پاورپوینت ساختار همبستگی بازده اوراق بهادار

صفحه 1:


صفحه 2:
multi-index models and averaging techniques = Multi-index models are an attempt to capture some of the nonmarket influences that cause securities to move together. The search for nonmarket influences is a search for a set of economic factors or structural groups (industries) that account for common movement in stock prices beyond that accounted for by the market index itself. Although it is easy to find a set of indexes that is associated with nonmarket effects over any period of time, as we will see, it is quite another matter to find a set that is successful in predicting covariances that are not market related. Averaging techniques are at the opposite end of the spectrum from multi-index models. Multi-index models introduce extra indexes in the hope of capturing additional information. The cost of ۱ introducing additional indexes is the chance that they are picking up random noise rather than real influences. Averaging techniques smooth the entries in the historical correlation matrix in an attempt to 0 out” random noise and so produce better forecasts. The potential disadvantage of averaging models is that real information may be lost in the averaging process 

صفحه 3:
General Multi-index Models Any additional sources of covariance among securities can be introduced into the equations for risk and return simply by adding these additional influences to the general return equation. Let us hypothesize that the return on any stock is a function of the return on the market, changes in the level of interest rates, and a set of industry indexes. 

صفحه 4:
This model can also be used if analysts supply estimates of the expected return for each stock, the variance of each stock’s returns, each index loading (bik between each stock i and each index k), and the means and variances of each index. This is the same number of inputs (2N 2L LN). However, the inputs are in more familiar terms. As discussed at several points in this book, the inputs needed to perform portfolio analysis are expected returns, variances, and correlation coefficients. By having the analysts estimate means and variances directly, it is clear that the only input derived from the estimates of the multiindex models is correlation coefficients. We stress this point because later in this chapter, we evaluate the ability of a multi-index model to aid in the selection of securities by examining its ability to forecast correlation coefficients. There is a certain type of multi-index model that has received a large amount of attention. This class of models restricts attention to market and industry influences. Alternative industry index models result from different assumptions about the behavior of returns and, hence, differ in the type and amount of input data needed. We now examine these models 

صفحه 5:
Industry Index Models Several authors have dealt with multi-index models that start with the basic single-index model and add indexes to capture industry effects. The early precedent for this work can be found in King (1966), who measured effects of common movement between securities beyond market effects and found that this extra market covariance was associated with industries. For example, two steel stocks had positive correlation between their returns, even after the effects of the market had been removed 

صفحه 6:
The assumption behind this model is that a firm’s return can be affected by the market plus several industries. For some companies this seems appropriate as their lines of business span several traditional industries. However, some companies gain the bulk of their return from activities in one industry and, perhaps of more importance, are viewed by investors as members of a particular industry. In this case, the effects on the firm’s return of indexes for industries to which they do not belong are likely to be small, and their inclusion may introduce more random noise into the process than the information they supply. This has prompted some authors to advocate a simpler form of the multi- index model: one that assumes that returns of each firm are affected only by a market index and one industry index. Furthermore, the model assumes that each industry index has been constructed to be uncorrelated with the market and with all other industry indexes 

صفحه 7:
AVERAGE CORRELATION MODELS The idea of averaging (smoothing) some of the data in the historical correlation matrix as a forecast of the future has been tested by Elton and Gruber (1973) and Elton, Gruber, and Urich (1978). The most aggregate type of averaging that can be done is to use the average of all pairwise correlation coefficients over some past period as a forecast of each pairwise correlation coefficient for the future. This is equivalent to the assumption that the past correlation matrix contains information about what the average correlation will be in the future but no information about individual differences from this average. This model can be thought of as a naive model against which more elaborate models should be judged. We refer to this model as the overall mean model 

صفحه 8:
A more disaggregate averaging model would be to assume that there was a common mean correlation within and among groups of stocks. For example, if we were to employ the idea of traditional industries as a method of grouping, we would assume that the correlation between any two steel stocks was the same as the correlation between any other two steel stocks and was equal to the average historical correlation among steel stocks. The averaging is done across all pairwise correlations among steel stocks in a historical period. Similarly, the correlation among any steel stocks and any chemical stocks is assumed to be equal to the correlation between any other steel stock and any other chemical stock and is set equal to the average of the correlations between each chemical and each steel stock. When this is done, with respect to traditional industry classifications, it will be referred to as the traditional mean model. The same technique has been used by Elton and Gruber (1973) with respect to pseudo- industries. 

صفحه 9:
FUNDAMENTAL MULTI-INDEX MODELS Two types of fundamental multi-index models have received a great deal of attention in the academic and practitioner literature. One set of models stems from the work of Fama and French (1993). The other stems from the work of Chen, Roll, and Ross (1986). 

صفحه 10:
Reasoning that both are proxies for risk, they found (in multivariate tests) that a cross section of average returns is negatively related to size and positively related to book to market ratios. In heen simple terms, small firms and firms with low book ‏ی‎ ‎to market are riskier than other firms, do they incorporate these variables into a multi-index time series model of returns? Components of the series, such as the book value of equity, are reported at most four times a year. For time series tests, we need at least monthly observations. Fama and French formulated three indexes to explain the difference between the return on any stock and the riskless rate of interest (30-day Treasury bill rate). The concept behind the size and book to market indexes is to form portfolios that will have returns that mimic the impact of the variables. By forming portfolios that have observable monthly returns, Fama and French convert a set of variables that ‏ی‎ ‎cannot be observed at frequent intervals into a set ‏ی نوی‎ of traded assets that have prices and returns that can be observed at any moment of time and over any interval. 

صفحه 11:
Chen, Roll, and Ross hypothesized a broad set of influences that could affect security returns. Their work is based on two concepts. The first is that the value of a share of stock is equal to the present heen value of future cash flows to the equity holder. Tanta ne Thus an influence that affects either the size of future cash flows or the function (discount rates) used to value cash flows impacts price. Once a set of variables that affects prices is identified, their second concept comes into play. They argue that because current beliefs about these variables are incorporated in price, only innovations or unexpected changes in these variables can affect return. In a series of articles, Burmeister, McElroy, Chen Roll and and others (1986, 1987, 1988) have continued the ۳ i development of a multi-index model building on Ross Model the work of Chen, Roll, and Ross. They find that five variables are sufficient to describe security returns. They employ two variables that are related gee 4 to the discount rate used to find the present value ‏و8‎ laid the groundwork for many of cash flows, one related to both the size of the cash flows and discount rates, one related only to cash flows, and a remaining variable that captures 

صفحه 12:
CONCLUSION THERE ARE AN INFINITE NUMBER OF SUCH MODELS. THUS WE CANNOT eee aa eee CM Mala See UL oa es TO SINGLE-INDEX MODELS. MANY OF THE RESULTS ARE PROMISING. THIS PROBABLY DOES NOT SURPRISE THE READER. WHAT SURPRISES MOST STUDENTS IS THE ABILITY OF SIMPLE MODELS, SUCH AS THE SINGLE- INDEX MODEL AND OVERALL MEAN, TO OUTPERFORM MORE COMPLEX MODELS IN MANY TESTS. ALTHOUGH COMPLEX MODELS BETTER DESCRIBE THE HISTORICAL CORRELATION, THEY OFTEN CONTAIN MORE NOISE THAN INFORMATION WITH RESPECT TO PREDICTION. THERE IS STILL A GREAT DEAL OF WORK TO BE DONE BEFORE COMPLICATED MODELS CONSISTENTLY OUTPERFORM SIMPLER ONES. 

The Correlation Structure of Security Returns—MultiIndex Models and Grouping Techniques multi-index models and averaging techniques ▪ Multi-index models are an attempt to capture some of the nonmarket influences that cause securities to move together. The search for nonmarket influences is a search for a set of economic factors or structural groups (industries) that account for common movement in stock prices beyond that accounted for by the market index itself. Although it is easy to find a set of indexes that is associated with nonmarket effects over any period of time, as we will see, it is quite another matter to find a set that is successful in predicting covariances that are not market related. ▪ Averaging techniques are at the opposite end of the spectrum from multi-index models. Multi-index models introduce extra indexes in the hope of capturing additional information. The cost of introducing additional indexes is the chance that they are picking up random noise rather than real influences. Averaging techniques smooth the entries in the historical correlation matrix in an attempt to “damp out” random noise and so produce better forecasts. The potential disadvantage of averaging models is that real information may be lost in the averaging process General Multi-index Models Any additional sources of covariance among securities can be introduced into the equations for risk and return simply by adding these additional influences to the general return equation. Let us hypothesize that the return on any stock is a function of the return on the market, changes in the level of interest rates, and a set of industry indexes. This model can also be used if analysts supply estimates of the expected return for each stock, the variance of each stock’s returns, each index loading (bik between each stock i and each index k), and the means and variances of each index. This is the same number of inputs (2N 2L LN). However, the inputs are in more familiar terms. As discussed at several points in this book, the inputs needed to perform portfolio analysis are expected returns, variances, and correlation coefficients. By having the analysts estimate means and variances directly, it is clear that the only input derived from the estimates of the multiindex models is correlation coefficients. We stress this point because later in this chapter, we evaluate the ability of a multi-index model to aid in the selection of securities by examining its ability to forecast correlation coefficients. There is a certain type of multi-index model that has received a large amount of attention. This class of models restricts attention to market and industry influences. Alternative industry index models result from different assumptions about the behavior of returns and, hence, differ in the type and amount of input data needed. We now examine these models Industry Index Models Several authors have dealt with multi-index models that start with the basic single-index model and add indexes to capture industry effects. The early precedent for this work can be found in King (1966), who measured effects of common movement between securities beyond market effects and found that this extra market covariance was associated with industries. For example, two steel stocks had positive correlation between their returns, even after the effects of the market had been removed The assumption behind this model is that a firm’s return can be affected by the market plus several industries. For some companies this seems appropriate as their lines of business span several traditional industries. However, some companies gain the bulk of their return from activities in one industry and, perhaps of more importance, are viewed by investors as members of a particular industry. In this case, the effects on the firm’s return of indexes for industries to which they do not belong are likely to be small, and their inclusion may introduce more random noise into the process than the information they supply. This has prompted some authors to advocate a simpler form of the multiindex model: one that assumes that returns of each firm are affected only by a market index and one industry index. Furthermore, the model assumes that each industry index has been constructed to be uncorrelated with the market and with all other industry indexes AVERAGE CORRELATION MODELS The idea of averaging (smoothing) some of the data in the historical correlation matrix as a forecast of the future has been tested by Elton and Gruber (1973) and Elton, Gruber, and Urich (1978). The most aggregate type of averaging that can be done is to use the average of all pairwise correlation coefficients over some past period as a forecast of each pairwise correlation coefficient for the future. This is equivalent to the assumption that the past correlation matrix contains information about what the average correlation will be in the future but no information about individual differences from this average. This model can be thought of as a naive model against which more elaborate models should be judged. We refer to this model as the overall mean model A more disaggregate averaging model would be to assume that there was a common mean correlation within and among groups of stocks. For example, if we were to employ the idea of traditional industries as a method of grouping, we would assume that the correlation between any two steel stocks was the same as the correlation between any other two steel stocks and was equal to the average historical correlation among steel stocks. The averaging is done across all pairwise correlations among steel stocks in a historical period. Similarly, the correlation among any steel stocks and any chemical stocks is assumed to be equal to the correlation between any other steel stock and any other chemical stock and is set equal to the average of the correlations between each chemical and each steel stock. When this is done, with respect to traditional industry classifications, it will be referred to as the traditional mean model. The same technique has been used by Elton and Gruber (1973) with respect to pseudoindustries. FUNDAMENTAL MULTI-INDEX MODELS Two types of fundamental multi-index models have received a great deal of attention in the academic and practitioner literature. One set of models stems from the work of Fama and French (1993). The other stems from the work of Chen, Roll, and Ross (1986). Reasoning that both are proxies for risk, they found (in multivariate tests) that a cross section of average returns is negatively related to size and positively related to book to market ratios. In simple terms, small firms and firms with low book to market are riskier than other firms, do they incorporate these variables into a multi-index time series model of returns? Components of the series, such as the book value of equity, are reported at most four times a year. For time series tests, we need at least monthly observations. Fama and French formulated three indexes to explain the difference between the return on any stock and the riskless rate of interest (30-day Treasury bill rate). The concept behind the size and book to market indexes is to form portfolios that will have returns that mimic the impact of the variables. By forming portfolios that have observable monthly returns, Fama and French convert a set of variables that cannot be observed at frequent intervals into a set of traded assets that have prices and returns that can be observed at any moment of time and over any interval. Fama–French Models Fama and French laid the basis for a multi-index model based on firm characteristics in a series of articles published in the early 1990s. They found that both size (market capitalization) and the ratio of book value of equity to the market value of equity have a strong role in determining the cross section of average return on common stocks Chen, Roll, and Ross hypothesized a broad set of influences that could affect security returns. Their work is based on two concepts. The first is that the value of a share of stock is equal to the present value of future cash flows to the equity holder. Thus an influence that affects either the size of future cash flows or the function (discount rates) used to value cash flows impacts price. Once a set of variables that affects prices is identified, their second concept comes into play. They argue that because current beliefs about these variables are incorporated in price, only innovations or unexpected changes in these variables can affect return. In a series of articles, Burmeister, McElroy, and others (1986, 1987, 1988) have continued the development of a multi-index model building on the work of Chen, Roll, and Ross. They find that five variables are sufficient to describe security returns. They employ two variables that are related to the discount rate used to find the present value of cash flows, one related to both the size of the cash flows and discount rates, one related only to cash flows, and a remaining variable that captures Chen, Roll, and Ross Model The second group of fundamental multi-index models of stock returns was published by Chen, Roll, and Ross (1986). Although the purpose of their article was to explain equilibrium returns (a subject we discuss at great length in Chapter 16), their analysis laid the groundwork for many of the models that were to follow CONCLUSION THERE ARE AN INFINITE NUMBER OF SUCH MODELS. THUS WE CANNOT GIVE DEFINITIVE ANSWERS CONCERNING THEIR PERFORMANCE RELATIVE TO SINGLE-INDEX MODELS. MANY OF THE RESULTS ARE PROMISING. THIS PROBABLY DOES NOT SURPRISE THE READER. WHAT SURPRISES MOST STUDENTS IS THE ABILITY OF SIMPLE MODELS, SUCH AS THE SINGLEINDEX MODEL AND OVERALL MEAN, TO OUTPERFORM MORE COMPLEX MODELS IN MANY TESTS. ALTHOUGH COMPLEX MODELS BETTER DESCRIBE THE HISTORICAL CORRELATION, THEY OFTEN CONTAIN MORE NOISE THAN INFORMATION WITH RESPECT TO PREDICTION. THERE IS STILL A GREAT DEAL OF WORK TO BE DONE BEFORE COMPLICATED MODELS CONSISTENTLY OUTPERFORM SIMPLER ONES.