Risk Management and Convex Return Profiles

John P. Hussman, Ph.D.
All rights reserved and actively enforced.
Reprint Policy

This week, I want to continue the analysis of "convexity," and its implications for how we should approach uncertainty in what seems likely to remain a very tenuous investment environment.

We continue to face two data sets here, which characterize two very different possibilities for the true state of the world. One possibility, which is clearly the one that Wall Street has subscribed to, is that the recent downturn was a standard, if somewhat more severe than normal, post-war recession; that the market's recent strength is an indication that it is looking forward to a full "V-shaped" recovery, and that the positive print for third-quarter GDP is a signal that the recession is officially over. Applying the post-war norms for stock market performance following the end of a recession, the implications are for further market strength and the elongation of the recent advance into a multi-year bull market.

The alternate possibility, which is the one that I personally subscribe to, is that the recent downturn was the initial phase of a more prolonged deleveraging cycle; that the advance we've observed in recent months most likely represents mean-reversion - qualitatively and quantitatively similar to the large and often abruptly terminated "clearing rallies" of past post-crash markets; that major credit losses are continuing quietly but are going unreported thanks to changes in accounting rules by the FASB this past spring, which allowed for "significant judgment" in accounting for loan losses and deterioration in the value of securitized mortgages; that a huge second-wave of mortgage losses can be expected from a reset schedule on Alt-A and Option-ARMs that has just started (following a lull in the reset schedule since March) and will continue into 2010 and 2011; that intrinsic economic activity remains abysmal; that recent GDP growth is an artifact of massive fiscal stimulus that is unlikely to have sustained follow-through; and that recent market valuations are not representative of those observed at the end of most post-war recessions, but are instead similar to those observed at major market peaks prior to the mid-1990's.

Our primary challenge here has been to reasonably integrate these two possibilities. As I noted last week, one possibility is to weight the data itself and treat the resulting average as if it were a single data set. However, by evaluating our investment approach on the assumption of each possible state of the world, and then weighting the resulting investment positions, we can obtain some benefit from "model diversification," which reduces our reliance on a single view or expectation being correct.

Of course, this is something of a simplification, and we can't take this argument too far. Though we care about expected returns, we also have to be concerned with risk; particularly individual outliers. For example, if there is an 80% chance of a 10% return and a 20% chance of a 50% loss, it is incomplete to think about the world as if the expected return is simply -2%. The potential risk in the event of extreme outliers still has to be acceptable.

In any event, our first rule of investment adjustment always holds: when moving from an existing holding to a new candidate, whether we are talking about two individual securities or a small modification in our investment approach, one should always attempt to part from the existing holding on relative strength, and acquire the new candidate on relative weakness. It is rarely optimal to shift from the holding to the candidate when the candidate has just enjoyed a positive outlier return and the holding has had a negative outlier.

For example, it is almost never a good idea for an investor to shift from one investment to another when the existing holding has declined and the candidate has already advanced. Similarly, it is usually a bad idea for a buy-and-hold investor to abandon that strategy for a bearish one only after the buy-and-hold approach has experienced a major loss and the bearish one has experienced a major gain. It may make sense to move incrementally, but selling low and buying high is generally not a successful approach. Unless the holding is clearly inappropriate, you always try to make your shifts on recent strength in the holding and recent weakness in the candidate.

This is how we shift stocks in our portfolios - we try to buy higher-ranked stocks on short-term weakness and sell lower-ranked holdings on short-term strength. If we can't do that, we prefer to move incrementally. The same rule applies to small, periodic improvements that we make to our investment approach. We try to implement any alterations on short-term strength in the existing model.

Suffice it to say that on last week's market weakness, we established a bit of net market exposure (about 5%), and also re-established a modest "contingent" position in call options that should allow us to generate exposure to market fluctuations in the event that the market advances further (without subjecting us to a great deal of downside risk if the market continues lower). Aside from those incremental changes, the Strategic Growth Fund remains well-hedged. If the market declines further, without marked deterioration in internals, I expect that we will add another incremental amount of market exposure. On any meaningful deterioration in market internals (with or without obvious weakness in the major indices), the Fund will shift to a fully defensive and tightly hedged stance.

In any event, I continue to place a great deal of focus on risk management here. I frankly don't trust the prevailing view that the economy is in a sustainable recovery. It's quite possible we'll see ourselves reversing last week's modest increase in exposure and moving back to a fully defensive stance, but we'll let the data drive that decision. For now, we're doing our best to maintain equanimity about market direction, while keeping defense as our primary concern.

What produces a convex return profile?

Last week, I introduced the concept of a "convex return profile." This is not a topic that you'll find in general finance textbooks, but it is extremely important in the context of long-term risk-management. A convex profile is shaped like the curve below. It has the feature that the average value of the function (the red dot) is greater than the function of the average value (the blue dot). A convex profile is desirable for a risk-managed investment strategy, because it implies that even when there is variation in market returns, the investment strategy tends toward a positive absolute return.

A convex profile is characteristic of an investment strategy that includes a strong risk-management component. It participates, on average (though certainly not always) in general market advances, and mutes losses in general market declines. It resembles the payoff profile of call options, except that call options generally have a negative intercept - they lose money if the market is unchanged. An effective profile should have a positive intercept, meaning that a zero market return should be generally associated with a positive expected strategy return, on average.

Below is a scatter plot of returns for the market (horizontal axis) and a convex investment strategy (vertical axis). The market returns reflect overlapping quarterly periods since 1940. You can assume that the strategy returns are hypothetical. This would be an example of an effective risk-managed investment approach, though it clearly has outliers. Note that a buy-and-hold strategy would be a perfect 45 degree line. In the plot below, the scatter of points away from the line implies a fairly low correlation between strategy returns and market returns, which would produce good diversification benefits, but also a good deal of "tracking risk" between the returns of the market and the returns of the strategy.

The convexity of this return profile is driven by the way that investment positions are determined. At each point in time, the investment position taken is essentially proportional to the average return-to-risk (over and above the risk-free rate) associated with the investment conditions prevailing at that time. But the market exposure drops to zero when the expected profile is negative. This is the source of the non-linearity. Conditions that have historically been associated with a high expected return per unit of risk, on average, are assigned a high investment allocation, while conditions having a poor expected return per unit of risk, on average, are assigned a small investment allocation. The profile of returns from the strategy is convex because the profile of exposures taken is basically convex.

To give you an idea of what is going on at the "micro" level, consider the graph below. Here, I've simulated random market returns from a set of ten different probability distributions, each having a different "mean" or expected return. For each distribution, we assign an investment position proportional to the average expected return of that distribution (zero if the average return is negative). The actual return in each instance, of course, can vary. So for example, when the expected return is high, a large investment allocation is selected. The actual return in each instance is drawn randomly from the corresponding distribution, and may be either positive or negative. If the actual market return was positive, the strategy posts a gain. If the actual market return was negative (even though the expected return was positive), the strategy posts a loss.

You can probably make out five lines of varying steepness, along with a horizontal line. The steepest scatter of points (largest gains, largest losses) is associated with the distribution that has the strongest expected return. It is steepest because a large investment allocation is taken when the market return is expected to be high, on average. In practice, of course, some of the actual returns will nevertheless be negative, and those unfortunate instances will be associated with the largest losses. Notice that there is also a long scatter of points along the horizontal axis. Along this line, the strategy return is zero even when the actual market return is significantly positive or negative. This is because the expected market return was negative, so the investment allocation was zero (resulting in a zero strategy return), regardless of the actual market return that occurred.

As you look at this scatter, it is important to notice that the relationship between the strategy return and the market return is always linear for any specific instance. For example, if you currently have a 50% allocation to market risk, your return, by definition, will be 50% of what the market return happens to be, whether that return is positive or negative. So there is a crucial distinction between how the strategy behaves instance-by-instance, and how it behaves on average.

Convexity is not a characteristic of any particular point or instance. Rather, a convex return profile emerges as a result of pooling numerous instances where, on average, the exposure to market risk is roughly proportional to the average return/risk profile associated with prevailing market conditions (falling to zero when that profile is negative).

When we average the above scatter plot by various levels of market return, the following relationship emerges.

The convex return profile that we observe doesn't mean that there were not individual losses. We can easily see from the scatter plot that there were. It also does not mean that there were not missed returns of significant magnitude. Those misses are also very clear in the scatter. Yet over time, averaging out all of the individual successes and misses, the overall profile is desirable.

In short, a convex return profile - what we view as the hallmark of an effective risk-managed investment strategy - is the result of repeatedly accepting greater levels of market risk in conditions that have historically been associated with strong return/risk profiles, and repeatedly avoiding market risk in conditions that have historically been associated with poor return/risk profiles.

Market Climate

As of last week, the Market Climate for stocks remained characterized by unfavorable valuations, with the S&P 500 still priced to deliver probable total returns of only about 6.5% annually over the coming decade. This is a level of valuation typically observed near major pre-1995 peaks, not troughs, and certainly not early bull markets. Market action last week remained mixed. Trends in the major indices, as well as breadth (advance-decline) remained generally intact, though we're observing some early divergences here. Meanwhile, the market remained generally overbought on intermediate-term measures, but short-term measures have cleared and are now somewhat oversold.

As noted above, we took some incremental exposure on market weakness late last week, but the operative word here is incremental - I continue to be primarily concerned about downside risk. At the same time, I do expect - on the basis of simulations - that when we are faced with investment conditions wholly outside of post-war experience (as we observed not only over the past year, but also in the late 1990's), allowing some "model diversification" as described above, will result in a moderate increase in full-cycle performance, a slight increase in short-term volatility, little or no increase in maximum drawdown, and a moderate improvement in our "tracking" of general market advances. In all, a general improvement in the return/risk characteristics of our approach.

In bonds, the Market Climate last week remained characterized by modestly unfavorable yield levels and moderately favorable yield pressures. The CIT bankruptcy filing over the weekend may prompt a bid in default-free Treasuries next week, and my expectations lean toward disappointment in the employment statistics, given that we are still observing well over a half-million new unemployment claims each week, and the likelihood of strong absorption from job creation isn't encouraging.

Although I continue to believe that the U.S. dollar will face enormous challenges in the coming years due to the massive issuance of U.S. government liabilities, the short-run likelihood is that any general increase in credit concerns may provide some support for the dollar - via demand for default-free assets - particularly given its current oversold condition. Conversely, commodities (which tend to move inverse to the value of the dollar) may come under some amount of pressure in the event that credit concerns increase. Mark Hulbert of the Hulbert Financial Digest notes that sentiment among investment advisors recommending gold has recently surged. For a good portion of the recent advance, gold sentiment has been fairly restrained. Though Hulbert's sentiment index is not quite as extreme as at other intermediate-term gold peaks in the past two years, the recent spike in bullish sentiment is notable, and our measures also suggest that the risk level and vulnerability of the precious metals market has increased lately.