Stationarity of Returns and Cumulative Returns

# Are cumulative returns stationary?


# Are cumulative returns stationary?












Log differenced returns, computed from stock prices, are known to be stationary. What about cumulative returns, are they also stationary? if not why not? Are there other properties, like non-i.i.d., that cumulative returns share with regular returns?

## Answer by demully (score 2, accepted)

https://quant.stackexchange.com/a/57484

In a nutshell...

- It's always prudent and conservative to assume that prices are non-stationary.

- But it's not actually as obvious this is true as it intuitively sounds. Intuitively, any random walk or any trend will always lead to a non-stationary process... but you'll practically struggle to prove the unit root (ie a random walk versus a slowly-correcting autoregressive process) more than intuition suggests. And if your "trend" is merely a co-integrated process with inflation/GDP etc., then the real price could indeed be stationary in nature ;-)



- Back in the real world, log-differencing prices to generate log-returns obviously removes any/much doubt here. These might exhibit some time-variation in means and in variance/volatility; but that is not prima facies evidence of non-stationarity. These might have sustained periods of high and of low, and of extremes of both (ie homoskedasticity); but unless you believe (and can demonstrate) that these tend to significantly correlate with respect to time, this is "omitted variable bias" in your model; not "non-stationary behavior". For the uncertainty about price behaviour above, looking at returns removes most "reasonable doubt" from the problem.

- So to "cumulative returns" - these come in two basic forms. You can have trailing returns from say 1 Jan this year; and you can have rolling hourly/daily/weekly/monthly returns.

- Put simply - the former are non-stationary. If I peg my starting point, then the variation of cumulative return will grow with respect to time, ie non-stationary. But if I look at rolling weekly, monthly, quarterly, or annual returns, these will remain stationary. Obviously, my last-12m return today will be very, very, highly correlated to the same yesterday or tomorrow. That doesn't matter - because the dropping off of the date last year generates an autoregressive process (that is stationary). My 12m returns today vs yesterday will always be very highly correlated... but this does not make them non-stationary. What matters here is not "distance in time" but "location in time". So if 12m today vs 12m yesterday get consistently more or less correlated as time goes by, then they're non-stationary.

- So as a general rule of thumb, it's OK to assume that returns (over most reasonable investible frequencies) are indeed stationary.

- Where this gets tricky is eg in bonds. At any given bond yield, any given change in yields will have a different "convexity effect". IE if yields structurally rise and fall over time, then the mean and the variance of associated bond returns will be affected; non-stationary. Likewise, however, economic regimes that might produce different PE (ie reciprocal earnings yield) regimes in stocks will have similar impacts. So stock returns might, likewise, be "trending" but nevertheless a "stationary" process. Because the returns are biased derivatives of a stationary risk-premium process :-(

Short answer:

- It's impossible to say (let alone prove) most of the time.

- It's always conservative, and never very wrong, to assume prices are non-stationary.

- It's usually sound to assume that returns are stationary.

- Rolling-cumulative returns as above; cumulative returns from a fixed starting point, non-stationary.

## Answer by mark leeds (score 6)

https://quant.stackexchange.com/a/57473

Hi: Even if returns were stationary ( which is probably dependent on the time series one is considering ), cumulative returns, where $n$ is not fixed ( as it in say a rolling sum with a fixed window size or a non-overlapping sum with a fixed window size ) definitely can't be stationary. Consider a pure noise process.

$logret_t = log(P_{t}) - log(P_{t-1}) = \epsilon_t$

Now take the cumulative sum of $ret_t$ over $n$ periods. This sum = $\sum_{i=1}^{n} \epsilon_{i}$.

A) The expectation of this sum is zero but the variance is increasing because it's $\sigma^2 \times n $ so the variance of the cumulative return is not constant.

B) In the case, where $n$ is fixed and the cumulative return is not overlapping, then stationarity follows because the variance is $\sigma^2 n$ with $n$ fixed.

C) In the case, where $n$ is fixed and the cumulative return is overlapping ( a rolling sum say ), I don't think it's stationary but the definition of what the mean of the process is, is not clear to me so I'm not certain about this case.

## Answer by Bob Jansen (score 4)

https://quant.stackexchange.com/a/57458

Stock prices are definitely not stationary as tomorrows closing price is strongly influenced by today's closing price and prices tend to change. Returns can be potentially stationary and are therefore a much better target for analysis. They don't need to be stationary and I don't believe they are. For example, historical returns exhibit heteroscedasticity. This answer explains more.

Cumulative returns don't need to be stationary either as their building blocks don't need to be. However, you could argue that if you aggregate returns some of the short term noise averages out and become more homoscedastic and thus become more stationary. On the other hand, it seems that the mean return varies over longer periods. The expected return over longer periods seems to vary though as the world economy moves through different stages. A varying mean again makes returns non-stationary.