Beta
Beta is a measure of systematic risk. It measures the responsiveness of a stock's expected return
to changes in the value of the whole stock market.
Let
R
p
= return on a portfolio of n assets
R
i
= return on asset i = 1, .
.., n
X
i
= proportion of the portfolio held in asset i
(1) R
p
= X
1
R
1
+ X
2
R
2
+ .
.. + X
n
R
n
The expected return on this portfolio equals
(2) E(R
p
) = E(X
1
R
1
) + E(X
2
R
2
) + .
.. + E(X
n
R
n
)
A measure of the risk for this portfolio is the standard deviation of the portfolio's return or its
squared value, the variance of the portfolio's return, s
p
2
.
(3) s
p
2
= E[R
p
 E(R
p
)]
2
This simplifies to
(4) s
p
2
= X
1
s
1p
+ X
2
s
2p
+ .
.. + X
n
s
np
,
where s
ip
equals the covariance of the return on asset i with the portfolio's return. It is a measure
of how the return on asset i changes with changes in the return of the whole portfolio.
Equation (4) tells us that the contribution to risk of asset i to the portfolio is X
i
s
ip
. So, the
proportion of risk contributed by asset i equals
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 Fall '09
 BradRifkin
 Variance, Personal Finance, Probability theory

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