# Expanding window vs Rolling window z-score

I wish to find the z-score of a value measure( e/g P/E ratio) to compare them across asset classes, currently i am using an expanding window z-score to calculate the long-term mean and standard deviation upto that point in time. But the problem with that is my z-scores vary by quite a lot if a start my backtesting from 1999 vs if I start my backtesting from 2001. If a value measure was completely mean-reverting this would not have ideally happened.

So currently I am thinking of calculating the half-life of a particular value measure and use a rolling window z-score with that half life. But i am sure that may be better ways of solving this problem (because the half life would also vary with time) and i'd appreciate some inputs.

• Aren't you trying to make stationary and ergodic a time series which is not due to its nature? You have prices in your time series - which of course will never converge to a single long term value, even if earnings would. – Lisa Ann Jun 3 '19 at 14:12
• P/E ratio is academically shown to mean revert, not price or the earnings themselves – Dhruv Mahajan Jun 3 '19 at 16:57
• I'm sorry, I've missed that proof. I've always thought that tradable homoscedastic mean reverting securities don't exist, otherwise they would be a free lunch. As long as earnings don't change that much over time, P/E is tradable. – Lisa Ann Jun 3 '19 at 20:12
• Confused about your objective. First, it's generally preferable to use harmonic means for price-based ratios as they're highly susceptible to skew (as you saw in 2001), or simply consider earnings yield rather than PE. Second, it isn't clear to me how or why you'd expect some kind of absolute distribution of, in this case, P/Es. It's understood, I think, that the market can be expensive or cheap overall at different periods of time, and even more so for individual names. I also don't know how you'd compare P/E across asset classes (did you mistype...currencies obviously don't have P/Es) – Chris Jun 4 '19 at 4:29
• I have other value measures for different asset classes gathered through past academic research ( e.g credit spread for corporate bonds ). And i am not computing means of P/E but E/P, sorry for the confusion, because higher E/P means undervaluation. What i need to do is compare the historical E/P (value) for a particular asset class with some other asset classes and rank them on monhtly basis, for that i need to calculate z-scores, because value measures are not directly comparable across asset classes ( e.g E/P and credit spread are not, bu thier z-scores are comparable) – Dhruv Mahajan Jun 4 '19 at 6:19

• I find it surprising you haven't read that p/e are mean reverting because that is the foundation of factor investing and value investing itself, I know someone who manages a $200 mil fund based on these value signals. Again that is not the point I'm trying to make, my point was if long term mean is unstable then maybe short term mean isn't and I can run rolling window zscore with rolling window to be the half life of the mean reversion, but thanks for your input anyway. – Dhruv Mahajan Jun 5 '19 at 4:53 • I mean, you're entitled to your opinion, but IME, the stationarity of valuation ratios has little to do with factor investing nor value investing specifically. As value is relative, mean of a ratio for some universe can and does vary, sometimes significantly, and you can still be thematically value-oriented. TBH, if you have a friend running a$200M fund doing this kind of thing, why not just ask him how he does this? – Chris Jun 5 '19 at 5:15