Relationships Between a Stock Market Price Index and Its Mutually Exclusive Subindices

IUST International Journal of Engineering Science, Vol . 19 , No .5-2, 200 8, Page 27- 31

RELATIONSHIPS BETWEEN A STOCK MARKET PRICE INDEX AND ITS MUTUALLY EXCLUSIVE SUBINDICES

Hamidreza Farhadi

Abstract : In this article we find relationships between a stock market price index and its mutually exclusive subindices. In particular, we show th at when moving from one time to another, the return of such an index is a convex combination of the returns of its mutually exclusive subindices if at most one base adjustment is done.

Keywords: stock market price index, mutually exclusive subindices, con vex combination

1. Introduction1 how optimistic or pessimistic people are about the This section is devoted to discuss some basics status of the economy in that period of time; therefore about the capital market indices, and to give the stock market indic es are used for economical motivation about the results to be discussed in the analyses too. section "Main Results" below. Those who want to get Generally a stock price index is in the form : to know more about the f ormulae and mechanisms of ca pital market price indices, could c onsult the M I = t × Base Po int ( 1) references [1] and [ 3]. In this section one may also see t B the term "Free Float", a concept that found its way to t where the capital markets in the recent years; to learn about it , consult th e references [1], [2], and [ 3]. However , • "Base Point" can be any positive integer (usually the free float concept does not play a crucial role in taken to be 100 ). This value is fixed at all times. this paper . • M t is the (free-float adjusted, or total share) A stock market index compares the total market value market capital of the constituents of the index. of the companies in the index portfolio with the total This number is equal to the sum market value at a base point in time. Fo r example, if n the index value at the initial time was set to 100 and if M t = ∑ pit q it the current index value is 120 , this means that the i=1 average increase in prices has cause the market value where, pit is the price of stock i at time t , and to increase by 20% . q it is the number of shares of company i at time Stock market indices are constructed for various t. portfoli os of stocks. Some are constructed for large companies while some are constructed for small • Bt , called "base market value", is a number that companies. Indices are also constructed for different changes only when there is a change in the (free - industry sectors . Over 40 sector indices are now float adjusted, or total) number of shares of a calculated in Tehran StockArchive Exchange. ofsecurity SID in the index (see below); its value at the Capital market indice s have a few applications. As an initial time t = 0 is set equal to M.0 important of such applications , they are used as Since the "Base Point" is fixed throughout, we may benchmark for the performance of investment combine this value with the value B t (by substituting portfolios. The mutual funds who invest in the stock Bt markets, evaluate their performances against the Bt by ) to write the index in the form Base Po int performance of a benchmark s uch as S &P500 . As M another important application , the move ment of the t . This convention will not affect the results I t = stock market index in a period of time is a reflection of Bt given below as "Main Results", as the result s of tha t I Paper first received July . 15 , 200 7, and in revised form Dec . section are mainly built on the quotient t in which 19 , 2009. I Hamidreza Farhadi , Departm ent of Mathematical Sciences, Sharif t −1 University of Technology , the " Base Poi nt" cancels out. *Dedicated to FEAS (Federation of Euro -Asian Stock Exchanges)

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Generally , two phenomena affect the total market "mutually exclusive subindices" of the index formed capital for a group of securities: one is the price on the set A of securities . So according to th is movement and the other is the change in the (fre e–float definition , "TOPIX Core 30 " and "TOPIX Large 70 " adjusted , or total) number of securities. Some of the are mutually exclusive subindices of "TOPIX 100 ". ways causing change in the number of shares are : 1) In this paper we are going to find relationships between adding a new security to the index , 2) deleting a a capital market index and its mut ually exclusive security from the index , 3) capital increase . A change subindices. Through oral communications with so me in M t caused by any of such events is not the result of people in the Tehran Stock Exchange back in July a fluctuation in the stock market. Therefore to remove 2005 , I noticed a wrong idea in them that if at some the effect of such change to M , a same proportional time the value of a main index falls between the values t of its mutually exclusive subindices, then from that change (increase or decrease, accordingly) needs to be time on, the value of the main index shoul d remain done to B.t For example, if a capital increase of ∆ is betwee n the values of those subindices. Providing an announced by a company at time −1t , the n the example below, we show that this idea is not generally increase in the value of the associated company causes true. T hen in proposition 1 of the section 2 we will the market cap M t−1 to increase to M t−1 ∆+ resulting prove the true relationship between a major index and its mutually exclusive subindice s. Subsequently, i n ∆ in a change of ×100 p ercent to M t−1 . Therefore proposition 2, we will give a generalization to the M t−1 counter example we are giving in the current section, the "base value" Bt−1 needs to increase accordingly to by showing that the values given in this example do not matter.  ∆    For the sake of simplicity throughout this article, we Bt = 1+ × Bt−1 to secure the continuity of th e  M t−1  assum e a m ajor index whose value at ti me t is denoted index when moving from time −1t to time t . The by I3,t , and its two mutually exclusive subindices value of ∆ could have been resulted by capital I and I. increase, deletion of a company from the index, 1,t 2,t addition of a company to the index, or some other ways. All such cases are treated similarly resulting in Example . Assume an index I3 together with its the identity mutually exclusive subi ndices I1 and I 2 whose values at time −1t are M+∆ M +∆ B=t−1 × B = t − 1 ( 2) tM t −1 I t−1 t − 1 I 1,t − 1 = 2000  I = 1000 M  2,t − 1 as I = t −1 . The value ∆ can be positive,  t −1 I = 1000 • Bt −1  3,t − 1 negative, or zero (a value of zero corresponds to a phenomenon such as "stoc k split" where no adjustment So I3,t−1 is between I and I . Suppose that at to the base value is needed). 1,t − 1 2,t − 1 We explore more details on indices needed for the rest that time, th e market caps and the base value associated of this article, by looking at some example, e.g. the with these three indices are Tokyo Stock Exchange. The main stock price index at Tokyo Stock Exchange (TSE) is TOPIX . It is a (free– M1,t− 1=2000 , B 1, t − 1 = 1 float adjusted) value –weighted index. Various sub -  M2,1t−=4000, B 2,1 t − = 4 indices of TOPIX are nowArchive calculated and published by of SID TSE. Depending on what sort of categorization is M3,1t−=6000 , B 3,1 t − = 6 • concerned, the companies forming a stock-market price index are grouped into non –overlapping subsets; then a Suppose that following time −1t and before the next price index is calculated on each group. For example, dissemination of the index at time t, a company in the the constituents of the "TOPIX 100" are those of the index I announces a capital increase of 10 unit of "TOPIX Core 30 " and "TOPIX Large 70 " combined. 1 As another example, the constituents of the TOPIX money. So the base value must be adjusted for time t. form three categories "Large -Si zed Stocks", "Medium - Since the same company is in the index I3 too, an

Sized Stocks", and "Small -Sized Stocks" on each of adjustment is needed for both B1 and B3 (see E q. which an index is calculated. "TOPIX Sector Indices" (2)) : is another way of grouping the constituents of TOPIX. In general if a set A of companies is divided into M +10 2000+ 10 smaller groups A , A , … on each of which an B=1,t − 1 ×= B ×=1 1.005 1 2 1,tM 1, t − 1 2000 index is formed, then we call these partial indices 1,t − 1

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M 3,t − 1 +10 6000+ 10 subindices. Suppose we have three indices I1 , I 2 and B3,t= ×= B 3,1 t − ×=6 6.01 • M 3,t − 1 6000 I3 with corresponding market caps M1 , M 2 and M 3

and the corresponding base values B1 , B2 and B3 . Suppose that the number of shares of the companies of To denote any of these at some ti me t, we just add a the index I 2 has not changed so that no adjustment is subscript t. needed for the case of I:2 Proposition 1. Suppose that between two times −1t and t (which are not necessarily consecutive times at B2,t= B 2,1 t − = 4 • which the indices are calculated), the base values B1 , B and B are either not adjusted or are adjusted at Further assume that at time t the prices hav e changed in 2 3 such a way that and . I 3,t M 1,t = 2050.2 M 2,t = 4160 most one time. Then the return is a convex Since the main group of the companies is partitioned I 3,t − 1 into two groups, we must obviously have I I combination of the returns 1,t and 2,t ; more I 1,t − 1 I 2,t − 1 M3,t= M 1, t + M 2 , t = 6210.2 • precisely, I I I Now then 3,t=λ 1, t +(1 − λ ) 2 , t ( 3) I3,t− 1 I 1, t − 1 I 2, t − 1 M 2050.2 I =1,t = = 2040 1,t where B1,t 1.005

M + ∆ M 4160 1,t − 1 1 I =2,t = = 1040 λ = 2,t M1,1t−+ M 2,1 t − +∆+∆ 1 2 B 2,t 4

and M 3,t 6210.2 I 3,t = = = 1033.31 B 3,t 6.01 M 2,t − 1+ ∆ 2 1 −λ = ( 4) M1,1t−+ M 2,1 t − +∆+∆ 1 2 how ing that the value of value the that showing I3 at time t falls out of the range determined by I and I. 1,t 2,t in which ∆ , ∆ , and ∆ denote the associated 1 2 3 changes in the capitals (a value of zero for ∆ would 2. Main Results correspond to the case of no adjustment to the Though we prove the following results for three associated base). See Fig . 1. indices, yet they can similarly be proved for a mother index and any number of its mutually exclusive Archive of SID

Fig. 1.

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Proof . First of all , note that since the indices I1 and I 2 Case I . . I1,1t−≤ I 3,1 t − < I 2,1 t − • have n o constituent in common, the equality ∆ 3 = ∆1 + ∆ 2 holds for the time when the adjustments In this case , take ∆1 = o . To decide on ∆ 2 , first note are done to the bases. Secondly, because of mutual I that 3,t − 1 is less than one, so is the term exclusiveness, the relationship I M= M + M holds among st the market caps of 2,t − 1 3,t 1, t 2 , t I these indices. So then (1− λ ) 3,t − 1 I 2,t − 1 (because 1− λ is b etween zero and one), where λ is M3,t MM 1, t+ 2, t IBIB 1, tttt ×+× 1, 2, 2 , I = = = as in the previous proposition. On the other hand, the 3,t B 3, t M3,-t1+∆ 3  M 3 ,- t 1 +∆ 3  term    I3,-t1  I 3 ,-1 t  I3,1t− M 1,1 t −+ ∆ 1 I 3,1 t − M1,1t−+∆ 1  M 2,1 t − +∆ 2  λ = I1,t×  + I 2 , t ×  IMM+ +∆+∆ I I  I  1,1ttt−−− 1,1 2,1 1 2 1,1 t − = 1,t− 1  2, t − 1  M + ∆  3,1t − 3 is close to zero when ∆ is large. So the sum of these   2 I 3,t − 1   two values is less than 1 when ∆ 2 is large:

I1,t I 2 , t ×()M1,1t− +∆+ 1 ×() M 2,1 t − +∆ 2 I1,t− 1 I 2, t − 1 I3,t− 1 I 3, t − 1 = • λ+(1 − λ ) < 1 ( 6)   • M 3,1t − + ∆ 3 I1,t− 1 I 2, t − 1   I 3,t − 1  Rearranging these terms, we will get : Dividing both sides by I , then collecting terms 3,t−1 I I  while noting that ∆ = ∆ + ∆ , one gets : I <1,t− 11(1) − − λ 3, t − 1 ( 7) 3 1 2 3,t − 1   • λ I 2,t − 1 

I1,t I 2 , t ×()M1,1t− +∆+ 1 ×() M 2,1 t − +∆ 2 I I I Now recall that λ contains only the information up to 3,t= 1,1 t− 2,1 t − •(5) th e point where the associated stock goes out of halt I M +∆ +∆ 3,1t− 3,112 t − (i.e., it contains no information from the next time t of disseminations of the indices). So at any time t after By s ubstituting the eq uality into Eq. that, it is possible to assume that the prices have moved M3,t− 1= M 1, t − 1 + M 2, t − 1 (5), the pro of is complete. █ in such a way that I1,t has fall en between the two numbers in (7):

We now show that it is possible to have I 3,t − 1 between I1,t− 1 I 3, t − 1  and (at time −1t ) but to have I out of I<< I 1(1) −− λ  ( 8) I 1,t − 1 I 2,t − 1 3,t 3,1t− 1 , t • λ I 2,t − 1  the range determined by I1,t and I 2,t , no matter what the initial values I , I and I are: The second inequality in ( 8) can be rewritten as 1,t − 1 Archive2,t − 1 3,t − 1 of SID I I  I <2,1t− 1 − λ 1 , t Proposition 2. If I 3,t − 1 is between the values 3,t − 1   1− λ I 1,t − 1  • I1,t−1 ≠ I 2,t−1 , then it is al ways possible to have capital increases ∆1 and ∆ 2 at time t (for I1 and I 2 Since the indices I1 and I 2 have no components in respectively) so that I3,t falls out of the interval with common, it is possible to assume that I 2 has moves in endpoints I1,t and I.2,t such a way that

Proof . Without loss of generality we assume I2,1t−  I 1 , t  I =1 − λ  ( 9) ; so then with at 2,t • I1,t− 1< I 2, t − 1 I1,1t−≤ I 3,1 t − ≤ I 2,1 t − 1− λ I 1,t − 1  least one of the inequalities being strict. Now one and only one of the following cases happen: From these two last observations we have

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I< I ( 10) 3,t− 1 2 , t • I2,t< I 3 , t • ( 16 )

But ( 9) can be rewritten as This inequality together with ( 13 ) and the assumption , give us I I I2,t= I 2, t − 1 λ1,t+(1 − λ ) 2 , t = 1 ( 11 ) • I1,t− 1 I 2, t − 1 I1,t< I 3 , t • ( 17 ) Based on the first proposition, this means that I 3,t =1, i.e. The inequalities (16) and ( 17 ) end the proof by I 3,t − 1 showing that I3,t has fallen out of the range

determined by I1,t and I 2,t . ( 12 ) I3,t= I 3, t − 1 • References Comparing this with ( 10 ), we will have I3,t< I 2 , t . [1] Dow Jones Indexes . "Dow Jones Global Indexes ", Again comparing (12 ) with the left hand side of ( 8) we documentation from the website of the Dow Jones Indexes at: 2007 , get . So I falls out of the range determined I3,t< I 1 , t 3,t http://averages.dowjones.com/mdsidx/downloads/DJGI_ Rulebook.pdf by I1,t and I 2,t proving the claim in this case. [2] S & P. "Rethinking Float Adjustment in the Context of Case II : I equals : 3,t − 1 I 2,t − 1 I1,1t−< I 3,1 t − = I 2,1 t − U.S. Indices ", documentation from the website of the Standard and Poor's at: 200 3, http :// www 2.standardandpoors.com /spf /pdf/ index /10270 In this case, take ∆1 = ∆ 2 = o . Since I1,t− 1< I 2, t − 1 , 3_us _float .pdf we may a ssume that the price at time t have moved only as much so as to have I1,t fallen strictly between [3] TSE ., "Tokyo Stock Exchange Index Guidebook", I and I documentation from the website of the Tokyo Stock 1,t-1 2,t-1: Exchange at: 2006 , ( 13 ) http://www.tse.or.jp/english/market/topix/e -yoryo.pdf I1,t− 1< I 1, t < I 2, t − 1 • On the other hand, it is not contradictory to ass ume that the prices of the components of I 2 have not changed while moving from time t -1 to time t, i.e.

I2,t= I 2,1 t − • ( 14 )

So far we have had the permission to assume that

I I1,t 2,t =1 and 1< . But according to the above I 2,t − 1 I1,t − 1

I 3,t I proposition, the number is between 2,t =1 I 3,t − 1 I 2,t − 1 I 1,t Archive of SID and , so that: I 1,t − 1

I3,t I 1 , t 1.< < • I3,t− 1 I 1, t − 1

The inequality on the left s ide reads

( 15 ) I3,t− 1< I 3 , t •

Since we already have I3,1t−= I 2,1 t − = I 2 , t , the inequality (15) reduces to