# Are S&P 500 monthly (or annual) returns a random walk?

I'm using financial software that assumes that yearly market returns are random and independent in their Monte Carlo analysis. Its not clear to me that this is the case. Is there an easy way for a "statistics challenged" person to test this?

• The most common model is random walk with a positive but small drift. Without the positive drift it will be pointless to invest in the stock market (assuming dividend payments included in the returns). Btw, returns are not independent, i.e. GARCH. – Cagdas Ozgenc Apr 4 '18 at 14:33
• @CagdasOzgenc, due to convexity in geometric Brownian motion there will be growth even without a drift. – Aksakal Apr 4 '18 at 14:37
• @Aksakal GBM leaks lower not higher, as -1% followed by +1% is negative or vice versa. – Cagdas Ozgenc Apr 4 '18 at 14:52
• Your title has an error. You almost certainly mean, "Can S&P 500 monthly (or annual) prices be reasonably modeled as a random walk?" If the steps are random, it is your path that is a random walk. – Matthew Gunn Apr 4 '18 at 15:45
• @MatthewGunn, if we're on the title, then it's sufficient to ask about monthly, because a random walk would mean that an annual return series must be a random walk too – Aksakal Apr 4 '18 at 16:04

Yes, this can be a reasonable assumption if the purpose of the model is not trading S&P index. No, there is no easy way to test it.

The assumption is not unreasonable for log returns, i.e. $\ln \frac{P_i}{P_{i-1}}$. Predictability of stock prices is a difficult subject. Short term returns are pretty much impossible to predict, there's some predictability in long-term prices, see e.g. Nobel prizes for Shiller and Fama. What is long and short is another subject. You could say that annual returns are somewhere in grey-ish area. Then you have high frequency trading, where things get even more interesting: although it's extremely short time frame, the actual mechanics of exchange operations start to play a role, so that you might be able to exploit them.

Testing predictability of stock returns is a very complicated matter. However, you can do some simple tests to get yourself comfortable with the assumptions. You'll see that it's a fairly reasonable assumption, which many people use in practice. For instance, get the price series, then extract log returns. Plot the histograms of annual returns, run ACF/PACF analysis, run spectral analysis, periodograms, return vs. squared or absolute returns etc. You'll run into a problem with the sample size: nonoverlapping annual returns are a very small sample, while overlapping ones lead to autocorrelation to deal with etc. There are a lot of little issues to care about in these kinds of tests

One thing to note though is the kurtosis. Price returns tend to have fat tails. That's why you may notice that the return distribution is not normal, and the price series are not Wiener process. This can be a random walk with non normal distribution

• "Short term returns are pretty much impossible to predict.." is a very vague statement that can be true or false depending on how short the time is. I predict 1-minute price returns with significantly positive R^2 (significantly lower RMSE of residuals compared to a naive model). – Alexey says Reinstate Monica Apr 4 '18 at 15:41
• @AlexeyBurnakov, R^2 doesn't mean anything unless you show how your algorithm works in backtesting using common methods used in statistical arbitrage field. – Aksakal Apr 4 '18 at 15:46
• "unless you show how your algorithm works in backtesting using common methods used in statistical arbitrage field". I do show it in my experiments. Don't ask me how many millions I made though (predictability does not equal profitability). – Alexey says Reinstate Monica Apr 4 '18 at 15:53
• @AlexeyBurnakov, no I wouldn't ask about millions. Neither of us would be here if we had working stock picking algos :) – Aksakal Apr 4 '18 at 16:01
• agreed. Let's say I can decrease residuals by 0.1% compared to residuals stemming from 0-return model (random walk), mean-return model (random walk with drift), or median-return model (that would probably be 0-return as well), but in order to beat associated costs, I would need to predict 10 times more accurately. That is my point. – Alexey says Reinstate Monica Apr 4 '18 at 16:06
• Let $P_t$ be the price of a market index at time $t$.
• Let $p_t = \log P_t$ be the log price.
• Let $r_{t \rightarrow t+1} = p_{t+1} - p_t$ be the log return.

A not insane starting point would be to assume log returns $r_{t\rightarrow t+1}$ are IID, drawn from some distribution that you estimate off the data. Can you easily falsify this model? Yes. What's obviously wrong with the IID model is that you have volatility clustering in real world data.

So you can get more and more sophisticated in your analysis. For risk analysis, I'd probably do some block bootstrap off the empirical distribution of log returns (if we're talking about the market return where we have a long time series).

--- Another comment --

Something to be aware of when running simulations is that you also have uncertainty over the simulation parameters. Eg. we don't have precise estimates as to the equity premium: the standard errors are huge. So if you're forming subjective probabilities over possible outcomes, there's even more uncertainty than a naive simulation would suggest.

• The issues with the log prices are discussed here: mathbabe.org/2011/08/30/why-log-returns – Cagdas Ozgenc Apr 4 '18 at 17:25
• @CagdasOzgenc The advantage of log returns is that repeated multiplication of gross returns becomes summation of log returns. That's convenient. The linked post indeed points out problems when assuming a simple parametric distribution for log returns, but you don't need to do that. If random variable X follows some distribution, then you can find the distribution for $\log X$. There's nothing fundamentally different working with $\log X$. Also, it's easy to point out problems with assuming gross returns follow the normal distribution etc.... Eg. it implies less than -100% returns are possible. – Matthew Gunn Apr 4 '18 at 17:43
• That's true. But in general people use basic regression models in finance, and they inevitably end up with a spherical distribution assumption. – Cagdas Ozgenc Apr 4 '18 at 17:52
• @CagdasOzgenc I agree people tend to make $\epsilon \sim \mathcal{N}(0,\sigma^2)$ type assumptions all over the place, but you also don't need to do that. – Matthew Gunn Apr 4 '18 at 17:57
• @CagdasOzgenc, just thoughts: as already mentioned above, any parametric distributional assumptions w.r.t. price returns are mostly pointless. E.g., "..when you assume a student-t distribution (a standard choice) of log returns...", you can just start with aligning an observed return PDF plot with that of t-distribution, given sample moments. Difference is striking. – Alexey says Reinstate Monica Apr 4 '18 at 18:06