Quit dragging me down

Although strategies to portfolio construction vary considerably across investor types, all approaches share a common consideration for the distribution of possible outcomes. The more extensive the list of possible ends, the more uncertain an investment becomes. In financial markets, we tend to tie uncertainty to the volatility of future returns. The importance of understanding the distribution of returns stems from the concept of "volatility drag." Now, I've referenced volatility drag in several pieces and figured it would be a great time to give folks a more detailed explanation for what's going on behind the scenes. So, of course, this post slightly deviates from the norm and will be far shorter than other posts.

Not all returns are equal.

At the risk of going back too far into the basics, to fully understand what volatility drag is, we first need to understand the difference between compounded and arithmetic returns. Geometric returns measure the average rate of return compounded over some time. Meanwhile, the arithmetic return is simply the average return over a given period. The crucial difference is that the geometric return will reflect reinvested returns, while the arithmetic return only reflects the average return. One interesting dynamic that falls out is that the geometric return will always be less than or equal to the arithmetic return.

Geometric return is what matters for terminal wealth. This compounding/reinvestment effect in returns is one of the main drivers of long-term wealth. Another driver of long-term wealth is volatility. To understand the importance of volatility in determining wealth, I think it's most helpful to use an example. Let's say you invest $100 in a company, and the stock doubles in the first year (100% return) and halves in the second year (-50% return). At the end of the first year, you have $200, but at the end of the second year, you are back to $100. Thus, your geometric return would be 0 percent. However, your arithmetic return would be 25 percent (100 - 50 / 2). The volatility drag is essentially the difference between the arithmetic return and the geometric return. It is important to note that these are two different calculations. The difference is justified, but understanding the impact volatility has on total wealth is crucial when discussing why investors care about volatility.

To hammer home this point, I ran a simple simulation using two portfolios. Both have an expected daily return of 10bps, but one portfolio has volatility of 10bps while the other has volatility of 50bps. The chart below shows the histogram of 100,000 portfolios simulated over 1,000 days and plots the distribution of the final wealth accumulated. We can see that the portfolio with the lower volatility has a relatively symmetric distribution, while the portfolio with the higher volatility has a more positive skew.

Furthermore, the chart below shows the cumulative distribution function for wealth at the end of the simulation. This line can be read as the probability that ending wealth is less than or equal to the value on the x-axis. Here we can see that over 50 percent of the time, wealth in the high volatility portfolio is less than the wealth in the low volatility portfolio. As volatility increases, the probability that wealth is below the low volatility portfolio increases. Of course, with that said, the upside of the higher volatility is massive, but the cost of being on the downside is also a lot higher.

This post may seem a bit more technical and abstract from what you're used to reading. Still, this concept is important to illustrate and set the context for the coming posts on volatility targeting and risk parity and why investors care about volatility.

Thanks for reading,

Tiago Figueiredo