*This commentary is available as a PDF download here.*

# Summary

- Even when an investment has a positive expected average growth rate, the experience of most individuals may be catastrophic.
- By focusing on the compound average growth rate, we can see the median realizations – which account for risk – are often more crucial decision points than ensemble averages, which are the focal point of Monte Carlo analysis.
- These arguments also provide a simple explanation for investor behavior that avoids the need for utility theory concepts that have been used for the past 200+ years.
- Since we can neither average our results with other investors nor average our results with potential copies of ourselves in infinite states of the world, the best we can do is try to average over time.
- Because we all live in a multi-period world where we have a single investment portfolio that compounds over time, managing risk can help us maximize our long-term growth rate even if it seems foolish in hindsight.

Pretend we come to you offering a new investment strategy. Each week, you earn 0.65% (such that over a year you earn 40%), but there is a 1-in-200 chance that you lose -95%. Would you invest?

If we simulate out a single trial, we can see that within a year, we may lose most of our money.

Of course, just because things went wrong in our singular example does not mean that this is necessarily a bad investment. In fact, if we evaluate the prospects of this investment by looking at the average experience, we end up with something far more attractive (the “Ensemble,” which is essentially a Monte-Carlo simulation of the strategy).

The math here is simple: 99.5% of the time we make 1.0065x our money and 0.5% of the time, we end up with 0.05x our money. On average, then, we end up with 1.0017x, or 1.092x annualized. While the average experience is not the 40% annualized we sought, the 9.2% return after a year is still nothing to scoff at.

Of course, the average is not actually achievable. There are not infinite variations of this investment strategy for you to allocate your capital across, nor, we suspect, do you have access to infinite versions of you living in parallel universes who can pool their risk.

Rather, you are forced to diversify your risk *over time*. Here we end up with a different picture.

Another series of unfortunate events?

Not so fast. You see, when we move to diversifying over time, we need to look at a *time-weighted *average. It is not the arithmetic mean we are after, but rather the *geometric* mean which will account for the effects of compounding. Calculating the geometric mean – 1.0065^{99.5%} x 0.05^{0.5%} – leaves us with a value of 0.9915, i.e. our wealth is expected to *decay* over time.

Wait.

How is it possible that *on average *the strategy is a winner if each and every path is expected to decay over time?

The simple answer: A few fortunate outliers make up for all decaying paths.

The slightly more complex answer: In this investment, our wealth can never go below $0 but we can theoretically make an infinite amount of money. Thus, over time, the average is dragged up.

**The Misleading Mean**

In many cases, the *average experience* can be entirely misleading for the experience you can expect. In the world of bell-curves and normal distributions, we typically expect experiences to be clustered around the average. For example, there are more people close to the average height than there are far away.

However, when other distributions apply, the average can be unlikely. Wealth distribution is a perfect example of this. In 2013 in the United States, the top 10% of families held 76% of the wealth while the bottom 50% held 1%. Using 2017 figures, if we divided net worth among the U.S. population – i.e. the “average” household wealth – it would come out to around $760,000 per family. The bottom 50%, however, have a net worth closer to $11,000 per family.

In other words, if you pick a random person off the street, their experience is likely much closer to $11,000 than $760,000. It’s the wealthy outliers that are pulling the average up.

A more applicable metric, in this case, might be the *median*, which will say, “50% of experiences are below this level and 50% are above.”

**The Role of Risk**

As it turns out, the median is important for those of us diversifying over time as well. If we consider our hypothetical investment strategy above, our intuition is that the median result is probably not great. *Eventually*, it feels like, everyone goes practically bankrupt. If we plot the median result, we see almost exactly that.

(As a side note, if you’re wondering why the median result exhibits a sawtooth pattern rather than the smoother results of the mean, the answer is the median is the actual result that sits at the 50^{th} percentile. Knowing that the probability of losing 95% of our wealth is 1-in-200, it takes time for enough individuals to experience a poor result for the median to drop.)

In fact, if we model investment wealth as a Geometric Brownian Motion (a commonly used stochastic process for modeling stock prices), then over the long run an investor’s compound growth rate approaches the *median*, not the mean.[1] The important difference between the two is that while volatility does not affect the expected level of wealth, it does drive the mean and median further apart. In fact, the median growth rate is the mean growth rate minus half the volatility squared (which you might recognize as being the common approximation for – drum roll please – the geometric growth rate).

In other words: volatility matters.

Most investors we speak with have an intuitive grasp of this concept. They know that when you lose 10% of your wealth, you need to gain 11.11% back to get to break even.

And when you lose 50%, and you need to earn 100% to get back to break even. Under compound results, feeling twice the pain from losses than the pleasure from gain makes complete sense. There are no individual and independent trials: results have consequences.

This is why taking less risk can actually lead to greater growth in wealth in the long run. If we take too little risk, we will will not participate, but too much risk can lead to ruin. For example, below we plot final wealth results after a 50% drop in market value and a 100% recovery depending on your *capture ratio*.

As an example of reading this graph, if we start with $1 and experience a 50% loss and a 100% gain, but are only 50% exposed to each of those movements (i.e. we lose 25% and then gain 50%), we end up with $1.125. At the far right of the graph, we can see that at 2x exposure, the first 50% move completely wipes out our capital.

**Common Sense Utility Theory**

What economists have found, however, is that even if we offer our investment as a one-off event – where the expected return is definitively positive – most would still forego it. To resolve this conundrum, economists have proposed utility theory.

The argument is that investors do not actually try to maximize their expected change in wealth, but rather try to maximize the expected *utility* of that change. The earliest formalization of this concept was in a paper written by Daniel Bernoulli in 1738, where he proposed a mathematical function that would correct the expected return to account for risk aversion.

Bernoulli’s originally proposed function was *log-utility.* And under log-utility, our investment strategy offering is no longer appealing: log(1.0065) x 99.5% + log(0.05) x 0.50% is a negative value. What’s interesting about log utility is that, due to the property of logarithms, it ends up creating the *identical* decision axiom as had we used our compound growth rate model.

log(1.0065) x 99.5% + log(0.05) x 0.50% = log(1.0065^{99.5%}) + log(0.05^{0.5%}) = log(1.0065^{99.5%} x 0.05^{0.5%})

So while utility theory is supposed to correct for behavioral foibles like “risk aversion,” what it really does is take a single-period bet and turn it into a multi-period, compound bet.

Under the context of multi-period, compounding results, “risk aversion” is not so foolish. If we have our arm mauled off by a lion on the African veldt, we cannot simply “average” our experience with others in the tribe and end up with 97% of an arm. We cannot “average” our experience across the infinite universes of other potential outcomes where we were not necessarily mauled. Rather, our state is permanently altered for life.

Similarly, if we lose 50% of our money, we cannot just “average” our results with other investors. Nor can we average our results with all the potential infinite alternate universes where we did not lose 50%. The best we can do is try to average over time, which means that our compound growth rate matters. And, as we demonstrated above, so does risk.

**Conclusion**

Ex-post, managing risk can often feel foolish. Almost exactly 9 years after the bottom of the 2008-2009 bear market, the S&P 500 has returned more than 380%. Asset class, geographic, and process diversification largely proved foolish relative to simple buy-and-hold.

Ex-ante, however, few would forgo risk management. Ask yourself this: would you sell everything today to buy only U.S. large-cap stocks? If not, then there is little to regret about not having done it in the past. While the narratives we spin often make realized results seem obvious in hindsight, the reality is that our collective crystal balls were just as cloudy back then as they are today.

Few lament that their house did not burn down when they buy fire insurance. We buy insurance “in case,” not because we want the risk to materialize.

We all live in a multi-period world where we have a single investment portfolio that compounds over time. In such a world, risk matters tremendously. A single, large loss can take us permanently off plan. Even small losses can put us off course when compounded in a streak of bad luck. While a focus on risk aversion may seem foolish in hindsight when risk does not materialize, going forward we know that managing risk can help us maximize our long-term growth rate.

[1] Derivations for this result can be found in our commentary *Growth Optimal Portfolios*

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