**What is Risk?**

Risk is a crucial concept for investors, but there are many different ways to think about risk. In financial theory, risk is most commonly associated with volatility, a measure of how widely the returns from a portfolio vary above and below its mean value. To many investors, equating volatility and risk does not seem to make a lot of sense. We like to think of surprises in returns that are above average as “good news” rather than being a manifestation of risk.

I have also seen risk defined as the permanent impairment of wealth (though I do not recall where). This definition means that the only real risk is that of losing wealth that can never be regained: investing in the next Enron is the real risk.

Equating volatility to risk makes sense, however, when we are unsure as to whether a loss can be recouped. Investors in Merrill Lynch (MER) have seen the price drop from about $90 at the end of May of 2007 to the high $20’s at the start of September of 2008. But can we know this is permanent? MER has a history of large price swings. This is where defining risk in terms of the permanent loss of wealth breaks down. Merrill’s investors have lost money, regardless of what happens from today forward.

With Lehman (LEH), investors are increasingly convinced that the firm won’t recover, but do you really have to wait to see the end game to conclude that a firm is risky? The answer, of course, is that there are hallmarks of the potential for “permanent impairment of wealth” in a range of market information—and forward estimates of volatility (generated by models and/or reflected in implied volatility) provide important signals of this risk.

**Volatility vs. Corporate Failure**

It is worthwhile to explore the link between default risk (permanent loss) and volatility. From the perspective of financial theory, volatility and default risk have a natural connection. The linkage comes in the study of financially distressed companies. As news becomes available that a company is in trouble, the stock price tends to become more volatile. This makes perfect sense: the market is increasingly uncertain about the company’s ability to perform. The volatility is a manifestation of the range of varying opinions regarding the company’s prospects.

There is a straightforward statistical way to think about how volatility relates to the probability of default. We can define default as a return of -100% (i.e. your investment becomes worthless). Statistical models of returns use volatility to predict the probabilities of the range of future returns. In the simplest approach, we might think of the return on a stock as having a normal (bell curve) distribution. If returns follow this distribution, we need only know the average return and the standard deviation of return to be able to compute the probability of a given return at some time horizon. Volatility is defined as the standard deviation in returns. If the average (expected) return remains the same, an increase in volatility increases the probability of a very large negative return (positive returns, too)—so the probability of a -100% return increases with volatility. Further, we can use the statistical model to predict the probability of a -100% return.

The theoretical connection between default probability and volatility means that a statistical model of the returns on a stock carries with it an implied default probability. In practical terms, this should mean that pricing of credit derivatives like credit default swaps should depend on volatility—and they do. The trailing and implied volatility on a stock is a determinant of the implied default risk in credit default swaps (source). There should also be a connection between volatility and default risk as measured by credit ratings of a firm—and we have explored this in earlier work (source). In this study, we showed that the projected probability of an extreme loss over one year generated by Quantext Portfolio Planner [QPP], a Monte Carlo model was closely related to Moody’s Market Implied Risk credit ratings for a series of firms. These results confirmed the close relationship between volatility and default risk.

The findings from our earlier analysis and work on the relationship of credit default swaps (in which counterparties are betting on the probability of default), Moody’s Market Implied Risk ratings, and projected volatility have important implications for how investors deal with investing in individual stocks. The evidence suggests that volatility and probability of default have a very definite relationship.

**Predicting Corporate Distress**

Another line of research in the work on corporate distress is the use of accounting and econometric statistics to predict stocks that are likely to become distressed or fail. Edward Altman, pioneering creator of the Z-score, has shown that a combination of accounting measures can predict the probability of bankruptcy remarkably well (source - *PDF file*). In a fairly recent article, In Search of Distress Risk, Campbell et al develop a more sophisticated regression model that includes both accounting data and econometric variables (source - *PDF file*). Their model predicts the probability of bankruptcy consistently. This study also develops statistical performance measures of stocks, sorted by their modeled assessment of distress. Their findings have important implications for investors:

1) Firms with a higher probability of bankruptcy tend to be substantially more volatile (and vice versa)

2) Firms with higher probability of bankruptcy tend to deliver substantially lower returns (and vice versa)

The authors sort companies in terms of their percentiles of projected probability of failure—starting with the lowest 5th percentile and going up to the 99th. There is a dominant relationship between the probability of failure and high volatility (see table below).

This chart shows the annualized volatilities of stocks in each percentile grouping of predicted failure risk. The companies with lowest probability of default (the left side of the chart) also have the lowest average volatilities. With increasing levels of default risk, volatility increases substantially. This result is entirely consistent with our earlier work. The reader may note that the relationship between volatility and probability of failure is quite flat until the 40th-60h percentile, at which point volatility and failure probability have a strong linkage.

This is also consistent with results in this article. The chart below shows data from this article (table labeled Moody’s Rating vs. QPP Outlook) for a series of stocks. This chart may be compared to the chart above to see the non-linear impact of volatility of probability of default. The table below shows results for a series of individual stocks (each bar is one stock), while the data from Campbell et al are for groups of stocks, sorted by default risk.

* Projected Probability of Default vs. Annualized Volatility for Stocks from **linked article*

The comparison between the two charts above is qualitative, of course, but gets the idea across. While QPP is a statistical model that captures some default risk via volatility, Campbell et al have developed a model that captures a range of other important dimensions of default.

Campbell et al go on to summarize their findings as follows:

“We show that stocks with a high risk of failure tend to deliver anomalously low average returns. We sort stocks by our 12-month-ahead estimate of failure risk, calculated from a model that uses only historically available data at each point in time. We calculate returns and risks on portfolios sorted by failure risk over the period 1981-2003. Distressed portfolios have low average returns, but high standard deviations, market betas, and loadings on Fama and French’s (1993) small-cap and value risk factors. They also tend to do poorly when market-wide implied volatility increases.”

This suggests that distressed stocks tend to exhibit momentum effects—things often get worse for quite a while before they improve. This is not entirely surprising, but it is worth keeping in mind—especially with the ongoing conjecture about when it makes sense to get back into financial stocks. The authors’ finding that these stocks tend to fare poorly as broad market risk (measured by VIX) increases makes sense if investors tend to flee high-risk equities as markets get more volatility (flight to quality).

This effect is also consistent with the increase in Credit Default Swap [CDS] spreads with market volatility noted earlier. Investors, broadly speaking, act in such a way that suggests that they recognize increasing default risk is associated with increased market and stock-specific volatility. This result is also consistent with how QPP projects risk. QPP links broad market volatility to the projected volatility for individual stocks—an increase in VIX will tend to increase default risk.

**Discussion**

So what does all of this mean for investors? In my earlier analysis on this topic (see this piece), I showed that projected volatility from QPP (a Monte Carlo model) mapped well to credit ratings from Moody’s. There was a clear relationship between projected probability of large losses and credit rating. In this discussion, I have brought in an analysis by Campbell et al that explicitly shows that volatility is well-correlated to probability of corporate failure. This study also shows that distressed firms tend to under-perform with some persistence: there appears to be momentum in the poor performance of these firms.

Very volatile stocks have elevated probability of failure and this risk is a very specific to a firm—what is called idiosyncratic risk. In my earlier analysis of this topic, I suggested that a reasonable rule of thumb for investors wishing to invest in individual stocks (while controlling the increased default risk associated with individual stocks) would be to avoid individual stocks which have projected one-year risk of loss of 50%-60% (or worse) at the first percentile. This guideline would also tend to keep investors out of companies that have credit ratings below investment grade (see this article).

The challenge that investors face is determining the relative attractiveness of owning every stock in a sector, as opposed to owning specific individual stocks. The risks associated with holding individual stocks are, I believe, substantially muted if controls are placed on the specific stocks under consideration. We know that the idiosyncratic risk (volatility) associated with individual stocks is substantially reduced as the number of holdings increases.

This does not tell the whole story, however. Investors in individual stocks need to be aware if they are investing in stocks with high default risk because these stocks have special properties: they may simply cease to exist. Further, conditions that drive distress at one firm may also impact other companies in a sector. Witness the recent declines in both share price and credit quality at a range of financial firms. Putting twenty of these together in a portfolio would not help much.

At the start of this article, I noted that many investors view risk in terms of a permanent loss of wealth and volatility as a less useful proxy. A body of research shows that there is a direct relationship between volatility and risk of corporate failure. This result has important implications for investors. Higher volatility represents the market’s assessment of uncertainty in the future prospects for a firm. The high correlation between volatility (both forward-looking and historical) and credit ratings and default risk models suggests that the volatility in a stock is pricing in distress risk fairly consistently. Thus, high volatility is a good indicator for investors to follow—especially large increases in volatility (source).

Further, the systematic relationship between volatility and failure risk has important implications for portfolio analysis. QPP (our portfolio tool) will tend to show an increase in failure risk with volatility in an individual stocks and will tend to show an increase in failure risk at individual firms with increasing market volatility [VIX]. These couplings are consistent with Campbell et al’s findings and we see these effects manifested in the bear market of 2008.

**Black Swan Events**

The rational coupling between volatility and failure risk is important in the debate over the impact of Black Swan events. Nassim Taleb has made the case that truly extreme events (Black Swans) are not properly captured by financial models. His argument largely centers on the idea that standard assumptions about portfolio returns fail in extreme events.

While his point is well taken, if standard portfolio modeling can predict the probability of corporate failure reasonably well, then corporate failure is not a Black Swan (by definition, a Black Swan is unpredictable). QPP is based on some fairly standard assumptions about statistical variability, but QPP’s predicted tails map well to the probability of failure. While there are certainly events that are so extreme that markets and models will incorrectly discount their probability (which would be Black Swans), if we can consistently capture the probability of corporate failure, I feel quite confident in these models.

**Summary**

To sum up, I want to examine a very simple case. Let’s imagine that an investor is comparing the relative merits of investing in Johnson and Johnson (NYSE:JNJ) vs. investing in the Russell 2000 (NYSEARCA:IWM). Standard reasoning would say that an investment in JNJ is inherently more risky than investing in IWM because the probability that JNJ will fail is surely higher than the probability that every stock in the Russell 2000 will simultaneously fail—and this is why investing in individual stocks is ill advised. But does this make sense?

From QPP’s projections, JNJ’s projected one-year 1% tail risk is a loss of -35%. For IWM, QPP projects a one-year 1% tail risk of -40%. The projected failure risk of JNJ is so low as to be effectively zero, as it is for an investment in IWM. This line of reasoning has very important implications for investors who have been taught that investing in an index that contains a wide mix of assets (including quite a few with high default risk) is always a more rational choice than investing in a much smaller number of individual securities because of default risk. To my mind, the evidence does not support this reasoning as long as the selection of individual securities does not include stocks with meaningful default risk.

As I have been editing this, we have seen the announcement of the government bailout of Fannie Mae (FNM) and Freddie Mac (FRE). It is, of course, of interest to see how the tails of these two stocks looked from the perspective of projected volatility from QPP. At the start of May 2008, FNM was trading at about $30 a share and FRE was trading at about $26 a share. Today (September 8), they both closed at less than $1 per share. Using default settings and data through the end of April 2008, QPP projected a one-year 1% return of -56% for FNM and a one-year 1% return of -82% (QPP users can easily verify these numbers). In March of 2008, I wrote :

*For QPP users who wish to purchase individual stocks but wish to mitigate the risk of bankruptcy in individual stocks, the 1-year 1% percentile return could be constrained to levels at or below 50%-60%.*

The projected volatility on these stocks showed a substantial probability of default—though of course if would have been even better if the model had predicted the default. My point is not that market volatility contains all the information, but rather that it contains some meaningful information. In the September 15, 2008 issue of Forbes, columnist Lisa Hess acknowledges that she recommended that readers purchase both Fannie Mae and Freddie Mac in the spring of 2008 . Investors who simply looked at a decent analysis of implied volatility from the market (either from the options markets or from a model like QPP) would have been given fair warning as the market’s assessment of the risks involved in such a bet.

**Disclosure:** The authors holds long positions in IWM and JNJ.