Random Walks: some comments re financial planning

We suppose that the price of a stock/mutual fund begins at the value P(0)
and, a day (or month or year) later has the value P(1), then P(2), then ... after N days (or months
or years), has the value P(N).
(Well ... uh, let's say it's "days" tho' you recognize
it could be months or years, okay?)
Fig. 1 
Anyway, we also suppose that, if today's price is P(n), then tomorrow's price is P(n+1) which
differs from today's price by an amount, say, b. So, we write:
P(n+1) = P(n) + b
If the daily prices did change by some number b, the chart
would look as it does on the left, depending upon whether b was
positive or negative. Not exactly the way stock prices behave, eh?

Okay, so if stock prices don't move in a straight line, how do they move?
Fig. 2 shows a picture of the Dow Jones Industrial Average. It's hardly a straight line growth
but if we plot the logarithm of the DOW values (instead of the values, themselves),
we get quite a nice straight line (Fig. 3). Well, sort of. It even straightens out a few wrinkles :)
Anyway, for now, we'll assume that the logarithms of our stock prices, P(n), do lie on a
straight line. That would mean that:
log P(n+1) = log P(n) + c
(Did I mention that we're considering the natural logarithm, not to the base 10 but to
the base e which has a value about 2.71828?)
For more on straight line fits to logplots,
see Best Fit.
We'll play with this, writing:
log P(n+1)  log P(n) = c
so
log(P(n+1)/P(n)) = c
soP(n+1)/P(n) = e^{c} 
Fig. 2
Fig. 3 
From this we get P(n+1) = P(n) e^{c} and if the daily gain
in stock price is r (r = 0.0123 means 1.23% gain), then we should have P(n+1) = P(n) (1 + r)
so we now see the significance of the funny e^{c}. We can write:
e^{c} = 1+r or c = log(1+r)
or e^{c} 1 = r
Because the daily gain is NOT a constant, but changes daytoday, that suggests that we
should modify c (or r) daily. But how?
We could, for example, choose for r the average daily gain over the previous umpteen days.
If we do modify r (or c) daily, we can write:
P(n+1) = P(n)(1+r) or, equivalently
P(n+1)  P(n) = P(n) r
Since P(n+1)  P(n) is the daily change in stock price, we'll call it
ΔP. (Mathtypes love to use Δthis
and Δthat to represent small changes
in this or that.) Anyway, that gives us our Random Walk prescription:
ΔP = P r
Each day we stare at today's stock price P, compute some daily gain r, then
increment our stock price by an amount ΔP giving us
the new stock price:
P + ΔP.
Then we do it all over again, using P + ΔP
as today's price!
Of course, if we choose a constant value for r we'd get a sequence of prices P(1), P(2), etc.
which looks like Fig. 4 (where we've chosen, for r, the average daily gain over the
time period indicated in the plot).
Uh ... one other thing: we can use the arithmetic mean of the daily gains:
{r(1)+r(2)+...+r(N)}/N
or we can use the geometric mean of the daily multipliers
(by daily multiplier I mean the ratio P(n+1)/P(n)):
{(1+r(1))(1+r(2))...(1+r(N))} ^{1/N}

Fig. 4 exponential growth

Surprise! The logarithm of this geometric mean is the
arithmetic mean of the logarithms:
{log(1+r(1))+log(1+r(2))+...+log(1+r(N))}/N
The two means give different results. The arithmetic mean overshoots the last number, P(N).
The geometric mean is dead on. Because of how the multipliers were defined,
namely P(n+1)=P(n){1+r(n)}, then the product {(1+r(1))(1+r(2))...(1+r(N))}
is none other than
{P(1)/P(0)}{P(2)/P(1)}...{P(N)/P(N1)} which is precisely P(N)/P(0).
Further, the arithmetic mean is ALWAYS larger than the geometric mean. (That's a deep mathie
theorem and applies to all sets of positive numbers, like the set: {1+r(n)} )
ANYWAY, it'd be neat if we could somehow incorporate the random fluctuations in stock
price (as in Fig. 2). That means r should vary, daytoday, in some meaningful fashion.
Fig. 5

To do this we select (for r) a value that has some kind of probability distribution.
For example, look at the daily percentage gains of the DOW over a three year period.
They're distributed very much like the
Normal Curve (that is, the infamous Bell curve  the red curve in Fig. 5). Anyway, it's close enough ... so we'll
assume that the daily gains r have a Normal distribution with some mean value and some
"spread" or "dispersion" about this mean ... as measured by the Standard Deviation.
for more on Standard Deviation, see
SD stuff
Suppose we have a collection of normally distributed random numbers with Mean = 0 and SD = 1.
(These tables abound!).
We'll call these numbers g(n) (with n=1, 2, 3, ....).

Then the numbers r(n) = D g(n) + A
will have Mean = A and Standard Deviation = D.
 That's because
multiplying a set of random numbers by D multiplies both the
Mean and the Standard Deviation by D. If the Mean is already zero, then the new set
(namely D g(n)) has Mean = 0 and SD = D. Now we add the constant A
which increases the Mean by amount A but doesn't change the SD. The result?
r(n) = D g(n) + A has Mean = A and SD = D.
Okay, we first generate a set of random numbers, Normally distributed with Mean = 0
and Standard Deviation = 1, then, to each number, we
(1) multiply by 1.15
(2) add 0.08
(3) divide by 100
(the last step, in order to change the numbers to a percentage). This should change our set of
Normally distributed numbers, namely the g(n) with Mean = 0 and SD = 1, to the set
r(n) with Mean = 0.08% and SD = 1.15% (just like our DOW, right?)
Both sets are shown in Fig. 6 and Fig. 7
Now we can play:
First, let's see if this Random Walk model of the DOW behaves anything like the DOW
(as in Fig. 2).
Each day we change our Market Index by some percentage by choosing one of the 100
random numbers. These percentages have a positive average value so we should see our Index
increase. But they also vary in the range 3% to +3% (roughly) and some are more apt to be
selected than others (according to Fig. 7) so we should get a few wriggles (or is it wiggles?).

Fig. 6
Fig. 7

Fig. 8 
Anyway, here's a sample (Fig. 8, where the real DOW is shown in gray).
Looks good, eh? Remember, each time we select a set of random gains (from our set of
100 ) we get a different picture of our RandomWalkDOW. Some may look like the
real DOW ... some won't.

Here are several different selections (from the same set of 100 random gains):
So, is the Random Walk good for predicting? Hardly. Of course, the above charts could
represent the growth (or decay?) of your stock portfolio, or, we could consider various
scenarios for a $50K portfolio over one year (250 market days):

Uh ... did I mention that these charts were generated with an Excel
spreadsheet which made the random selection of daily gains (from the set of 100 Normally
distributed gains) and plotted a chart each time I did an F9 recalculation?
There's a moral here:
With volatile investments a Random Walk says that just about
anything can happen (and occasionally does).
Now for a change of pace:
Suppose there are just two possible daily portfolio multipliers: 1.01 (meaning a 1%
increase in assets) or 1/1.01 = 0.990099 (about 1%) meaning a loss which completely
eliminates the 1% gain. We choose between the gain and the loss by flipping a coin. Heads
means we multiply our assets by 1.01. Tails we divide by 1.01  okay? Below, a few charts
of a $50K portfolio (over one year) using the Heads & Tails probability distribution and
to the right, a Heads/Tails (+1% or 1%) distribution (50% probability of each, having Mean = 0%
and Standard Deviation = 1%). 

Although anything can happen, we somehow feel comfortable in saying that,
over the long haul, we should neither lose nor gain (because we trust that there will be
pretty well the same number of heads as tails). In other words, over the long haul,
the most probable outcome is a $50K portfolio. This "most probable" chart
is just a horizontal line at $50K, eliminating all volatility and wiggles ... nice and smoooth.
This is what one does with Random Walks.
One expects that, although anything can happen, over the long haul, we would expect
a certain "most probable" outcome from our Random Walks. Guess what the "most probable" chart
would look like?
One other thing:
If we do a Random Walk a jillion times, each time generating a possible future evolution of
the market(s), then analyze these jillion scenarios ... we're using a technique called
Monte Carlo. If'n y'all want to do Monte Carlo,
try this.
Here's something to think about: 
Over the past fifty years there were 600 monthly closing prices for, say, the S&P 500 index.
From this set we extract two (count 'em, two) parameters:
some Mean Monthly Return and the Standard Deviation. Armed with these parameters
and the assumption of some Probability Distribution
(which supposedly characterizes the past fifty years), we use a Random Walk and make conclusions
about this or that. Do these conclusions apply to the future ... or the past? Surely not the
future. Who can predict the future? Besides, "... past behaviour does not imply
future evolution". So, it's the past. Why, then,
wouldn't we just use all the information available (for fifty years)? For example:
The annualized return of the S&P 500 was 9.2% (meaning that a 9.2% annual increase in this
index would generate the Jan, 2000 value, starting with the Jan, 1950 value). If we had $100K
invested in this index (without trading fees or foreign exchange losses or whatever) and made
monthly withdrawals at the annual rate of 7% (meaning .07 x $100K/12 = $583/month), this amount
increasing each year with inflation, then how would our portfolio have evolved?
Why, then, would we use Random Walks? Beats me ...
And why would one assume that the Distribution of Returns (with its Mean and Standard
Deviation) characterizes the past umpteen years? Reordering the Returns wouldn't affect this
Distribution at all, but would certainly affect many conclusions which were based upon the
Distribution.
I think there's a moral here:
When possible, use the data ... and not a mathematical proxy 

Oh ... one other thing: you might take a peek at
Investing which also
involves using past performance to guesstimate future evolution.
One last thing: (Have I said that before?)
If you think that examining the past, extracting a few parameters
>(such as Mean Monthly Gain & Standard Deviation) and assuming
some probability distribution for the monthly gains, then using that data to divine the future evolution of
the market  if you think that's a reasonable thing to do  take a peek at the S&P500
... and it's actual evolution
during the various decades since 1950
>... and the actual variatons in
Mean Monthly Gain and Standard Deviation  from decade to decade
One last thing:
Here's a neat Excel spreadsheet.
> RightClick here to download a .ZIPd version.
Every time you press F9 (to recalculate) you get another 10year graph, generated by choosing
120 monthly gains (that's 10 years, eh?). These 120 gains are chosen, at random,
from the 600 monthly gains of the S&P500, in the period Jan 1, 1950 to Jan 1, 2000.
This "random" graph flashes across a darkened sky like a bolt of white lightning (with the
"five decade" chart, above, as backdrop).
Three random bolts are shown here
>but y'all only git one bolt, per F9, in the spreadsheet


You'll have noticed that the 120 selected (bolt of lightning) gains are chosen from a set
which has the identical Mean, Standard Deviation and distribution as the S&P500
... 'cause it is the S&P500!
>So what can I expect for the next 10 years, if I invested in the S&P500?
Nothing.
>But ... if I applied some sophisticated mathematical prescription for identifying the most probable ...
Nothing.
>But, what if ...?
Nothing.
Examine the bolts of lightning, apply sexy mathematical/statistical methods and try to
predict the location of the next bolt.
Would you feel safe in avoiding that location? Hardly. What you can do with certainty is to make statements like:
"Had I applied technique X, for the period Jan, 1980 to Jan, 2000, then this or that
result would have been obtained.
If'n yer happy with that ... good.
>But ... but ... what about most probable or statistically expected ...
If you were to be reincarnated, a thousand times, and repeated the same investment scheme, then
you may be interested in the most probable result.
Do you intend to be reincarnated, a thousand times?
>Yes.
Then, by all means, place your faith in mathematical predictions of the future. In fact, try
a Monte Carlotype scenario.
Here's a spreadsheet you can play with:
Rightclick here and Save Target or Save Link
to download spreadsheet as a .ZIPd file.
Other related spreadsheets concerning Portfolio growth (and decay!) are described
here and here.
