*The arguments in the following article have been made before and are well known among finance professionals and academics. Volatility as a convenient risk measure and -definition remains very popular, despite its many drawbacks. Given the importance of risk for investment decisions, this topic is worth discussing in more detail.*

## Overview

When discussing finance and investments, the term *risk *is almost always involved. There is talk about risk vs. return, value at risk, risk free rates and high-risk opportunities. Given the central role of risk in finance, it is remarkable how little thought is put into establishing what the term precisely means.

In non-academic finance, risk is mostly synonymous with volatility (the standard deviation of returns). However, is measuring the variability of returns really what investors care about?

The Merriam-Webster encyclopaedia dictionary defines risk as:

The possibility that something bad or unpleasant (such as an injury or a loss) will happen.

The important word is the word *loss*. Increased variability of returns is only a bad thing if the chance of losses increases. The classic example comparing 2 portfolios:

- A portfolio containing a risk free bond.
- A portfolio containing the same bond plus a call option on an underlying with uncertain future value.

At maturity, the first portfolio will be worth the par value of the bond. The second portfolio will be worth the par value of the bond plus the potential payoff of the option. Given the higher variability of possible outcomes, the second portfolio may end up having a higher volatility. It is doubtful whether it is also more risky from an investor's perspective, as in any case the value of portfolio #2 is greater than to equal to the value of portfolio #1.

Michael Keppler makes a similar argument in his well-known 1989 article Risk is Not The Same as Volatility (translated from the German article Risiko ist nicht gleich Volatilität).

Suppose the price of a stock goes up 10 percent in one month, 5 percent the next, and 15 percent in the third month. The standard deviation would be five with a return of 32.8 percent. Compare this to a stock that declines 15 percent three months in a row. The standard deviation would be zero with a loss of 38.6 percent. An investor holding the falling stock might find solace knowing that the loss was incurred completely “risk-free.”

These examples illustrate an important point: Volatility treats changes in returns to the upside the same as changes to the downside. If returns are distributed symmetrical around 0, this may make some sense. However, option payoffs clearly violate this assumption. As long as these returns are sufficiently skewed to the upside, a higher volatility is desirable for the investor.

Also, any backward looking measure faces issues when markets are in an extraordinarily quiet period. If there is indeed a new paradigm of low volatility and high returns, then everything is fine. On the other hand, if it turns out to be the calm before the storm (i.e. The New Economy or The Great Moderation), then basing investment decisions on historical volatilities may lead to suboptimal portfolios.

In fact, overheating markets, which are well in bubble territory often have such high returns combined with low volatility. This leads to further counter-intuitive results. A stock that has had spectacular low-volatility returns for the past 2 years will be seen as relatively risk-free. Even though it may be overvalued. If a crash happens and the stock price drops by 20% then the stock will have a higher (historical) volatility than before and hence more risk.

## Risk Measures

Semi-variance seems more plausible than variance as a measure of risk, since it is concerned only with adverse deviations.

After that, the question arises: What is a sensible definition of risk?

A number of other measures are common, here a short, non-exhaustive list of examples:

- Semi-variance (as mentioned above), or its square root semi-volatility. Instead of looking at all changes in the returns, only the negative ones are looked at. It was never very popular.
- Value at Risk (VaR) - Essentially a confidence interval, that determines the probability that the mark-to-market loss on a portfolio over a given time horizon exceeds a set value (i.e. a 5% probability that the loss on the portfolio will exceed USD1mm over 1 day). Note that there is no information on the expected size of the of the loss, should it exceed the threshold. It is very widely used, but also has many critics that consider its methodology flawed or at least insufficient. It is often used as a measure in risk management and risk controlling. Rick Bookstaber mentions in is blog post The Fat-Tailed Straw Man that Risk Managers are well aware of the limitations of VaR, and that the general critique on its use ignores the fact that it is typically just one number of many that risk managers are looking at:

So, to recap, we all know that there are fat tails; it doesn’t do any good to state the mantra over and over again that securities do not follow a Normal distribution. Really, we all get it.

- Conditional Value at Risk (CVaR) - CVaR tries to overcome the shortcomings of VaR, by measuring the expected shortfall. It is sometimes used in conjunction with VaR and sometimes by itself, typically in portfolio management problems.

The obvious common feature of the three examples is the asymmetric nature of measurement - only the bad outcomes are considered.

A more thorough analysis of VaR and CVaR is beyond the scope of this first post on risk and may be done at a later stage.

## Permanent Loss of Capital

*permanent loss of capital*. This definition is more in line with intuition and works well qualitatively. However, it is more difficult to do quantitative analysis this way. Most investors do not wait until an asset has gone to zero before they sell. Hence, modelling the realisation of losses has to take into account idiosyncratic thresholds, where loss-making positions are closed out. Depending on the time horizon, the liquidity and the conviction of an investor about a position this level may change over time, which further complicates any analysis.

## Stress Tests

- What happens if equity markets crash by more than x% this year?
- What happens if GDP falls by z% in a given year?
- What happens if interest rates go up by at least y%?
- What if half the instruments in the portfolio terminate their contracts in the fifth year?
- What happens if oil prices rise by 200%?

## Conclusion

The focus on volatility as a risk measure is not the result of some sinister conspiracy of high finance. It exists mainly because of a certain inertia of both banks and clients. Banks that want to sell products and services to clients have to speak their language. Portfolio managers that do not manage their own money have to speak the language of their bosses, investors and shareholders. In a way, even abstract concepts like volatility have a networking effect. Regardless of its real world merits, volatility as a concept becomes more useful, the more people know and use it. So everyone is using it because everyone else is using it.

Depending on the application, different risk definitions may be appropriate. Investors must be thoughtful when choosing, by which measure they estimate the risk they are taking. Keeping Peter Drucker's famous quote in mind:

What gets measured, gets managed.

Looking at different risk measures and thorough analysis of the actual positions can help in understanding the likelihood of negative outcomes. In the end, having a good grasp of the risk of a portfolio is as much art as it is science.