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The personalization of security selection: An application of fuzzy set theory
Fuzzy Sets and Systems 5 (1981) 1-9 © North-Holland Publishing Company
THE P E R S O N A L I Z A T I O N OF S E C U R I T Y SELECTION: A N A P P L I C A T I O N OF F U Z Z Y SET T H E O R Y Irene M. H A M M E R B A C H E R and Ronald R. Y A G E R Iona College, New Rochelle, N.Y. 10801, U.S.A. Received May 1978 Revised September 1979 We apply the theory of fuzzy subsets to the multiple objective decision problem of stock selection. We allow our objectives to have varying degrees of importance. We discuss various criteria used it, selecting stocks. We indicate some procedures for subjectively evaluating the membership functions associated with these criteria.
Keywords: Multiple objective decisions, Financial analysis, Subjective assessments, Security analysis, Fuzzy sets.
1. Fuzzy set theory In a series of articles beginning in 1965, Zadeh [ 1, 9, 10, 11, 12] has presented the concept of fuzzy sets which enables one to deal with ideas that are imprecise and/or subjective. In this section, we shall present those concepts of fuzzy set theory which are relevant to the investor's decision-making problem. Assume we have a set of alternative stocks, X, and a group of objectives Ct, C 2 , . . . , Cp. We can use the idea of a fuzzy set to define these objectives in terms of the alternatives. That is, each objective can be represented as a fuzzy subset over the set X. In this representation the grade of membership of any x ~ X in Ci indicates the degree to which x satisfies the condition specified by objective Ci. This grade of membership is necessarily a subjective evaluation by the decision maker. A procedure for evaluation has been suggested by Saaty [5]. Having represented the objectives as fuzzy subsets the next question arises as to the combining of the objectives into a decision function D. If a decision maker states that what he wants is an alternative that best satisfies Ct and C2 and . . . and Cp, Bellman and Zadeh [1] have suggested that what the decision maker is doing is 'anding' his objectives. Using the linguistic characterizations available in fuzzy sets we can express this formulation as D. That is, using n to indicate 'and' our decision function becomes: D =
n c2 n
n . . . n cp,
where the C~ are the fuzzy sets representing the objectives. The question of which form of fuzzy intersection to use to represent 'and' is much discussed in the literature [7]. The choice of this operation depends upon
2
I.M. Hammerbacher, R.R. Yager
our interpretation of 'and'. We shall in this paper follow Zadeh's premise that lacking further information, the choice for 'and' is the min operation [ 1, 10]. That is, D = q n Ca o (73 o - • • n cp is defined as:
D(x)=min{q(x), C 2 ( x ) , . . . , Cp(x)}. The decision membership function D(x) can be interpreted as the degree to which each alternative satisfies the overall set of objectives. The optimal alternative is selected as the alternative that has the highest degree of membership in D. In order for a decision process to truly reflect what the investor considers desirable, some methodology should be included to incorporate a procedure for allowing an investor to input his subjective evaluations of the importance of each of the objectives. Yager [8] has presented a methodology for including these values in a hierarchical fashion. This involves selecting for each objective a number a 1>0 indicative of its importance. The higher the a, the more important the objective. Then each objective is raised to a power, which is its importance. A decision function D is now formulated as follows:
D= cT, nq
n-., nc2.
The c~'s are determined subjectively, in a procedure which incorporates the tradeoffs the decision maker is willing to accept between his objectives. This procedure has the effect of making bad alternatives in important objectives appear much worse. It should be noted that if A is a fuzzy subset with membership function A(x), the A ~ is always a fuzzy subset with membership function (A(x)) ~. Saaty [4, 5, 6] has developed a procedure for obtaining a ratio scale for a group of p elements based upon paired comparisons. This procedure has been used by Yager [8] to elicit from a decision maker the importances to be used in our model.
2. An example Consider a potential investor who describes her investment preferences for common stocks in the following manner: (1) she is seeking a high return in the form of capital gains; dividend income is irrelevant to her; (2) she would like to protect her invested capital, but recognizes that one must accept some risk in order to obtain a 'high return'; (3) she would rather own stocks whose prices display minimum short-term fluctuations; (4) she recognizes that the correlation between stock price and company earnings is observable but not statistically large and therefore she would prefer companies whose sales and earnings have exhibited steady long term growth to companies which have grown erratically. She believes that each stock she purchases should be valued individually and re-evaluated approximately every 6 months. She uses other investment media (gold, bonds, and an occasional fling in commodities futures) and considers these
The personalization of security selection
3
four to be the components of her portfolio. Presently, she is considering adding to her common stock holdings, and is searching for a security which appears to be the most satisfactory for her. Her stated preferences cannot be measured directly, and available variables must be used as proxies for them. In some instances, more than one available variable would be an appropriate indicator that the stock would or would not satisfy her preferences. For instance, the search for 'capital gains' requires present and future data, so the appropriate proxy variable could be any one of a dozen. To name just a few of the variables which could be considered omens of capital gains to come, one could use: the price-earnings multiple, on the theory that low multiple stocks are the ones which have historically increased the most in price; or the rate of return on assets, on the theory that substantial profitability induces marketability, which produces capital gains; or the rate of growth of earnings, on the theory that a high rate of growth should continue for a sound company, other things remaining constant, and that this growth should drive up the stock price; or an index of advisory recommendations to buy the stock, on the theory that an increase in such recommendations should increase the demand for the stock and therefore raise its price. As another example, when the investor specifies 'minimum short term price fluctuations', she could determine whether or not a stock satisfied that description by examining any of the following variables: Beta, a measure of stock price volatility relative to the general market; or the coefficient of variation of the stock price; or a measure of the frequency with which a company appears on the list of most active stocks; or some measure of the range of a stock's price over some approp~mte time period. Whichever' preference is being considered, no one variable has tested out satisfactorily as a reliable indicator, for all (or even most) stocks over multiple time periods. The literature of finance and the performance of professional investors has borne this out for decades. Therefore, a selection of variables that the investor considers appropriate, coupled with her assessment of what a 'good' value for each variable might be, constitutes the investor's objectives. The fact that these objectives may be uncertain, subjective, imprecise, and/or may be subject to change over time does not preclude their use in our rational decision-making process. An objective is uncertain when we cannot obtain a guaranteed forecast of a known, precise variable. All that is necessary for fuzzy set theory to be applied to stock selection, however, is that we have the investor's forecast of that variable and his evaluation of that forecast as good or bad on a scale of 1 to 0. We have already indicated the degree of subjectivity that enters into the selection of a variable to satisfy the investor's preferences. We need only add, at this point, that it is also possible to allow for exceptions for particular companies in special circumstances. An investor might, for example, change his 1 to 0 evaluation of a particular stock's capital gains potential (whatever the variable he is using to indicate that potential), if rumors that the company is about to be taken over by another circulate. An objectivemsuch as 'high dividend yield'--is subject to change over time. While a 3% yield might be common at one point in time and therefore be
4
LM. Hammerbac~er, R.R. Yager
assigned a membership grade of 0.5, in a different set of economic circumstances a 6% dividend yield might only be graded 0.4. Fuzzy set theory provides the ideal medium in which to evaluate preferences which are precise or imprecise, su0jective or objective, certain or uncertain, and constant or subject to change. Let us proceed with an example of our application. Assume our investor has refined her preferences and established the following objectives: (1) a low price-earnings multiple; (2) short-term price stability; (3) a dividend yield which is either low or very high; (4) a low debt ratio; (5) a high short interest; (6) few recent insider sales of the stock. To avoid unduly time-consuming consideration of the five-thousand-plus common stocks on the market, our investor uses the traditional type of filtering system I to select a group of promising stocks for evaluation. Her filter eliminates the stocks of companies in unattractive industries, companies whose earnings growth has not been better than the rate of inflation (on the average) over the past 5 years, companies which showed losses in either of the last 2 years, companies for which inadequate data were available, and companies which she would not want to consider for any other personal reason whatever, e.g., companies doing business with South Africa. In essence, the filter is a fuzzy set with membership grade 1 or 0 over the set of all stocks. The determination of membership is subjectively evaluated by the investor, and is determined by the rule 'I do or do not want to consider this stock as a potential investment'. This filtering procedure leaves her with the set X of n possible alternative investments. The next step is to evaluate each of the objectives over X. That is, for each of the objectives, she creates a fuzzy subset over X, whose membership function indicates the degree to which a stock satisfies the objective. (1) Low price-earnings ratio. The ratio of stock price per share to earnings per share is a standard statistic associated with each stock. It is primarily a measure of the attractiveness of the stock, and a low pie for an investment-quality stock is desirable because it is a strong indicator of a potential capital gain. The problem, then, is for our decision-maker to subjectively evaluate the grade of membership (the degree of desirability) of the set 'a low pie'. Because pie can fluctuate slightly every time the stock is traded, one possible procedure for obtaining the membership grade could be to locate the most recent pie on the following graph and read off the membership grade from the Y-axis (Fig. 1). In effect, our hypothetical investor has decided that if a stock is selling for less than the earnings per share of the company, there is something wrong with it, and she d~esn't want to invest in it. Therefore, for pie less than one, the degree of membership is zero. She also believes that for most companies at the present time, a pie in the range of 6 to 10 is 'normal'. Therefore, stocks with low pie's i One could revert to Classical Investment Theory and use, for example, Nerlove's seven variables [2] or any three of Renwick's four variables [3] as the components of one's filter, but when tested, both provided lists of potential investments which were much too long to be practical.
The personalization o[ security selection
5
emberhi 1 pl
Degree of
0
t 6
i 10
pie
Fig. 1.
(less than 6) are underpriced and desirable; stocks with high p/e's are overpriced and undesirable. At this juncture, it is necessary to re-emphasize a previous point: in a different market, our investor might set up a different scale, and in any market, a different investor would set up a different scale. (2) Short term price stability. Beta, a measure of the stock's price varEation relative to the market's variation, has become a generally accepted statistic within the last 5 years. Our hypothetical investor considers B e t a = 1.00 as the ideal because B e t a < l . 0 0 indicates that the stock moves less than the markel and would preclude any substantial price appreciation, while Beta > 1.00 indicates that the stock moves more than the market. Our decision maker could subjectively evaluate the grade of membership in the set 'a Beta near 1.00' for specific stocks by using a transformation rule form Beta to [0, 1] as shown in Fig. 2. Again, the shape of this figure would change with the degree of an investor's risk aversion (or risk-preference) at a given point in time. (3) Low or very high dividend yield. A low dividend yield is an indication of the company's ability and/or willingness to grow internally. A high dividend yield is attractive to income seeking investors, and this basic source of demand for the stock will, in time, drive the price up and the yield back down. One possible procedure is to use the dividend yield in two fuzzy sets. In the first set, high-yielding stocks would be graded close to 0. In the second set, the reverse would be true. The decision maker would then incorporate both fuzzy sets into his decision. Another possible procedure is to recognize that a 5% dividend yield is currently the norm, and significant deviations from the norm in either direction are desirable. This would allow our investor to obtain membership grade by using a U-shaped curve as a transformation rule, where a 5% dividend yield is graded close to 0 and low and high yields are graded close to 1. We have assumed that our investor prefers this second procedure. (4) A low debt ratio. Since we have conjured up a conservative investor, we will Degree of membershi p |
1
Fig. 2.
Beta
6
LM. Hammerbacher, R.R. Yager
assume she is adverse to financial risk and will therefore attempt to avoid stock issued by heavily leveraged companies. If we define leverage as the r~tio of long term debt to total long term capital, we have a variable which ranges from 0 to 1. Since our investor is debt-averse, she will use the transformation 'membership grade = 1.0 - debt ratio'. (5) A high short interest. Short interest is a measure of the number of shares of stock that have been sold short on the market. Short selling itself is a bearish sign; one only sells short when one expects the price to move down in the near future. But every short sale has to be covered, i.e., the stock has to be bought back at a later date. Therefore, a large amount of short sales is a bullish sign, because a large number of shares will have to be bought in the near future. In order to get a comparative measure of short interest for different size companies, the short interest ratio has been constructed. This is the ratio of the number of shares of stock which were sold short to the average number of shares traded per day. For the vast majority of companies in a normal market, the short interest ratio is less than one. Only when the ratio gets significantly above one does it become a bullish signal. Therefore, a possible procedure for obtaining the membership grade could be to use the transformation rule which states: (1) if the short interest ratio is between 0 and 4.0, then the membership grade is 25% of the ratio; (2) if the short interest ratio is greater than 4.0, then the membership grade is 1.0. (6) Few recent insider sales. Our hypothetical investor is a very suspicious person. She believes, as does a sizable segment of the market, that when corporate insiders (officers or shareholders owning 10% or more of the outstanding stock) are buying the company's stock, they know something good that she doesn't know and the price of the stock is going to go up. When they sell the stock, they know something bad that she doesn't know, and the price is going to go down. The net change in the number of shares held by insiders is a readily available monthly statistic. Again, because of the varying sizes of companies, it must be converted into a ratie The ratio of insider net change (purchases less sales) per month to total outstanding shares is by nature very small. For the majority of companies, it is zero much of the time. It never reaches or even approaches plus or minus one, so a possible procedure for transforming insider trading to a fuzzy set might be: (1) if the ratio (T) is 0.5<~T~<1.0, then t'~c membership grade is 1.0; (2) if the ratio is - 0 . 5 <~ T ~<+0.5, then the membership grade is T + 0 . 5 ; (3) if the ratio is - 1.0~ < T~<-0.5, then the membership grade is 0. Having quantified our investor's objectives, and having created a fuzzy set for each, the next step is to determine the importance of the various objectives to her. The goal here is to obtain a scale which ranks the objectives as to their importance. First we construct a matrix of paired comparisons. The investor constructs the matrix by comparing, on a scale of 1 to 9, the ratio of importance of one objective to the next. In the matrix below, the ith j entry indicates the ratio of importance of objective i to objective/. The objectives are, as indicated: the price-earnings ratio (P), Beta (B), the Dividend yield (Y), the Debt ratio (D), the short interest ratio (S), and the insider trading ratio (T).
The personalization o[ security selection
Y D 7 5 7 5 1 1 ~1 ! 3 i 1 5 5 1 +~ 1 ~1
P P -1 B 1 Y I D 1 S i T_v
S 3 3 .~I s1 1 ~1
B 1 1
!
7
T 77 1 3 5 1_
This matrix indicates that, to our hypothetical investor, the price-earnings ratio and the Beta are equally important, but the price-earnings ratio is much more important than the dividend yield. Using a program which evaluates the maximum eigenvalue for a matrix, we get the following eigenvector representing our hypothetical investor's importance ratings: -1.8486 1.8486 0.2172 0.2172 1.2834 _0.5856
= ap= aB =av =aD = aS =aT-
Next, we selected a samp~,e of eight stocks 2 from our investor's filtered list, to complete the example in a reasonable amount of space. For each stock, we determined its grade of membership in each fuzzy set according to the procedures described above. Our fuzzy sets are:
{0.70
065 0+0 040 07° ,,6,, / 9
Xi
,
'
X2
~'3
'
X4
'
'
X5
'
X6
X7
'
X8
I
B = / 0"46, 0.50 0.38 0.46, 0 . 6 4 0.65 0.54 0.4__8/ [
X!
X2
X3
X4
X5
X6
X7
X8
)
y = { 0.03, 0.19 ,0.63 ,0.82 ,0.79 ,0.66 ,0.02 .0.79/__ x i
D
x2
x3
- -
[
s:{O
~
'J
X1
X2
X3
,
,
,
X2
x5
x6
x 7
x8
J"
0.86 0.55 0.50 0.72 0.82 0.85 /
0.61,0.56 ~
x4
X3
,
X4
,
'~
X5
,
X6
X7
0,3 00 0 0+ 0,s i ,
X4
,
X5
,
X6
,
X7
,
X8
J
,
X8
T = / 0 . 4 0 , 0 + 6 0 0.50 0.40 0.58 0.40 0.60 0.40} ,
I, X 1
X2
,
X3
,
X4
,
X5
,
X6
,
X7
•
X8
Our decision set now becomes: D = P~PN B~,N Y ~ r i D % n s~s NT ~+ 2 The companies are Bausch and Lomb, Allegheny Ludlum, Coca Cola, W.R. Grace, White Consolidated Industries, Southland Corp., Mobil Oil, and Eltra Corp., respectively. The data have been obtained from the Value Line Investment Survey and Media General's Financial Weekly.
I.M. Hammerbacher, R.R. Yager
8
where W
p.-..
Q
B y D
Ot B
.-.
oty
---
¢I¢D
~..
S T
~S
Ot T
--.
{0.70, 0.55, 0.20, 0.65, 0.80, 0.40, 0.70, 0.60} 1"8486 {0.517, 0.331, 0.051, 0.451, 0.662, 0.184, 0.517, 0.389}, {0.46, 0.50, 0.38, 0.46, 0.64, 0.65, 0.54, 0.48}1"s486 {0.238, 0.278, 0.167, 0.238, 0.438, 0.451, 0.320, 0.257}, {0.03, 0.19, 0.63, 0.82, 0.79, 0.66, 0.02, 0.79}°'2172 {0.467, 0.697, 0.905, 0.958, 0.950, 0.914, 0.428, 0.950}, {0.61, 0.56, 0.86, 0.55, 0.50, 0.72, 0.82, 0.85}°'2172 {0.898, 0.882, 0.968, 0.878, 0.860, 0.931, 0.9588, 0.965}, {0, 0.13, 0.20, 0.13, 0.05, 0, 0.85, 0 . 1 8 } 1"2834 {0, 0.073, 0.128, 0.073, 0.021, 0, 0.812, 0.110}, {0.40, 0.60, 0.50, 0.40, 0.50, 0.40, 0.60, 0 . 4 0 } 0.5856 {0.585, 0.741, 0.666, 0.585, 0.666, 0.585, 0.741, 0.585}.
Thus, we determine that: D={0, 0.073, 0.051, 0.073, 0.021, 0, 0.184, 0.110}. It is noted that the elements of D are the price-earnings ratio and the short interest ratio, which is to be expected since, for our investor, these were two of the three most important variables. She would then select that security which best satisfies D, which is x7 in this case.
3. Conclusion
Using fuzzy sets to select securities is a procedure which permits investors to logically and consistently choose the most appropriate investment for themselves by including their muh,ple objectives and their varying perceived importances. The traditional stock valuation models attempt to help a generalized investor to select the 'best' stocks in a rational market, but our approach has been far more personalized than this. The application of fuzzy set theory stresses the uniqueness of the investor and his perceptions, and provides a tool which will help him to cope with the decision making process.
References [1] R.E. Bellman, and L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (1970) B141-B164. [2] M. Nerlove, Factors affecting differences among rates of return on investments in individual common stocks, Rev. Econ. Statist. 50 (1968) 122-136. [3] F.B. Renwick, Asset management and investor protfolio behavior: theory and practice, J. Finance 24 (1969) 181-206.
The personalization of security selection
9
[4] T.L. Saaty, A scaling method for priorities in hierarchical structure, J. Math. Psychoi. (1977) 234-281.
[5] T.L. Saaty, Exploring the interface between hierarchies, multiple objectives, and fuzzy sets, Fuzzy Sets and Systems 1 (1978) 45-66.
[6] T.L. Saaty and M. Khouja, A measure of world influence, J. Peace Sci. 2 (1976) 31-48. [7] U. Thole, H.J. Zimmermann and P. Zysno, On the suitability of minimum and product operators for the intersection of fuzzy sets, Fuzzy Sets and Systems 2 (1979) 167-180.
[8] R.R. Yager, Multiple objective decision making using fuzzy sets, Int. J. Man-Machine Studies (1977).
[9] L.A. Zadeh, ~ uzzy sets, Information and Control 8 (1965) 338-353. [10] L.A. Zadeb, Dutline of a new approach to analysis of complex systems and decision processes, IEEE Tranz Systems, Man and Cybernet. SMC-3 (1973) 28-44.
[11] L.A. Zadeh, Calculus of fuzzy restrictions, in: L.A. Zadeh, K. S Fu, K. Tanaka and M. Shimura, [12]
eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (Academic Press, New York 1975) pp. 1-39. L.A. Zadeh, Theory of fuzzy sets, in: J. Belzer, A. Holzman and A. Kent, eds., Encyclopedia of Computer Science and Technology (Marcel Dekker, New York, 1977).
THE P E R S O N A L I Z A T I O N OF S E C U R I T Y SELECTION: A N A P P L I C A T I O N OF F U Z Z Y SET T H E O R Y Irene M. H A M M E R B A C H E R and Ronald R. Y A G E R Iona College, New Rochelle, N.Y. 10801, U.S.A. Received May 1978 Revised September 1979 We apply the theory of fuzzy subsets to the multiple objective decision problem of stock selection. We allow our objectives to have varying degrees of importance. We discuss various criteria used it, selecting stocks. We indicate some procedures for subjectively evaluating the membership functions associated with these criteria.
Keywords: Multiple objective decisions, Financial analysis, Subjective assessments, Security analysis, Fuzzy sets.
1. Fuzzy set theory In a series of articles beginning in 1965, Zadeh [ 1, 9, 10, 11, 12] has presented the concept of fuzzy sets which enables one to deal with ideas that are imprecise and/or subjective. In this section, we shall present those concepts of fuzzy set theory which are relevant to the investor's decision-making problem. Assume we have a set of alternative stocks, X, and a group of objectives Ct, C 2 , . . . , Cp. We can use the idea of a fuzzy set to define these objectives in terms of the alternatives. That is, each objective can be represented as a fuzzy subset over the set X. In this representation the grade of membership of any x ~ X in Ci indicates the degree to which x satisfies the condition specified by objective Ci. This grade of membership is necessarily a subjective evaluation by the decision maker. A procedure for evaluation has been suggested by Saaty [5]. Having represented the objectives as fuzzy subsets the next question arises as to the combining of the objectives into a decision function D. If a decision maker states that what he wants is an alternative that best satisfies Ct and C2 and . . . and Cp, Bellman and Zadeh [1] have suggested that what the decision maker is doing is 'anding' his objectives. Using the linguistic characterizations available in fuzzy sets we can express this formulation as D. That is, using n to indicate 'and' our decision function becomes: D =
n c2 n
n . . . n cp,
where the C~ are the fuzzy sets representing the objectives. The question of which form of fuzzy intersection to use to represent 'and' is much discussed in the literature [7]. The choice of this operation depends upon
2
I.M. Hammerbacher, R.R. Yager
our interpretation of 'and'. We shall in this paper follow Zadeh's premise that lacking further information, the choice for 'and' is the min operation [ 1, 10]. That is, D = q n Ca o (73 o - • • n cp is defined as:
D(x)=min{q(x), C 2 ( x ) , . . . , Cp(x)}. The decision membership function D(x) can be interpreted as the degree to which each alternative satisfies the overall set of objectives. The optimal alternative is selected as the alternative that has the highest degree of membership in D. In order for a decision process to truly reflect what the investor considers desirable, some methodology should be included to incorporate a procedure for allowing an investor to input his subjective evaluations of the importance of each of the objectives. Yager [8] has presented a methodology for including these values in a hierarchical fashion. This involves selecting for each objective a number a 1>0 indicative of its importance. The higher the a, the more important the objective. Then each objective is raised to a power, which is its importance. A decision function D is now formulated as follows:
D= cT, nq
n-., nc2.
The c~'s are determined subjectively, in a procedure which incorporates the tradeoffs the decision maker is willing to accept between his objectives. This procedure has the effect of making bad alternatives in important objectives appear much worse. It should be noted that if A is a fuzzy subset with membership function A(x), the A ~ is always a fuzzy subset with membership function (A(x)) ~. Saaty [4, 5, 6] has developed a procedure for obtaining a ratio scale for a group of p elements based upon paired comparisons. This procedure has been used by Yager [8] to elicit from a decision maker the importances to be used in our model.
2. An example Consider a potential investor who describes her investment preferences for common stocks in the following manner: (1) she is seeking a high return in the form of capital gains; dividend income is irrelevant to her; (2) she would like to protect her invested capital, but recognizes that one must accept some risk in order to obtain a 'high return'; (3) she would rather own stocks whose prices display minimum short-term fluctuations; (4) she recognizes that the correlation between stock price and company earnings is observable but not statistically large and therefore she would prefer companies whose sales and earnings have exhibited steady long term growth to companies which have grown erratically. She believes that each stock she purchases should be valued individually and re-evaluated approximately every 6 months. She uses other investment media (gold, bonds, and an occasional fling in commodities futures) and considers these
The personalization of security selection
3
four to be the components of her portfolio. Presently, she is considering adding to her common stock holdings, and is searching for a security which appears to be the most satisfactory for her. Her stated preferences cannot be measured directly, and available variables must be used as proxies for them. In some instances, more than one available variable would be an appropriate indicator that the stock would or would not satisfy her preferences. For instance, the search for 'capital gains' requires present and future data, so the appropriate proxy variable could be any one of a dozen. To name just a few of the variables which could be considered omens of capital gains to come, one could use: the price-earnings multiple, on the theory that low multiple stocks are the ones which have historically increased the most in price; or the rate of return on assets, on the theory that substantial profitability induces marketability, which produces capital gains; or the rate of growth of earnings, on the theory that a high rate of growth should continue for a sound company, other things remaining constant, and that this growth should drive up the stock price; or an index of advisory recommendations to buy the stock, on the theory that an increase in such recommendations should increase the demand for the stock and therefore raise its price. As another example, when the investor specifies 'minimum short term price fluctuations', she could determine whether or not a stock satisfied that description by examining any of the following variables: Beta, a measure of stock price volatility relative to the general market; or the coefficient of variation of the stock price; or a measure of the frequency with which a company appears on the list of most active stocks; or some measure of the range of a stock's price over some approp~mte time period. Whichever' preference is being considered, no one variable has tested out satisfactorily as a reliable indicator, for all (or even most) stocks over multiple time periods. The literature of finance and the performance of professional investors has borne this out for decades. Therefore, a selection of variables that the investor considers appropriate, coupled with her assessment of what a 'good' value for each variable might be, constitutes the investor's objectives. The fact that these objectives may be uncertain, subjective, imprecise, and/or may be subject to change over time does not preclude their use in our rational decision-making process. An objective is uncertain when we cannot obtain a guaranteed forecast of a known, precise variable. All that is necessary for fuzzy set theory to be applied to stock selection, however, is that we have the investor's forecast of that variable and his evaluation of that forecast as good or bad on a scale of 1 to 0. We have already indicated the degree of subjectivity that enters into the selection of a variable to satisfy the investor's preferences. We need only add, at this point, that it is also possible to allow for exceptions for particular companies in special circumstances. An investor might, for example, change his 1 to 0 evaluation of a particular stock's capital gains potential (whatever the variable he is using to indicate that potential), if rumors that the company is about to be taken over by another circulate. An objectivemsuch as 'high dividend yield'--is subject to change over time. While a 3% yield might be common at one point in time and therefore be
4
LM. Hammerbac~er, R.R. Yager
assigned a membership grade of 0.5, in a different set of economic circumstances a 6% dividend yield might only be graded 0.4. Fuzzy set theory provides the ideal medium in which to evaluate preferences which are precise or imprecise, su0jective or objective, certain or uncertain, and constant or subject to change. Let us proceed with an example of our application. Assume our investor has refined her preferences and established the following objectives: (1) a low price-earnings multiple; (2) short-term price stability; (3) a dividend yield which is either low or very high; (4) a low debt ratio; (5) a high short interest; (6) few recent insider sales of the stock. To avoid unduly time-consuming consideration of the five-thousand-plus common stocks on the market, our investor uses the traditional type of filtering system I to select a group of promising stocks for evaluation. Her filter eliminates the stocks of companies in unattractive industries, companies whose earnings growth has not been better than the rate of inflation (on the average) over the past 5 years, companies which showed losses in either of the last 2 years, companies for which inadequate data were available, and companies which she would not want to consider for any other personal reason whatever, e.g., companies doing business with South Africa. In essence, the filter is a fuzzy set with membership grade 1 or 0 over the set of all stocks. The determination of membership is subjectively evaluated by the investor, and is determined by the rule 'I do or do not want to consider this stock as a potential investment'. This filtering procedure leaves her with the set X of n possible alternative investments. The next step is to evaluate each of the objectives over X. That is, for each of the objectives, she creates a fuzzy subset over X, whose membership function indicates the degree to which a stock satisfies the objective. (1) Low price-earnings ratio. The ratio of stock price per share to earnings per share is a standard statistic associated with each stock. It is primarily a measure of the attractiveness of the stock, and a low pie for an investment-quality stock is desirable because it is a strong indicator of a potential capital gain. The problem, then, is for our decision-maker to subjectively evaluate the grade of membership (the degree of desirability) of the set 'a low pie'. Because pie can fluctuate slightly every time the stock is traded, one possible procedure for obtaining the membership grade could be to locate the most recent pie on the following graph and read off the membership grade from the Y-axis (Fig. 1). In effect, our hypothetical investor has decided that if a stock is selling for less than the earnings per share of the company, there is something wrong with it, and she d~esn't want to invest in it. Therefore, for pie less than one, the degree of membership is zero. She also believes that for most companies at the present time, a pie in the range of 6 to 10 is 'normal'. Therefore, stocks with low pie's i One could revert to Classical Investment Theory and use, for example, Nerlove's seven variables [2] or any three of Renwick's four variables [3] as the components of one's filter, but when tested, both provided lists of potential investments which were much too long to be practical.
The personalization o[ security selection
5
emberhi 1 pl
Degree of
0
t 6
i 10
pie
Fig. 1.
(less than 6) are underpriced and desirable; stocks with high p/e's are overpriced and undesirable. At this juncture, it is necessary to re-emphasize a previous point: in a different market, our investor might set up a different scale, and in any market, a different investor would set up a different scale. (2) Short term price stability. Beta, a measure of the stock's price varEation relative to the market's variation, has become a generally accepted statistic within the last 5 years. Our hypothetical investor considers B e t a = 1.00 as the ideal because B e t a < l . 0 0 indicates that the stock moves less than the markel and would preclude any substantial price appreciation, while Beta > 1.00 indicates that the stock moves more than the market. Our decision maker could subjectively evaluate the grade of membership in the set 'a Beta near 1.00' for specific stocks by using a transformation rule form Beta to [0, 1] as shown in Fig. 2. Again, the shape of this figure would change with the degree of an investor's risk aversion (or risk-preference) at a given point in time. (3) Low or very high dividend yield. A low dividend yield is an indication of the company's ability and/or willingness to grow internally. A high dividend yield is attractive to income seeking investors, and this basic source of demand for the stock will, in time, drive the price up and the yield back down. One possible procedure is to use the dividend yield in two fuzzy sets. In the first set, high-yielding stocks would be graded close to 0. In the second set, the reverse would be true. The decision maker would then incorporate both fuzzy sets into his decision. Another possible procedure is to recognize that a 5% dividend yield is currently the norm, and significant deviations from the norm in either direction are desirable. This would allow our investor to obtain membership grade by using a U-shaped curve as a transformation rule, where a 5% dividend yield is graded close to 0 and low and high yields are graded close to 1. We have assumed that our investor prefers this second procedure. (4) A low debt ratio. Since we have conjured up a conservative investor, we will Degree of membershi p |
1
Fig. 2.
Beta
6
LM. Hammerbacher, R.R. Yager
assume she is adverse to financial risk and will therefore attempt to avoid stock issued by heavily leveraged companies. If we define leverage as the r~tio of long term debt to total long term capital, we have a variable which ranges from 0 to 1. Since our investor is debt-averse, she will use the transformation 'membership grade = 1.0 - debt ratio'. (5) A high short interest. Short interest is a measure of the number of shares of stock that have been sold short on the market. Short selling itself is a bearish sign; one only sells short when one expects the price to move down in the near future. But every short sale has to be covered, i.e., the stock has to be bought back at a later date. Therefore, a large amount of short sales is a bullish sign, because a large number of shares will have to be bought in the near future. In order to get a comparative measure of short interest for different size companies, the short interest ratio has been constructed. This is the ratio of the number of shares of stock which were sold short to the average number of shares traded per day. For the vast majority of companies in a normal market, the short interest ratio is less than one. Only when the ratio gets significantly above one does it become a bullish signal. Therefore, a possible procedure for obtaining the membership grade could be to use the transformation rule which states: (1) if the short interest ratio is between 0 and 4.0, then the membership grade is 25% of the ratio; (2) if the short interest ratio is greater than 4.0, then the membership grade is 1.0. (6) Few recent insider sales. Our hypothetical investor is a very suspicious person. She believes, as does a sizable segment of the market, that when corporate insiders (officers or shareholders owning 10% or more of the outstanding stock) are buying the company's stock, they know something good that she doesn't know and the price of the stock is going to go up. When they sell the stock, they know something bad that she doesn't know, and the price is going to go down. The net change in the number of shares held by insiders is a readily available monthly statistic. Again, because of the varying sizes of companies, it must be converted into a ratie The ratio of insider net change (purchases less sales) per month to total outstanding shares is by nature very small. For the majority of companies, it is zero much of the time. It never reaches or even approaches plus or minus one, so a possible procedure for transforming insider trading to a fuzzy set might be: (1) if the ratio (T) is 0.5<~T~<1.0, then t'~c membership grade is 1.0; (2) if the ratio is - 0 . 5 <~ T ~<+0.5, then the membership grade is T + 0 . 5 ; (3) if the ratio is - 1.0~ < T~<-0.5, then the membership grade is 0. Having quantified our investor's objectives, and having created a fuzzy set for each, the next step is to determine the importance of the various objectives to her. The goal here is to obtain a scale which ranks the objectives as to their importance. First we construct a matrix of paired comparisons. The investor constructs the matrix by comparing, on a scale of 1 to 9, the ratio of importance of one objective to the next. In the matrix below, the ith j entry indicates the ratio of importance of objective i to objective/. The objectives are, as indicated: the price-earnings ratio (P), Beta (B), the Dividend yield (Y), the Debt ratio (D), the short interest ratio (S), and the insider trading ratio (T).
The personalization o[ security selection
Y D 7 5 7 5 1 1 ~1 ! 3 i 1 5 5 1 +~ 1 ~1
P P -1 B 1 Y I D 1 S i T_v
S 3 3 .~I s1 1 ~1
B 1 1
!
7
T 77 1 3 5 1_
This matrix indicates that, to our hypothetical investor, the price-earnings ratio and the Beta are equally important, but the price-earnings ratio is much more important than the dividend yield. Using a program which evaluates the maximum eigenvalue for a matrix, we get the following eigenvector representing our hypothetical investor's importance ratings: -1.8486 1.8486 0.2172 0.2172 1.2834 _0.5856
= ap= aB =av =aD = aS =aT-
Next, we selected a samp~,e of eight stocks 2 from our investor's filtered list, to complete the example in a reasonable amount of space. For each stock, we determined its grade of membership in each fuzzy set according to the procedures described above. Our fuzzy sets are:
{0.70
065 0+0 040 07° ,,6,, / 9
Xi
,
'
X2
~'3
'
X4
'
'
X5
'
X6
X7
'
X8
I
B = / 0"46, 0.50 0.38 0.46, 0 . 6 4 0.65 0.54 0.4__8/ [
X!
X2
X3
X4
X5
X6
X7
X8
)
y = { 0.03, 0.19 ,0.63 ,0.82 ,0.79 ,0.66 ,0.02 .0.79/__ x i
D
x2
x3
- -
[
s:{O
~
'J
X1
X2
X3
,
,
,
X2
x5
x6
x 7
x8
J"
0.86 0.55 0.50 0.72 0.82 0.85 /
0.61,0.56 ~
x4
X3
,
X4
,
'~
X5
,
X6
X7
0,3 00 0 0+ 0,s i ,
X4
,
X5
,
X6
,
X7
,
X8
J
,
X8
T = / 0 . 4 0 , 0 + 6 0 0.50 0.40 0.58 0.40 0.60 0.40} ,
I, X 1
X2
,
X3
,
X4
,
X5
,
X6
,
X7
•
X8
Our decision set now becomes: D = P~PN B~,N Y ~ r i D % n s~s NT ~+ 2 The companies are Bausch and Lomb, Allegheny Ludlum, Coca Cola, W.R. Grace, White Consolidated Industries, Southland Corp., Mobil Oil, and Eltra Corp., respectively. The data have been obtained from the Value Line Investment Survey and Media General's Financial Weekly.
I.M. Hammerbacher, R.R. Yager
8
where W
p.-..
Q
B y D
Ot B
.-.
oty
---
¢I¢D
~..
S T
~S
Ot T
--.
{0.70, 0.55, 0.20, 0.65, 0.80, 0.40, 0.70, 0.60} 1"8486 {0.517, 0.331, 0.051, 0.451, 0.662, 0.184, 0.517, 0.389}, {0.46, 0.50, 0.38, 0.46, 0.64, 0.65, 0.54, 0.48}1"s486 {0.238, 0.278, 0.167, 0.238, 0.438, 0.451, 0.320, 0.257}, {0.03, 0.19, 0.63, 0.82, 0.79, 0.66, 0.02, 0.79}°'2172 {0.467, 0.697, 0.905, 0.958, 0.950, 0.914, 0.428, 0.950}, {0.61, 0.56, 0.86, 0.55, 0.50, 0.72, 0.82, 0.85}°'2172 {0.898, 0.882, 0.968, 0.878, 0.860, 0.931, 0.9588, 0.965}, {0, 0.13, 0.20, 0.13, 0.05, 0, 0.85, 0 . 1 8 } 1"2834 {0, 0.073, 0.128, 0.073, 0.021, 0, 0.812, 0.110}, {0.40, 0.60, 0.50, 0.40, 0.50, 0.40, 0.60, 0 . 4 0 } 0.5856 {0.585, 0.741, 0.666, 0.585, 0.666, 0.585, 0.741, 0.585}.
Thus, we determine that: D={0, 0.073, 0.051, 0.073, 0.021, 0, 0.184, 0.110}. It is noted that the elements of D are the price-earnings ratio and the short interest ratio, which is to be expected since, for our investor, these were two of the three most important variables. She would then select that security which best satisfies D, which is x7 in this case.
3. Conclusion
Using fuzzy sets to select securities is a procedure which permits investors to logically and consistently choose the most appropriate investment for themselves by including their muh,ple objectives and their varying perceived importances. The traditional stock valuation models attempt to help a generalized investor to select the 'best' stocks in a rational market, but our approach has been far more personalized than this. The application of fuzzy set theory stresses the uniqueness of the investor and his perceptions, and provides a tool which will help him to cope with the decision making process.
References [1] R.E. Bellman, and L.A. Zadeh, Decision making in a fuzzy environment, Management Sci. 17 (1970) B141-B164. [2] M. Nerlove, Factors affecting differences among rates of return on investments in individual common stocks, Rev. Econ. Statist. 50 (1968) 122-136. [3] F.B. Renwick, Asset management and investor protfolio behavior: theory and practice, J. Finance 24 (1969) 181-206.
The personalization of security selection
9
[4] T.L. Saaty, A scaling method for priorities in hierarchical structure, J. Math. Psychoi. (1977) 234-281.
[5] T.L. Saaty, Exploring the interface between hierarchies, multiple objectives, and fuzzy sets, Fuzzy Sets and Systems 1 (1978) 45-66.
[6] T.L. Saaty and M. Khouja, A measure of world influence, J. Peace Sci. 2 (1976) 31-48. [7] U. Thole, H.J. Zimmermann and P. Zysno, On the suitability of minimum and product operators for the intersection of fuzzy sets, Fuzzy Sets and Systems 2 (1979) 167-180.
[8] R.R. Yager, Multiple objective decision making using fuzzy sets, Int. J. Man-Machine Studies (1977).
[9] L.A. Zadeh, ~ uzzy sets, Information and Control 8 (1965) 338-353. [10] L.A. Zadeb, Dutline of a new approach to analysis of complex systems and decision processes, IEEE Tranz Systems, Man and Cybernet. SMC-3 (1973) 28-44.
[11] L.A. Zadeh, Calculus of fuzzy restrictions, in: L.A. Zadeh, K. S Fu, K. Tanaka and M. Shimura, [12]
eds., Fuzzy Sets and Their Applications to Cognitive and Decision Processes (Academic Press, New York 1975) pp. 1-39. L.A. Zadeh, Theory of fuzzy sets, in: J. Belzer, A. Holzman and A. Kent, eds., Encyclopedia of Computer Science and Technology (Marcel Dekker, New York, 1977).