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(7 of 20) Ch.13 - Calculation of expected return, variance, & st. dev.: example with 2 stocks - YouTube
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In the next examples, we will compare investments
into two different stocks.
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Okay, and we will again, stick to three possible
states of the economy; boom, normal, recession.
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In some problems in this class, for example
in the homework assignments, you may see only
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two states.
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Let's say good and bad.
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Or boom and recession.
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Or three states, or four states.
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I don't think I recall seeing more than four
states of the economy.
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Although you can technically have as many
states of the economy as you like.
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Okay.
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So, let's look at this problem.
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We have a 20% of the boom state, and 50% of
the normal state, and a 30% of the recession
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state.
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The return on stock of firm A is 30% in the
boom, 12% in normal, minus 10% in recession.
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The return on the stock of firm B is minus
5% in boom, 7% in normal, 15% in recession.
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Separately, let's focus on stock A and calculate
its expected return variance and standard
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deviation.
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And then, we'll do that separate for stock
B.
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For stock B, just like in the earlier examples,
we would need to look at the boom state and
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multiply the probability of the boom state
by the return on stock A in the boom state.
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In decimals, that's .2 times .3.
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Then, we add a similar product for the normal
state.
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The probability of normal is 50%.
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The return on stock A in the normal state
is 12%.
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So, I'm adding .5 times .12.
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And then, I'm adding the same product for
the last, the recession state, 30% probability
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or .3 multiplied by negative 10% return, or
a negative .1 in decimals.
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The sum of these three products gives me .09
in decimals, or a 9% expected return.
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So, I'm expecting my money to grow on average
by 9% every year in the future, based on what
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I know about the possibilities and how likely
they are.
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Then, using the formulas from before, I can
calculate the variance of stock A returns,
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which gives me .0201.
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The square root of variance gives me the standard
deviation for stock A, .1418 in decimals,
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which is 14.18% standard deviation.
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Then, I need to do the same math, but for
stock of company B. So, what information will
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I need to use, the first column, the probabilities,
of boom, normal, and recession; 20%, 50% and
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30%, respectively.
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And the information on stock B. I don't need
information on stock A. For stock B, minus
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5% in boom, 7% return in normal, 15% return
in recession.
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So, just like I did it for stock A, I'm multiplying
the probability by the return in that same
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state of the economy.
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And then, I'm adding those products across
all states, which gives me a 7% expected return.
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The variance is 0.0048 when I use all numbers
and decimals.
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And the square root of my variance is 0.0693
in decimals, which is 6.93% standard deviation.
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Okay, so let's answer the following question,
which stock would you say is more attractive
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to investors?
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Stock A or stock B?
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Usually the answer I hear are its stock A
because you make on average more money, and
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that's true.
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Your expected return stock A is 9%, your expected
return every year on stock B is 7%.
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So, 9% is better than 7% for the expected
annual return.
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However, you can also compare the amounts
of risk of the two stocks.
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If you, you know, will compare the values
we got for the standard deviations.
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So, this standard deviation for stock A is
14.18%.
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And the standard deviation for stock B returns
is only 6.93%.
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Which one is better?
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It's lower for stock B, which means lower
fluctuations in actual returns, which means
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less uncertainty about what the return may
be every year.
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So, the returns are kind of closer together
right?
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You can almost see, if you look at the table
with the given information.
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The returns on stock A are kind of much wider
apart.
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So, the range goes between minus 10% and 30%.
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Whereas in stock B, the range is between minus
5% and only 15%.
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So, the three returns on stock B are much
closer together than the three returns on
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stock A. And mathematically, we can you know,
prove that by calculating the standard deviations,
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and double checking that it is a smaller number
for stock B than for stock A. And so, overall,
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which stock would you buy?
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A, with the better return, but a worse amount
of risk?
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Or, stock B with a lower return but a much
better you know lower amount of risk.
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There is no correct answer to this kind of
question.
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It's basically up to you, it's a matter of
personal preference, whether you're okay with,
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you you're your money going up and down by
a lot every year.
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But on average, you're guaranteed a better
you know, increasing your money every year.
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Or, whether you are a more risk-averse person.
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So, you don't want your money to fluctuate
too much, you prefer a lower standard deviation.
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But of course, if you don't risk you don't
win.
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Right.
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If you're not taking your chances, then you
will not be rewarded with a high return.
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Your return every year on average will be
only 7%.
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