Author: Matthew Hoff
License:
MIT
This package provides functions & documentation for solving “The Bill Miller Problem” presented within theoretical physicist & mathematician Leonard Mlodinow’s book entitled The Drunkard’s Walk: How Randomness Rules Our Lives. More generally, the functions herein can be used to solve for - either analytically or by simulation - the likelihood of obtaining a winning streak of given length within a given number of attempts, as attempted by a specified number of individuals.
This package can be installed directly from GitHub via the remotes
package.
remotes::install_github("mghoff/billmillr")The story goes that Bill Miller (financier & hedge fund manager) was perceived as the premier stock picker after having performed incredibly well over 15 consecutive years (i.e. by beating the S&P-500 stock index each year). As a result, he was celebrated and acclaimed by the likes of Forbes and others, who estimated and published statistics on the odds of his success. They estimated that the likelihood of his being this performant by random chance alone was 1 in 32,768 or ~0.0032%. This estimate is roughly true if one considers only one individual - Bill Miller, in this case - picking stocks. In other words, they claimed his 15-year winning streak is very likely not driven by random chance alone, but instead by his knowledge and intuition of the market - allowing him to skillfully pick winning stocks seemingly at will.
However, what Dr. Mlodinow understood and illustrated in his book is that there are/were many hedge fund managers all picking stocks. Based on this fact, he poses the first of two refinements to the above estimation: “Out of 1000 stock pickers (coin tossers), what are the odds that at least 1 of them beats the market every year over 15 consecutive years?” The answer to that question is roughly 3% - far greater than the original estimate of 0.0032%. This is trivial to verify.
The second and final refinement Dr. Mlodinow poses considers the scenario of beating the market 15 years consecutively or longer within a 40 year period; i.e. over 40 years and with 1000 traders, what is the probability that at least 1 trader will obtain a winning streak of at least 15 years given that the odds of winning (beating the S&P-500) in any given year are equal to 0.5 (a fair coin toss)? On this additional refinement, Dr. Mlodinow claims the odds are roughly 3 out of 4, or 75%; however, he provides no proof for this claim.
Using the functions within this package, one can calculate - again, both analytically and by numerical simulation - these odds within a high degree of accuracy. It is found that the odds estimate of these two refinements is roughly ~33.7%… quite different still from the claimed 75%.
One must compute the odds of getting a run (i.e. streak) of at least k heads out of N coin tosses where p (q = 1-p) is the probability of obtaining heads (tails) from the toss of a coin.
Mathematically,
which can be broken down recursively into the following sum of terms:
This sum of terms is provided by odds_of_streak().
For more information, see this Ask A Mathematician post.
To calculate the likelihood that at least j out of M people will obtain a streak of at least k heads out of N coin tosses, one must perform the following:
- Calculate the PDF:
Again, this is provided by odds_of_streak().
- Next, calculate the CDF:
- And lastly, calculate the final result:
which is provided by prob_of_at_least_k().
Load Package…
library(billmillr)In the context of the Bill Miller problem, we calculate the probability of obtaining a winning streak of at least 15 heads out of 40 coin tosses, given that the probability p (q) of heads (tails) is fair, i.e. p = q = 0.5.
tictoc::tic()
pS <- odds_of_streak(num_coins = 40, min_heads = 15, prob_heads = 0.5)
pS
#> [1] 0.000411981018260121
tictoc::toc(func.toc = msg.toc)
#> 16.474 hours elapsed
sessionInfo()
#> R version 4.1.2 (2021-11-01)
#> Platform: aarch64-apple-darwin20 (64-bit)
#> Running under: macOS Monterey 12.3.1
#> Matrix products: default
#> LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib
#> locale:
#> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#> attached base packages:
#> [1] stats graphics grDevices utils datasets methods base
#> other attached packages:
#> [1] billmillr_0.2.5
#> loaded via a namespace (and not attached):
#> [1] compiler_4.1.2 tictoc_1.0.1 tools_4.1.2 Using pS, we can now calculate the probability that at least 1 person
out of 1000 people will obtain such a winning streak.
pK <- prob_of_at_least_k(N = 1000, K = 1, p = pS)
pK
#> [1] 0.3377194Simulate and return resulting data
set.seed(1203)
tictoc::tic()
sim_data <- run_simulation(
iters = 2000,
trials = 1000, # Number of traders
sample_space = c("H", "T"),
sample_size = 40, # Number of years
run_value = "H",
run_length = 15 # Number of consecutive winning years
)
tictoc::toc()
#> 39.241 sec elapsed
sim_data[2000, 3:4]
#> prob_of_zero prob_of_ge_one
#> 2000 0.666 0.334
And finally, run a small bootstrap sampling of the simulation…
# Number of times to run the simulation
bsn <- 500
# Build an empty matrix of proper dimensions to capture results
bs_sim_mtx <- matrix(
data = 0, nrow = bsn, ncol = 2,
dimnames = list(c(), c("prob_of_zero", "prob_of_ge_one"))
)
# Run and time a bootstrap sampling estimate of the above simulation
tictoc::tic()
for (bsi in 1:bsn) {
dat <- run_simulation(
iters = 2000,
trials = 1000, # Number of traders
sample_space = c("H", "T"),
sample_size = 40, # Number of years
run_value = "H",
run_length = 15 # Number of consecutive winning years
)
# Take the last row from simulation data above as i-th entry into matrix
bs_sim_mtx[bsi, ] <- as.matrix(dat[bsn, 3:4])
}
tictoc::toc(func.toc = msg.toc)
#> 5.148 hours elapsed
colMeans(bs_sim_mtx)
#> prob_of_zero prob_of_ge_one
#> 0.66252 0.33748