How to annualize Sharpe Ratio if monthly returns are serially correlated? Calculation of autocorrelations

Question

I am looking at a data set of 60 monthly returns (last 5 years) and want to calculate an annualized Sharpe Ratio.

The usual way of doing this is to calculate the monthly Sharpe Ratio first, and then multiply it by a scaling factor. This scaling factor is the square root of 12 if returns are not serially correlated.

In my data set however, the returns exhibit statistically significant autocorrelations. I am aware that Lo (2002) suggests to use a scaling factor which accounts for the first 11 autocorrelations (specifically, autocorrelations for time lags 1 to 11).

My question revolves around the calculation of the autocorrelations for the purpose of annualization: Do I calculate the first 11 autocorrelations for the biggest possible periods (in my case it would be 60 - 11 = 49 months)? Or do I calculate autocorrelations for 12 month periods?

I tried to retrieve the correct way to do this from Lo (2002), but this uncertainty remains for me after reading through the paper and similar Q&A threads I found.

Answer 1

Whilst autocorrelation does not affect the sharpe ratio when using purely returns, positive autocorrelation does reduce the sharpe ratio when considering the price space - all else equal.

Autocorrelation affects the ending distribution of the asset price. Higher autocorrelation = higher volatility in ending prices. An example to illustrate this: with -1 autocorrelation, all returns are followed by a return of equal and opposite size & direction. So the returns will have the same volatility as the I.I.D process, but the price will be fixed at s_0 - hence 0 volatility.

Carol Alexander writes about this in her second book on market risk analysis. The channel bionic turtle has a video on the change in scaling factor from just sqrt(t).

Here is the link to the video: https://youtu.be/Ms3uu9TKcgA

Then it is a matter of multiplying the volatility in the denominator by the scale factor.

Hope it helps.

Answer 2

The following simulations indicate that autocorrelation does not bias the Sharpe ratio:

ope <- 252     # (trading) days per year
mu <- 0.001
sg <- 0.0130
zeta <- sqrt(ope) * mu / sg
print(paste0("Annualized Sharpe is around ",zeta,"\n"))
n <- 3*ope     # simulate 3 years of data
simit <- function(n,mu,sg,rho=0.0) {
    # compute y which have mean mu, whose marginals have standard deviation sg, and autocorrelation rho
    y <- mu + sg * sqrt(1-rho^2) * arima.sim(model=list(ar=c(rho)),n=n,rand.gen=rnorm) 
    # return the sharpe of the same
    mean(y)/sd(y)
}
set.seed(1234)
vals <- replicate(100000,simit(n,mu,sg,rho=0.9))
print(paste0("empirical Sharpe and population SNR are: ",round(mean(vals),5)," and ",round(mu/sg,5),"\n"))

I get results:

[1] "empirical Sharpe and population SNR are: 0.07744 and 0.07692\n"

Corresponding to a bias of less than 1%.

The calculations for why this is the case are detailed in Short Sharpe Course linked above (and the very expensive book form of the same).