Volatility acts like a drain on the compounding of wealth over the long-term. We can see this statistically by comparing the distributions of terminal wealth for two different investments: 1. 10% expected return with 20% volatility 2. 8% expected return with 4% volatility Running 10,000 simulations we can show the distribution of terminal wealth for $100 invested in the 10%/8% investment for ten years. The result is a positively skewed distribution with the most likely ending wealth being $180: ![[Pasted image 20231111155820.png]] Investing $100 in the 8%/4% investment for ten years you are more likely to end up with more wealth at the end of the ten years as your most likely outcome is an ending wealth of $194. You are also less likely to lose money but your average ending wealth is not as high as the first strategy because there is less positive skew. This is a great example of how [[Jensen's Inequality|averages can be misleading.]] ![[Pasted image 20231111155946.png]] ## Terminal Wealth The formula for terminal wealth given starting wealth $P_{0}$, expected return $r$, volatility $\sigma$, and time $T$ is: $TW=P_{0} \times e^{(r-\frac{\sigma^{2}}{2}) \times T + \sigma \times \sqrt{T} \times Z}$ where $Z$ is the standard normal random variable. If we look closely at this formula we will see that the expected return is reduced by volatility. In fact, the expected return, which is the arithmetic mean return is reduced by volatility to give us the geometric mean return: $ GR = r - \frac{\sigma^{2}}{2}$ The arithmetic mean is a measure of central tendency of a set of variables and as we have seen above is very sensitive to outliers. It is also a single-period concept whereas the geometric mean is a measure of central tendency of a set of variables that are *multiplied together*. This is important because whenever we talk about compounding wealth we are talking about multiplication. So it is here we can explicitly see how volatility *reduces* the growth of your wealth over time. Said differently, volatility distorts our view of reality. ## The Magic of Compounding If you invest $100 in something that is expected to return 10% *per year* you will expect to end up with $271.83 at the end of ten years. Instead if you earned 10% *over* ten years you would end up with $110. Terminal wealth includes compounding, which is when your money is making money for you, and is a multiple period concept which is why we [[Multiplication Principle|multiply]] our returns and focus on the geometric return over the arithmetic return. As we have already seen the geometric return that we actually realize is dependent on the volatility of the expected returns and a higher volatility will reduce the geometric mean. Returning the our two investments above we can use the equation above to determine what the geometric mean return is for each investment: $ 0.08 = 0.10 - \frac{0.20^{2}}{2}$ $0.079 = 0.08 - \frac{0.04^{2}}{2}$ We can see from this that different combinations of expected returns and volatility can deliver similar terminal wealth outcomes. This is important because of the two investments above one isn't necessarily **better** than the other; they are just different. The magic of compounding works in both directions. We don't necessarily need a higher return to increase wealth, we need the right combination of returns and volatility which is why people focus so much on the Sharpe Ratio: $ SR = \frac{r}{\sigma}$ To illustrate how compounding works in both ways consider a \$100 investment that returns 15% in the first year leaving you with \$115. Then the second year you lose 15%. On the surface you might average year one and two and conclude you broke even but the reality is that you actually lost \$2.25 because: $$115 \times 0.85 = $97.75 $ But once again this is a single period concept, so we need to consider how this multiplies over time. We can adjust the above equation for time as such: $ SR = \frac{r \times T}{\sigma \sqrt{T}}$ What this reveals is another magic trick of compounding: [returns scale linearly but risk does not](https://moontowermeta.com/if-you-make-money-every-day-youre-not-maximizing/). We saw this in the pictures of the distributions above as the positive skew for the more volatile investment. It also reveals a limitation of the Sharpe Ratio: a higher Sharpe does not ensure a higher terminal wealth. Astute readers will have noticed that the second investment in our example above had a higher Sharpe than the first, but also that the mean and median terminal wealth for the lower Sharpe investment were higher. A higher Sharpe increases the probability of a better wealth outcome, but on average does not necessarily give you a higher terminal wealth. To achieve that you need a relatively high expected return. Consider a third investment with a 10% expected return and 6.67% volatility (1.5x Sharpe) we can see below in the distribution of outcomes that it gives us an equivalent mean, but higher median, and mode of terminal wealth to the first investment with a 0.50x Sharpe. ![[Pasted image 20231112183225.png]] ### Don't Pay the Stupid Tax All this explains the concept of the interplay between volatility and compound returns in purely mathematical terms and leaves out the philosophical considerations of risk and return. We have seen how volatility can act against compounding but that it doesn't scale the same way that returns compound. But volatility has another negative consequence and it is the impact is has on our emotions. When things don't turn out the way we expect, and we have already shown how volatility distorts our expectations of terminal wealth, volatility exposes us to the stupid tax. This happens when we choose to opt out of the compounding mechanism of the market by selling early. Volatility puts stress on our emotions and causes us to make poor decisions which impact our terminal wealth outcomes. There is a difference between knowledge and wisdom and volatility accounts for the difference. Volatility tests our [[Faith]] and when we resort to trying to control our outcomes it narrows our potential. ### Path Dependency Terminal wealth is dependent on the path you take which is dependent on your choices along the way. When you experience greater volatility the choices you make count for more in determining your terminal wealth. **Cases:** - [CHG Issue #130: Discerning Experts](https://open.substack.com/pub/cedarshillgroup/p/chg-issue-130-discerning-experts) - [CHG Issue #124: Path Dependency](https://cedarshillgroup.substack.com/p/chg-issue-124-path-dependency) Explore Further: [[Portfolio Construction]] | [[Risk]] Tags: #portfolio-management Your support for Cedars Hill Group is greatly appreciated <form action="https://www.paypal.com/donate" method="post" target="_top"> <input type="hidden" name="hosted_button_id" value="74PGN8ZXHQVHS" /> <input type="image" src="https://www.paypalobjects.com/en_US/i/btn/btn_donate_LG.gif" border="0" name="submit" title="PayPal - The safer, easier way to pay online!" alt="Donate with PayPal button" /> <img alt="" border="0" src="https://www.paypal.com/en_US/i/scr/pixel.gif" width="1" height="1" /> </form>