Why Financial Time Series Models Often Use Returns Instead of Prices
Summary
Financial prices often behave as nonstationary series: their statistical properties can depend on the price level and past observations. Returns, calculated as percentage changes or differences in log prices, are more often treated as stationary. That makes returns easier to compare across assets and more suitable for many statistical modeling methods, which generally rely on stable properties over time.
The discussion suggests examining autocorrelation across lags: prices typically show strong persistence, while returns often have much smaller correlations. Log prices can also be represented as an initial log price plus accumulated log returns, linking return models to price changes. These are useful starting points rather than universal rules. Returns can exhibit changing volatility and volatility clustering, and stationarity assumptions should be checked for the data and application. The claim that returns are predictable while prices are not is presented as an oversimplification, not a general modeling result.
Key ideas
- Prices often behave as integrated, nonstationary series, while returns are more often modeled as stationary.
- Percentage changes and log differences can reduce dependence on the prior price level.
- Prices tend to show stronger autocorrelation than returns, though the pattern depends on the data.
- Log price changes accumulate to form log prices, connecting return models to price evolution.
- Return series can still have changing volatility and clustering, so stationarity is not guaranteed.
Tags
Full text
# Why do we usually model returns and not prices?
# Why do we usually model returns and not prices?
I think this is a quite similar question for most of you, however it is not completely understandable for me at the moment:
Why do we usually use returns and not prices to model financial data in time series analysis,...?
## Answer by John (score 18, accepted)
https://quant.stackexchange.com/a/16484
Basically, prices usually have a unit root, while returns can be assumed to be stationary. This is also called order of integration, a unit root means integrated of order 1, I(1), while stationary is order 0, I(0). Time series that are stationary have a lot of convenient properties for analysis. When a time series is non-stationary, then that means the moments will change over time. For instance, for prices, the mean and variance would both depend on the previous period's price. Taking the percent change (or log difference), more often than not, removes this effect.
One easy way to visualize the difference between the two is to calculate the correlation with lagged values of a time series with itself (also called the autocorrelation function). Plot this for a number of lags. Returns should have correlations near zero, while prices should start out very high and exponentially decay.
## Answer by user3264325 (score 8)
https://quant.stackexchange.com/a/16512
Perhaps overly simplistic and repeating the pt above, but when doing statistics, ideally we want to compare like with like.
Returns can be comparable with each other.
Prices on the other hand always depend on the previous price.
## Answer by Richi Wa (score 4)
https://quant.stackexchange.com/a/16516
Just a bit of illustration added to @John's answer. Look at log prices $\log(P_t)$, assume that you know $P_0$ then $$ \log(P_t) = \log(P_0) + r_1 + \cdots r_t $$ where $r_i = \log(P_i)-\log(P_{i-1})$ are the log returns. By modelling the log-returns (which as already said take values on the whole real line which is a nice property for modelling) we model the 'atoms' of the prices.
There things as changing volatility (of returns) and volatility clustering but as a starting point we can assume the volatility of $r$ as constant even then the volatility of $\log(P_t)$ would be increasing as more and more summands enter. Imagine what happens if we look at reality where the volatility of $r$ is stochastic or at least changing (hetero-skedasticity).
Meucci in his work looks at invariant quantities in financial markets. Following his illustrations you can convince yourself that returns can more easily be assumed to be invariant than prices. @John has already answered a similar question quite exactly 2 years ago.
## Answer by Orvar Korvar (score 0)
https://quant.stackexchange.com/a/70854
Somewhat sloppy, you could say this.
Prices are totally unpredictable and follow a brownian motion. Therefore you can not predict prices. However, returns have structure so you can predict returns.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.