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 stay-invested-all-the-time 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 stay-invested-all-the-time 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 always-invested 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 "always-invested". 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 one-year window. See that "very very worst" month, Oct/87? That's the green guy that looks like the CN Tower. It moved into our one-year 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 always-invested annual "gain factor", we multiply twelve monthly gain factors together, like so (which happens to be for the year ending Jan 1, 2000):


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:


So, what's the ratio (Timer portfolio)/(Always-Invested portfolio)?
Divide and get, quite simply, 1/1.063 = .941 or 94.1%, that is:
B = (Timer_who_misses_the_Best_month portfolio)/(Always-Invested 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)/(Always-Invested 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 Always-Invested.
  • 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 Always-Invested.
  • 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 Always-Invested), 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.


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 Always-Invested) 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 fifty-one 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 Fifty-four 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 fifty-one years of the S&P 500 and order the months from Minimum Gain to Maximum Gain. If we had invested in the Worst Sixty-nine 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 re-order 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 stay-invested-all-the-time) ... 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 always-invested 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 500DOW
Omit the best 10 months
Omit the worst 10 months
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
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 always-invested?
    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 out-then-in, 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 Always-Invested (i.e. Buy-and-Hold). I know of one such strategy: VMA

Volume-weighted 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 5-day moving average stock price is 25%* below the 200-day Volume Weighted Moving Average (VMA), then invests it all. When the 5-day average rises 25%* above the 200-day 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 ... RIGHT-click 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. Always-Invested 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 mid-February ... 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. Always-Invested'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. Always-Invested.

After this less-than-exhaustive analysis, I now have a conjecture concerning some

Universal Truths re Market Timing
  1. For each Timing Strategy there is an optimal set of parameters.
  2. The use of these optimal parameters will always result in a portfolio gain which exceeds the Buy-and-Hold gain.
  3. Optimal parameters will depend upon future evolution of the Market/Stock/MutualFund, hence cannot be known!
  4. The probability of choosing optimal parameters by examining past market performance is ZERO.
  5. Market Timing is a strategy for those with appropriate genetic configuration.
  6. Market Timing is more fun than Buy-and-Hold!

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