WEBVTT
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In this segment, we make a
connection between the
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correlation coefficient and
some fairly realistic real
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world situations.
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The bottom line will be that
the presence or absence of
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correlations can make
a huge difference.
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Suppose that you run an
investment company that
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invests in real estate, and
you have 100 million of
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capital that you
want to invest.
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Now you have learned or believe
that it helps to
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diversify, to not put all of
your eggs in the same basket.
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And for that reason, you're
going to invest some of your
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money into different states.
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You will be investing in 10
different states, and in each
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state, you will invest 10
million so that your total
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investment is spread between
those 10 states.
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For each state, you have a model
that tells you that the
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return on your investment,
that is your profit--
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It's, of course, random, but you
expect it to be 1 million
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on the average, that is, in
terms of the expected value,
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but there's also a fair amount
of randomness, and so the
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standard deviation is 1.3.
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Now, if you look at one state
in isolation, it would be a
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pretty risky investment because
the standard deviation
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is comparable to the mean.
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It's not an unlikely event to
have a return that's one
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standard deviation
below the mean.
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And if that happens, your
return is going to be
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negative, and you're
losing money.
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But then you argue that
you're investing in
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10 different states.
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Yes, you might lose money in
some of them, but overall, you
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would expect to have a pretty
high confidence that you will
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end up having a positive
return.
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Is this correct or not?
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Let us do some calculations.
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We will look at the variance
of your total return.
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The variance of the sum of
random variables is given by
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the formula that we
have developed.
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It's the sum of the variances.
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But then you also have a bunch
of covariance terms that have
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to do with the relation of the
different random variables.
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Now, you make the assumption
that the different states are
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different markets--
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one doesn't affect the other--
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so that the Xi's are
uncorrelated.
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In that case, in this variance
formula, the covariance terms
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are all 0, and they disappear
and you're left with the sum
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of 10 variance terms.
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Now, each one of these variances
is equal to the
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square of the standard
deviation.
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And we have a variance
of 16.9.
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You then take the square root to
find the standard deviation
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and the square root of
this number is 4.1.
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Now, your expected return is
equal to 10, which is 2 and
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1/2 standard deviations.
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You will only lose money if
the outcome is 2 and 1/2
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standard deviations
below the mean.
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And that's a fairly unlikely
outcome, and so in this
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situation you feel very
confident that you will have a
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positive profit.
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Suppose, however, that your
assumption is wrong, and that
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actually the different Xi's are
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correlated with each other.
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And suppose that the
correlation is
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pretty high, 0.9.
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Essentially, this means that the
real estate market in one
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state is strongly related to the
behavior of the market in
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another state.
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And that could be, perhaps,
because the markets in
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different states are affected by
some more global phenomenon
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that operates on a
national level.
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So in this case, the covariance
of Xi with Xj is
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going to be the correlation
coefficient times the standard
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deviation of Xi times the
standard deviation of Xj,
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which is 0.9 times
1.3 times 1.3.
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And so the co-variance
turns out to be 1.52.
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And in that case, the variance
of the sum, using this formula
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here, is going to be equal to 10
times the variance that you
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have in each state, which is 1.3
squared, plus you have a
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bunch of terms here.
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How many terms?
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There's 90 of them, and
each one of these
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terms is equal to 1.52.
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And the variance turns
out to be 154.
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Now you take the square root
of that, and you find a
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standard deviation of 12.4.
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Now, your expected profit
is 10, but the standard
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deviation is 12.4.
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And if you happen to be one
standard deviation below the
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expectation, which is something
that has a sizable
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probability of occurring,
then your profit
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is going to be negative.
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So in the uncorrelated case,
you're pretty certain that you
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will have a positive profit,
but if the correlations
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actually turn out to be
significant, then you're
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facing a very risky situation.
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To some extent, this is similar
to what happened
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during the great financial
crisis.
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That is, many investment
companies thought that they
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were secure by diversifying and
by investing in different
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housing markets in different
states, but then when the
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economy moved as a whole, it
turned out that there were
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high correlations between the
different states, and so the
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unthinkable, that is large
losses, actually did occur.