What a Difference a Month Makes: some comments on missing the best or worst months

I've read (somewhere, sometime ... like this, provided by
bylo)
statements about what'd happen if you missed the "best umpteen
months" or maybe the "worst umpteen months", and how that'd affect your portfolio and what
this implies re Market Timing.
So, I decided to investigate this myself:
We consider investing $1.00 in the S&P 500, on Jan 1, 1950. (We could consider other markets
or indexes, but this one'll illustrate the point ... and the S&P data is readily available.)
By Jan 1, 2000 that $1.00 would be worth $81.79, a significant gain since the S&P 500 went from
17.05 to 1394.46 during that fifty year period: a 9.2% annualized gain. However, during that
period, one month (Sep/74) had a gain of 16.3% and another month (Oct/87, what else?) had a gain
of 21.8% (that's a negative gain!).
Suppose we had missed the Sep/74 gain? (That is, we just cashed out our S&P investment and
reinvested at the beginning of the the following month.) Our portfolio would then
be worth only $70.32 by Jan 1, 2000. The omission of that single "best month" resulted in a
decrease of 14.0% in our portfolio! ... and an annualized gain that decreases
from 9.2% to 8.9%

Fig. 1

Aah, but what if we had missed that "worst month": Oct/87 (the only market crash since 1929)?
Our final, Jan 1/00 portfolio would be worth $104.54, an increase of 27.8%
(over the stayinvestedallthetime portfolio) ... and an annualized
gain that increases from 9.2% to 9.7%
Of course, the best month  or worst month  could occur at some time other than
Sep/74 or Oct/87, but that wouldn't affect the above numbers: we just multiply all 600 monthly
"gain factors" (a 2.3% monthly gain means a monthly "gain factor" of 1.023) to get the
stayinvestedallthetime portfolio. We'd just leave out the "best" or
"worst" monthly gain factors to get the other two figures; the ordering is unimportant.
What's more important? Missing the "best" month or missing the "worst" month? Let's start at
Jan 1, 1951 and see how our portfolio would have fared had we missed the "best" month for the
previous year (compared to an alwaysinvested portfolio). Then we do the same for
the year ending on Feb 1, 1951. Then the year ending Mar 1, 1951 etc. etc. While we're at
it we'll also consider how our portfolio would have fared had we missed the "worst" month of the
previous year (for the years ending Feb 1, 1951 then Mar 1, 1951 etc. etc.).
Fig. 1A
So what can we conclude from Fig. 1A? If we had invested for one year only, starting some time from
Jan/50 to Jan/99, missing the "best" month of the year, then, on average (whatever that means!),
your portfolio would be 6.4% lower than being "alwaysinvested". Had you invested for one
year, missing the "worst" month, the average gain would have been 6.5% higher. For me, I can
draw no conclusion from this ... but remember, we're assuming you do not add to your portfolio.
You start with $umpteen and just take it out or put it in (to miss the "best" or "worst" months).
P.S. The charts look like the Toronto skyline cuz the "GOOD" and "BAD" months often
hang around for a year before they move out of the oneyear window. See that "very very worst"
month, Oct/87? That's the green guy that looks like the
CN Tower. It moved into our oneyear window
on Nov 1, 1987 and hung around until we moved to the year ending Nov 1, 1988 !!
Now here's an interesting item: to get the alwaysinvested annual "gain factor", we
multiply twelve monthly gain factors together, like so (which happens to be for the year
ending Jan 1, 2000):
(0.968)(1.039)(1.038)(0.975)(1.054)(0.968)(0.994)(0.971)(1.063)(1.019)(1.058)(0.949)
Now we replace the "best" monthly gain factor, namely 1.063, by 1.000 ('cause that's what the
Timer gets if he cashes out and misses this "best" month). That gives an annual gain
factor, for the Timer, of:
(0.968)(1.039)(1.038)(0.975)(1.054)(0.968)(0.994)(0.971)(1.000)(1.019)(1.058)(0.949)
So, what's the ratio (Timer portfolio)/(AlwaysInvested portfolio)?
Divide and get, quite
simply, 1/1.063 = .941 or 94.1%, that is:
B = (Timer_who_misses_the_Best_month portfolio)/(AlwaysInvested portfolio) = 1/best_month_gain_factor

If we do the same for the Timer who misses the worst month, we'd get
W = (Timer_who_misses_the_Worst_month portfolio)/(AlwaysInvested portfolio) = 1/worst_month_gain_factor

which happens to be (for our example): 1/.949 = 1.054 or 105.4%
Some observations:
 It's possible that the "best" and "worst" month gain factors are both less than 1
(it was a lousy year) so both Timers would do better than AlwaysInvested.
 It's possible that the "best" and "worst" month gain factors are both greater than 1
(a bull market!) so both Timers would do worser than AlwaysInvested.
 The ratio of Timer portfolios, (Miss_the_Worst)/(Miss_the_Best), is just the ratio
of gain factors:
W/B =(Best_month)/(Worst_month).
 The sensitivity (to market conditions) of the Timer portfolios
(compared to AlwaysInvested), depends upon the reciprocal of the "extreme"
gain factors (either the MAXimum or MINimum gain factors, like 1.063 and .949 in the above
example). It's a fact of nature (!) that the smaller the number, the greater the influence that
number has on its reciprocal.
(What!)
Well, what I mean is ... small changes in x make much larger changes in 1/x, if x
is small. On the other hand, if x is large, small changes in its value make small
changes in 1/x. And when do we go from the former to the latter? At x = 1.
Ain't that nice? Our "gain factors" are close to 1.
However, the "worst" months
(usually^{*})
have a gain factor less than 1, so missing the "worst" month
(usually^{*}) has a
greater influence on the Timer's portfolio (compared to AlwaysInvested) than
missing the "best" month (where the gain factor is
usually^{*} greater than 1). Perhaps that's
what the Timer is after: the chance to miss the "worst" while running the risk of
missing the "best".
^{*} I say "usually", but in every single
twelve month period which began on the first of the month (from Jan 1, 1950 to Jan 1, 2000), the
"worst" month gain factor was less than 1 ... and 100% of the "best" months had a
gain factor greater than 1.
Nuff said ... except for one other thingy:
Suppose we look at fiftyone years of the S&P 500, from Jan/50 to Dec/00, and order the months
from Maximum Gain to Minimum Gain. The Maximum monthly gain was during
October, 1974 and the Minimum was (what else?) October, 1987. If we had just invested in the Best Fiftyfour
months, we'd have achieved ALL of the gains for the period Jan/50  Dec/00. The other 558 months
would have given us 0% gain ... the positive gains cancelling the negative!
And now, for a change of pace, suppose we look at those fiftyone years of the S&P 500
and order the months from Minimum Gain to Maximum Gain. If we had invested
in the Worst Sixtynine months, we'd have lost 99% of our investment.
Aah, but we've been assuming we invest and sit back.
NOW, suppose we were investing monthly, say $100 per month. The months when these "best"
or "worst" occur are VERY important! In fact, if the worst month were the very
first month, we'd lose very little because we'd only have $100 invested (and we'd lose 21.8% of
$100 ... peanuts, eh?).
On the other hand, if the worst month were the last month, we'd lose 21.8% of (are
you ready for this?) over a million dollars!! (I might point out that $100 per month invested in
the S&P 500 over this fifty year period would have resulted in a Jan 1, 2000 portfolio of
$1,058,594. I might also point out that we're ignoring any losses due to broker
commissions or foreign exchange losses or foreign content penalties or ...)
Fig. 2 
Okay, now suppose we invest that $100 per month and omit the N "best months" or
"worst months". What'd we get is illustrated in Fig. 2 (over that particular fifty year
period ... with the "best"
and "worst" months occuring where they actually did occur, remembering that the results
can be wildly different if we reorder the monthly gains, putting the "best" or "worst" months
at the beginning or, perhaps, at the end of the period).
Personally, I like the blue curve.
Note, too, that the results can also
be wildly different if we choose another time period (other than Jan/50 to Jan/00).
Note, too, that the results can also
be wildly different if we choose another market or an individual stock or mutual fund or ...

Anyway,
Fig. 2 shows how radically different the results would have been for the S&P 500
(compared to stayinvestedallthetime)
... for the period Jan 1, 1950 to Jan 1, 2000.
We can, however, consider shorter time intervals (which still contain those "very best" and
"very worst" months of Sep/74 and Oct/87 ... as well as an additional assortment of nine "best" and "worst"
which change, of course, with the time period being considered).
In Fig. 3, the effect of avoiding the "worst" months is shown; it decreases with the time
interval. Avoiding the "best" months? That increases. Each gets closer to 92.2%; that's when
there's just ten months in our time interval and alwaysinvested gets
an 8.2% gain (hence a "gain factor" of 1.082) and we avoid 'em all get a 1.000 gain factor
so the ratio is 1.000/1.082 = .922 or 92.2%, see?
Note(s): A major influence on the numbers is the location
of these "best" and "worst" months. (example: closer to the beginning of the interval.)
Of course, shortening the time interval also leaves less and less time for the effects of
"best" and "worst" to sink in.
(What on earth does THAT mean!?)
Well, uh, I mean if there were only eleven months and we eliminate ten (either the "best" or
"worst" ten), then we've got a single monthly gain (or loss) but 11 x $100 contributions to our
investment portfolio ... and how much damage can one month do?
(Don't ask!)

Fig. 3

Anyway, because these results are so sensitive to where these "best" or "worst" months occur,
when the time period starts (and ends), the length of the period, the particular index or
stock or mutual fund ^{*}
... I place little significance on the actual numbers
used to illustrate What a Difference N Months Make, but I
am now very aware of how radically different the results can be!
(Sorta like "exponential growth"; forget the numbers, but be impressed with the result!)
for the period Jan/70 to Jan/00
comparing to an always invested portfolio 
S&P 500  DOW 
Omit the best 10 months 
58%  55% 
Omit the worst 10 months 
199%  230% 
Table 1.

^{*} For example, if we consider avoiding the
"best" and "worst" ten months for the DOW, for the thirty year period from Jan,1970 to
Jan,2000 we'd get a different result as illustrated in Table 1 where the percentages are
what our portfolio would have been, had we avoided the "best" and "worst" ten month periods
... as a percentage of an "always invested" portfolio.

For example, by missing the
best ten months of the DOW, we'd have a portfolio worth just 55% of an always invested
portfolio.
(... and the locations of these "best" and "worst" months, for the S&P 500 and DOW??)
Fig. 4
If'n yer countin' the number of reds and greens, to see if there are ten of each,
don't bother. Sometimes a "best" month occurs immediately after a "worst" month and the
two vertical lines are one atop t'other (graphically speaking). For example, although Sep/74
was the "very best" month for the S&P500, Aug/74 was one of the ten "worst" months!
That's shown in Fig. 5 where we've blown up this piece of the chart: After months
(and months and months) of lousy market returns, there's significant
volatility and a couple of the "worst" are followed by a couple of the "best". The S&P 500
had been falling for almost two years (well ... maybe 1.75 years ... see
the 1973/74 bear market)
and was recovering.

Fig. 5 
Note(s):
Remember: By "missing umpteen months" we mean that we didn't suffer any losses or
accumulate any gains for each of these umpteen months  our "gain factor" was 1.00  we sold our stocks,
sat on the cash for a month, then reinvested at the start of the following month, repeating this
ritual for each of the missing months. Clear as mud, eh?
Remember, too, that we're assuming a fixed monthly investment (say $100, tho' it could
be any fixed amount; the percentages we've given won't change). Even for the "missing months",
we stick our $100 into cash, reinvesting at the start of the following month (assuming, of
course, that the following month ain't one of those "missing months", else we'd wait for another
month and ... well, you get the idea.)
So, what does all this mean re Market Timing?
 Presumably, adopting a Buy & Hold philosophy eliminates the Risk of missing the "best"
months. You get 'em all.
 By adopting a Market Timing strategy, the investor presumably accepts that Risk in
order to achieve a possibly greater Reward ... by missing the "worst" months.
 What are the chances of the Timer achieving greater returns?
 Is it valid to assume some random selection of months for the Timer to miss,
while playing her game? (Hardly. Though the chance of missing one of the 10 "worst" months in
thirty years is maybe 3%, the chance of missing them all is ... uh,
0.00000000000000001% or thereabouts. Besides, with random selection, one must also assume that
the Timer exercises no cerebral function.)
How 'bout missing  by chance  both the "best" AND the "worst" ten months? :)
From Jan/50 to Jan/00? We'd be ahead, compared to alwaysinvested? Here's the picture:
Fig. 6
 Is there some (other) way of quantifying this Risk/Reward scenario? (See Risk/Reward Ratios.)
 Is there a way of incorporating the cost {such as brokerage fees, foreign exchange losses
(if any), foreign content penalties (if any)} of getting outthenin, for each month the
Timer plays this game? As a fixed cost? A percentage  which grows with the portfolio
size ... or decreases?
I have no idea
... yet.
Perhaps it's best to analyze various "timing strategies" to see which ones would have done
better or worse than AlwaysInvested (i.e. BuyandHold). I know of one such
strategy: VMA
Volumeweighted Moving Average
In this strategy, we'll assume we begin with $10K in cash. Mr. Buy_and_Hold invests it all in a
stock. (Here we'll use CBR as an example 'cause I got me the data for that guy.) Ms. Timer waits
until the 5day moving average stock price is
25%* below the
200day Volume Weighted Moving Average (VMA), then
invests it all. When the 5day average rises
25%*
above the 200day VMA she sells it all. How would
each have fared over the 250 days
ending May 8, 2000? (Why 250 days? Cuz that's the data I happen
to have.)
Fig. 7
Aha! Ms. Timer wins, hands down! In fact, CBR hardly changed a whit over that
time interval yet she made a bundle. (I knew she would!) A 64% gain.
* Why 25%? Why not? You might also use the
Standard Deviation of the daily
gains as a percentage of the Mean daily gain. That'd be 17% and that would've given Ms. Timer
a gain of 42%. Not too shabby.
So what does this prove? That Timing is good for your financial health? (Hardly. But you
can try VMA yourself, on your favourite Index/Stock/MutualFund with this spreadsheet:
VMA.ZIP ... RIGHTclick and Save Target or Link)
But I'll tell ya somethin' ... Timing is much more fun!
Do I use VMA? Sometimes ... look here.
Okay, let's try another, more popular (!) timing strategy: Bollinger Bands
Every day we compute, for the previous 20 days, the average A and the Standard
Deviation SD of stock prices. Then we compute the Upper
and Lower Bollingers:
U = A + 2*SD and
L = A  2*SD
If we plot these Upper and Lower Bollinger bands (for a few years, ending May, 2000) we'd get:
Fig. 8A 
Fig. 8B 
where Fig. 8B gives a closeup of the last few months.
The Timing Strategy we'll use is this:
 Whenever the stock price exceeds U, we sell.
 Whenever the stock price drops below L, we buy.
How'd we do? Starting with a $100K portfolio (on April 1, 1997) Mr. AlwaysInvested would
have $179,335 by May 8, 2000.
And Ms. Timer? She'd have $205,531. Their portfolios, for the last
few months, is shown in Fig. 9. Notice that Ms. Timer was entirely in cash until early January, 2000
when the stock price fell below the Lower Bollinger,
so she bought (with all the cash she had) and didn't sell until midFebruary ... etc. etc.
While she was entirely in cash, her portfolio remained constant.

Fig. 9 
Of course, Ms. Timer should really have put her cash into Money Market ... and paid
transaction charges for the trading she did. And who says the Average and Standard Deviation
should be computed over 20 days? And why U = A + 2*SD?
Why not some number other than 2?
Why not U = A + 1.5*SD?
For a multiplier of 1.5, Ms. Timer would have had a final
portfolio of $176,267 ... somewhat worser, yet already, than Mr. AlwaysInvested's $179,335.
However, for an even smaller multiplier, just 1.0, her a final portfolio would be
better: $208,463! Mamma mia! Who's to know what multiplier to use?
It's clear that Ms. Timer must select an appropriate set of
parameters in order beat Mr. AlwaysInvested.
After this lessthanexhaustive analysis, I now have a conjecture concerning some
Universal Truths re Market Timing
 For each Timing Strategy there is an optimal set of
parameters.
 The use of these optimal parameters will always
result in a portfolio gain which exceeds the BuyandHold gain.
 Optimal parameters will depend upon
future evolution
of the Market/Stock/MutualFund, hence cannot be known!
 The probability of choosing optimal
parameters by examining past market performance is ZERO.
 Market Timing is a strategy for those with appropriate genetic configuration.
 Market Timing is more fun than BuyandHold!

P.S. For more bumpf on Market Timing (sort of):
just
and you may (or may not) want to look at this
Technical Analysis stuff
... for the adrenaline rush
or maybe download a spreadsheet or three
