Jeff Sommer's recent article in the New York Times, "Fear of Inflation Prompts Big Losses in Stocks" makes a big deal about the recent one-week decline of 3.2% in the Dow Jones Industrials. But if you think about it, it is not particularly unusual.
Roughly, the DJI has a mean weekly return of about zero. Its annual standard deviation of return is about 15%, more or less. Assuming weekly returns are independent as an approximation, a 15% annual volatility corresponds to a 2.08% weekly volatility. A weekly return of -3.2% is only 1.54 standard deviations from the mean.
Assuming normality, the probability of a result 1.54 standard deviations below the mean or worse is 6.2%. This sounds pretty low, but Mr. Sommer did not pick this week at random. He scoured recent history for the worst week and found that it was the worst since about a year ago.
The real question is how likely is at least one weekly decline of at 3.2% or worse in a year.
Roughly this size negative return or worse should occur about three times a year (0.062*52=3.2). In fact, the probability that it would occur at least one week a year is about (1-(1-0.062)^52)=0.964, or about 96.4%.
Something that is expected to happen at least once a year with probability 96.4% is not unusual.
It is tempting to blame Mr. Somer for either incompetence or grand standing. However, we should remember that journalists and reporters are trained as journalists and reporters, and cannot be expected to understand useful things (note the grand standing :-)).
3 comments:
DJIA volatility is close to 10% now, not 15%. This should reduce the probability of this kind of drop happening once a year to roughly 50%. Your conclusion still holds.
Nice post. But what do you mean by "Roughly, the DJI has a mean weekly return of about zero."? Is it right to say anything is roughly zero? Or, do you mean thatthe weekly mean return is but smaller than the std deviation? Yes, this is a serious question, thank you.
If you think of stocks returns as a mean plus a random disturbance, then, with the usual assumptions, mean return per period is proportional to the length of the period. An annual return of 10% converts to a quarterly return of 2.5% for example. The corresponding weekly mean return is small enough to ignore in this example.
In contrast annual standard deviation of return converts to 1/squareroot(52) of the annual standard deviation.
The result is that for very short periods, all that you notice is the standard deviation, i.e., the noise, because the short period signal to noise ratio is so small.
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