Aggregation of Forecasts and Recommendations of Financial Analysts in the Framework of Evidence Theory Ekaterina Kutynina Alexander Lepskiy National Research University - Higher School of Economics, Moscow, Russia The 10 th conference of the European Society for Fuzzy Logic and Technology - EUSFLAT 2017, September 11 - 15, 2017, Warszawa, Poland Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 1 / 24
Preamble The Problem of Aggregation of Recommendations and Forecasts Among the tasks associated with evaluating the recommendations of financial analysts, the important challenge is to aggregate recommendations and forecasts. We have: a number of financial analysts’ recommendations and forecasts regarding the expected potential for growth in the prices of shares of a particular company (”sell”, ”hold”, ”buy” + target prices of shares); the data on the real value of the shares of this company in the certain period in the past; the history of financial analysts’ recommendations in the past. How we can aggregate financial analysts’ recommendations in the best way? Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 2 / 24
Preamble Some Studies on the Aggregation of Analysts’ Recommendations in [Berkman &Yang 2016] it was demonstrated that taking into account the recommendations aggregated at the country level improves profitability on the international stock market; in [Howe et al. 2009] it is shown that the inclusion of changes in aggregated recommendations on average positively affects for income and profit; in [Huang et al. 2009] was showed the importance of the recommendations combining and forecasts of the target price for building a profitable investment strategy; in [Kim et al. 2001] it was argued that the average value is inefficient for aggregating forecasts. This inefficiency increases with the growth in the number of forecasts in aggregation. Also there were offered some procedures for selecting ”good” forecasts for aggregation. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 3 / 24
Preamble Evidence Theory in Financial and Economic Analysis A weighted consensus forecast is usually used as a general method of aggregation. The actual problem is to choose such aggregation procedures, which most fully take into account information from individual analysts ( uncertainty ), the history of their forecasts ( reliability ), their correlation ( conflict ), etc. All the noted features can be described within the theory of evidence. The theory of evidence is widely used in financial and economic analysis, for example: in forecasting of investments’ profitability on the basis of interval expert assessments [Utkin 2006]; in marketing analysis of data [Kanjanatarakul et al. 2014]; in forecasting of income on the stock market [Autchariyapanitkul et al. 2014]; in foresight studies [Xu et al. 2014], etc. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 4 / 24
Preamble Outline of Presentation the background of evidence theory; the survey of financial analysts’ data on the Russian Stock Market; the construction of bodies of evidence of financial analysts’ and implementation of different aggregation procedure; experimental results. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 5 / 24
Evidence Theory and Combining Rules Background of Evidence Theory Ω be a some universal set of all possibility values of experimental results, P (Ω) be a powerset of Ω; a mass function m : P (Ω) → [0 , 1], � A ∈P (Ω) m ( A ) = 1; A ⊆ Ω is called a focal element if m ( A ) > 0; the pair F = ( A , m ) is called a body of evidence, BE ; F (Ω) be a set of all BE on Ω. BE is said to be categorical (is denoted as F A = ( A, 1)) if it has only one focal element; BE F Ω = (Ω , 1) is said to be vacuous ; if F j = ( A j , m j ) ∈ F (Ω) and � j α j = 1, α j ∈ [0 , 1] ∀ j , then F = � j α j F j = ( A , m ) ∈ F (Ω), where A = � j A j , m ( A ) = � j α j m j ( A ); we have F = � A ∈A m ( A ) F A ∀ F = ( A , m ); BE is said to be simple if it has the form F ω A = (1 − ω ) F A + ωF Ω , ω ∈ [0 , 1]. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 6 / 24
Evidence Theory and Combining Rules Combining Rules Suppose there are two independent groups of experts who provide their forecasts, given in the form of two bodies of evidence F 1 = ( A 1 , m 1 ) and F 2 = ( A 2 , m 2 ) on Ω. We want to combine these two BE in one BE. There are a few popular combining rules. The mass function m = m 1 ⊕ m 2 of new BE obtained with the help of Dempster’s rule is calculated by the formula 1 � A � = ∅ , m ( A ) = m 1 ( A ) m 2 ( B ) , 1 − K B ∩ C = A where K = K ( F 1 , F 2 ) = m ( ∅ ) = � B ∩ C = ∅ m 1 ( A ) m 2 ( B ). The value K characterizes the amount of conflict in two information sources. If K = 1 then it means that information sources are absolutely conflict and Dempster’s rule cannot be applied. Since the combining rule ⊕ is an associative operation, then any finite number of BE can be combined. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 7 / 24
Evidence Theory and Combining Rules Boundaries of Expectation and Discounting Let all focal elements of BE F = ( A , m ) are bounded sets in R . Then the lower and upper boundaries of expectation of belonging of true alternative can be calculated: � � E[ F ] = A ∈A m ( A ) inf { A } , E[ F ] = A ∈A m ( A ) sup { A } . The degree of reliability of information source be taken into account with the help of discount coefficient α ∈ [0 , 1] [Shafer, 1976]: m α ( A ) = (1 − α ) m ( A ) m α (Ω) = α + (1 − α ) m (Ω) . ∀ A � = Ω , If α = 1, then it means that information source is absolutely not reliable. If α = 0, then information source is absolutely reliable. The some combining rule applied after discounting of initial BE. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 8 / 24
The Survey Data The Survey Data As a rule, an expert estimation of the share price behaviour consists of two indicators: the target price and direct recommendation . Analysts’ information: the target price is the share price expected by the expert at the end of the forecast period; recommendations of analysts can take the values ”sell”, ”hold”, ”buy”; 7 Russian banks and 3 analytical companies that provide their annual forecasts for 16 Russian companies represented on the Russian stock market during January 2010 – May 2016; the data on the real value of the shares of these companies in the period from January 2010 to May 2016. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 9 / 24
The Survey Data Determination of Focal Elements 1. We used the relative target price target price of the share stock Crv ( stock, t )= actual price of stock on the date of the forecast t. 2. Boundary values of focal elements are calculated as a solution to the problem of minimization an error in the incorrect classification of recommendations Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 10 / 24
Determination of BE Determination of Bodies of Analysts’ Evidence During one year one analytical company gives several recommendations. For each analytical company i and each stock BE can be constructed F i,stock = ( A i,stock , m i,stock ). Each BE has not more than three focal elements S i,stock , H i,stock , B i,stock , and mass functions m i,stock ( A ) equal to relative frequency of recommendation. The set Ω is added to the set focal elements in the case of discounting. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 11 / 24
Determination of BE The Problem of Finding the Optimal BE Let we have n categorical BE (recommendation of i -th source in during a year) that ordered by the time F A s , where A s ∈ { S i,stock , H i,stock , B i,stock } , s = 1 , . . . , n . We will consider a BE s =1 F α s F ( α 1 , . . . , α n ) = ⊕ n 1 ≥ α 1 ≥ · · · ≥ α n ≥ 0 A s , for finding of recommendation of i -th source on the end a year with account of revision of forecasts. Here F α s A s = (1 − α s ) F A s + α s F Ω . Criteria for optimization C ( α 1 , . . . , α n ) = (E 0 [ F ( α 1 , . . . , α n )] − p ) 2 → min , where p is an actual last ”pre-forecast” relative price of the share, E 0 [ F ] = 1 2 (E[ F ] + E[ F ]) is the middle value of the interval of expectation of the forecast price. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 12 / 24
Determination of BE Example Let n = 4, F 1 = F 2 = F 4 = F S (”sell”) and F 3 = F H (”hold”). Then F ( α 1 , α 2 , α 3 , α 4 )= F α 1 S ⊕ F α 2 S ⊕ F α 3 H ⊕ F α 4 S = m ( S ) F S + m ( H ) F H + m (Ω) F Ω . F α 1 S , F α 2 S , F α 3 H , F α 4 � � The conflict of discounting BE is equal K = K = S =(1 − α 3 )(1 − α 1 α 2 α 4 ); the values of a mass function are equal: m ( S )= α 3 (1 − α 1 α 2 α 4 ) , m ( H )= α 1 α 2 (1 − α 3 ) α 4 , m (Ω)= α 1 α 2 α 3 α 4 . 1 − K 1 − K 1 − K Consequently, we have C ( α 1 , α 2 , α 3 , α 4 ) = (E 0 [ F ( α 1 , α 2 , α 3 , α 4 )] − p ) 2 = � 2 � α 3 (1 − α 1 α 2 α 4 ) S 0 + α 1 α 2 (1 − α 3 ) α 4 H 0 + α 1 α 2 α 3 α 4 Ω 0 − p , α 3 + α 1 α 2 α 4 − α 1 α 2 α 3 α 4 where S 0 , H 0 , Ω 0 are middles of intervals of relative prices. Kutynina & Lepskiy (HSE) Aggregation of Forecasts EUSFLAT 2017 13 / 24
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