This article is devoted to the analysis of coherence of financial recommendations with respect to securities of the Russian companies. The study is based on the analysis of approximately 4000 recommendations and forecasts of 23 investment banks with respect to around forty securities of Russian stock market over the period of 2012-2014 years. The predictive history of each of the investment bank was considered as evidence in the framework of evidence theory. The coherence of recommendations was evaluated with the help of the so-called conflict measure between the evidence, which determined on the subsets of the set of all evidence. Then the study of coherence was reduced to analysis of values of the conflict measure. This analysis was performed with the help of game-theoretic methods (Shapley index, interaction index), network analysis methods (centralities), fuzzy relation methods, hierarchical clustering methods.
The forecasts and recommendations of financial analysts' (of investment banks) are
the important sources of information in decision making by the participants of the
financial market. The different aspects connected to the recommendations of financial
analysts' are reflected in the research literature. The influence of forecasts of financial
analysts' on the investors and the reaction of market on these forecasts is estimated in
[
The analysis of the coherence of forecasts and recommendations is one of the
important directions of research. The coherence of recommendations is determined as a
rule as similarity of recommendations that is given by different analysts with respect
to the same securities. The level of coherence of the recommendations is evaluated
more often as an average of all recommendations for a particular security. For
example, the dependence of coherence of the forecasts from a number of the shares
characteristics was investigated in [
In this study, analysis of the coherence of financial analysts' recommendations
about the value of the shares of Russian companies in 2012-2014 will be performed in
the framework of evidence theory (Dempster-Shafer theory, [
The work is structured as follows. The main notions of the evidence theory, the notion of the conflict measure are given in Section 2. The axiomatic of the conflict measure is discussed in this section too. The research database is described in Section 3. Section 4 is devoted to the description of evidence corresponding to database and the used conflict measure in the term of evidence. Section 5 is the main part of the work, in which study the coherence by the different methods. Finally, some conclusions from research are presented in Section 6. 2
Let X be a finite set and 2X be a powerset of X . The mass function and the focal
element are the fundamental notions in evidence theory. The mass function is a set
function m : 2X → [
The value m( A) characterizes the degree that true alternative from X belongs to
the set A ∈ 2X . The subset A ∈ 2X is called a focal element if m( A) > 0 . Let
A == {A} be a set of all focal elements. Then the pair F = (A , m) is called a body of
evidence. Let F ( X ) be a set of all bodies of evidence on X . Note that the body of
evidence can be considered for an arbitrary nonempty set X , if the set function
m : L → [
Let we have two bodies of evidence F1 = (A1, m1) and F2 = (A 2 , m2 ) . For example,
these evidences can be obtained from two sources of information. Then we have a
question about the conflict between the two evidences. Historically the function
K0 (F1, F2 ) connected with Dempster’s combining rule [
B∩C=∅, B∈A1,C∈A2
The axioms of the conflict measure are considered in [
The notion of a conflict measure (and corresponding axioms) was generalized in
[
K0 ({Fi1 ,..., Fik }) =
∑ Ai1 ∩...∩Aik =∅ mi1 ( Ai1 )...mik ( Aik ), Fis = ({Ais }, mis ) , s = 1,..., k , (2) satisfies the monotonicity condition: K(Bʹ) ≤ K(Bʹʹ) , if Bʹ ⊆ Bʹʹ and Bʹ, Bʹʹ ∈ 2M . This means that the adding of new evidence to the set of evidence does not reduce the conflict measure. 3
The conflictness (and coherence as the dual concept of) of the evidences about analysts' forecasts (investment banks) is investigated in this article. The conflictness characterizes in this case the degree of non coherence of forecasts for some set of experts.
The study is based on the analysis of approximately 4000 recommendations and forecasts of 23 investment banks with respect to around forty securities of Russian stock market over the period of 2012-2014 years. The databases of the agencies Bloomberg and RBC are the sources of information. The forecasts are presented by experts of the world's largest investment banks including such renowned companies as Goldman Sachs, Credit Suisse, UBS, Deutsche Bank and others.
Each investment bank makes recommendations of three types to sell/hold/buy with forecast of target price of the security. The target prices of forecasts are recalculated into the so-called relative values of target prices. The relative value of a target price is a ratio of the predicted price to the quotation of the security on the date of the forecast.
The boundaries of relative prices between the recommendations of various types were determined by maximizing number of recommendations that fall into the "corresponding" intervals: [0, 0.97), [0.97, 1.22), [1.22, +∞). Thus, we have nine sets, each of which represents the interval and a label of recommendation type: A(t) = [0, 0.97) , 1 A(t) = [0.97,1.2) , A3(t) = [1.2, +∞) , t = 1, 2, 3 , where t = 1 ‒ to sell, t = 2 – to hold, 2 t = 3 – to buy. 4
The belonging of the relative price of the forecast of a certain type (to buy/hold/sell) to one of the three intervals can be considered as an evidence of the investment bank. Then we can found the body of evidence for given investment bank. Let we fixed the i -th investment bank, i = 1,..., l ( l is a number of investment banks), ci(kt) is a number of belonging of relative price to interval Ak(t) , Ni is a general number of forecasts for i -th investment bank. Then mi ( Ak(t) ) = ci(kt) Ni is a frequency of belonging of relative price to interval Ak(t). The mass function mi satisfies the normalization condition: ∑t ∑k mi ( Ak(t) ) = 1 for all i = 1,..., l . Then Fi = ( Ak(t) , mi ( Ak(t) )) k,t evidence of i -th investment bank, i = 1,..., l . We can consider that all evidences have is a body of the same set of focal elements (even if mi ( Ak(t) ) = 0 for certain indexes) and all different focal elements A(t) are pairwise disjoint. Thus, the vector m = (m(s) )9s=1 , k m(k +3(t−1)) = m( Ak(t) ) , k = 1, 2, 3 , t = 1, 2, 3 corresponds bijectively to the body of evidence F = ( Ak(t) , m( Ak(t) )) . The set of all such evidence forms a simplex k,t S = {m(s) : m(s) ≥ 0 ∀s,
∑9s=1 m(s) = 1}.
The formula (2) for calculation of conflict measure K0 (F1,..., Fl ) can be simplified.
Proposition 1 [
The following measure
K(B) = 1− ∑kim:Fii∈nB mi ( Ak ), (3) satisfies also the monotonicity condition and will be considered as a conflict measure below instead of measure K0 in this paper.
Let mi(k) = mi ( Ak ) ∀k , i = 1,..., l and we denote Fi ∈ B ⊆ M for shot i ∈ B . We denote the measure K(Fi1 ,..., Fis ) as K(mi1 ,...,mis ) , if mip ↔ Fip , p = 1,..., s with consideration of the vector representation of evidence.
We will consider a coherence measure C = 1− K which is defined on 2M together with a conflict measure K . This measure characterized the degree of coherence of financial analysts' recommendations.
Below, we are interested in estimation of increments of the individual analysts' contribution in the total conflict: Δi K(B) = K(B ∪{i}) − K(B) , B ⊆ M \ {i} , Δij K (B) = K (B ∪{i, j}) − K(B ∪{i}) − K(B ∪{ j}) + K(B) , B ⊆ M \ {i, j} . Let (t)+ = {t0,,tt ≥<00,. The following proposition is true for measure (3) and the increments Δi K (B) and Δij K (B) .
Proposition 2. The following equalities are true for any mi , m j ∈ S and B ⊆ S : 1) Kij = K (mi ,m j ) = ∑k max{mi(k) , m(jk)} −1 = 12 ∑ k mi(k) − m(jk) ; 2) Δi K(∅) = 0 and Δi K(B) = ∑k max{mins∈B ms(k) , mi(k)} −1 = ∑k (mins∈B ms(k) − mi(k) )+ , if ∅ ≠ B ⊆ M \{i} ; 3) Δij K (∅) = Kij and Δij K (B) = −∑k (mins∈B ms(k) − max{mi(k) , m(jk)}) , if ∅ ≠ B ⊆ M \ {i, j} . +
Remark 1. The equality 1) in Proposition 2 shows us that the measure of pair conflict K(⋅,⋅) is a metric on the simplex S .
Remark 2. All pair increments of the conflict measure with non empty coalitions are not positive as follows from 3): Δij K (B) ≤ 0 ∀B ≠ ∅ , B ⊆ M \ {i, j} . This means that the inclusion of any analyst in the greater coalition increases the conflict measure by a smaller amount than the inclusion of the analyst in the smaller coalition.
5.1
If the monotone measure K is defined on the set of all subsets of M then we can
determine the contribution of i-th analyst in general conflict K(M ) of the set of all
analysts M as the difference K(M ) − K (M \ {i}) . More accurately the contribution
of i-th analyst in general conflict can be determined as a average contribution in the
conflict of the group (coalition) of analysts B : Δi K(B) = K(B ∪{i}) − K(B) , where
the average is computed for all groups (coalitions) of analysts B , B ⊆ M \ {i}. In this
case we will get so called Shapley value [
Proposition 3. The following formula is true for Shapley values of conflict measure (3): vi = ∑∅≠B⊆M \{i}α l ( B ,1) ∑k max{mk(i) , ms∈iBn mk(s)} − l −l1 , i = 1,..., l .
The contributions of all investment banks in the general conflictness of recommendations in period 2012-2014 are shown in the Fig. 1. These contributions were estimated with the help of Shapley values. The general conflictness for all 23 investment banks is equal 0.625 . , 0.1 Remark 3. The following denotations of investment banks are used on Fig. 1–4, in Tables 1– 2: 1 ‒ Alfa-Bank, 2 ‒ Aton Bank, 3 ‒ BCS, 4 ‒ Veles Capital, 5 ‒ VTB Capital, 6 ‒ Gazprombank, 7 ‒ Metropol Bank, 8 ‒ Discovery Bank, 9 ‒ Renaissance Capital, 10 ‒ Uralsib Capital, 11 ‒ Finam, 12 ‒ Barclays, 13 ‒ Citi group, 14 ‒ Credit suisse, 15 ‒ Deutsche Bank, 16 ‒ Goldman Sachs, 17 ‒ HSBC, 18 ‒ J.P. Morgan, 19 ‒ Morgan Stanley, 20 ‒ Raiffeisen, 21 ‒ Rye. Man&GorSecurities, 22 ‒ Sberbank CIB, 23 ‒ UBS.
The interrelation between the Shapley values of investment banks and the
profitability of forecasts was analyzed in [
Proposition 4. The following formula is true for pair interaction index of the conflict measure (3) Iij = l−11 Kij −
∅≠B⊆∑M \{i, j}α l ( B , 2) ∑k(mins∈B ms(k) − max{mi(k) , m(jk)})+ .
The values of the interaction index Iij that characterized the contributions of pairs of investment banks in the general conflict of forecasts about the value of shares of Russian companies in period 2012-2014 are shown in Table 1. The values for which Iij ≥ 0.013 are indicated only in the table. interesting for us in Table 1. It is the pairs (in decreasing order of Iij ): (12,14), (7,21), (6,11), (11,21), (12,20), (13,23). 5.3
We consider the coherence graph of recommendations G = (N,C) on the set of all investment banks, where N = {ni} be a set of all nodes (investment banks), C = {Сij } be a set of edges with weights Сij = 1− Kij and Kij be a value of conflict measure between the i -th and j -th investment banks, which calculated by formula (3). We can consider the “roughenned” coherence graph instead of the graph G for a better visualization with weights Сij = ⎨⎩⎧10,, KKijij <≥hh,, where h is a threshold value. The such graph, which constructed by the data of value of shares of Russian companies in period 2012-2014, is shown in the Fig. 2 for h = 0.15 .
The matrix of pair coherence of recommendations C = {Сij } is a symmetric and
non-negative. We investigate the problem of finding such investment banks, which
have a most influence on coherence of recommendations. We will consider the
socalled eigenvector centrality [
We have λmax = 17.9 for the data of value of shares of Russian companies in period 2012-2014. The values of coordinates of corresponding eigenvector (centrality vector) are shown in the Fig. 3. As can be seen from this figure, the greatest influence on the coherence of the recommendations in accordance with the values coordinates of the centrality vector have the banks (in descending order of influence) 9,1,18,15,23, etc.
The centrality vector correlated greatly and negatively with the Shapley vector. The corresponding correlation coefficient is equal to −0.86 .
However, pairwise coherencies of recommendations do not give a complete picture of the more complex (not pairwise) interactions. This kind of interaction can be revealed with the help of analysis of the cluster structures of relations on the set of evidence, which is given by a conflict measure. 5.4 Let
M ={=F1,..., Fl } be a set of evidence. Then the pair conflict
measure
Kij = K (Fi , Fj ) and the corresponding coherence measure Сij = 1 − Kij can be
considered as binary fuzzy relations, which are given on the Cartesian square M 2 . The
relation C = (Cij ) is a similarity relation (i.e. reflexive and symmetric fuzzy relation)
[
Thus, the matrix (Kˆ ij ) can considered as a matrix of distances between the analysts. The corresponding matrix of coherence Cˆ = (Cˆij ) can considered as a similarity matrix between the investment banks.
The structure of coherence of investment bank recommendations can be analyzed with the help of the α-cut Cˆα = {(F,G) : Cˆ (F,G) ≥α }, α ∈ (0,1] of the fuzzy similarity relation Cˆ . For every fixed α ∈ (0,1] the set Cˆα defines the equivalence relation, which induces a partition of evidence M on the equivalence classes.
The equivalence classes of coherence indicated in Table 2 (only not singletons) for some values of α ∈ (0,1] for the data of value of shares of Russian companies in period 2012-2014. Each of these classes represents set of analysts, whose recommendations have а large degree of coherence. This degree is defined by threshold α.
We consider the matrix Kˆ = 1− Cˆ , where Cˆ is a transitive closure of similarity relation C = 1 − K , K = (Kij ) and Kij is a value of conflict measure between the i -th and j -th investment banks, which calculated by formula (4). A conflict measure considered on the set of evidence M ={=F1,..., Fl } .
The cluster analysis of coherence of analysts' recommendations will be performed
using one of the methods of hierarchical clustering. For example, we will use the
Unweighted Pair-Group Method Using Arithmetic Averages (UPGMA) [
In this paper, the coherence of investment bank recommendations was studied for the data of value of shares of Russian companies in period 2012-2014. The specific of the study consists in using the conflict measure defined in the framework of the belief function theory for determination of the coherence of recommendations. The analysis of coherence was reduced to analysis of values of the conflict measure. This analysis was performed with the help of game-theoretic methods (Shapley index, interaction index), network analysis methods (centralities), fuzzy relation methods, hierarchical clustering methods.
The following results were obtained: ─ the ranking of investment banks with respect to their contribution to the overall coherence of the recommendations using the Shapley value was obtained; ─ the contributions of the separate pairs of investment banks in the total conflict of recommendations of the Russian companies with the help of the interaction index were evaluated; ─ the investment banks rendering the greatest influence on the coherence of the recommendations were detected with the help of the analysis of the centrality; ─ the sets of analysts whose recommendations have a greater degree of coherence were identified with the help of analysis of fuzzy similarity relations generated by the coherence measure; ─ the main cluster structures of investment banks with respect to coherence of the recommendations were identified by the method of hierarchical clustering; ─ the expressions for some of the calculated parameters (Shapley values, interaction index) were obtained in the terms of evidence.
In addition, we have shown that the set of the key investment banks, have made the greatest contribution to the overall coherence of the recommendations obtained with the help of Shapley values and the methods of analysis of the centrality, are close together. Similarly, the cluster structures of analysts, whose recommendations have a greater degree of coherence, obtained by the methods of analysis of the fuzzy similarity relations and methods of the hierarchical clustering, are close to each other. Indirectly, this confirms the importance of the results.