An Example of an Investment Model That Makes Something Out of Nothing (Sort of): Implications for Building Applied Mathematical Models ()
1. Introduction
1.1. The Model Development Process
The development of applied mathematical models takes place within a framework that is simple, elegant, and sometimes under-appreciated [1]. To summarize this process:
Step 1: Make simplifying assumptions about the system being modeled. The modeler acknowledges at the start of the process that these assumptions will not be fully realistic, but instead are intended to reflect the most important features of the system being studied.
Step 2: Develop a model around those simplifying assumptions.
Step 3: Collect data and assess the fit of the model to the data. Ideally, this assessment will involve multiple datasets, collected under diverse circumstances. Explore a) where the model fits the data well; and b) where the model fits the data poorly. If the model fits enough of the data sufficiently well then it is tentatively accepted and conclusions about the system being studied are drawn. If not, focus on the discrepancies (e.g., in which experiments did the model fail?) revise the model assumptions —typically, in the direction of an incrementally more complex description of the system being studied—and begin step 3 again.
1.2. What Makes a Good Model?
A “good” model typically is parsimonious, yet fits the data sufficiently well. Moreover, it captures the most important features of the system under study. It is this combination of both empirical and theoretical evidence that gives the modeler confidence that the model might generalize to new data sets.
1.3. What Can Go Wrong?
Especially when models are derived in an interdisciplinary context, a particularly common danger is that a mathematically elegant model will be created that “provides the right answer to the wrong question”—in other words, that the model either embeds an unsound understanding of the system in question and/or focuses on elements of the system that aren’t fundamentally important. This danger is particularly acute when the modeler is a mathematician or statistician without deep expertise in the system under study and the substantive experts are unfamiliar with the nuances of the modeling.
Another danger in modeling is that the experiments used to test the model aren’t sufficiently rigorous. To use a medical example, one component of personalized medicine is the development of predictive models—for example, a logistic regression model that predicts the probability of a patient having a serious complication from chemotherapy. If this model is originally developed using patients from tertiary care medical centers and all of the validation studies are performed using similar patients, it would be reasonable for a user to be concerned that the model might not perform as well when applied to patients receiving care in less specialized facilities.
2. Example
We provide an example of how making a seemingly trivial modification to a simple economic model and then performing the evaluation of that model using the current state of the economic science provides surprising results —so surprising, indeed, that they appear to violate one of the most basic tenets of investment theory and thus appear to “make something out of nothing”.
2.1. Conceptual Framework
The conceptual framework for the example is the simple errors in variable model under which classical portfolio theory is derived. In this model, the returns for an investment are assumed to be derived from independent random samples from the same statistical distribution. For example, the annual returns for an index fund that tracks the overall stock market might be modeled as a sequence of independently generated Gaussian variables, with mean 0.10 and standard deviation 0.15.
Economist don’t always assume that the distribution of investment returns is Gaussian—for example, these returns are often assigned log-normal, Levy or other types of distributions. For the purposes of understanding the model development process, what is much more important is not the shape of the distribution of returns but rather the fact that the returns from year to year are assumed to be independent. In fact, the average return isn’t central to the present argument. Accordingly, we can assume that the price of the investment at any point in time, denoted by Yi, is:
.
Here, the Ei are independently, identically distributed Gaussian random variables with mean 0 and standard deviation σE. Under this highly simplified model, stock prices will follow a random walk.
The link between the features of the system presumed to be most important and the model is provided by Ei. Specifically, if it is assumed that the stock market is a “perfect market” with large numbers of well-informed investors, none of whom are large enough to affect prices, then it follows that the current stock price Yi represents all available information. (Otherwise, arbitrage will push the price toward a value that reflects this information.) Moreover, it follows that at the next time point stock prices will only change in response to the new information Ei+1. This new information—for example, a tsunami, a change in government policy—is unpredictable, since otherwise it would have already have been reflected by the previous Yi. Accordingly, the series of perturbations (i.e., “shocks”, “errors”) are independent. Once this errors in variables structure is assumed, the other main elements of classical portfolio theory become logical consequences [2].
The assumption that the stock market is a perfect market—and thus that the Eis are independent—illustrates steps 1 and 2 of the modeling process. Certainly, economists don’t believe that any actual market is “perfect” in the sense of the above. However, it might be plausible to posit, as a first approximation to describing an exceedingly complex system, that the stock market is perfect and then to explore how much (or little) harm is done by such an assumption.
2.2. Relationship between Risk and Return
Apart from the assumption about independence, the above model embeds within Ei a second critical feature of the system—namely the variability in stock prices as quantified by σE. When σE is small, as is the case for established companies with dependable prospects, the stochastic path followed by the stock price is less variable than for more speculative enterprises (for which σE is larger).
In classical portfolio theory, volatility in prices is considered to be synonymous with risk. The rationale— which is ultimately derived from the assumption that the system being studied is a perfect market—is that all other information, including information about “business risk”, must already been accounted for in the current stock price.
Importantly, one of the implications of classical portfolio theory is that “the only way to obtain higher (expected) returns is to accept an increased amount of risk (volatility)”. Without recapitulating the argument in detail, its basis is arbitrage - for example, if two stocks had identical expected returns but different levels of risk (volatility) investors would prefer to hold that stock rather than its more volatile partner, tend to buy the one and sell the other, and thus cause their prices (and expected future returns) to re-adjust.
The conclusion that higher (expected) returns require accepting additional risk is highly intuitive and, indeed, one of the original benefits of classical portfolio theory was that it provided a mathematical justification to support this intuition.
2.3. Behavioral Finance Critique
Proponents of behavioral finance argue that changes in stock prices are predictable, at least in some cases. As an example, they argue that when a fundamentally sound company such as Johnson & Johnson (JNJ) suffers temporary reversals in the performance of its businesses investors will engage in a predictable over-reaction and sell the stock en masse. One proposed explanation for this behavior is that this selling is an evolutionarily-selected behavior—our surviving ancestors reacted to the “bad news” that the sound in the bushes might be a lion by running first and asking questions later. Another proposed explanation is that the hedge fund managers and other large speculators that set stock prices in the short run have financial incentives to engage in herding behavior. Regardless of the explanation, the selling causes the price of JNJ’s stock to become temporarily underpriced, eventually leading to a (somewhat) predictable counter-reaction whereby the returns for this stock are better than would be otherwise anticipated. The strategy of “value investing” attempts to take advantage of this putative predictability in market behavior.
Value investing is but one example of the sort of predictions that are generated by the conceptual model(s) underlying behavioral finance. To modify the above model to take into account value investing requires adding complexity to its structure.
2.4. A Revised Model for “Safe Haven Companies”
The debate between supporters of modern portfolio theory and those of value investing essentially amounts to one between nihilists and fundamentalists. The assertion that all available information is contained in the current stock price is nihilistic—as it implies that for the purposes of predicting future stock prices the fundamental elements of the company (e.g., its products, its business model, its management) don’t matter. Fundamentalists argue the reverse—not only does this information matter but, although it might be ignored by the market in the short-term, truth about long-term corporate performance will eventually win out.
There is one point about which these two schools of thought agree—namely, that stocks of “safe-haven” companies such as Johnson & Johnson (JNJ) have lower risk than average. The nihilists base their opinion on the low level of volatility in their stock prices. Fundamentalists base their opinion on the fact that these are large, wellfinanced companies that offer products that consumers must continually repurchase (e.g., band aids), and thus they have a lower level of “business risk” than average.
In practice, the prices of safe-haven companies evidence a stronger tendency to regress towards the mean than do most others. One explanation for this phenomenon is that, because their stream of earnings and dividends is so predictable, their “intrinsic value” can be estimated with a much higher degree of accuracy than is the case for other companies. Indeed, it can be postulated that the prices of safe haven stocks most often depart from this intrinsic value because of the view that large institutional investors have about the other stocks in the market. When these investors are feeling speculative, they sell the safe haven stocks in order to raise money to buy other companies. When these investors are stricken with panic, they do the opposite. This amounts to a tendency for historically low prices to regress toward their mean—a trend that becomes increasingly strong as the price diverges from its intrinsic value.
Accordingly, a model of the behavior of the prices of safe-haven stocks can be specified as follows: 
, where α > 0 and T represents the company’s “intrinsic value”. When the price at time i - 1 is less than intrinsic value, 
will exceed 0, and thus its impact will be to push the next price toward intrinsic value.
To comment on the process of model formulation, although the “intrinsic value” T for a safe haven company can’t be observed, it is a quantity about which most investors would roughly agree—thus, the intention is that T quantifies one of the fundamental properties of the system under study. The term α is intended to represent another such fundamental property; namely, the tendency for (some) stock prices to regress toward a (generally understood) target value. T and α represent an incremental increase in the complexity of the original model; moreover, when α = 0 the original model is recovered.
2.5. A Value Investment Strategy
The goal of “safe-haven value investors” is to buy safehaven companies at prices below their intrinsic value, to hold these companies until their prices exceed their intrinsic values, and thus to obtain superior risk-adjusted returns. Nihilists deny that it is possible for any strategy to obtain superior risk-adjusted returns.
2.6. Design of an Experiment
On first examination, an empirical test of the “value investment hypothesis” should be straightforward. For example, suppose that the basic premise of value investing is operationalized as follows: The value investor is seeking mature, fundamentally sound companies whose stock price has suffered a temporary reversal. Many mature companies pay dividends, and when the stock price falls the dividend yield (i.e., the dividend divided by the stock price) rises. Thus, stocks with (temporarily) high dividend yields should out-perform the market (after appropriate adjustment for risk).
The steps in the algorithm to test this hypothesis are as follows:
1) Define the strategy in sufficient detail to be implemented by computer (e.g., select all stocks whose current dividend yield is at least 5%.
2) Using a database, select (and pretend to buy) all stocks with the desired characteristics as of a certain date such as 1/1/2000.
3) Hold the stocks for a specified period such as 1 year.
4) Using the same database, pretend to sell these stocks as of a certain date such as 12/31/2000.
5) Based on the difference between the purchase and sales prices (plus any dividends received), calculate a hypothetical rate of return for the time period in question.
6) Repeat the process for multiple time periods (e.g., 1/1/2001-12/31/2001, 1/1/2002-12/31/2002, etc.).
7) Using the data from all the time periods, calculate an estimated rate of return, plus a standard deviation.
8) Use this latter standard deviation as an input to riskadjust the estimated rate of return 9) Compare the risk-adjusted return of the strategy with that of the overall market. If the strategy being tested outperforms the market the theory is supported—otherwise, not.
To date, the experiments that have tested the value investment hypothesis have all taken this form (albeit with different criteria for operationalizing the value investment strategy). The results haven’t been definitive. In general, and consistent with a broader literature on behavioral finance, such value-based strategies do tend to outperform the market, but by only a modest amount. This relative outperformance supports the positions of both parties. The fundamentalists point to the outperformance and, indeed, an entire industry has developed around the idea of behavioral-financed-based investment. The nihilists point to the modest levels of outperformance as evidence of how difficult it is to “beat the market”, and hold out the possibility that more sophisticated methods of risk-adjustment would cause the apparent outperformance to disappear altogether.
2.7. Current Critique of the Critical Experiment
It turns out that the above experiment, although easy to describe and perform, is difficult to interpret. This critique isn’t central to the present argument, and thus is relegated to Appendix 1. For now, we will stipulate that the experiment can be performed and interpreted as intended.
2.8. A New Critique of the Critical Experiment
Perusal of Appendix 1 demonstrates that the investment literature contains criticisms of every step in the above algorithm with the exception of the innocent-sounding step 3 (hold the stocks for a specified period such as a year). Apart from being intuitively natural, the original motivation for fixing the holding period was based on economic theory. Specifically, stocks are often held until a specific date—for example, an “investment horizon” might close on the date of a scheduled retirement, on the date that the investor expects to enter college, etc. Much of investment theory begins by fixing the investment horizon.
However, in actual practice investors seldom buy and sell stocks on pre-specified dates. They try to time their buying in order to obtain bargains, and they try to wait until the stocks they hold become over-valued before they sell. If they succeed in “timing the market”, their returns will exceed those of the strategy of buying and selling at pre-specified dates. Indeed, the greater the volatility in the stocks that they hold, the greater is the potential increase in returns associated with buying and selling opportunistically, and the greater is the loss in statistical power associated with performing assessments in the usual fashion.
This inconsistency between the behavior of actual investors and the assumption that investors buy and sell at pre-specified time periods (i.e., an assumption of the test of the models, albeit not of the models themselves) turns out to have unexpected consequences.
3. Methods
3.1. A Simulation-Based Test of a Strategy for Safe Haven Stocks
To design a test of the value investment hypothesis, assume that the investor has a relatively long investment horizon such as 10 years, each of which consists of 250 trading days. On trading day 0, the investor purchases a stock at its intrinsic value of $100. The holding period is indefinite—the investor will sell the stock when its return is R. For example, if R = 4% the investor will sell the first time the stock price reaches $104. Once the stock is sold, another stock with similar characteristics is bought and the process is repeated. At the end of the investment horizon, the stock that the investor owns is sold, regardless of price.
Importantly, this test embeds within it one of the primary assumptions held by the nihilists; namely, that investors are unable to consistently identify those situations where the price of a stock is less than its intrinsic value. Instead, what is being assumed is that investors are merely no worse than average in selecting among safe-haven companies. Accordingly, what is being tested is a strategy that is weaker than what value investors are actually attempting to accomplish.
3.2. Expected Returns for the Strategy
The intrinsic value of actual safe-haven companies is expected to increase over time, due to the compounded growth of their earnings. Here, for simplicity, we have assumed that the intrinsic value remains constant. Accordingly, if the nihilists are correct the expected return of this strategy, estimated by following the returns of a hypothetical large cohort of investors each having an investment horizon of 10 years, should be 0.
On the other hand, if the observed return exceeds 0, we will have demonstrated that the returns of safe-haven companies can be enhanced by taking advantage of the stochastic nature of their prices. This enhancement will only be observed for the minority of stocks demonstrating mean-reversion. Accordingly, the returns of the safehaven companies will exceed those of some of the other companies in the market with higher levels of risk. In the limited sense defined above, this will provide a counterexample to the investment maxim that in order to obtain increased gains investors must always accept increased risk.
4. Results
Table 1 presents results under the following scenarios. The investment is purchased at the intrinsic value of $100. The distribution of the errors is Gaussian with mean 0 and standard deviation 1. The parameter representing regression toward the mean, α, is 0.01. The investment horizon is 10 years (2500 trading days). If the price exceeds the threshold value, a profit is taken and the simulation is re-set—that is, another stock is purchased at $100. On day 2500 the stock is sold. For example, setting the threshold value to $101, on average the value of the investment at day 2500 is $224, for an absolute return of $124 (i.e., $224 minus the original $100).