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At 11:33 AM -0500 12/21/01, Paul Altman wrote:
>Can someone please explain to me what the heck standard deviation is?
>Yes, I've got the formula. Yes, I understand that it's a popular
>measurement of deviation. But the various components of the formula
>seem kind of convoluted to me, and I can't really follow the "common
>sense" idea of all the parts of the formula.
There is a mathematical reason for the standard deviation.
You can easily show that the periodic returns from any investment are
not constant but have a distribution of values. This distribution
closely follows the "bell-shaped curve" which is called the "normal
distribution" in statistics.
(If you are compounding your returns, that is, investing in the next
period, the value of your account at the end of the previous period,
you need to use logarithmic returns to account for the compounding
but this is a small refinement.)
The Central Limit Theorem of statistics tells us that we would expect
the distributions of returns to have such a normal distribution and
it is easy to demonstrate this with examples. (It is not exactly a
normal distribution for complex reasons but it is pretty close.)
A normal distribution has two parameters that describe it's shape:
> The average value - the value at the center on the distribution
> The standard deviation - the width ("fatness") of the distribution.
If you invested in a mutual fund (Fund A) that had an average annual
return of 20% and an annualized standard deviation of 15%, you would
expect that that in about two thirds of the years, your yearly return
would be between 5% and 35%. A person who sees that this fund had an
annual return of 35% in 2001 might invest and be very surprised when
the return for next year is only 5% but statistically, it would not
be unusual.
If you invested in a mutual fund (Fund B) that had an average annual
return of 20% and an annualized standard deviation of 5%, you would
expect that that in about two thirds of the years, your yearly return
would be between 15% and 25%.
Obviously, the smaller standard deviation indicates much more
consistent performance even though the long-term average return of
both funds is the same.
The Sharpe Ratio combines the return and the standard deviation into
a single measure of risk-adjusted return. To calculate the Sharpe
Ratio, you need to use the Excess Return. This is the actual annual
return, less what you could have made on simply keeping the money in
T-Bills or a CD (assume 5% for an example).
Thus, for both of the above funds, the annual return was 20% and so
the Excess Return was 15%.
Sharpe Ratio = Excess_Return / Standard_deviation (all on an
annual basis)
so the Sharpe Ratio in the two cases is:
Fund A Sharpe = 15% / 15% = 1.0
Fund B Sharpe = 15% / 5% = 3.0
We see that Fund B has a three times higher risk-adjusted return.
If you double your bet-size, your excess return and your standard
deviation will both double but the Sharpe Ratio will remain the same.
People always talk about the "returns" of their investments but never
talk about the other critical component, the standard deviation of
those returns. Considering one without the other is meaningless since
you can get any return you want by increasing leverage (bet-size).
>Why not just cut a mean through the data and measure the average
>(absolute) deviations from the mean? I.e., what is the advantage of
>the SD formula over a simple average deviation?
If the distribution is a "normal" distribution the average deviation
is equal to about 80% of the standard deviation. If the distribution
is not "normal" the relationship is different.
Bob Fulks
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