Stocks Up and Down

One often hears the statement that stock prices increase most of the time and ...

>You mean they don't?
Well, not always. In fact, I always thought that prices increase, maybe 60% of the time.
Of course, that doesn't mean much if the weekly increases average 0.1% whereas the decreases average 5%.

>So you decided to check it out, right?
Right ... and here's what I found:

  1. We download ten years worth of weekly prices for ten stocks. (That's about 500 prices for each stock.)
  2. We determine the percentage of times the weekly prices increase or decrease, and the average of the increases and decreases.
  3. We generate pretty charts, like this:

>That doesn't mean much if the weekly increases average 0.1% whereas ...
Yes, yes I know! Here are the average increases / decreases, called UP return % and DN return %:

>Wanna hear my conclusion?
>We can expect the weekly returns to be positive more than 50% of the time and the average positive return is greater than the average negative return.
And you base that on these ten stocks over that 10-year period, eh?
>Why not?We're talking future here and that's fuzzy so my conclusion is ....

Anyway, you may like to play with a spreadsheet where you enter ten Yahoo stock symbols and click a button and get some pretty charts:

Click on picture to download spreadsheet

You can get charts for a ALL stocks ... or just the percentages for a single stock.
>Then what?
Then decide for yourself whether or not you'll say: "Stocks increase most of the time."
>So, what about daily returns? How many UP days then?
Here's another chart:

>And stocks other than the S&P index? What about them?
I have no idea. I got those daily results here (as suggested by JWR).
However, the spreadsheet can do ten years worth of daily stuff:

>Hey! Those daily UP percentages are almost 50%. How about monthly?
Yeah. Good idea. Here they are:

Note that, for the Nasdaq, only 52% of the months were UP-months.
However, the average UP return% was significantly larger than the DN return% and the Nasdaq gained about 6.5% over this 10-year time period.

Indeed, ^IXIC looked like this:  

I make the pretty pictures ... you make the conclusions.

>Can't you prove anything?
Sure. Easy. We'll make some unjustified assumptions and arrive at an unwarranted conclusion, like so:
  • We'll assume a stock with a fixed Mean annual return of R% and fixed Standard Deviation of S%.
  • We'll assume that, dividing a year into N equal parts, the returns over this shorter time period average R/N%.
    (That's assuming returns vary with the length of the time period.)
  • The Standard Deviation over this shorter time period is S/sqrt(N)%.
    (That's assuming Standard Deviation varies with the square root of the time period.)
  • We'll assume that the returns (daily, weekly, monthly ... whatever) are all Normally distributed.
Then, if F(R), the cumulative probability distribution for annual returns, looks like Figure 1b, the fraction of positive returns is 1 - F(0).

When we divide the annual period into N subperiods, the shape of the distribution changes, but the fraction of positive returns is still 1 - F(0)

See the dots in Figure 1a? Them's F(0) and they increase with N. ... so 1 - F(0) decreases.

>You're talking R = 8% and S = 20%?
Yes ... for now.

Figure 1a
In Figure 1b, (where we consider N = 1 meaning the time period is 1 year), 1 - F(0) = 1 - 0.345 = 0.655 or 65.5% positive returns.

As a function of N, 1 - F(0) looks like Figure 1c:

Figure 1c

Figure 1b
>Aha! So daily returns are more like 50%-50% than monthly or yearly, eh?
Yeah, and hourly returns are even more like 50%-50%, and secondly returns ...
That's a technical term.

However, this decrease in the fraction of positive returns (as the time period decreases) is often quoted.
Indeed, in one book the author notes that, for an annual 10% return and 15% volatility (or Standard Deviation), the fraction of positive returns is 93% for a year but just 67% for monthly returns.

One can generate his results using (in Excel):     1 - NORMDIST(0,R/N, S/N,1)     with R = 0.10, S = 0.15 and N increasing from 1 to 400:

That's Excel's command for a normal distribution.
>And that's a proof that the fraction of positive returns gets to be more like 50% as the time period decreases, right?
Do you accept the assumptions?

>Certainly not!
Then it's not a proof.
Indeed, over the past ten years the fraction of UP returns for the S&P500 was 52% on a daily scale and 54% for weekly and 60% for monthly.
Not so dramatic as suggested by making all them assumptions, eh?

>For that particular time period, from Aug/96 - Aug/06, right?
Yes, but over the period Aug/86 - Aug/96 the fraction of UP returns for the S&P500 was 54% on a daily scale and 59% for weekly and 67% for monthly.

>And that proves ... what?
I give up. What?

>Okay, I'll accept that one should expect roughly 50% UP-days, over a short time period like a day, but what about ...?
Two UP-days in a row, or three or four?
If we assume the day-to-day returns are independent (meaning today's returns is not influenced by yesterday's return or the day before that),
then for N days there are 2N possible sequences of UP and DOWN days.

For 3 days you could have DDD, DDU, DUD, DUU, UDD, UDU, UUD, UUU and that makes 8 which is 23 ... and just one of these is UUU.
For N days, N-in-a-row would be just one of these 2N, so the chances of getting all UP-days is 1/2N.

>And that's a proof, right? I mean, that's exact and ...
Only if we assume independence of successive returns and we consider a sequence of jillions.
Like tossing 10 coins. You could easily get 70% heads.
Aah, but toss 10 jillion coins and it'd be very close to 50% heads.
Anyway, for 20 years of the DOW (some 5000 daily returns) we get this