In the previous post, I went through a simple exercise which, to me, clearly demonsrtates that 60% out of sample guess rate (on daily basis) for S&P 500 will generate ridiculous returns. From the feedback I got, it seemed that my example was somewhat unconvincing. Let’s dig a bit further then.

Let’s add Sharpe ratio and maximum drawdown to the CAGR and compute all three for each sample.

return.mc = function(rets, samples=1000, size=252) { require(PerformanceAnalytics) # The annualized return for each sample result = data.frame(cagr=rep(NA, samples), sharpe.ratio=NA, max.dd=NA) for(ii in 1:samples) { # Sample the indexes aa = sample(1:NROW(rets), size=size) # All days we guessed wrong bb = -abs(rets) # On the days in the sample we guessed correctly bb[aa] = abs(bb[aa]) cc = as.numeric(bb) # Compute the statistics of interest for this sample. result[ii,1] = Return.annualized(cc,scale=252) result[ii,2] = SharpeRatio.annualized(bb,scale=252) result[ii,3] = maxDrawdown(cc) } return(result) }

Let’s look at some summary statistics:

require(quantmod) gspc = getSymbols("^GSPC", from="1900-01-01", auto.assign=F) rets = ROC(Cl(gspc),type="discrete",na.pad=F)["1994/2013"] df = return.mc(rets, size=as.integer(0.6*NROW(rets))) summary(df,digits=2) # cagr sharpe.ratio max.dd # Min. :0.34 Min. :1.8 Min. :0.13 # 1st Qu.:0.45 1st Qu.:2.3 1st Qu.:0.22 # Median :0.48 Median :2.5 Median :0.26 # Mean :0.48 Mean :2.5 Mean :0.27 # 3rd Qu.:0.51 3rd Qu.:2.7 3rd Qu.:0.31 # Max. :0.67 Max. :3.5 Max. :0.63

The picture is clearer now. Lowest Sharpe ratio of 1.8 among all samples, and a mean at 2.5? Yeah, right.

The results were similar for other asset classes as well – bonds, oil, etc. All in all, in financial markets, like in a casino, a small edge translates into massive wealth, and most practitioners understand that intuitively.

Thanks for sharing. I think you meant to put bb under the performance arguments, not cc (as it does not exist).

Something kind of interesting, is that the actual realized bias was about 54% over the rets period, with a CAGR of about 7%. If you plug 54% under the size argument of the mc function, you find that 7% was in the very low end of the range of results! Imagine some oracle had a crystal ball and could predict 54% daily returns over all those years, and as a result, he might have expected something like 16% Mean CAGR, but only achieved the lowest end of the mc CAGR results, and the high end of the drawdown results — talk about bad luck on top of edge. Maybe that tells us something cautionary about trying to optimize solely around hit rates.

df = return.mc(rets, samples=1000, size=as.integer(0.54*NROW(rets)))

> summary(df,digits=2)

cagr sharpe.ratio max.dd

Min. :0.058 Min. :0.30 Min. :0.21

1st Qu.:0.134 1st Qu.:0.70 1st Qu.:0.33

Median :0.158 Median :0.82 Median :0.38

Mean :0.160 Mean :0.83 Mean :0.40

3rd Qu.:0.185 3rd Qu.:0.96 3rd Qu.:0.46

Max. :0.307 Max. :1.60 Max. :0.74

> Return.annualized(rets,scale=252)

GSPC.Close

Annualized Return 0.0713289

> maxDrawdown(rets)

[1] 0.5677539

Thanks for finding the “cc” bug – it was meant to be “cc = as.numeric(bb)”.

About the buy and hold – although it seems “lucky” in terms of guess rate (54%), it is quite “unlucky”, as you have observed, in terms of returns. The asymmetry of the returns (returns in a bear vs returns in a bull) is a plausible explanation. Likewise, at a 50% guess rate, one loses money on average.

Last but not least, luck is a huge factor indeed on the Drawdown front as well. Imagine running into the 40% losing days right off the bat – the drawdown will be nearly 100%. The loses are also massive if one guessed correctly, at 60%, the days with the smallest returns.

Thank you for sharing.

I simulated different success rates over at my blog “Data Shenanigans”, mainly to introduce people to parallel computing, but also to show how a succcess rate of roughly 55% results in an exploding value.

If you want to find more, please visit: https://datashenanigan.wordpress.com/2015/09/23/simulating-backtests-of-stock-returns-using-monte-carlo-and-snowfall-in-parallel/

David