Download STOCK Beta

Last Study Topics
• Measuring Portfolio Risk
• Measuring Risk Variability
• Unique Risk vs Market Risk
Today’s Study Topics
• Portfolio Risk
• Market Risk Is Measured by Beta
• Beta as a Portfolio Risk Measurement
Case: Lambeth Walk
• Example: Lambeth Walk invests 60% of his
funds in stock I and the balance in stock J. The
standard deviation of returns on I is 10%, and
on J it is 20%. Calculate the variance of
portfolio returns and Standard deviations,
assuming;
– a. The correlation between the returns is 1.0.
– b. The correlation is .5.
– c. The correlation is 0.
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• a. The correlation between the returns is 1.0.
Portfolio Valriance  [(.60) 2 x(.10) 2 ]
 [(.40) 2 x(0.20) 2 ]
 2(.60x.40x 1x0.10x0.2 0)  0.0196
Standard Deviation  0.0196  14 %
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• b. The correlation is .5.
Portfolio Valriance  [(.60) 2 x(.10) 2 ]
 [(.40) 2 x(0.20) 2 ]
 2(.60x.40x 0.50x0.10x 0.20)  0.0196
Standard Deviation  0.0148  12.16 %
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• c. The correlation is 0.
Portfolio Valriance  [(.60) 2 x(.10) 2 ]
 [(.40) 2 x(0.20) 2 ]
 2(.60x.40x 0x0.10x0.2 0)  0.0100
Standard Deviation  0.0100  10 %
General Formula for Computing Portfolio Risk
• The method for calculating portfolio risk can
easily be extended to portfolios of three or
more securities.
• We just have to fill in a larger number of
boxes. Each of those down the diagonal—the
shaded boxes - contains the variance weighted
by the square of the proportion invested, on
the next slide.
Portfolio Risk
The shaded boxes contain variance terms; the remainder
contain covariance terms.
1
2
3
STOCK
To calculate
portfolio variance
add up the boxes
4
5
6
N
1
2
3
4
5
6
STOCK
N
Understanding
• Notice that as N increases, the portfolio
variance steadily approaches the average
covariance. If the average covariance were
zero, it would be possible to eliminate all risk
by holding a sufficient number of securities.
– Unfortunately common stocks move together, not
independently. Thus most of the stocks that the
investor can actually buy are tied together in a
web of positive covariance which set the limit to
the benefits of diversification.
HOW INDIVIDUAL SECURITIES AFFECT PORTFOLIO
RISK
• The risk of a well diversified portfolio depends
on the market risk of the securities included in
the portfolio.
• Wise investors don’t put all their eggs into just
one basket:
• They reduce their risk by diversification.
• They are therefore interested in the effect that
each stock will have on the risk of their
portfolio.
Betas for selected U.S common stock
STOCK
Beta (B)
STOCK
Beta (B)
Amazon
3.25
Boeing
.56
Coca-Cola
.74
Dell Computer
2.21
Exxon Mobile
.40
General Electric
1.18
General Motors
.91
McDonald’s
.68
Pfizer
.71
Reebok
.69
Market Risk Is Measured by Beta
• If you want to know the contribution of an
individual security to the risk of a well
diversified portfolio, it is no good thinking
about how risky that security is if held in
isolation—you need to measure its market
risk, and that boils down to measuring how
sensitive it is to market movements.
• This sensitivity is called beta (B).
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• Stocks with betas greater than 1.0 tend to
amplify the overall movements of the market.
• Stocks with betas between 0 and 1.0 tend to
move in the same direction as the market, but
not as far.
• Of course, the market is the portfolio of all
stocks, so the “average” stock has a beta of
1.0.
Dell Computer
• Dell Computer had a beta of 2.21. If the future
resembles the past, this means that on
average when the market rises an extra 1%,
Dell’s stock price will rise by an extra 2.21
percent.
– When the market falls an extra 2%, Dell’s stock
prices will fall an extra 2 x 2.21 %= 4.42 %.
– Thus a line fitted to a plot of Dell’s returns versus
market returns has a slope of 2.21.
Return on Dell Computer %
Expected
stock
return
2.21%
+
1.0%
Expected
market
return
Beta and Unique Risk
Market Portfolio - Portfolio of all assets in the
economy. In practice a broad stock market
index, such as the S&P Composite, is used
to represent the market.
Beta - Sensitivity of a stock’s return to the
return on the market portfolio.
Portfolio standard deviation
Measuring Risk
Unique
risk
Market risk
0
5
10
Number of Securities
15
Unique Risk
• Of course Dell’s stock returns are not perfectly
correlated with market returns.
• The company is also subject to unique risk, so
the actual returns will be scattered about the
line in Figure.
• Sometimes Dell will head south while the
market goes north, and vice versa.
Why Security Betas Determine Portfolio
Risk
• Let’s review the two crucial points about
security risk and portfolio risk;
– 1) Market risk accounts for most of the risk of a
well-diversified portfolio.
– 2) The beta of an individual security measures its
sensitivity to market movements.
Explanation 1: Where’s Bedrock?
• Where’s bedrock? It depends on the average
beta of the securities selected.
– The portfolio beta would be 1.0, and the
correlation with the market would be 1.0.
– If the standard deviation of the market were 20 %
(roughly its average for 1926–2000), then the
portfolio standard deviation would also be 20%.
• What if the portfolio beta would be 1.5?
– What would be the portfolio’s S.D?
Continue
• The general point is this:
– The risk of a well-diversified portfolio is
proportional to the portfolio beta, which equals
the average beta of the securities included in the
portfolio.
• This shows how portfolio risk is driven by
security betas.
Beta and Covariances
• A statistician would define the beta of stock i
as;
 im
Bi  2
m
– It turns out that this ratio of covariance to
variance measures a stock’s contribution to
portfolio risk.
Beta and Unique Risk
 im
Bi  2
m
Covariance with the
market
Variance of the market
Coca-Cola & Reebok
• Remember that the risk of this portfolio was
the sum of the following cells:
Coca - Cola
Coca - Cola
Reebok
x 12 σ12  (.65) 2  (31.5) 2
x 1 x 2 ρ12 σ1σ 2  .65  .35
 .2  31.5  58.5
Reebok
x 1 x 2 ρ12 σ1σ 2  .65  .35
 .2  31.5  58.5
x 22 σ 22  (.35) 2  (58.5) 2
• If we add each row of cells, we can see how
much of the portfolio’s risk comes from CocaCola and how much comes from Reebok.
Continue
• Coca-Cola’s contribution to portfolio risk
depends on its relative importance in the
portfolio (.65) and its average covariance with
the stocks in the portfolio (774.0).
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• The proportion of the risk that comes from the
Coca-Cola holding is;
• Similarly, Reebok’s contribution to portfolio
risk depends on its relative importance in the
portfolio (.35) and its average covariance with
the stocks in the portfolio (1,437.3).
•
=.35 x 1,437.3 = .35 x 1.43 = .5
•
1,006.1
Understanding
• In each case the proportion depends on two
numbers:
– The relative size of the holding (.65 or .35) and a
measure of the effect of that holding on portfolio
risk (.77 or 1.43).
• The latter values are the betas of Coca-Cola
and Reebok relative to that portfolio.
– On average, an extra 1% change in the value of the
portfolio would be associated with an extra 0.77%
change in the value of Coca-Cola and a 1.43%
change in the value of Reebok.
Does Standard Deviation Relevant?
• Example: Lonesome Gulch Mines has a
standard deviation of 42% per year and a beta
of .10. Amalgamated Copper has a standard
deviation of 31% a year and a beta of .66.
Explain why Lonesome Gulch is the safer
investment for a diversified investor?
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• Solution: In the context of a well-diversified
portfolio, the only risk characteristic of a
single security that matters is the security’s
contribution to the overall portfolio risk. This
contribution is measured by beta. Lonesome
Gulch is the safer investment for a diversified
investor because its beta (+0.10) is lower than
the beta of Amalgamated Copper (+0.66). For
a diversified investor, the standard deviations
are irrelevant.
Summary
• Portfolio Risk
• Market Risk Is Measured by Beta
• Beta as a Portfolio Risk Measurement