Why do you lose money on sound investments?

Sep 19, 2013


Revisiting risk management tools 

Consider the following investment opportunity. Every month, the value of your investment increases by 15% with a 51% probability, or decreases by 15%. With more chances to gain than to lose value, your advisor confirms that the investment is sound, and will yield an expected return of 3.5% per year (mathematically-inclined readers are invited to verify this prediction). Confident in your estimations, you decide to invest. To your surprise, after a couple of years your investment lost 18% of its initial value instead of yielding the expected 7% return (Figure 1). Even worse, it keeps going down. What went wrong?




This example, although simplified, presents features similar to real-life investments. In particular, it is multiplicative in nature: the investment is multiplied by some factor between successive periods. Many famous and widely used financial models behave similarly, such as the Black-Scholes model for financial markets containing certain derivative investment instruments.

Resolving the paradox

The issue lies in the tools we use to analyze random events. We have been trained to think in terms of ‘ensemble averages’ to estimate future outcomes. An ensemble average processes all possible outcomes weighted by their occurrence probabilities. It measures the mean value observed over many investors. However, atypical high gain, low probability events can dominate its value. This means that it doesn’t always reflect the typical outcome of an individual investment.

By contrast, the so-called ‘time average’ measures the expected return on a typical realization of the investment.1 In the present case, it takes the value -9% on a yearly basis, revealing the likelihood of a negative return on investment. Armed with this knowledge, you could have avoided this bad investment.

The concept of time average is, unfortunately, relatively unfamiliar to many financial economists and risk managers. One possible reason is that the two types of averages coincide in many situations, resulting in the ensemble average being automatically used. However, for multiplicative processes, which govern the evolution of many financial instruments, the two averages differ and give a very different view of return on investment and, more generally, of risk management.

Practical implications

Incorporating time averages in the toolset will naturally improve wealth management. They provide a better risk-return measure than currently calculated expected returns. Indeed, they reflect what happens on typical outcomes, which can be financially detrimental, as revealed by the above example. This is crucial for long-term projections, especially in cases where leverage is high or option strategies are used.
Because time averages are more sensitive to risks than ensemble averages, they provide a natural tool for position sizing (i.e. optimizing return and managing risk by adjusting how much we should wager). It is easy to come up with examples where ensembles averages would push us to wager 100% of our wealth (and almost certain financial death) based on a positive return, whereas a time average approach would balance return and risk into a finite optimal position size. Of course, no investor looks at return in isolation, but time averages intrinsically account for a series of risk factors.
Beyond individual asset owners, time averages also affect investment strategies in general. For example, since the financial crisis of 2008, we already know intuitively that there is a point at which leverage becomes toxic. Time averages might help us better understand where that point may lie. Some high-leveraged investment strategies could turn out to be unattractive no matter how small the position size.

Other financial questions can be revisited in light of the time average measure. It could generate new insights into issues such as what the margin requirements or minimum capital levels should be, or into the maximum permissible loan-to-value ratios for mortgages. A deeper understanding of these factors could lead to improved regulations.


Improved risk management requires a better risk framework, better risk governance and better tools. This article contributes to the ‘better tools’ category. It tries to bring awareness and clarity to certain statistical subtleties that nonetheless have major consequences for our understanding of risk. A better understanding of risk can only result in improved investment strategies and wealth management.



1The notion of typicality can be made mathematically precise. 

Further reading

1. Towers Watson, The irreversibility of time (2012).
2. S. Redner, Random multiplicative processes: An elementary tutorial, Am. J. Phys. 58 (1990).
3. O. Peters, Optimal leverage from non-ergodicity, Quantitative Finance 11, 1593 (2011).



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