APT Risk Model Theory 
Background
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Market Risk
There is a need to estimate the market risk inherent
in portfolios of securities.
If excess returns above the riskfree rate are sought,
which many investors do, risk must be taken. If the
market is largely efficient then there are very few free
lunches, and excess returns will only be broadly
available when some additional risk is borne.
The kind of risk that should carry a risk premium, i.e.
risks on which the market requires a higher payout in
return for intentionally bearing the risk, is market
risk. Market risk is the risk of unexpected variability
in the price of securities.
Other risks are sometimes discussed in relation to
investing, including liquidity risk, interest rate risk,
operational risk, credit risk, legal risk, brand risk, etc.
Risk is something of a buzzword and some of these
‘risks’ do not really describe a single, coherent,
consistent concept. Others are capable of being
redefined as an element of market risk, e.g. credit
default risks cause prices to change, and should be
described as part of a complete market risk
framework.
We need to know if the return being sought is
sufficient for the level of risk required to achieve it. A
better risk model will give us a better estimate of the
risk, and our actions will be more efficient.
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Risk as variance
The central measure in risk control is variance (or
sometimes standard deviation,
volatility
, which is the
square root of variance). The core supporting concept
is the covariance matrix. The covariance matrix is a
table that includes the relative variance of every
security with respect to every other one. Popularised
by Harry Markowitz in the 1950’s, his insight into its
use was so important that it earned him the Nobel
Prize.
(Note: Some people propose the use of
semivariance, which only measures negative
deviations. Underpinning such measures is the
belief that we should only be interested in
‘downside’ measures of risk. This is in itself a naïve
interest.
When a fund manager delivers significant
underperformance, all investors are unhappy. When
the fund manager delivers outperformance beyond
their stated aim or risk limits, everyone should
equally be unhappy. It means the process was not
really under control, and the manager was only
lucky that the performance was up rather than
down.
Risk measures are principally about how much the
value of some securities can vary, and placing
limits on that variance. Whether performance is up
This discussion focuses on why estimating market risk is important, what
kind of variables can be used to capture market risk, the problems with
variance  the main risk measure, how the problems are resolved by using
factor models, and why some kinds of factor model are more suitable
than others.
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or down is a function of how good their predictions
of
return
were, and the risk system should not be
partial to positive or negative deviations.)
If we have an accurate covariance matrix, we can
compute most (but not quite all) of the important
risk statistics.
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 Fall '09
 RichardMarin
 Capital Asset Pricing Model, Risk in finance, Covariance matrix

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