Examining the Random Walk Hypothesis

The Random Walk Hypothesis is a financial theory that states the prices of securities and assets in a stock market are random and not influenced by past events. In this post, we will examine whether the Random Walk Hypothesis applies to the Mediocre Social Network Apps’ stock price (fictional company, but real asset price data). In the end, we will also try to forecast Mediocre’s price for the first 10 trading days of October.

More About Random Walk

The Random Walk Hypothesis is named after the book “A Random Walk Down Wall Street” written by American economist Burton Malkiel who argued the stock price movements are uncorrelated with another and have the same probability distribution. In other words, in the context of daily prices, Random Walk implies that tomorrow’s stock price is equal to today’s price with a random step up or down. The Random Walk Hypothesis is also consistent with the Efficient Market Hypothesis (EMH). In particular, if EMH holds, that is, all available information about future price movements is already priced into the market, then future price movements will follow a Random Walk as new and unpredictable information emerges.

Mediocre’s Stock Price

We will analyze the daily stock price history of Mediocre Social Network Apps through the lens of the Random Walk Hypothesis. Here’s the daily price data of the company from 02 Jan 2015 to 30 Sep 2019.

Daily Stock Price of Mediocre Social Network Apps from 2015-2019

The company has been struggling in the last few years as reflected in the net downward trend of the price. There are sudden and unpredictable changes in the direction of the price. Furthermore, we see lots of short-term volatility and some larger-scale peaks and valleys. What if we instead look at the Daily Log Returns \(\bigg(log\bigg[\frac{P_{t}}{P_{t-1}}\bigg]\bigg)\) ?

Daily Log Returns from 2015-2019

This looks very much like noise, that is, the volatility (variance) seems constant over time and the day-to-day changes are almost completely random. Is the mean of the daily changes significantly different from zero, which is to say, is there a significant day-to-day drift over this period? Well, the mean is calculated to be -0.0005 and the standard deviation of the mean is 0.0003 (roughly 3% of a cent) over the whole period which implies the mean is not significantly different from zero. Since the daily changes seem uncorrelated and have mean zero it’s reasonable to say that the stock price of Mediocre Social Network Apps Incorporated is well approximated by a random walk without drift.

Forecasting using Random Walk

We saw that random walk without drift model fits well on Mediocre’s stock price data from 02 Jan 2015 - 30 Sep 2019. Now we can try forecasting the price 10 days beyond 30 Sep, that is, for the first 10 trading days of Oct 2019. A random walk’s forecasts for the future are equal to the last observation, as future movements are unpredictable, and are equally likely to be up or down. Therefore, for Mediocre’s stock price we predict $20.48, the last observed price on Sep 30, for the first 10 trading days of October as indicated by the horizontal red line below.

Mediocre’s Price Forecasts for the first 10 Trading Days of October

The green shaded regions (sideways parabola) represent uncertainty bands around the forecasts for 10-days ahead and we see that bands grow wider with the forecast horizon and the uncertainty increases by a lot in forecasting more than one day ahead. This implies that short-run changes in Mediocre’s stock prices cannot be reliably predicted in advance. A much wider implication of stocks following Random Walks is that an investor can’t outperform the market on the whole without taking on large amounts of additional risk. And so that investor might be better off by investing in passive funds, such as mutual funds, for a chance to realize profits rather than amplifying his/her risks by trading individual stocks.