# Why asset price volatility increases wealth inequality

This blog post will cover my paper which explains why an increase in asset price volatility will lead to an increase in wealth inequality.

Wealth inequality, and inequality in general, have recently come to the forefront of the economic research agenda (Pikkety 2015, Saez & Zucman 2016). When it comes to the role of the stock market in generating wealth inequality, most authors (Adam & Tzamourani 2016, Colciago et al. 2019) have focussed on the effect of changes in the level of stock prices and reached a consensus that rising stock prices lead to increased wealth inequality while falling stock prices have the opposite effect. Bringing in a different focus, Campbell, Ramadorai, and Ranish 2019 find that heterogeneity in log stock market returns increased equity wealth inequality in Indian stock markets through two channels. The first is that some undiversified portfolios randomly do well, while others randomly do poorly. The second channel is that wealthy investors tend to have lower variance, which will increase their log returns relative to non-diversified (poor) investors. These channels are relatively unexplored.

In this paper, I

if some undiversified investors randomly do well, while others randomly do poorly, this would generate inequality on its own.

In this paper, I build on that assertion and ask whether stock price volatility will increase wealth inequality, since it is well known that stock price volatility magnifies both trading returns and losses. Following the logic of Campbell, Ramadorai, and Ranish 2019, if stock price volatility increases the heterogeneity of log returns, this should, in turn, increase wealth inequality.

Testing this assumption is difficult because it requires data on individual stock holdings. This type of data is typically not available. Furthermore, to test the influence of volatility in isolation, one would like to operate in a controlled setting.

To do the latter, I turn to simulation modelling. To generate individual level data, I employ an Agent-Based Model (ABM). Because ABMs simulate individual interactions over time, they can generate individual level data.

My ABM consists of a hundred noise traders that hold a portfolio that contains stocks and money. Each agent forms expectations about the future stock price based on a white noise parameter. This means that every simulation period agents will randomly expect the price to move up or down. When they expect that the price will go up the next period, they buy now. Agents that expect the price to go down will try to sell stocks.

Before they can send buy or sell orders to the matching algorithm, they will need to determine a price. To keep things simple, they will set their price equal to their expected value. Using this price, they will determine the volume they want to sell based on their wealth. Given their return expectations with the current price, they determine how much stocks and money they want to hold using mean-variance portfolio optimization, where they use historical variance as a proxy for future variation. The difference between the amount of stock the traders want to hold and actually hold will determine the volume and sign of the order they will create. If the volume is positive, it will be a buy order and vice versa.

How will these orders be matched? For this, I use a limit-order book matching algorithm that is very similar to those employed by real stock markets. The algorithm tries to match the buy orders with the highest price to the sell order with the lowest price, as long as the price of the buy order is higher or equal to that of the sell order. When two orders are matched, a transaction will take place between the owners of these orders.

I simulate this model using Monte Carlo methods in which the model is simulated multiple times with different random seeds. Using this technique, the following data is generated.

What do these picture depict? Let’s start with the top-left picture. The black line represents the Monte Carlo average stock price that is generated by the interactions of the agents. The dotted lines indicate the 95% confidence intervals. The top-right picture shows, in bars, the average volume traded every period.

The two bottom panels depict the evolution of inequality over the course of the simulations. The leftmost bottom panel depicts the evolution of the Gini coefficient. The Gini coefficient is the most commonly used measure of inequality. It can take values between 0 and 1, where 0 represents perfect equality and 1 represents perfect inequality. The rightmost bottom panel depicts that dynamics of the Palma coefficient. The Palma ratio measures the ratio of the richest 10% of the agent’s share of wealth over the share of the poorest 40% of the agents. Both measures increases over time.

My hypothesis is that this is caused by heterogeneity in log returns. Unlike simple returns, log returns are additive over time. This means that if log returns are different for rich agents than they are for poor agents, this should impact inequality over time. The next picture plots all returns versus the level of wealth of an agent of the previous period, the level of wealth that generated that return.

The top panel shows the relationship between returns and wealth. You can see that there is actually a negative relationship. Bigger wealth tends to generate less extreme simple returns, or smaller absolute returns (top-right panel). But since these returns are non-additive over time, they do not contain information about the evolution of wealth inequality. For that, log returns are needed.

The relationship between log returns and wealth is shown in the bottom panel. Here, we see a totally different picture. Log returns tend to become more volatile with wealth (bottom-left panel) and absolute log returns tend to become bigger with wealth. What do these results mean?

They mean that there tends to be a split between richer traders. They either make a lot of money, or they fall to the bottom of the wealth distribution. Once at the bottom, it is very difficult to reach the top again since lower wealth traders earn lower log returns. The story of Campbell, Ramadorai, and Ranish 2019 is consistent with these findings.

However, this is not the point of my paper. I wanted to find out whether increasing volatility, increases heterogeneity in log returns, and in turn, increases wealth inequality. To do that, a simulation experiment is needed.

Luckily, the ABM can facilitate this because it allows us to vary volatility while leaving all other properties the same. However, this being an ABM, I cannot change price volatility directly because the trading between the autonomous agents will generate the price and its volatility. That being said, in this particular model with noise traders, the parameter which regulates the intensity of noisy expectations will directly impact price volatility. When traders have more noise expectations, they will create more noisy prices.

In this simulation experiment, I increase the noise parameter from 0.05 to 0.5 in steps of 0.05. To make sure that randomness does not affect our results, I will simulate 10 Monte Carlo simulations for every step. This means, there will be 9 * 10 = 90 simulations in total. At the end of every simulation, I measure the Gini and Palma coefficient.

So, what happens if we plot the volatility of expectations against the Monte Carlo average price volatility, heterogeneity of returns and (end of simulation) Gini and Palma? Let’s see:

There is a lot to unpack here. Let’s start with the top-left panel. The picture shows that, as the volatility of expectations increases (X-axis), the average volatility of the price (Y-axis) increases. The confidence interval bands are not too wide, showing that this effect is reliable. This confirms that the parameter we chose for our experiment was correct.

Next, we move to the second part of the transmission channel: heterogeneity of log returns. I measure heterogeneity in log returns as the standard deviation of individual log returns. The top-right picture shows that increasing volatility of expectations indeed increases the heterogeneity of log returns, as expected.

Finally, I consider the final part of the channel: inequality. Increasing the volatility of expectations, did indeed increase price volatility and the heterogeneity of log returns. The bottom two panels confirm that it also increases inequality. Both the general measure of inequality (Gini coefficient, bottom-left panel) and the measure of inequality at the extremes (Palma ratio, bottom-right panel) increase as volatility increases.

Therefore, the experiment does not contradict our hypothesis. Increasing volatility seems to increase heterogeneous log returns and wealth inequality. This is the main result of the paper.

But, how do we know that this result is not a consequence of the specific model parameters that I have chosen? What happens if other parameters were chosen for sample size, risk aversion, the spread of the agents, and horizon they use to calculate volatility? Does price volatility still increase inequality under different parameter variations?

The answer to this question can be obtained by doing a what is known as a *global sensitivity analysis*. In such an analysis, parameters are first varied, yielding several parameter combinations. Then, for each combination, I run Monte Carlo simulations and record the level of inequality (Gini) at the end of the simulation.

To make sense of this new data, I use a technique known as Extended Fourier Amplitude Sensitivity Test or **eFast**. Running eFast over the results produces two measures. The total and direct effects changes of parameters have on the Gini coefficient. The following picture shows the results:

The picture clearly shows that the volatility parameter has the biggest effect on inequality. Even though all of these parameters were varied, volatility is still associated with a large variance in the end-of-simulation Gini coefficient. This means that the results of the simulations are robust.

So, where does that leave us? The results produced by the simple simulation model are in line with the theory that stock price simulation increase heterogeneity of log returns, which in turn increases wealth inequality. Now, we might wonder, will these results translate to the real world? In other words, is the model an accurate representation of the real world.

Obviously, the model is a highly stylized version of reality. It only considers a simple stock market with very naive traders. Why are these results useful for those of us who are trying to understand the real world? The reason is that making the model more realistic would not change the conclusion. For example, I tried adding more realistic trading strategies that produce more or less price volatility. The result that volatility increases inequality did not change.

Furthermore, I could add other markets, but still this basic channel would be the same. That would introduce new channels that might offset this effect but would not invalidate this channel. What we have done here is confirmed the possibility of the channel through which stock price volatility can increase wealth inequality. That is the contribution of this paper.

What are the practical implications of this though? Since various policy makers are concerned with the levels of inequality, they should take this asset volatility channel into account. Especially since the traders that win are not more deserving than the losers, all traders have random expectations. So, not only does volatility increase inequality, it doesn’t even necessarily reward the most skilled traders.

*At this moment, this blog is still evolving. I will update it over time*