Bayesian Networks: A Decision Tool to Improve Portfolio Risk Analysis
Riza Demirer Department of Economics & Finance Southern Illinois University-Edwardsville School of Business, Alumni Hall 3132 Edwardsville, IL 62026-1102 618-650–2939
[email protected] Ronald R. Mau University of Kansas, School of Business 1300 Sunnyside Ave. Lawrence, KS 66045-7585 785-864-3838
[email protected] Catherine Shenoy † University of Kansas, School of Business 1300 Sunnyside Ave. Lawrence, KS 66045-7585 785-864-7519 785-864-5328 (FAX)
[email protected]
June, 2005 Please do not quote without permission.
†
Corresponding Author.
Bayesian Networks: A Decision Tool to Improve Portfolio Risk Analysis Abstract
This paper demonstrates how Bayesian Networks can aid decisions of individual security analysts and portfolio managers. We present a decision tool to improve analysts’ forecasts, portfolio decision-making, and risk analysis. Our paper is related to findings in behavioral finance that show buying and selling behavior that is consistent with biased decision-making. However, most behavioral finance literature is descriptive, not normative. Our goal is to improve rational financial decision making by helping analysts de-bias their probability assessments and systematically improve value-at-risk forecasts. We illustrate how to combine historical quantitative information with qualitative information in a systematic way using a graphical modeling tool called Bayesian networks.
1. Introduction Portfolio management is a very special problem in engineering, of determining the most reliable and efficient way of reaching a specified goal, given a set of policy constraints, and working within a remarkably uncertain, probabilistic,
always
changing
world
of
partial
information
and
misinformation, all filtered through the inexact prism of human interpretation Ellis(1985).
Security analysts evaluate a variety of information to decide whether to buy, sell, or hold a security; however, most research in security analysis has not concentrated on the individual decision maker. Research in security analysis has primarily concentrated in two different areas. The first is pricing and valuation models. The second area examines the relationship between firm or economic variables and earnings’ forecasts. In this paper we use a Bayesian network to model economic relationships to arrive at an earnings forecast for each stock in the portfolio and a return distribution for the portfolio. The output of the model is a probability distribution which combines historical information with current news. Traditional pricing models such as the Capital Asset Pricing Model or the Arbitrage Pricing Theory describe the relationship between economic variables, firm characteristics, and stock returns. The models are based on historical, quantitative data, and the results are intended for the average or typical firm. Most analysts use this historical, quantitative analysis as a complement to a more holistic approach that includes a wider variety of information. Analysts typically concentrate on special situation and individual cases, not the average. Their information includes historical data and qualitative, imprecise evidence that may affect a firm. For example, an analyst may consider some of the following questions: How effective or trustworthy is a
firm’s management? What is the effect of China’s entry into the WTO on a particular line of business? How reliable are a firm’s financial statements? We show how to integrate this type of information with historical, quantitative data. In this paper we apply a graphical, decision-modeling tool, Bayesian networks, to security analysis. In portfolio management, analysts must assess a large amount of sometimes conflicting information to make a decision based on uncertain information. We suggest that Bayesian networks are especially well suited for this task. This tool helps experts represent uncertain, ambiguous or incomplete knowledge that portfolio managers and analysts often deal with in their analyses. The output of the model is a probability distribution for portfolio value. Since we focus on modeling the entire distribution, measures of risk and Value-at-risk (VAR) are modeled naturally. In addition, when new evidence is added to the model, the effect of that new evidence on other variables in the model and the effect of the evidence on risk is also computed. There are two types of inputs to the model – the graphical relationship and a set of equations and conditional probability distributions described by the graph. The conditional probability relationships can be estimated from historical data or from expert judgment. Shenoy and Shenoy (2000) show how traditional expected return models such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) can be used to model relationships in a Bayesian network. We illustrate how to combine quantitative data with qualitative or soft information in a systematic way. The major portion of this paper provides an example of how to combine macroeconomic factors and firm specific factors. However, the methodology we propose is intended to be flexible enough to reflect an individual analyst’s decision-making process. This paper makes
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several contributions. First, we suggest the use of Bayesian networks as a compact and intuitive decision-modeling tool to represent different aspects of uncertainty inherent in security analysis. Bayesian networks have been used in different decision support system contexts that combine qualitative and quantitative information. 1 We extend the use of this decision tool to security analysis. Second, we provide a methodology to combine economic and firm specific variables. Our example contains a range of factors that determine the stock returns at the macro-economic, industry, as well as the firm level. Third and most importantly, we illustrate a systematic way to add new information (or evidence) to current security analysis techniques and to see the effect of that evidence on the valuation. This feature of Bayesian networks helps analysts understand the effect of new information on their models. They must explicitly model new information. The model then illustrates the effect of the new information on their earnings estimate. Finally, the framework we present can be used to create an expert system in which learning takes place. The paper proceeds in the following way. Section 2 provides a brief outline of several results in behavioral finance and introduces Bayesian networks as an effective tool for security analysis. Section 3 presents a general overview of how to set up a Bayesian network for security analysts. Section 4 presents an example Bayesian network that combines macro-economic and firmspecific variables that an analyst might use in a top-down analysis in the biomedical device industry. Section 5 explains how a decision maker can use our framework to add new qualitative or quantitative information as evidence to the model. Finally, Section 6 concludes the paper and discusses possible extensions to Value at Risk analysis for portfolios.
1
For some examples see Jensen (1996)
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2. Security Analysis and Decision Making Under Uncertainty This section relates some results in behavioral finance to security analysis. We also discuss how some findings from the uncertainty in artificial intelligence literature relate to security analysis. In particular we argue that a Bayesian network is an important tool to improve security analysis by helping the analyst improve forecasts and eliminate bias. Behavioral finance research has shown that some aggregate, market-pricing patterns are consistent with human cognitive patterns that are considered ‘irrational’ or non-value maximizing. Most behavioral finance articles test the Efficient Market Hypothesis versus one or more cognitive shortcomings that humans are prone to experience when trying to assess probabilities. Cognitive psychologists first identified some of these biases and showed the difficultly of achieving rational economic decisions. 2 In a broad review of finance literature, Hirshleifer (2001) reviews and summarizes a range of behavioral finance articles and finds that “ … expected returns are determined by both risk and misvaluation.” He attributes this misvaluation to various cognitive and information processing biases. Research into improving probability assessment and decision-making has produced some tools and results that could successfully be applied to improving security analysis. For example, Alpert and Raiffa (1982) demonstrate most people consistently underestimate standard deviations in probability distributions. If standard deviations are underestimated, then surprises are over-estimated. In their experiments, groups were asked to estimate the value of surprising outcomes, defined as the 1st and 99th percentiles. Initially, 46% of the outcomes fell in the surprising category; however, the surprises fell to 23% with one round of feedback. The
2
See Kahneman, Slovic, and Tversky (1982) and Kahneman and Tversky (1986) for some examples.
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estimations improved but remained well above the 2 percent level. An additional round of feedback resulted in much better probability assessments. In an automated Bayesian network decision support system, we can capture the probability judgments analysts make over time, compare to actual data when it becomes available, and then provide feedback on a timely basis. Based on the work by Alpert and Raiffa, this feedback should improve analysts’ probability assessments which should lead to improved future performance. Hagstrom (1999) observed: “Whether or not they recognize it, virtually all decisions that investors make are exercises in probability. For them to succeed, it is critical that their probability statement combines the historical record with the most recent data available. And that is Bayesian analysis in action.” A Bayesian network is a tool to help experts represent uncertain, ambiguous or incomplete knowledge. Bayesian networks use probability theory to represent uncertain knowledge 3 . A Bayesian network consists of two parts - a qualitative graphical structure of the relationships in the model and a quantitative structure represented by the probability distributions that are indicated by the graph. In a Bayesian network, historical information can provide a framework or baseline model to develop prior distributions. New quantitative information, qualitative information, or evidence can be added to the network as appropriate to develop posterior probabilities. Bayesian networks are not black box expert systems. We propose a flexible decision support system that provides an analyst with a tool to improve individual decision making, to facilitate learning, and to improve performance. The network provides a simple method of incorporating probabilities into the decision maker’s analysis.
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Decision makers in many different contexts combine quantitative data and qualitative information. Bayesian networks have been applied in a wide variety of decision-making contexts. Some examples are venture capital financing (Kemmerer et al 2001), auditing (Gillett and Srivastava 2000), medical diagnosis (Bielza et al 1999), and software design (Horvitz et al 1998), among many. There are three important steps in building a Bayesian network – the graphical representation, specifying the numerical relationships among the variables, and making inferences or decisions based on new evidence. Of the three, several researchers have shown that the graphical representation may be the most critical. Henrion et al., 1994 and Darwiche and Goldszmidt, 1994 show that results are more sensitive to the qualitative structure of the model and that the results are robust with respect to the numerical specification. Generally, it is better to have a very good graphical representation of the decision problem with approximately correct probability distributions than to have a poor graphical model with very precise probability distributions. Several commercial software tools estimate Bayesian networks. The tools allow the user to enter the graph and specify the numerical relationships among the variables. The software calculates the inferences based on these inputs. The inference results are shown graphically as probability distributions for the network. Hugin (www.hugin.com) and Netica (www.norsys.com) can both be used to automate the process. Netica is used for the example presented in this paper.
3
See Jensen (1996) for a review.
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3. Developing a Bayesian Network for Security Analysts This section presents a general overview of how to set up a Bayesian network to help security analysts make better decision. We assume that the analyst is using a top-down approach starting with economic and industry analysis before proceeding to a firm analysis. In section 4, we provide an extended example. We describe the three basic steps in the Bayesian network construction. The first step is developing the graphical model. This step includes identifying the relevant variables and specifying whether they are independent, or not. The second step is the specification of the numerical relationship between the variables that are not independent. Finally we describe how to update the model to add new evidence. 3.1.
Graphical Representation
As mentioned in section 2, the first step in constructing a Bayesian network is the graphical model. The graphical model is a directed, acyclic graph where nodes represent variables and directed arcs (arrows) represent the conditional probability relationships assumed in the model. The variables and the arcs between the variables are the inputs to the graph.
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Stock 1 Factors Industry 1 Factors
Stock 1
Stock 2 Factors Economic Factors
Stock 2
Portfolio
…
Industry 2 Factors
Stock n Stock n Factors
Figure 1. General Graphical Portfolio Model In a complicated model, it is useful to group variables and relationship in order to have an overview of the structure. Figure 1 shows a general graphical structure to model stock prices and portfolio value. The shaded nodes in Figure 1 represent groups of variables. In this model multiple economic factors affect multiple industry factors. For example, the arc from “Economic Factors” to “Industry 1 Factors” indicates that the probability distribution of the industry factors is conditioned on the economic factors. The first step in a general top-down security analysis is to identify the specific economic, industry, and firm specific variables that make up the Bayesian network. Usually, an analyst would already have a number of specific variables identified. Academic research can also provide some examples. The Capital Asset Pricing Model is a very simple example of a firm variable and an economic variable – the market return and a security return. The arcs leading to the stocks indicate that the stock return distributions are conditioned on industry factors and stock-specific factors. The arcs also indicate which type of probability
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distribution must be estimated. A node with arcs leading out only indicates a marginal probability distribution. A node with arcs leading into it indicates a conditional relationship. The absence of a directed arc from a node is also meaningful because the absence indicates independence. For example, the portfolio node has directed arcs leading from each of the stocks and not from any economic, industry or firm-specific factor because we assume that the portfolio distribution is independent of all other factors given the distributions of stock returns. The absence of a directed arc denotes conditional independence between two nodes. Here is an example of a situation that might be modeled. Suppose an analyst wishes to forecast pretax income using a proforma income statement. The relation among the income statement variables may be graphically represented as follows: Revenue Segment 1
Revenue Segment 2
Revenue Segment 3 Operating Expenses
COGS Segment 1
COGS Segment 2
COGS Segment 3 Other Income or Expense
Gross Margin Pretax Income
Figure 2. Example Graphical Model for Pretax Income The graph in Figure 2 shows that the decision maker assumes that revenue and cost of goods sold (COGS) for each segment are independent, but that the revenue and cost of goods sold for each segment are related to the revenue for that segment. This graphical representation also
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assumes that operating expenses are independent of each segment and other income and expenses. An example of the situation shown in Figure 2 is LSB Industries. LSB manufactures chemicals, climate control products, and provides industrial engineering services. Each of the segments operates in a totally different market. An example of where this representation is less appropriate is Limited Brands. They classify their two main segments of business as the apparel businesses and the intimate brands business. They attempt to locate stores in the two lines of business in the same malls, and their target market overlaps in many cases. In the Limited Brands case, the revenue from each segment is not independent. Figure 3 shows a graphical representation of the Limited Brands situation. Revenue Segment 1
Revenue Segment 2
COGS Segment 1
COGS Segment 2
Operating Expenses
Other Income or Expense
Gross Margin Pretax Income
Figure 3. Alternative Pretax Income Graphical Model The analyst benefits in several ways from the graphical construction of a Bayesian network. First, the construction of the graphical portion of the network helps the analyst clarify and refine his view of the relationships among the variables. Next, the analyst may not always have a good understanding of how a decision is reached. They fully understand which variables are used, but how the variables are combined and weighted to come up with a decision is not always well
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understood or systematic. In Shenoy (2001), almost all analysts agreed that qualitative information was important, but when questioned about how it was incorporated in a decision, most analysts could not be specific. 3.2.
Determining Numerical Links between Variables
Each node in the Bayesian network is a variable that is described either as a constant value, a probability distribution, or as a function of other variables. In a Bayesian network, the primary focus is on determining the probability distribution of the relevant nodes. In spreadsheet analysis, the primary focus is on determining variable values and the functional relationships among the variables. Specialized software tools for spreadsheets, such as @Risk and Crystal Ball, also combine constant, functional relationships and probability distributions using simulation. Bayesian networks use an efficient algorithm called local computation to find the probability distribution of interest. Local computation leads to fast updates of probabilities and is especially important in the case of decision problems where a large number of macro and micro-economic variables are involved. In addition, new information can easily be incorporated into the network, and the new information updates the entire network. In this section we illustrate some of the different ways to find the probability distributions in the network and show how these can be combined with constant values or functional relationships. There are two primary ways to find probability distributions for the nodes in the network. One way is to use historical data. The other is to use subjective probability judgments. The two methods can also be combined. There are also two types of probability distributions in a Bayesian network – marginal distributions and conditional distributions. A marginal distribution for a node is one that does not
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have an arrow leading into it. For example, in Figure 3, Revenue-Segment 1, Operating Expense, and Other Income or Expense variables are all marginal probability distributions. In Figure 2, however, there are more marginal distributions – one for each segment’s revenue and also for Operating Expense, and Other Income or Expense. A conditional probability distribution is indicated by a node that has at least one arrow leading into it. In Figure 3, COGS for both segments are conditional distributions, as is Gross Margin and Pretax Income. A familiar way to use historical data to find a conditional probability distribution is using regression analysis. The dependent variable in a normal form regression is normally distributed with the mean estimated by the regression line given the dependent variables. For example, COGS may be estimated using the following regression formula, COGS1 = α + β ( Re venue1 ) + ε1
The conditional return distribution for COGS is a normal distribution with conditional mean, E ( COGS1 | Re venue1 ) = α + β ( Re venue1 ) , and standard deviation, σ ε .
The regression method uses historical data to estimate the relation between variables in the model. Several other methods to elicit subjective conditional probability distributions have also been used. Howard (1989), Spetzler and Staël von Holstein (1975), and Kemmerer et. al. (2001) all relate methods to help elicit subjective conditional probability distributions. In the extended example in Section 4, we use both regression and subjective conditional probability distributions. Another way to determine the numerical inputs for the network is through simulation. A spreadsheet simulation model can be used to determine the probabilities and state values for the firm-specific variables. Simulation software such as @Risk or Crystal Ball can be used with a spreadsheet model to generate a distribution for earnings per share. Figure 4 shows a simplified
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spreadsheet used to generate values of selected items that determine earnings per share growth. An analyst can use historical data on income statement items like revenue, gross margin, and SG&A to generate random values and a distribution for these variables 4 . In this way, we obtain a probability distribution for the earnings growth. One advantage of this method is that it allows the analyst to incorporate assumptions for any items in the financial statements, and link it to the earnings growth node in the Bayesian network.
Figure 4. Generating distributions through simulation 4. An Illustrative Example This section presents an example Bayesian network that combines several macro-economic and firm-specific variables that an analyst might use in a top-down analysis in the biomedical device industry. The output of the network is the portfolio value in thousands of dollars. Note that the model presented in this section is only one way of incorporating macro and micro variables to security analysis. Here we discuss the choice of variables used in the model and the calculations
4
We simply assume normal distributions for these variables with mean and standard deviations calculated using historical numbers from the financial statements.
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for different probability distributions. In the next section we show how to update the model with new evidence. Figure 5 shows the example Bayesian network for four firms in the biomedical device industry. The ultimate goal is to model the probability distribution for the portfolio value in the lower right hand corner. In this example we use the software Netica to draw and compute the model.
Figure 5. Example Biomedical Device Industry Bayesian Network The upper left hand corner of Figure 5 contains four macro-economic variables that affect market return. We use these variables based on a series of papers by Berry, Burmeister, and
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McElroy (1988), Burmeister, Wall and Hamilton (1986), and Burmeister and Wall (1986). They describe and test a multi-factor return model using a default risk premium (DP), a maturity premium (MP), unexpected changes in inflation (INF), and unexpected changes in productivity (PDT) to determine the expected market return (MKT). In this example, the return on the biomedical industry (IND) is conditioned on the market return. The four stocks that we used are: Guidant Corporation (GDT), St. John’s Medical (STJ), Boston Scientific (BSX), and Medtronic, Inc (MDT). For each of the four individual stock returns, we model the required rate of return distribution as a conditional probability distribution that depends on the industry return distribution. For each firm we determine a P/E distribution based on earnings per share (EPS) growth and the required rate of return. EPS is determined by prior EPS and earnings growth. Price is simply EPS times the P/E ratio.
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10.2 ± 4.9
Figure 6. Prior probability for Portfolio Value Node The portfolio value is price times number of shares for each stock. Figure 6 shows the extended view of the prior probability of the node – Portfolio Value. The numbers along the left-hand column of the node show the state values. The first value, 0, represents the range of portfolio values from 0 to $1,000, and the last value, 19, represents the range of portfolio values above $19,000. The next column represents the probability of each state given the price distributions of the four stocks and the number of shares owned. Since the number of shares owned is a constant, they are not shown. We assume that the portfolio consists of 100 shares of each stock although the calculation of the optimal number of shares can be easily incorporated to the model. Given this prior probability distribution, the value of the portfolio is $10,200 with a standard deviation of $4,900. The Value-at-Risk (VaR) at the 0.03 level is between 0 and $1,000. The 0.05 level falls between $1,000 and $2,000.
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Table 1 shows the descriptions of macro economic and firm-specific variables we used in the Bayesian Network of Figure 5. Calculation details of probability distributions are provided in the Appendix. Table 1. Description and Construction of Economic Firm Variables Variable
Probability Distribution
Description
Data Source
Macro economic and Industry Variables Default Risk Premium (DP)
Long-term corporate bond return less long-term government bond return.
Marginal
Ibbotson and Associates
Maturity Premium (MP)
Long-term corporate bond return less one-month treasury bill return
Marginal
Ibbotson and Associates
Unexpected Change in Inflation (INF)
Expected inflation at the beginning of the month less actual inflation at the end of the month
Marginal
Ibbotson and Associates
Unexpected Change in Productivity (PDT)
Expected growth in real final sales at the beginning of the month less actual real final sales at the end of the month
Marginal
National Income Accounts
Other Market Effects (MKT)
Expected market return using the previous 4 factors less actual market return
Industry effects (IND)
Return on equally weighted bio-medical supply index
Conditional on variables {DP,MP, INF,PDT}
S&P 500 Index
Conditional on variable {MKT}
Constructed based on returns for six biomedical supply firms
Conditional on {IND}
Historical security returns
Firm-specific Variables (For each security) Security_k
Required return on security
Growth_security
Growth in earnings
Conditional on {k}
EPS_security
Earnings per share
Deterministic, EPS0*(1+growth)
PE_security
Price to earnings ratio
Conditional on {k, growth}
Price_security
Price
Deterministic, PE*EPS Deterministic, weighted average of security values
Portfolio Value
An important modeling decision is to decide whether to use discrete probability models or a combination of the both discrete and continuous probability models for the variables. Continuous distributions are normally used to estimate economic data and returns; however, a major drawback to using continuous probability distributions is that it complicates the process of adding evidence. Evidence is most naturally added to a discrete model even when the probability distribution is continuous. For example, when asked about expectations about inflation, the analyst is most likely to give a point estimate or a less precise estimate ‘higher than normal’.
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In this example, we choose to model the economic variables as discrete variables for two reasons. First, inferences based on the graphical structure of the model are relatively robust to the form of the inputs (Henrion et al., 1994). Second, and most importantly, we want to model the effect of new evidence on the stock price or portfolio value, so we decide to make this model more amenable to the addition of evidence. 5. Adding Evidence Perhaps the most important feature of the Bayesian network framework is that it allows the decision maker to add new qualitative or quantitative information as evidence to the model. Our model allows the user to add evidence in several ways. First, the decision maker can use expert judgment on any variables to update the probabilities assigned to certain states of those variables. This is the traditional form of adding evidence to a Bayesian network. Evidence is defined as any information that changes the probability distribution in the network. For example, if the user is bullish for her expectations for the market overall, she can update the probabilities assigned to specified states for the node representing market return (MKT). In this way, re-compiling the model leads to re-calculation of all related conditional probabilities that go into final portfolio value. Another example of adding evidence to the network comes from a news report on July 10, 2001. Guidant failed to win FDA panel backing for its pacemaker-like heart-failure device. Based on this information, we assumed that earnings for the company would be negatively affected. Therefore, we added evidence to the Bayesian network indicating that earnings growth for GDT would fall in the low state. Figure 7 shows the GDT and portfolio probability
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distributions before adding evidence, and Figure 8 shows the same view with the addition of the new evidence.
10.2 ± 4.9
Figure 7. GDT and Portfolio Distribution before adding evidence
10.1 ± 4.9
Figure 8. GDT and Portfolio Distribution after adding evidence Incorporating this evidence into the model resulted in a predicted 18 percent price decline. The actual market reaction to this announcement was a 13.75 percent decline from the close on July 9, 2001 to the close on July 10. However, the announcement was made at 12:37 on the July10th (Bloomberg news). The total decline through July 11th was 22.96 percent.
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Alternatively, evidence can be added in terms of updating the actual values of states of variables. In this example, we rely on historical data to identify the low, medium, and high ranges of chance variables. For example, in the case of macro economic factors, we set our range values based on historical data for inflation, the maturity premium, etc. However, if an analyst believes that a structural change has occurred in the probability distributions for those variables 5 , then she can specify other, new value ranges for these variables. In Figure 8, the VaR for the portfolio has not changed, even though the probability of some lower states has increased and the probability of some of the higher states had decreased. For example the probability of the portfolio value being between 10 and 11 thousand dollars has declined from 0.152 to 0.139. The very lowest states from 0 to 4 have not changes, but state 5 has become slightly more likely.
9.2 ± 5.0
Figure 9. Portfolio Distribution with additional Evidence
5
Structural changes may be due to the expectation of a crash, a major default, or any other event that potentially leads to a shift in the probability distributions for variables used in the model.
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Figure 9 shows another possible after some very negative evidence has been observed. In this instance, we see that again that VaR has not changed. However, the mean has decreased significantly and the value of the
6. Limitations, Extensions, and Conclusion This paper introduces a decision support system for equity analysts using a Bayesian network. The proposed framework provides a systematic method to assist a decision maker in representing uncertain, ambiguous, incomplete and conflicting information that security analysts and portfolio managers routinely analyze. It allows for both quantitative data and qualitative data to be incorporated in the decision-making process and uses macroeconomic and firm-specific data to develop industry returns, equity valuations, and portfolio value. Finally, the Bayesian network allows the analyst to see the effect of new evidence on various outcomes. A Bayesian network can also provide feedback to an analyst on forecasts and help improve learning and reduce bias in those forecasts. We present an example to demonstrate how the analysis could be performed and updated as new information becomes available. The example deals with a limited portfolio consisting of four firms in the medical devices industry and a news event that had an impact on GDT’s future growth. In Section 5, we showed that updating the model by reducing the growth estimates resulted a prediction similar to the market reaction. The network is adaptable to various situations and different types of evidence can be added to the network. The model could be extended and used in other industries as well as for diversified portfolios. The model can also be extended to asset allocation in which an optimization module is incorporated such that optimal weights are calculated and revised as evidence is added.
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The proposed framework is also a risk management tool since the output consists of probability distributions for the value of the portfolio. A logical extension of the network is incorporating value at risk (VaR) into the network as an additional tool for portfolio managers. When new evidence is added, the effects on the values and standard deviations to calculate revised VaR estimates as new evidence is available. Simulations could also be performed to determine the value at risk under various scenarios. Incorporating VaR would also provide portfolio managers likely scenarios for the value of the portfolio as new evidence becomes available. The next step is to test the Bayesian network approach using analysts who are making everyday forecasts to determine how much bias is reduced and whether forecasts are more accurate. The Bayesian network should assist the user in making unbiased decisions that are not based on emotional/behavioral aspects of investing but are based on conditional probabilities that incorporate both quantitative and qualitative data in an explicit manner. "The investor who permits himself to be stampeded or unduly worried by unjustified market declines in his holdings is perversely transforming his basic advantage into a basic disadvantage. That man would be better off if his stocks had no market quotation at all, for he would then be spared the mental anguish caused him by other persons' mistakes of judgment." Graham (1973).
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Appendix Macro-economic inputs The top portion of Table 1 provides a definition of each of the macro-economic factors and their construction. All variables are measured as a monthly percentage. We discretize the data based on a modification of the bracket-median method. 6 The states are defined as ‘low’, ‘medium, or ‘high’. The data set for the market and the industry return were then used to estimate conditional probability distributions P(MKT|DP,MP,INF,PDT) and P(IND|MKT). For example the conditional probability of MKT given variables {DP, MP, INF, PDT} contains 81 (34) possible states as each of the conditioning variables has three possible states. Thus, the macro-economic portion of the network contains a total of six probability distributions; four marginal distributions and two conditional distributions. Table 2 shows a portion of the conditional probabilities for the variable MKT. 7 Table 2. Conditional probabilities for MKT. DP Low Low Low
P(MKT|DP,MP,INF,PDT) MP INF PDT Low Low Low Low Low Medium Low Low High
Low 0.40 0.50 0.60
MKT Medium 0.60 0.20 0.20
High 0.00 0.30 0.20
Firm-specific inputs The relationship among the firm-specific variables is also a conditional probability model. The lower portion of Table 1 provides a list of firm-specific variables used in the example. We distill the inputs obtained from the earnings statement to expected earnings growth (g), and use this variable and its probability distribution in the Bayesian network. We use analyst forecasts of growth in earnings to determine the probability distribution. The link from the macro-economic model comes through the required rate of return which is conditioned on the industry required
6
We examined three discretization techniques to fit three-state discrete probability distributions for variables DP, MP, INF, and PDT: The bracket-median method (Clemen, 1991), the bracket-mean method (McNamee and Celona, 1987), and the extended Pearson-Tukey (Keefer and Bodily, 1983). The reader is referred to Smith (1993) for a comparison of these methods. 7 Our method is a discretization of the method used in Berry et al’s (1988) multi-factor model. In the continuous case, regression provides estimates of the conditional mean and standard deviation assuming that a variable is normally distributed with the estimated conditional parameters.
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rate of return. The price earnings ratio (PE) is a conditional probability distribution based on earnings growth (g) and the required rate of return on the stock (k). Based on the data, we identify a range of values for each possible state of the variables g, EPS, and PE. We estimate two probability distributions P(g|k) and P(PE|k,g). There are two deterministic relationships – EPS and price. EPS is based on historical EPS and growth. State values for the price earnings ratio variable are calculated using a modification of the constant growth model as described in Elton and Gruber (1991), Chapter 18. The constant growth model is a simple discounted cash flow model that assumes growth (g) will continue at the same rate into the indefinite future. Generally, the analysts interviewed indicated constant growth was assumed for the time period analyzed. n
PE ratio = ∑ t =1
where
k g
( )
1+ g t 1+ k
= required rate of return, and = earnings growth.
Finally, price earnings ratio and earnings per share are used to determine a probability distribution for the price estimate on the firm’s stock. The last step is to use the price estimates for each individual firm to calculate a probability distribution on the portfolio value. We assumed that the portfolio holds 100 shares of each stock and defined value ranges in thousands of dollars for each possible state of portfolio value.
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