# Why prices are usually not stationary, but returns are more likely to be stationary?

I read in a course material for time-series that

Daily stock prices $$X_t$$ are in general not stationary but the daily returns defined by $$Y_t := \frac{X_t - X_{t-1}}{X_{t-1}}$$ may be stationary.

Would you please answer me why $$X_t$$ is usually not stationary, and $$Y_t$$ is more likely to be stationary?

The return $$Y_t$$ represents the increase in the value of the stock as a percentage of its previous value. This return fluctuates a great deal in an economy, but in a properly functioning economy, it does tend to fluctuate around a small positive value. Consequently, the total stock price for a company tends to grow roughly exponentially over time. The same thing can be observed in stock indices that aggregate stocks from a large number of different companies.

To see what I am talking about, consider the following chart showing the S&P500 stock index from 1835-2015 (source here). The chart shows the total index; you can see that the index follows roughly exponential growth (i.e., it is roughly linearly increasing when show on the logarithmic scale). Returns fluctuate substantially, but over the long term they give rise to roughly exponential growth.$$^\dagger$$ The returns over time are arguably stationary, and you can fit the returns reasonably well with stationary time-series models such as ARMA, ARCH, GARCH, etc. Even if these returns are not exactly stationary (e.g., exhibiting long-term cycles or changes) they are certainly much closer to being stationary than the stock price itself, since the latter has roughly exponential growth.

$$^\dagger$$ Note that in the case of a stock index it is a bit more complicated, since underperforming companies leave the index and better performing companies are added, so the return on the index tends to be higher than what you would expect for an individual company.

• Why do you say that returns are "certainly" not stationary? Are you saying the chart supports this? As far as I can tell, they could easily be stationary. May 10, 2020 at 21:44
• Autocorrelation doesn't imply non-stationarity. May 11, 2020 at 2:19
• @Ben.. Much fluctuation does not rule out stationarity? In fact ARCH models are stationary but model high volatility. May 13, 2020 at 1:33
• Thank-you all for the feedbakc in comments. I have amended the answer to give a better discussion of the stationarity of returns. The mere fact that you can apply stationary models to these is not proof of their stationarity, but they are at least reasonably well modelled by some stationary models.
– Ben
May 13, 2020 at 2:13

Stock prices can be thought of being a cumulative sum of mean-independent increments due to economic (and other types of) shocks. This is per definition a process with a unit root: $$X_t=X_{t-1}+\varepsilon_t=(X_{t-2}+\varepsilon_{t-1})+\varepsilon_t=\dots=\sum_{\tau=0}^t\varepsilon_\tau.$$ (After the first equality, the coefficient in front of $$X_{t-1}$$ is unity; this is the unit root.) Meanwhile, the increments $$\varepsilon_\tau$$ are mean-independent and "almost stationary", but their scale grows with the level of the stock price (hence not stationary). When you divide them by the level, $$\frac{\varepsilon_\tau}{X_{\tau-1}}$$, you get an approximately stationary process.

This is a generalization but I think it is useful to think of the price of the stock as

$$X_t = E_t P_t$$

where $$E_t$$ is a company's earnings and $$P_t$$ the multiple of earnings investors are willing to pay for the stock (also known as a P/E ratio).

$$E_t$$ is non-stationary since earnings tend to grow over time due to economic growth and inflation. On the other hand it is somewhat reasonable to assume $$P_t$$ stationary since the passage of time should not affect the multiple of earnings investors are willing to pay for a stock.

Putting this together $$X_t$$ is non-stationary since it has a time-dependent mean function.

Now, looking at returns we can rearrange the equation as

$$Y_t=\frac{X_t-X_{t-1}}{X_{t-1}}=\frac{E_tP_t-E_{t-1}P_{t-1}}{E_{t-1}P_{t-1}}=\frac{E_tP_t}{E_{t-1}P_{t-1}}-1$$

In this form it would appear that the fraction $$\frac{E_tP_t}{E_{t-1}P_{t-1}}$$ does not have a time dependent mean function, since the dependence which $$E_t$$ has on time is negated by having it on the numerator and denominator of that equation.

For example assuming $$P_t,E_t$$ independent for all $$t$$, and assuming the expected growth rate of earnings is 2%, we see that $$E[Y_t]=E\Big[\frac{E_tP_t}{E_{t-1}P_{t-1}}-1\Big]=E\Big[\frac{E_t}{E_{t-1}}\Big]E\Big[\frac{P_t}{P_{t-1}}\Big]-1=1.02E\Big[\frac{P_t}{P_{t-1}}\Big]-1$$

Now since $$P_t$$ is unlikely to have a mean function that is dependent on time, the mean function of $$\frac{P_t}{P_{t-1}}$$ should also be independent of time leading to $$E[Y_t]$$ to be independent of time (one of the conditions of stationarity).