I want to talk about a total Portfolio gain, over N years (or days or months), and how
it depends upon the MEAN return and the distribution of returns and ...
>Like Normal of Lognormal stuff?
Yes. Suppose that the N returns are denoted by
x_{1}, x_{2}, x_{3}, ... x_{N} where, for a return of 12.3% we
put x = 0.123, okay?
>So far, so good, but what about this kurtosis thing?
Patience.
Now, for N such returns, the Gain Factor is
(1)
G = (1+x_{1})(1+x_{2})(1+x_{3})...(1+x_{N})
Since it's not easy to deal with products, we take logarithms and get:
(2)
log(G) = log(1+x_{1})+log(1+x_{2})+log(1+x_{3})+ ... +log(1+x_{N})
Now let's talk about adding a bunch of terms in a SUM ... like (2), above.
If there are 100 terms in a SUM and 50
have the value 12,
and 30
have the value 15,
and 20
have the value 17 then
we can evaluate the SUM as
50*12
+ 30*15
+ 20*17, right?
>So far, so good, but my doctor warned me about kurtosis and ...
Pay attention. We now look again at the SUM in (2), above. We suppose that many of the returns are the same
... or at least close to each other. In particular, we suppose that all returns lie between, say, 100%
and +100% and we divide that interval into small subintervals of
length Δx
and a fraction of the returns lie in each interval ... between x
and x+Δx.
>Example?
Well, we suppose that, for N = 100
returns:
11 lie in a small interval near r = 5% and
14 lie in a small interval near r = 6% and
13 lie in a small interval near r = 7% and ...
>A picture would be good here.
Here's the distribution
If you add up all the blue numbers you'll get 100,
but many are the same (or, at least they lie in
the same small intervals
... near 5%, 6%, 7%, etc.).
 Figure 1

It'd mean that the SUM of logarithms (as indicated by (2), above) has terms like:
... +
11*log(1+0.05) +
14*log(1+0.06) +
13*log(1+0.07) +
...
which is a SUM of terms which look like:
n(x)*log(1+x)
where
n(x)
is the number of terms which have the same xvalue.
For simplicity, we'll write the SUM as
Σ
n(x)*log(1+x)
Is that okay, for an example?
>Yes, please continue.
Okay, in general we have, continuing from (2) above:
(3)
log(G) = Σ
n(x)*log(1+x).
Now, if the various values of x are small
(remembering that a 9% return would have x = 0.09),
then
(4)
log(1+x) = x  x^{2}/2 + x^{3}/3  x^{4}/4 + x^{5}/5 
>You're assuming natural logarithms, to the base e?
Of course. Is there any other kind?
Notice how well the terms up to x^{4}/4 approximate log(1+x)
Now we can rewrite equation (3) like so:
 Figure 2

(5)
log(G) = Σ
x n(x) 
(1/2)Σ
x^{2} n(x) +
(1/3)Σ
x^{3} n(x) 
(1/4)Σ
x^{4} n(x) +
(1/5)Σ
x^{5} n(x)  ...
If the total number of returns is N, then the SUM of all the
n(x) is N and, in particular,
n(x)/N is the fraction of terms
associated with the same xvalue. We'll call this fraction
f(x), so
(6)
f(x) = n(x)/N.
Now we divide equation (5) by N and get:
(7)
log(G)/N =
log(G^{1/N}) = Σ
x f(x) 
(1/2)Σ
x^{2} f(x) +
(1/3)Σ
x^{3} f(x) 
(1/4)Σ
x^{4} f(x) +
(1/5)Σ
x^{5} f(x)  ...
so we have:
(8)
G^{1/N} = EXP{Σ
x f(x) 
(1/2)Σ
x^{2} f(x) +
(1/3)Σ
x^{3} f(x) 
(1/4)Σ
x^{4} f(x) + ... }
Figure 2A shows a picture of the relative size of the terms.
>zzzZZZ
Wait! We're almost there! If the xvalues are annual returns:
 G^{1/N} is the Annualized Return
 Σ
x f(x), the first moment, is the MEAN of the returns: M
 Σ
x^{2} f(x), the second moment, will give the VARIANCE
... or the square of the Standard Deviation: S
 Σ
x^{3} f(x), the third moment, will give the SKEW
 Σ
x^{4} f(x), the fourth moment, will give the KURTOSIS
 Figure 2A

My stock has Kurtosis
A healthy man called Moses
Had a stock with Halitosis
He gave it a pill
But it got very ill
And its chart got thoracic Kyphosis.
His stock underwent Hypnosis
To cure the painful Kyphosis.
But the market tanked
And his stock sanked
And Moses got ill with Leukosis.


>Will give? What does that mean? Will give ... what ...?
One usually doesn't use the raw moments, except for the MEAN, but normalizes or standardizes them.
For example, here's a couple of distributions which look identical except that the MEAN has changed by +30%
(or 0.30) ... and that has shifted the chart due East by the same amount:
Figure 3
In order to ignore this shift when the MEAN changes (since we're interested in the "shape" of the distribution, not
where it's located), we can replace x by (x  M) so we're measuring the deviation of the returns from their MEAN.
Then, when we change the MEAN (by, for example, adding a constant amount to each return), the chart will continue
to look just like the left chart above (where the MEAN is 0).
>And if we change the Standard Deviation? What then?
I was just getting to that. Here's what'd happen when we change the value of S:
Figure 4
>What're the yellow stripes?
They're one Standard Deviation, S, above and below the MEAN. You'll notice that by dividing by
N, as in f(x) = n(x)/N,
we've changed the number of returns lying near an xvalue to the fraction of returns lying
near an xvalue. That makes our analysis independent of the number of returns. In fact, the vertical scale in our
distribution charts always run from 0 to 1 (or 0% to 100%).
>N could be 100 or 1000 or 10,000 or ...
Yes, yes, but now we should do likewise for the horizontal scale. We need some standardized horizontal
measure. After all, people use this stuff when the horizontal numbers aren't annual returns but maybe degrees Fahrenheit
or kilograms or maybe miles or hours or ...
>How about choosing "1" as the horizontal standard?
Well, that's close to what we'll do. When x = M + S then ...
>So x is at a yellow stripe?
Yes. Then we'll call that "1" unit from the MEAN. To do this we not only replace x by (xM) ... to shift the chart so the "new"
mean is zero ... but we'll also replace x by y = (xM)/S so that ...
>So that when x=M+S then it's "1" unit from the MEAN.
Well, y = 1, where y = (xM)/S is our new horizontal scale.
It has the advantage (over x) in that it's dimensionless. It'll have the same value whether x is measured in
centimetres, miles or lightyears.
Indeed, when generating a distribution from scratch, one often
defines a standardized distribution for the y's, then use x = M + y*S to calculate x. For example,
if we want to generate a Normal distribution with MEAN = M and Standard Deviation = S, then we get a bunch of y's from
a standard Normal distribution (with M=0, S = 1), then get the x's via x = M + y*S and
the x's will have MEAN = M and SD = S.
 Figure 4A

In MS Excel, the function NORMSINV(RAND()) returns a standard Normal distribution
(with Mean = 0, Standard Deviation = 1) whereas M+S*NORMSINV(RAND()) returns Normally distributed
numbers with Mean=M and SD = S.
For example, suppose we plot the fraction of numbers (x's or y's) with a particular value, we'd get two curves
(as in Figure 4A where the y's are a standard Normal distribution and
the x's have M = 3 and S = 2).
The right curve is the left curve shifted right by 3 and the horizontal
dimension stretched by a factor 2. The vertical dimension, which measures the
fraction of x's or y's which have a particular value, doesn't change.
>I thought the area under the curve must be "1", no?
Well, there are the same number of x's as y's (for example: 100 of each) so the fraction (or percentage) having a
particular value is the same ... and that's the vertical value. On the other hand, one usually sees the vertical dimension
as the probability density and ...
>Yeah! That's what I'm talking about! The density!
Okay. If we have jillions (or an infinity) of numbers then we might want to consider the fraction (or percentage)
lying within some interval along the horizontal axis (as opposed to the fraction having one of N = 100 values).
>Like: "How many lie between x = 1.0 and x = 1.2", eh?
Well, not "How many?". The answer to that question might be "Infinity.
More like "What percentage?". In that case the area under the curve
must be "1" so, when we stretch the horizontal distance by S, we must shrink the vertical dimension by S as we have in
Figure 4B which shows the probability density.
(See Distributions
and/or Normal/Lognormal.) It's like, if you have a rectangle and you
increase the width by some factor, then you must decrease the height by the same factor in order to retain the same area.
The use of density is appropriate when you've got lots of numbers. For example, if we have a bunch of beads
on a string we might plot the mass of the beads, as we move along the string. If, instead, we have a bar of metal with
variable density (like: aluminum at the ends changing continuously to gold at the centre), then we'd be better off considering
the change of density (meaning the mass per unit length) as we move along the bar.
>I'll take the centre!
Are you paying attention?

Figure 4B
Figure 4C

For a Normal distribution (Mean = M and SD = S), the density is given by:
A: f(x) = {1/(sqrt(2π) S)} e^{{(xM)/S}2/2}
After standardization (Mean = 0 and SD = 1) we'd have:
B: f(y) = 1/(sqrt(2π)) e^{y2/2}
Notice the division by S, in (A).
(This f is not to be confused with the earlier f which ain't a density !)
>That is really confusing, I mean ...
The earlier f is okay for measuring the mass
distribution for beads on a string. The f(x) and f(y) in A and B, above, are appropriate for continuous mass distributions.
>Okay, why should the Standard Deviation be used as a horizontal distance? Why not a vertical ... ?
Excellent point. I forgot to mention that S, the Standard Deviation, is a measure of how far the
xvalues are from their MEAN. In fact, S^{2} is the average squared distance from the MEAN:
(9)
(1/N) {
(x_{1}M)^{2} + (x_{2}M)^{2} +
... + (x_{N}M)^{2} } = S^{2}
or, in terms of our new standardized numbers:
(10)
(1/N) {
(x_{1}M)^{2}/S^{2} + (x_{2}M)^{2}/S^{2} +
... + (x_{N}M)^{2}/S^{2} } =
(1/N) {y_{1} ^{2} + y_{2} ^{2} + ... +y_{N} ^{2}} = 1
Notice that (according to equation 10), points with coordinates
(y_{1}, y_{2}, y_{3}, ... y_{N}), in Ndimensional space,
lie on a hypersphere of radius SQRT(N). The greater the number of points, the greater the dimension N, the farther from
the origin. That observation is connected to Einstein's analysis of a random walk or Brownian Motion and how far
you'd expect to be from the origin (where you started) after N random steps and ...
>Let's not get off track, okay? What's the Standard Deviation of these new numbers, y_{1} and y_{2} and so on?
One.
>One what?
Equation (10) says it all.
The MEAN of these new y's is the number "0" (because we've subtracted M from all the x's)
... and the Variance (hence the Standard Deviation) of the y's is the number "1" (because we've divided by S).
Note, too, that if x were measured in kilometres (instead of return percentages) then M and S would be in kilometres and they'd all change
if we changed the units to miles. However, y = (xM)/S would have the same value in kilometres as in miles. Nice, eh?
>And if x were in degrees Fahrenheit then ...
Then we could change to Celsius and, although the xvalues would change, the yvalues wouldn't.
If we change to yvariables in Figure 1,
where y = (x0.064)/0.029, and we change the vertical coordinate to percentages
(so 14outof100 becomes 14%), we'd get Figure 5:
Figure 5
The right chart is standardized.
If we add some number (like 15%) to every x, the right chart wouldn't change.
If we multiply every x by some number (like 12), the right chart wouldn't change.
If x was the high temperature each Christmas for the past 100 years, measured in degrees Celsius, and we changed to degrees Fahrenheit, the right ...
>The right chart wouldn't change. Yeah, right, okay ... I get it!
Good for you.
>Okay, so what's KURTOSIS?
 M = (1/N) {x_{1} + x_{1} + ... +x_{1}} is the MEAN of the x's.
 S = SquareRoot[ (1/N) {
(x_{1}M)^{2} + (x_{2}M)^{2} +
... + (x_{N}M)^{2} } ] is the Standard Deviation of the x's.
 then SKEW =
(1/N)Σ y_{k}^{3} =
(1/N) [
((x_{1}M)/S)^{3} +
((x_{2}M)/S)^{3} + ... +
((x_{N}M)/S)^{3}]
 and KURTOSIS =
(1/N)Σ y_{k}^{4} =
(1/N) [
((x_{1}M)/S)^{4} +
((x_{2}M)/S)^{4} + ... +
((x_{N}M)/S)^{4}]
Notice that, after changing from the x's to the y's, we'd have a set of y's with
MEAN = 0 (that's the first moment of the y's) and
Standard Deviation = 1 (that's the second moment of the y's) and
the third and fourth moments of the y's
give the SKEW and KURTOSIS for the y's ... and for the x's !
Look carefully at the SKEW. If the xdistribution is symmetrical about the MEAN, then the distribution of
xvalues above the MEAN is the same as below. This gives rise to positive and negative yvalues, hence
positive and negative values for y^{3}, and they cancel out, making SKEW = 0. That's the case for the
Normal Distribution. On the other hand, if the distribution is more heavily weighted below the MEAN, the
negative values for y^{3} predominate and that'll give a negative SKEW. More above and you get a positive SKEW.
Look also at the KURTOSIS definition. It involves the 4^{th} power of the deviation from the MEAN (so
they're always positive) and, if there are more and more xvalues far from the MEAN, they contribute significantly to the sum of the
y^{4}values ... and the KURTOSIS increases. KURTOSIS is then a measure of how big the "tails" are.
Remember Figure 1? Well, the MEAN and Standard Deviation for the x's are: M = 6.4% and S = 2.9%
and if we calculate the SKEW and KURTOSIS, then make some changes in the xvalues far from the MEAN ... voila!
Figure 6
>Your charts are all inventions. Haven't you got a real, live ...?
On the right are General Electric stock prices, involving about a thousand daily returns.
Below, the distribution of these daily returns for GE, along with a Normal and Lognormal distribution
with the same MEAN and Standard Deviation  for comparison purposes. (Nobuddy is suggesting that the GE distribution
is one or t'other, but it's interesting to note that there's little to choose between normal and lognormal distributions, though
neither is a very good representation for the GE distribution):
 Figure 7A

Figure 7B
The left chart gives the actual distribution: KURTOSIS = 6.86
See the outliers those returns way out on the tails of the distribution, at 15% and 12% and even one at about +20%?
>20% for a daily return? You're kidding, right?
It was October 19, 2000. GE had closed at $51.75 on the 18th and at $61.88 on the 19th. That's almost a 20% jump
(shown as a red dot in Figure 7A). This, together with a few more large changes, make for a big KURTOSIS.
However, if we replace a half dozen of these wayout daily returns by 0%, we get the right chart.
The outliers are gone, there are more returns at 0% and a we've got a smaller KURTOSIS.
>At 3.60 it still seems big to me.
Actually, for the Normal Distribution (which is perfectly symmetrical about the MEAN), KURTOSIS = 3.0 and, for
this reason, KURTOSIS is sometimes defined to be:
(11) KURTOSIS =
(1/N)Σ y_{k}^{4}  3 =
(1/N)Σ {(x_{k}M)/S}^{4}  3
which, by subtracting "3", gives the deviation from the value for the Normal Distribution.
With this definition, the Normal Distribution has a KURTOSIS = 0.
>Which one do you use?
I don't use either. I just regurgitate what others say ... just in case you run across one or the other in your travels.
>Why the two definitions?
Why are you asking me? But I can tell you, there are other definitions too. Want to see them?
>Definitely not!
Okay, here's another. The MS Excel definition:
>Gee, thanks. I needed that.
However, for large values of N, it looks like which you'll recognize from (11) above, right?
>Wrong! And by the way, what about Σ y^{5} ... that fifth moment? What's its name?
I have no idea.
>What? All this and you don't have a name for it? You should really look up the
name before you start writing tutorials. I can't believe you'd go through all that stuff and not
have a name. People like names. They're important! Haven't you noticed that? You see a strange bird and everybody
says "What is it?" and they're not satisfied until you say, "a South American Twiddlewot"
then they're all happy, they have a name and they're satisfied and they don't need any more information about the Twiddlewot, just a name because ...
zzzZZZ
Here are some examples of Skew and Kurtosis ... some stocks on the DOW, using eight years of weekly returns:
>And what formula are you using for ...
For Skew & Kurtosis? Excel's formulas:
>How many formulas for skew and kurtosis are there?
How high can you count? For Kurtosis, some subtract 3 so that a Normal distribution has Kurtosis = 0.
In the Standard Deviation, some divide by (n1) and some divide by n. However, for large n, they all reduce to equation (11), above.
Anyway, when ordered, they go like this:

Stock  Skew  Kurtosis 
AA  0.25  1.17 
AIG  0.75  3.92 
AXP  0.18  2.89 
BA  0.78  6.65 
C  0.45  2.73 
CAT  0.77  3.91 
DD  0.25  1.44 
DIS  0.08  2.82 
GE  0.21  3.41 
GM  0.02  1.92 
HD  0.34  4.33 
HON  0.55  7.67 
HPQ  0.01  1.24 
IBM  0.37  2.66 
INTC  0.20  2.08 
JNJ  0.47  4.72 
JPM  0.18  1.65 
KO  0.30  1.81 
MCD  0.01  1.01 
MMM  0.33  2.54 
MO  0.08  4.43 
MRK  0.42  3.13 
MSFT  0.37  2.73 
PFE  0.01  1.52 
PG  2.67  25.19 
T  0.68  4.20 
UTX  1.30  11.42 
VZ  0.39  2.44 
WMT  0.43  2.02 
XOM  0.14  1.13 

>Mamma mia! That PG is really skrewed, eh?
Procter & Gamble? It looks that way. The return distribution is SKEWed toward returns less than the Mean.
Indeed, there were dramatically poor returns, opening at $117 on Jan 3, 2000 and closing at $94 by the end of the month.
Note, too, that the "tails" are big compared to the symmetric Normal distribution (as measured by Kurtosis),
and they're larger than Normal (since Kurtosis is positive) and ...
>But MCD is more normal, wouldn't you say?
Yes, I guess McDonald's is normal.
Here's some more (mostly) Vanguard funds, using annual returns from 1972 to 2006 (obtained here):
Note that here, Kurtosis is sometimes larger and sometimes smaller than Normal
(so the spread or "tails" may be larger or smaller)
and VISGX, Vanguard's Small Cap Growth Index, is quite ... uh normal.

Fund  Skew  Kurtosis 
BRSIX  0.05  0.43 
EM15  0.19  0.23 
NAESX  0.14  0.41 
PCRIX  1.52  4.94 
VBMFX  0.78  1.27 
VDMIX  0.27  0.17 
VEIEX  0.04  1.29 
VEURX  0.62  1.48 
VFINX  0.54  0.52 
VFISX  0.64  0.69 
VFITX  0.79  1.24 
VGSIX  0.46  0.13 
VIGRX  0.31  0.77 
VIMSX  0.30  0.33 
VIPSX  0.02  0.22 
VISGX  0.04  0.01 
VISVX  0.36  0.24 
VIVAX  0.11  0.40 
VMPXX  0.65  0.81 
VPACX  0.91  1.35 
VTRIX  0.07  0.84 
VTSMX  0.55  0.52 
VUSTX  0.85  0.53 
VWELX  0.51  0.12 
VWINX  0.03  0.61 
VWNDX  0.37  0.56 

>Yeah, so where's the spreadsheet?
Just click on the picture below to download:
to continue
