to Market swings. A change in the Market return is not associated with change in LM’s price.

General Electric, with an estimated beta less than one (bGE=.55), is a low risk investment. GE returns dampen Market swings. With a percent increase in the Market, we expect to see an average increase of .55% in GE’s price.

Apple stock, with an estimated beta of one (bApple=1.22) is riskier than LM or GE, and mirrors Market movement. With a percent increase in the Market, we expect to see an average increase of about one percent, 1.22%, in Apple’s price.

IBM is the riskiest investment of the four, with an estimated beta greater than one (bIBM=1.45). IBM returns exaggerate Market swings. With a percent increase in the Market, we expect to see an average increase of 1.45% in IBM’s price.”

157

(.010) (.21) (.017) (.35)

6.4 A Portfolio’s Expected Return, Risk and Beta Are Weighted