### Random Walk Part 1 – A Random Walk down a Dead-end Street

Advisor Perspectives* welcomes guest contributions. The views presented here do not necessarily represent those of *Advisor Perspectives*.*

*This is the first of a four-part empirical research study into the fallacy of the “random walk” view on investment reward and risk. The first two parts focus on the inefficacy of such a view on asset price behaviors. Part 3 deals with random walk's deficiency in characterizing risk. The final paper presents a new framework for defining and managing investment reward and risk.*

Modern finance blossomed after the 1960s and brought us Nobel Prize-winning ideas such as Modern Portfolio Theory, Capital Asset Pricing Model, Arbitrage Pricing Theory, the Black-Scholes option formula, Efficient Market Hypothesis and others. They all have one common theme – asset prices move in a random walk, a term popularized by Burton Malkiel's classic book entitled "A Random Walk Down Wall Street."

What is a random walk? How does one prove that prices follow a random walk? Are short-term random walks different from long-term ones? Do prices have memories? Do returns scale with time and volatilities scale with the square root of time? Can prices still be random if they do not follow a random walk? Why are prices so difficult to model? Are some passive index fund advocates like Jack Bogle, Burton Malkiel and Eugene Fama insincere when they favor certain regions, assets or factors over others if they truly believe in random walk and efficient markets?

I address some of these questions using the daily closes of the Dow Jones Industrial Average (DJIA) from 1900 to 2016. In Part 2, I extend the quest to six asset classes – large-caps (the S&P 500), small-caps (the Russell 2000), emerging markets, gold, the dollar and the 10-year Treasury bond. I will challenge many modern finance doctrines that are based on the random walk paradigm.

**What is a random walk?**

In 1900, Louis Bachelier planted the mathematical seeds of random walk. Some 50 years later, Harry Markowitz and others cultivated his seeds into a blooming field called modern finance. Bachelier theorized that prices fluctuated in a Brownian motion – a term used by Albert Einstein in the title of his 1905 paper. Einstein reasoned that molecules diffuse in a manner resembling the way pollens jiggle in water – an observation first made by botanist Robert Brown in 1827.

The terms random walk, Brownian motion (arithmetic and geometric), and Gaussian distribution (normal and log-normal) have subtle mathematical differences. In modern finance, however, these terms are used interchangeably by Markowitz, Osborne and Fama. In this and the three follow-up articles, the term random walk refers to Gaussian statistics (bell-shaped distributions).

Modern finance is built on the notion that rational investors speculate in an efficient market like pollens and molecules wander mindlessly in a fluid. Hence, distributions of all asset price returns should follow the bell-shaped probability density functions (PDFs). To find out how well the random walk model reflects reality, I compare the theoretical PDFs to actual return histograms over various time horizons from one day to 10 years using the daily closing prices of the DJIA from 1900 to 2016. In all time horizons, two types of histogram are used – linear and logarithmic. A tutorial on the basics of probability density function and the construction of both linear and logarithmic histograms are presented in the Appendix.