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This is the third of my four-part empirical research into the fallacy of the random walk view of investment reward and risk. Part 1 and Part 2 discussed the random walk's failure in portraying asset returns. A random walk depicts risk as volatility.

This article explains why this view is problematic. Risk and volatility are different conceptually. The bell-curve understates actual risks by huge amounts. For asset returns that are not bell-shaped, volatility has no meaning. Finally, volatility is akin to noise, which alleviates instead of elevating risk.

In part 4, I will define investment reward and risk mathematically. I will demonstrate how this probability-based framework enables investors to beat the S&P 500 total-return with less risk.

**Risk and volatility are conceptually different**

Volatility is a measure of uncertainty in a bell-shaped distribution. The academics define risk as uncertainty primarily for mathematical convenience. Glyn Holton argued that if risk were akin to uncertainty, then a man jumping out of an airplane without a parachute would face no risk because his death was 100% certain. Figure 1 illustrates Holton's argument in an investment context. The green dune at the bottom right is an investment with an expected value of 100% and a volatility of 10%. The pink spike on the left is another investment with an expected value of -99% and a volatility of 1%.

From a risk perspective, modern finance favors the pink investment that offers a 10 times lower volatility than the green one. Investors, however, would intuitively avoid the pink investment because they see the 100% odds of a total loss with no chance of any gain. The academics view the green investment as more risky because it is 10-times more volatile than the pink one. Investors would jump on the green one because they see near 100% odds of doubling their money with zero chance of any loss.

Figure 1 illustrates the conceptual flaw in the random walk notion of risk. Outcome uncertainty is not necessarily risk. Risk is an unacceptable loss.

**Random walk grossly underestimates risk**

Benoit Mandelbrot points out in The Misbehavior of Markets that Gaussian statistics (random walks) that are behind modern finance grossly underestimate the probabilities of many stock market crashes. To find out how way-off the random walk predictions are, I computed the probability density function (PDF) of the daily returns of the Dow Jones Industrial Average (DJIA) using a measured mean of 0.03% and a standard deviation of 1.24% (from Figure 7 in Part 1). The dark blue curve in figure 2 is the random walk PDF showing the model's predictions of the DJIA's one-day price changes in 116 years (data sources: MetaStock and Yahoo Finance).