The Search For The Rational Investor
Thomas E. Berghage
Before we set out on our search for a truly rational investor, we think a few comments are order regarding human investment decision-making. The behavioral finance people have gone to great length to discredit human decision-makers. They provide study after study that demonstrates irrational behavior when humans are presented with decisions involving uncertainty. Humans seem to be swayed in their decision making by a number of extraneous factors such as the probability of gain or loss, the context in which the information is presented, prior decisions, and a host of other irrelevant information. The suggestion is that humans are so poor at mental accounting that it is virtually impossible for them to make rational decisions.
Well, there is another way of looking at these results and it puts the burden on the system designer rather than the poor human decision maker. This alternative view suggests that if a system is properly designed to deliver appropriate information to the human decision maker, rational decisions can be made. The problem is that none of our current security analysis techniques or systems are capable of digesting the available information and presenting it to us in a form that overcomes the weaknesses of human decision-making. Perhaps we should be spending more time on designing and building analysis systems that compensate for human idiosyncrasies and allow humans to make rational decisions.
In our last book (Berghage and Berghage, 2002) we introduced the idea of using the theory of Signal Detection (TSD)in the evaluation of investment system performance. Originally developed for use in perceptual psychology by Swets, Tanner, and Birdsall (1961) and Green (1960) the theory was intended to characterize not only the acuity of an individual’s discrimination but also the psychological factors that bias the individual’s judgments. The same rational can be applied to security analysis and selection. The investment process involves first, being able to discriminate among good and poor performing investment opportunities and second, avoiding those preconceived, unsubstantiated, biases dealt with in Behavioral Finance, that interfere with rational decision making. I believe that with minor modifications TSD has wide application in the financial field and can be used to quantify the performance of financial analysts, whether they are human or machine. The theory can also be used to mathematically describe the rational investor, the concept that is so fundamental to current financial theory.
The Theory Of Signal Detection (TSD)
The theory of signal detection has two parts of quite different origins. The first comes from mathematical statistics and is a translation of the theory of statistical decisions. The major contribution of this part of the theory is that it permits a determination of the individual’s discriminative capacity, or sensitivity, that is independent of the judgmental bias or decision criterion the individual may have had when the discrimination was made. The second part of the theory comes from the study of electronic communications. It provides a means of calculating for simple signals, such as tones and lights, the best discrimination that can be attained. The prediction is based upon physical measurements of the signals and their interfering noise. The opportunity to compare the sensitivity of human observers with the sensitivity of an “ideal observer” for a variety of signals is of considerable usefulness and of growing interest, in sensory psychology.
Signal detection theory has been applied to several topics in experimental psychology in which separation of intrinsic discriminability from decision factors is desirable. Included are attention, imagery, learning, conceptual judgment, personality, reaction time, manual control, and speech.
The analytical apparatus of the theory has been of value in the evaluation of the performance of systems that make decisions based on uncertain information. Such systems may involve only people, or people and machines together, or only machines. Examples come from medical diagnosis on a physical examination, or on an x-ray image, or where machines make diagnoses, perhaps by counting blood cells of various types. To our knowledge the application of Signal Detection Theory in security analysis has not been attempted, but its relevance seems obvious.
Basic Principals of TSD Related To Security Analysis
Decision Theory involves four main concepts: (1) evaluation, (2) likelihood ratio, (3) decision rule, and (4) criterion.
Evaluation: Refers to the analyst’s assessment of the financial and operational performance of a corporation in an attempt to select companies whose equity (stock) will perform well (appreciate over time) in the security markets. It is assumed that the valuation results in a continuous variable that ranges from extremely unattractive to extremely attractive.
It is assumed that numerous companies will be subjected to the evaluation process, some of which will actually perform well in the markets over time (SN, Signal plus Noise), and some that will not (N, Noise). The distributions associated with these two conditions are usually designated as fSN(x), which represents the probability density function of x given the occurrence of signal plus noise and fN(x), which represents the probability density function of x given the occurrence of noise alone.
The distributions for the two conditions are shown in Figure 9-1. The evaluation continuum (x) is shown as the abscissa. The amount of separation between the two distribution means will depend on the strength of the signal and the ability of the analyst’s evaluation to detect the signal and discriminate between SN and N. Now, given a company evaluation (x), the problem becomes one of deciding whether the company comes from distribution fN(x) or distribution fSN(x). For every evaluation, there is some probability density that it resulted from noise alone and similarly, some probability density that it was due to signal plus noise. These two density functions allow us to define a new quantity, the likelihood ratio, l(x) .
fSN(x)
fN(x)
Likelihood Ratio: The likelihood ratio, l(x), expresses the likelihood (probability density) that the company evaluation (x) arose from SN (signal plus noise) relative to the likelihood that it arose from N (noise alone). The likelihood ratio is a number, not a probability.
Decision Rule: The decision rule is a formal statement that indicates the basis on which the analyst should make his decision about a particular company. In terms of signal detection theory, the rule can be stated as follows: Choose SN if l(x) ³ 1. This statement defines the decision rule and specifies the choice to be made by selecting or designating a particular number; in this case, number “one.” This number is called the likelihood ratio criterion, or simply the criterion.
Criterion: The criterion should be appropriate to the circumstances, but is generally set based on the profit/loss associated with the four possible outcomes (Table 9-1). The payout matrix is used along with the a priori probabilities to define the rational investor, a concept we will define here shortly. In the original Theory of Signal Detection the criterion was referred to as Beta (b), but because of the confusion this may cause in financial cycles we prefer to call the criterion Psi (y). It seems appropriate in that it describes the response bias of the analyst.
The basic idea in the Theory of Signal Detection is that all signals are imbedded in a background of noise and it is the job of the observer/analyst to determine when he/she is confronted by noise alone (N) and when there is a signal present (SN). To translate this into our financial analysis problem, the analyst (observer) must determine if the company he/she is looking at will appreciate in value (SN) over some given time period, i.e. 3,6,12,24, or 36 months or is the company just part of the background economic noise (N). To accomplish this task and make the decision necessary analysts use a number of different techniques, some of which are useful and some of which are unfortunately are just part of the noise. Regardless of whether the analysts quantify their techniques or just act on their gut reaction they end up with a continuum such as that shown in Figure 9-1, stretching from very negative to very positive. On this continuum you can plot two frequency distributions, one for those companies that actually appreciated over the selected time frame (SN) and one for those companies that failed to produce a positive return (N). The degree of separation between these two distributions (d’) is a good measure of the effectiveness of the analyst’ discriminative capability. A big overlap between these two distributions indicates a problem, and the need to take a long hard look at the analysis process and perhaps explore alternative approaches that will further separate the distributions and increase the size of the mean difference. The goal of the analysis process is to increase the size of d’, our measure of discriminability.
Figure 9-1
Probability Density Functions Of
Noise and Signal Plus Noise
As you can see from Figure 9-1, d’ is the difference between the means of the two density functions expressed in terms of their standard deviations. Those readers that remember their Statistics 101 class will recognize the following equation as a t-test between two sample means.
d’ = MfSN - MfN
sD
d’ = Measure of discriminability
MfSN = Mean of the Signal plus Noise Distribution
MfN = Mean of the Noise Distribution
sD = Square Root of(s21 + s22)
Another way of understanding the results shown in Figure 9-1 is to tabulate the figures in a 2 X 2 matrix (Table 9-1). The binary decision task of whether to buy or not buy a stock produces four possible outcomes, (1) a “Hit” - when the analyst recommends buying the stock and it produced a positive return (Y/SN), (2) a “Miss” - where the analyst passed on the stock and it produced a positive return (N/SN), (3) a “Correct Rejection” – where the analyst passed on the stock and it did not produce a positive return (N/N), and (4) a “False Alarm” – when the analyst recommends buying the stock and it fails to produce a positive return (Y/N). These four outcomes are shown in Table 9-1. The two probability density functions that are used to calculate d’ are P(Y/N) and P(Y/SN). The other two cells in the matrix are important in evaluating the response bias of the analyst.
Table 9-1
Matrix of Response Alternatives
Actual Stock Performance
|
|
|
Negative Return (N) |
Positive Return (SN) |
|
Analyst |
Buy (Y) |
Y/N “False Alarm”
|
Y/SN “Hit” |
|
Forecast |
Pass (N) |
N/N “Correction Rejection |
N/SN “Miss” |
There are two probability relationships that occur within this matrix that are independent of each other, P(Y/N)+P(N/N)=1.0 and P(Y/SN)+P(N/SN)=1.0. These two relationships always hold true even though the probabilities in the four cells can be adjusted depending on how willing the analyst is to make an error (how high an analyst sets his/her criterion for saying “Buy”). There is an optimum criterion to use and it depends on the payouts associated with the cells in the matrix. In the original Signal Detection Theory the optimum criterion was called the “Ideal Observer” and as we will see in the next section the optimum criterion in financial circles is called the “Rational Investor.”
The traditional way of looking at the observer (analyst) response bias (y) is with a plot similar to that shown in Figure 9-2. In perceptual psychology it is generally referred to as a Receiver Operating Characteristic (ROC) curve and shows the willingness of the operator (analyst) to accept additional “False Alarms” in order to increase the number of “Hits.” The diagonal line in Figure 9-2 that goes from the bottom left hand corner to the top right hand corner is for responses that are due to chance when the chance of a “Hit” or “False Alarm” are equally probable. As I pointed out in Chapter 2, the chance probability of a “Hit” differs for various sectors of the market. It also changes over time depending on economic conditions. For the market as a whole during the years 1998 – 2000 the chance probability of a “Hit” was approximately 51.6%. The changing probability of a “Hit” means that the diagonal line describing chance responses will move either to the right or left on the graph depending on the a priori probability of the event occurring (the inherent risk). This shifting diagonal chance-line is important because it is the bases for evaluating performance over chance levels. As we will find in the next section it has an impact on the response bias of the “Rational Investor.”
Figure 9-2
Analyst Operating Characteristic (AOC) Curve*
* In TSD this figure is called the ROC curve for Receiver Operating Characteristic. These curves depict the willingness of a decision maker to accept more false alarms (errors) to increase the number of hits (the y criterion being used by the analyst). The relationship between hits and false alarms is dependent on the ability of the decision maker to discriminate between the two conditions, signal plus noise, and noise alone.
The Capital Asset Pricing Model (CAPM)
As discussed in Chapter 7, the Capital Asset Pricing Model (CAPM) is based upon a number of assumptions such as risk aversion; identical time horizons and expectations of all investors with respect to each financial asset; identical borrowing and lending rates; neither taxes nor transaction costs; and, finally, rational investors who seek to hold efficient portfolios. Most of these assumptions are understandable and had to be made because economic and investment conditions vary so widely among investors. The ideas of “risk aversion” and “rational investors” are behavioral assumptions that can be dealt with by reducing the influence of the human element in the investment process.
The assumption of risk aversion seems quite reasonable. The extraordinary general tendency of investors, especially institutional investors, to hold portfolios of assets rather than a single asset with the greatest expected return is prima facie evidence of risk aversion. Although there may be a few investors who enjoy and value risk per se, it is difficult to believe that they are numerous or important in understanding the investment process. Certainly, financial institutions, which account for most of the activity in the securities markets, are not managing investments on behalf of their beneficiaries or clients for the fun of taking chances. The assumption of risk aversion seems well founded.
The “rational investor” assumption however is not on such solid ground. In fact, the Behavioral Finance community has done a pretty good job of destroying the concept altogether. As the research in the previous chapter outlines, the human investor, regardless if they are investing for themselves or an institution often times act in an irrational manner. Without this important assumption of rational investing the Capital Asset Pricing Model (CAPM) loses some of its relevance. The CAPM is based on sound logic and good mathematics so anything we can do to shore up its effectiveness and make it a useful tool is a step in the right direction. Toward this end let’s add the Theory of Signal Detection to the Capital Asset Pricing Model and use the mathematical description of the Ideal Observer to describe the Rational Investor.
Alignment Of Two Theories
The decision made by an analyst will be affected by several factors:
The analyst uses information about these factors to create a dichotomy between those companies he reports as “Buys” and those he recommends passing on. The dichotomy is determined by a criterion (y) that can be adjusted by the analyst. Through an appropriate analysis of the data, a quantitative estimate of this criterion can be obtained (to view the effect of changing these parameters the reader is directed to the web page at http://psychlab1.hanover.edu. The Rational Investor is simply a mathematical expression, which specifies the maximal performance that may be obtained in a particular situation.
Assuming that an analyst has some type of evaluation system that produces a continuum, from bad to good, for selecting stocks and has some feel for the distributions described in Figure 9-1, he/she still has to establish a criterion (y) or cutoff point along this continuum that separates his “Buys” from his “Don’t Buys.” This criterion point establishes the probabilities of a “Hit” and a “False Alarm” described in Figure 9-2. There is an optimum place for this criterion point that would be used by the totally Rational Investor. This point is mathematically defined by the a priori probabilities for the occurrence of signal plus noise P(SN),and noise alone P(N) in Table 9-1, along with the profit and loss associated with each of the four response alternatives. The Rational Investor’s criterion point is defined as:
P(N)(VN.N + KY.N)
L =
P(SN)(VY.SN + KN.SN)
Where:
P(N) is the a priori probability of noise alone,
P(SN) is the a priori probability of signal plus
noise,
VN.N is the value or gain of a correction
rejection,
VY.SN is the value of a correct decision (a
“Hit”),
KY.N is the cost of a false alarm, and
KN.SN is the opportunity cost of a miss
* Remember, we are using y for what the psychology community calls b.
The investment universe being used establishes the a priori probabilities and the values and costs in a pay-off matrix weight the impact of these probabilities. The values in the pay-off matrix change as the criterion (y) changes
By using TSD we can quantify the Rational Investor and evaluate the characteristics of a security analysis system regardless of whether it is human or machine. We can separate the impact of the system’s sensitivity and its ability to discriminate from any response bias that exists within the system. With a mathematical description of the Rational Investor it is now possible to design and build non-human security analysis systems that are truly rational. The next step is enhancing the ability of analysis systems to discriminate between good and poor performing securities. The remaining chapters will focus on this task.