It’s been widely reported that quant equity funds have been pummeled in the months following the equity market lows reached in early March. During this period a rally of historic proportions has transpired with leadership emanating from the most beaten-down names in financials, industrials and consumer cyclicals.^{i} Dubbed the “junk rally” by some, the outperformance of financially shaky, beaten-down stocks relative to more financially stable firms has been impressive by any historic standard. While the overall market is up over 30%, a large number of these beaten-down stocks have doubled or tripled in value since hitting multi-year lows in early March.
So, what exactly is the root cause of the underperformance experienced by quant equity funds and how does it relate to the junk rally? Is the junk rally simply an aberration or will this phenomena persist for months or even years into the future? Moreover, how can quant portfolio managers adjust their portfolio construction processes to achieve better performance under such challenging market circumstances?
This article attempts to answer these questions by shedding light on a new source of risk now pervasive in equity portfolios.
Quant fund underperformance: Wrong way factor bets?
One could hypothesize that the underperformance of quant equity funds is simply a result of wrong-way bets placed on long favored factors which happened to move violently against them (say, going short leverage and long momentum or long earnings yield). While this is most likely true, I think there is more going on here than that. A new dynamic is at play with new sources of risk for equity portfolios. The nature of this risk is asymmetric and quant equity portfolio construction processes generally aren’t designed to control for asymmetric risk. In essence, quant funds have been caught off-guard by a type of risk for which they do not have the tools in place to adequately manage.
Understanding Equity Portfolio Risk Management
Of paramount importance within the portfolio management process is the notion of controlling risk. This usually takes the form of conforming to tracking error limits for a benchmarked portfolio or standard deviation limits for an absolute return portfolio.
In any given quant shop, it’s par for the course to accomplish this through the discerning use of a risk model. Portfolios are constructed and rebalanced using mean-variance optimization, a process designed to find optimal portfolios by balancing risk with return expectations; achieving diversification while positioning the portfolio to take advantage of alpha opportunities. The current state of affairs in equity markets has thrown a serious wrench into this machinery however. Even the most robust of quant risk management processes have experienced portfolio standard deviation and tracking error significantly higher than forecast. Yet it’s not just a mis-estimation of portfolio standard devation that’s a problem. It’s a mis-estimation of the type of risk.
This isn’t the first time this has happened of course.
The Quant Meltdown Summer 2007
We all know the story of how quant funds were taught a painful lesson in the summer of 2007. During the early days of the credit crises the rapid shift in economic landscape manifested itself in forced liquidations and rapid de-leveraging among large quantitative hedge funds.^{ii} The situation was amplified for quant hedge funds as it turned out many of them were employing similar trading strategies. In essence, everyone was betting on the same factors. The rush to exit positions triggered an avalanche of unwinding in crowded trades which in turn pushed quant factors into extreme territory within a matter of days. As Litterman at GSAM is quoted as saying “It was like turning a light on in a dark room to see the amount of crowdedness…”.^{iii}
The summer of 2007 was a lesson in crowded trades and the necessity of diversifying sources of alpha. This type of risk was obviously not captured by your typical risk model. The current situation is similar to 2007 in that we are experiencing risk not captured by a typical risk model, albeit there is a different lesson to be learned here. Today’s lesson is about capital structure theory and the asymmetric return profile it generates in certain market environments.
I will state the conclusion first: Given the current market environment, the risk of an equity portfolio is similar to the risk of a book of options with large gamma.
While this may sound esoteric for those unfamiliar with options pricing theory, I encourage the reader to continue reading as the ramifications for equity portfolio management are considerable. As much as possible I will attempt to translate the material into a vernacular more familiar to those with a background in equity models.
Risk Models Reviewed
At this point it’s worthwhile to quickly review the world of equity risk modeling. In an equity risk model, equity returns are attributed to underlying “factors” that influence the entire universe of equities, thereby defining the co-movement of their returns. Factors can be economic in nature (such as changes in interest rates and commodity prices), or fundamental factors (such as leverage, industry, size and value) or statistical factors which have no easy interpretation.
Regardless of the factors in the model, the general idea is to control risk at the second moment of the return distribution (i.e. portfolio standard deviation).^{iv} Operationally, this amounts to forecasting a variance-covariance matrix of factor returns, the exposure of each stock to these factors, and the specific risk of individual stocks. Once we have this data in hand, we can crank the computational machinery to compute a forecast of portfolio risk.
This approach to risk management has served remarkably well and has stood the test of time.^{v} Stock returns have been reasonably symmetric for most of the past 20 years and managing risk to the second moment has so far been adequate. The problem with this approach however is that the return distribution for many stocks has become extremely asymmetric as of late. In fact, we should expect positive skewness (i.e. a much higher probability of experiencing large positive returns) in stock returns. It’s the third moment of the return distribution we need to be concerned about now.
Fat tails and Black Swans, right? Wrong
Yes, we all know about fat tails and black swans and yes, this is a big risk that more often than not goes woefully unheeded by many in our field. This isn’t the phenomena I am referring to in this article though. Contemplate this assertion: Even if asset returns fit a normal distribution the return to many individual stocks should be asymmetric, even in theory.^{vi}
To understand why this is the case, it’s important to distinguish between “asset” returns and “equity” returns. The former being the returns on the assets of the firm (i.e. the left hand side of the balance sheet), the latter being the returns to stocks (i.e. the shareholders equity on the right hand side of the balance sheet). We must recognize that equity is a leveraged play on the assets of the firm, or more technically, equity is a call option on the assets of the firm. As we all know, option returns are far from symmetric.
This issue has surfaced only recently because the economic downturn has left more companies than ever in a state of undercapitalization. In options pricing parlance, this broad based undercapitalization means that equity has been moving from in-the-money territory to at-the-money territory. This inevitably results in greater asymmetry in stock returns.
Alas, we may have gotten ahead of ourselves so let us take a step back. To best understand the current situation we need the perspective of a model which can explicitly take into account the capital structure of a company. The best model for this is the the structural credit modeling framework first proposed by Merton in 1974 and expanded upon by many researchers in the years since.
Structural Credit Models 101
While many flavors of structural models abound, the basic idea is that equity is a call option on the assets of a firm, with a strike price equal to the value of the firm’s debt. The reasoning behind this as follows: If at some time in the future the value of the firm’s assets is enough to satisfy the claims of the firm’s creditors, equity shareholders are privy to whatever value is left over after satisfying creditor claims (i.e. any upside return on the assets).^{vii} If the value of the firm’s assets is less than what is required to satisfy the claims of creditors, the shareholders have the option of defaulting on debt repayment. Where does this option come from? It’s a direct result of the limited liability legal structure inherent to equity ownership. Equity shareholders can never lose more than the amount of their investment.
If one draws the payoff to equity it is indeed exactly the same payoff as a call option and can be modeled as such.
Now that we have established the option-like nature of equity, the pertinent question to ask is “what drives option value?”. According to options pricing theory the value of an option is derived from the value of the underlying (i.e. firm assets) relative to the strike price (i.e. firm liabilities), as well as the volatility of those assets.^{viii} While I won’t go into the details of how one creates or estimate such a structural model, it’s enough to say that structural models have become popular among corporate bond portfolio managers as a useful framework for forecasting default probabilities of public companies.
If we look at the current situation through the prism of the structural model, it quickly becomes apparent that a large cross-section of the US equity universe is closer than it has ever been in the past to being an at-the-money call option.
Beware the Gamma
So, what do we know about at-the-money call options? First, gamma is highest and positive for at-the-money call options. Technically, option gamma is the second derivative of price with respect to changes in the underlying, while delta is the first derivative. Large positive gamma means that the value of equity increases at a quickly increasing rate as the value of the firm’s assets increases. Reinterpreted in the context of an equity risk model, we can say that stock Beta and specific risk will increase at an increasing rate as the firm experiences positive asset returns (and vice versa for negative returns). In this translation, option delta is synonymous with stock Beta, while option gamma is the rate of change in stock Beta for changes in the overall market (think of option gamma as representing non-linearity in stock Beta). This behavior is most pronounced in highly leveraged firms. Gamma is also highest for firms with highly volatile assets. Thus, firms with highly volatile assets should experience this behavior more than firm’s with low volatility assets.
The above two observations can explain the explosive returns observed in banking stocks during the recent rally. According to the theory, observing these type of explosive upside returns is more probable than ever given that banks have witnessed unprecedented volatility in banking assets, coupled with the fact that most banking stocks are very close to being at-the-money.^{ix} This combination has resulted in large positive gamma for banks stocks which, from a probabilistic perspective, means a greater likelihood of achieving large positive returns.^{x} Put another way, the Beta of banking stocks increases quickly as the market rallies.
I encourage the interested reader to investigate further aspects of the model to better understand the recent rally in beaten-down, highly leveraged stocks.^{xi} Suffice to say that at this unique juncture in equity markets history, an equity portfolio should be viewed very much as a portfolio of options. The core implications of this are significant: Managing portfolio risk by only looking at portfolio standard deviation (or tracking error) is a recipe for disaster.
I suspect this is what’s happening to quant equity funds these days. While on the surface it appears that certain time honored factors have underperformed, what is really happening is a manifestation of the option-like nature of equity. It’s not uncommon for quant funds to be short the very firms that have the highest gamma and long the firms that have the lowest gamma. While many of these portfolios appear to be factor neutral, the embedded gamma causes these portfolios to become quickly unhinged.
As this author can attest, managing the risk of an options portfolio is orders of magnitude more complex than managing the risk of an equity portfolio. The skewness of the distribution alone necessitate more powerful risk modeling techniques.
Again, while all the above concepts may sound esoteric the ramifications are highly consequential and should be heeded by all equity portfolio managers. Moreover, it’s reasonable to assume that this phenomena will persist for several years (or for as long as it takes for firms to clean up their balance sheets).^{xii}
Recommendations
Managing risk and building optimal portfolios in the context of an asymmetric return distribution is an advanced subject that can involve tools such as portfolio simulation and multi-target optimization. The process could become quite complex to say the least.
Should every quant fund scramble to put such tools in place? I’ll leave it to the reader to decide on what’s best for their particular situation. However, I will offer a suggestion that may involve the least amount of disruption to an existing portfolio construction process.
The approach can be considered a “quick and dirty” solution (and by “by quick and dirty” I mean an excruciatingly painful (but not too painful) modification to an existing mean-variance portfolio optimization process ^{xiii}).
The premise here is to neutralize the portfolio’s exposure to asymmetric risk using the same method that is typically used to neutralize portfolio exposure to other factors. This method is commonly employed by market neutral funds to neutralize against industries for example. The method amounts to adjusting positions to ensure zero exposure at the portfolio level. In our context, the basic steps of this process are as follows:
Forecast asymmetric risk at the individual company level by estimating a structural model (or at least approximating it using the main drivers that go into a structural model). The gamma of equity in the structural model is the appropriate measure to compute.
Create an “Assymetric Risk Factor”. Each stock’s exposure to this risk factor is equal to its option gamma computed above.
Impose an equality constraint in the mean-variance optimizer to set the portfolio’s exposure to this factor to zero.
While not a perfect solution, the procedure described above will result in a portfolio that is gamma neutral and allows for the existing mean-variance optimization process to remain intact. Risk forecasts will no longer be biased and the portfolio will not experience asymmetric returns due to the option-like nature of equity.
Conclusion
The current economic situation has resulted in a state of broad based undercapitization in US equity markets. Coupled with the unprecedented volatility experienced in markets as of late, equity portfolios are behaving much like option portfolios. As a result, equity portfolios are subject to new sources of asymmetric risk that are rarely addressed in current portfolio risk management processes. Equity quant funds must modify their existing portfolio construction processes or risk experiencing continued poor performance as the economy recovers over the next several years.
End Notes
i The blog “Zero hedge” has reported extensively on the junk rally. See: “Bill Miller Correct: Value Funds Burying Quants,…For Now“,and “Open Letter To Quant Funds“, and “Renaissance Underperforms S&P by 17% In April“, and “Market Dispersion has Collapsed”
ii For an account of this see the NYT article by Joe Nocera
iii See the Bloomberg article “Goldman’s Global Alpha Gains 19% After 2007 Plunge”
iv By “second moment” I’m referring to the mathematical moments of a probability distribution. The most important of which are the first moment (mean), the second moment (standard deviation), the third moment (skewness), and the fourth moment (kurtosis).
v While Nassim Taleb may disagree with this statement, the fact remains that these models are still the de facto standard in the equity portfolio risk management world.
vi This fact is often swept under the carpet in the equity risk modeling world.
vii Practitioners often model these options as perpetual options since firms can roll over their debt.
viii Of course, interest rates and time-to-maturity also drive option value according to the classic theory. However, our modified version of the model drops maturity as an input since equity is a perpetual option. As for interest rates, for the purposes of this discussion rates are so low that we can safely ignore this input.
ix The past year has witnessed bank Capital Ratios drop to severally low levels.
x As an aside, the structural model has much to say regarding the current controversy surrounding the solvency of the US banking sector. Critics claim that assets of firms such as Citigroup or Bank of America will likely not be enough to cover the firm’s liabilities (given the significant write down in asset values). Even assuming this is true for a moment, the equity of these firms would still be highly valuable since even out-of-the money call options are quite valuable, especially those with highly volatile underlyings. While no option intrinsic value may exist, the option will always have time value, at least until an option expiry date arises. This could be an eternity so long as Geithner and the Fed continue to divert taxpayer money into ensuring equity shareholders never have to face the expiry date.
xi For example, the model has much to say about Vega, that is, how stocks will react to changes in market volatility.
xii Judging by history this will probably take a few years. Corporate default rates tend to peak one to two years following an equity market downturn.
xiii Alas, nothing in quant portfolio management is ever easy, only simple or complex. Anyone who has ever dealt with financial data understands exactly what I mean by this.
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