# 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,...?

• Related question: quant.stackexchange.com/questions/8875/…
– John
Feb 7 '15 at 16:53
• Prices are bounded to be nonnegative, while log-returns can have any value, which makes them easier to model. Feb 8 '15 at 10:10
• I dont have time to formulate an answer right now but I recommend the "Quest for Invariance" article by Meucci. The basic principle is: You need to look for an iid distributed "invariant". This obviously cant be the prices. Most people consider stock returns as more or less iid, thats why we use them. They are the invariants. Others use time series models to explain the distribution in more detail. Here, its the innovations $\epsilon$ who are the invariants (iid). For options for example, returns are of little use. We use the implied vol surface to get one of the invariants. Feb 10 '15 at 10:21

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.

• I dont think this is the correct answer, because returns can also be non-stationary. You would need to test prices and returns for stationarity to decide which one to use according to your answer (which I dont think is correct). Feb 10 '15 at 14:52
• @emcor I tried to pick my words carefully on purpose. I wrote that returns can be assumed to be stationary rather than that they were actually stationary. Of course, if there is volatility clustering or some other effect, then returns aren't technically stationary (variance changing over time). Nevertheless, stock prices almost always reject the Dickey Fuller test and returns almost always do not reject the Dickey Fuller test. Thus, it is useful to operate on the theory that the prices will have a unit root and returns do not.
– John
Feb 10 '15 at 15:11
• Maybe the benefit of additivity of log-returns to project invariants to the investment horizon could be described more prominently. Feb 12 '15 at 11:11
• @RndmSymbl I think Richard's answer covers that. However, I think the discussion of log returns vs. arithmetic returns is not particularly relevant to why to use prices versus returns. Not that it's unimportant, but just not the first thing I think of.
– John
Feb 18 '15 at 4:02

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.

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.

• I dont see the point of your answer, you can aswell rewrite returns in terms of prices, and "returns are much more invariant (whatever this exactly means)" is a really low comment. Feb 10 '15 at 14:56
• Dear @emcor. You are right about this not very insightful comment. Of course if one reads Meucci's work one sees what is meant. So I edited the questions. Thanks for your valuable input.
– Ric
Feb 10 '15 at 15:02
• And @emcor. In case you ever heard of white noise and random walks it might be natural to consider returns, maybe aggregate them for longer perios and if needed multiply them by some initial price.
– Ric
Feb 10 '15 at 15:04