Nassim Taleb on the signal to noise ratio (and why you shouldn't read the news)

Nassim Taleb has made the point that by sampling an information source very frequently you will end up seeing more noise than signal.

In particular, in Fooled by Randomness (2nd edition, p 65), he wrote:

A 15% return with a 10% volatility (or uncertainty) per annum translates into a 93% probability of success in any given year. But seen at a narrow time scale, this translates into a mere 50.02% probability of success over any given second as shown in Table 3.1.

This is shown in the output of the following R function. The vector time contains the number of units into which one year is broken. Thus, the 15% return is broken into 15%/4 each quarter, while the standard deviation over the same interval is divided by the square root of the number of quarters (because the variance should be divided by the number of intervals, and the standard deviation is the square root of the variance).

fbr.table31 <- function(){
  mean <- 15                            # 15% return
  sd <- 10                              # 10% error rate per annum
  time <- c(1, 4, 12, 365, 365*24, 365*24*60, 365*24*60*60)
  label <- c("year", "quarter", "month", "day", "hour", "minute", "second")
  ### what fraction of distribution is > 0?
  data.frame(year.fraction = time,
             probability = pnorm(0, mean=mean/time, sd=sd/sqrt(time), lower.tail=FALSE),
             row.names = label) }
        year.fraction probability
year                1   0.9331928
quarter             4   0.7733726
month              12   0.6674972
day               365   0.5312902
hour             8760   0.5063934
minute         525600   0.5008254
second       31536000   0.5001066

Similarly, in Antifragility (p 126), he wrote:

Assume further that for what you are observing, at a yearly frequency, the ratio of signal to noise is about one to one (half noise, half signal)—this means that about half the changes are real improvements or degradations, the other half come from randomness. This ratio is what you get from yearly observations. But if you look at the very same data on a daily basis, the composition would change to 95 percent noise, 5 percent signal. And if you observe data on an hourly basis, as people immersed in the news and market price variations do, the split becomes 99.5 percent noise to 0.5 percent signal.

af.p126 <- function(){
  time <- c(1, 365, 365*24)
  label <- c("year", "day", "hour")
  data.frame(year.fraction = time, noise = 1 - sqrt(1/time), row.names = label) }
     year.fraction     noise
year             1 0.0000000
day            365 0.9476576
hour          8760 0.9893157

Author: Steve Bagley

Created: 2016-04-30 Sat 15:43