that, for example, the CPI or SPI has increased by 100 g%,. × end of story. ..... (
2010, p. 5) state in their publication Index Mathematics: Index Methodology that ...
Indices (2010, pp. 5-6) as ..... We have made adjustments that occur from
company addition and deletion ..... THE DYNAMICS OF NEW RESOURCE
PROJECTS A.
ECONOMICS
VOLATILITY AND STOCK PRICE INDEXES
by
Kenneth W Clements and H Y Izan and Yihui Lan Business School The University of Western Australia
DISCUSSION PAPER 11.16
August 2011
VOLATILITY AND STOCK PRICE INDEXES*
by Kenneth W. Clements, H.Y. Izan, and Yihui Lan Business School The University of Western Australia
DISCUSSION PAPER 11.16
Abstract The stochastic approach to index numbers has been successfully applied to the estimation of the Consumer Price Index (CPI). One distinct advantage of this approach is that it provides the whole distribution of the index, not simply a point estimate of the rate of inflation. In this paper, we extend the application of the stochastic approach to the estimation of a stock market index. We demonstrate how this approach can be used to identify “redundant stocks” that do not contribute significantly to the overall index. For index tracking purposes, these stocks can be safely excluded.
*
Kenneth Clements is a Winthrop Professor of Economics and BHP Billiton Research Fellow, H.Y. Izan is a Winthrop Professor in Accounting and Finance and Yihui Lan is an Associate Professor in Finance, Business School, UWA. This research was supported in part by BHP Billiton and the Australian Research Council. We thank Grace Gao for helpful discussion and research assistance. We have also benefited from the research assistance of Liang Li, Haiyan Liu, Jiawei Si and Stephane Verani.
1. Introduction Of the whole universe of economic data, the consumer price index (CPI) and the stock price index (SPI) probably attract the most attention. While the CPI is vital for the setting of interest rates by the central bank, for deflating nominal incomes into their real counterparts, and as the basis of “cost escalation” clauses in many contracts, the SPI is used by individual and institutional investors as a benchmark against which to evaluate the performance of their portfolios, by technical analysts or chartists when they make buy and sell recommendations, and by economists to study the relationship between financial markets and economic activity. But the SPI also plays a broader role in society. When stock markets are down, retirees’ incomes are squeezed, investment bankers’ bonuses evaporate, endowment earnings of universities fall, companies cut back on expansion, and even governments feel the pressure by the generalised anxiety caused by depressed equity prices. Given their value to such a wide group of users, it is surprising that the theory underlying indices of stock prices has not been examined more thoroughly. Index numbers are summary measures of the information on individual prices and quantities. The traditional approaches to index numbers begin and end with a single number, so that, for example, the CPI or SPI has increased by 100 g%, end of story. We argue that this is a limited way to summarise price behavior. Suppose price share i grows at rate g i , so that the transition from base period 0 to current period 1 takes the form pi1 1 g i pi0 . Consider the
extreme case when all changes are identical, g i g,i 1, , n stocks. Here, as all prices change proportionately, there can be no disputation that the index also increases by g. Panel A of Figure 1 illustrates this scenario with n=5 component prices. The graph on the far left of this panel shows that the scatter of the current- against base-period prices lies on a straight line passing through the origin with slope 1 g. The growth rate g is the change in the overall index, as can be seen from the frequency distribution on the far right of Panel A. In reality however, prices do not move proportionally and the graph on the far left of Panel B illustrates this case. Here, some prices grow faster than average, others slower, but the average growth rate, g, is the same as in Panel A. The index clearly lies somewhere in the middle of individual prices, as shown in the far right of Panel B. In this case, we have pi1 1 g pi0 error , where the pricing error results from disproportionate behavior. While the average growth is the same in Panels A and B, they represent fundamentally different experiences. The disproptionate price movement of Panel B implies that the overall index is less well-defined -- we can have less confidence that prices grow on average by 100 g% in this case, as compared to Panel A when prices move in tandem. While 1
this issue is at the heart of the index-number problem, surprisingly the statistical implications have not been fully appreciated. In the past few decades, index-number theory and practice have gone through a period of renewed interest and new insights. One development is the stochastic approach, which has been resurrected by Balk (1980), Clements and Izan (1981, 1987), and Selvanathan and Rao (1994).1 The overarching feature of the stochastic approach is that it provides the whole distribution of the index, not just the point estimate. Each price change is treated as a noisy reading on the common trend in all prices, with this trend playing the role of the index. The least-squares method provides an elegant solution to the problem of how best to combine the individual price changes so as to minimize the noise. Accordingly, the stochastic approach offers a richer framework that opens up new opportunities; and it does this by using the same basic information of traditional approaches. With randomness playing a central role, the stochastic approach is an appealing way to analyse stock prices. This paper investigates the use of the stochastic approach in forming a stock price index and contributes in three aspects. First, in Section 2, we set out the basics of index-number theory and relate it to conventional indexes of stock prices. Second, we apply the stochastic approach to share prices in Section 3. Finally, in Section 4, we apply the framework to the issue of portfolio tracking and investigate whether it is possible to ignore certain stocks on the basis of their contribution to the index. Concluding remarks are provided in Section 5. 2. Index Numbers This section sets the scene for what is to follow by providing the necessary basics of index-number theory. The presentation is generic and stylised in order to highlight just essential elements of the index-number approach. To focus on the analytics, we deliberately ignore a number of practical issues, such as the chaining of index numbers, a procedure that is widely used by statistical agencies.
1
According to Diewert (2008), the stochastic approach is now one of the four main approaches to index number theory. The other three are the fixed basket approach, the test approach and the economic approach. Applications of the stochastic approach include the measurement of inflation (Clements and Izan, 1981, 1987 and Selvanathan and Selvanathan, 2006), world interest rates (Ong et al., 1999) and international competitiveness (Clements et al., 2008). See Clements et al. (2006) for a comprehensive review. 2
Three Price Indices
Suppose there are n goods in two periods, the base period and the current period, to be denoted by 0 and 1. Let pit be the price of good i in period t, i 1, , n, t 0,1. A price index addresses the question of how best to summarise the difference between the vector of the n prices in the base period, p10 , , p n 0 , and that in the current period, p11 , , p n1 . We could consider an unweighted average of the n price relatives, p11 p10 , , p n1 p n 0 , that is, 1 n in1 pi1 pi0 . But to recognise that some goods are more important than others, some form of a weighted average is more appealing as that makes the index more representative of underlying expenditure patterns. A number of popular indices are formulated in this manner and below we consider some alternatives. Let q i0 be the quantity consumed of i in 0, M 0 in1 pi0 q i0 be total expenditure in 0 and w i0 pi0 q i0 M 0 be the budget share of i in 0, which has a unit sum. The Laspeyres index uses as weights these base-period budget shares: n p P L w i0 i1 . i 1 pi0
This index can also be expressed as in1 pi1q i0 M 0 , which is the ratio of two costs of the consumption basket. The first, in1 pi1q i0 , is the cost of the base-period basket, q10 , , q n 0 , at current-period prices, p10 , , p n1. The second, M 0 , is the cost of the same basket evaluated at base-period prices. It is convenient to define M tt in1 pit q it as the basket in period t costed at
prices prevailing in t, where t , t 0,1 , so that observed base-period expenditure, M 0 , now becomes M 00 . Accordingly, Laspeyres can equivalently be written as P L M10 M 00 . This dating notation for prices and quantities can also be applied to the budget shares, so that w itt pit q it M tt , with in1 w itt 1, and w i00 w i0 . Thus, we have PL
(1)
n p M10 w i00 i1 . 00 M i 1 pi0
The Paasche price index replaces base-period quantities with their current-period counterparts, so that (2)
PP
n p M11 w i01 i1 . 01 M i 1 pi0
3
A further way of expressing this is as follows: Write the term after the first equals sign in equation (2) as M11 1 01 11 01 M M M
1
n
p i 1
q
11
i0 i1
M
1
p q n
i1 i1
i 1
11
M
p
pi1
i0
1 n
w p i 1
11 i
i1
pi0
1
.
This shows that Paasche is a weighted harmonic mean of the n price relatives: 1
1 n P 11 p i1 P w i . p i 1 i0
(2)
Here, the weights are the current-period budget shares, so that in this form Paasche is more “similar” to Laspeyres. Rather than using either base- or current-period quantities, the Fisher index strikes a symmetric balance by taking the geometric mean of Laspeyres and Paasche: pi1 pi0 i 1 . 1 n p i1 w11 i p i 1 i0 n
PF PLPP
(3)
10
11
M M
M 00 M 01
w
00 i
Volume Indices
The three indices above refer to prices. The corresponding volume, or quantity, indices are
(4)
01 n Q L M w 00 q i1 , Q P i M 00 i 1 q i0 QF QL QP
1
1 M11 n 11 q i1 w i , M10 i 1 q i0
M 01M11 M 00 M10
.
The observed ratio of total expenditure in the two periods is M11 M 00 , which is also known as the value ratio. Indices (1) -(4) satisfy the following value-preserving identities: (5)
P L QP P P QL P FQF
M11 , M 00
so that the product of the relevant pair of price and volume indices is the expenditure ratio. Equation (5) shows that deflation of the expenditure ratio by the Laspeyres (Paasche) price index gives the Paasche (Laspeyres) volume index, while deflation by Fisher gives back Fisher:
4
M11 M 00 M11 M 00 M11 M 00 P L Q , Q , QF , L P F P P P
(6)
which again reveals the symmetric nature of Fisher. Revealing Inequalities
It follows from equations (1) and (4) that P L QL
M10 M 01 , M 00 M 00
so that P L Q L Z1
(7)
M11 M11 M 00 1. , with Z= M 00 M10 M 01
As Z1 1, P L Q L M11 M 00 . This shows Laspeyres is subject to an upward bias as the product of the price and volume indices exceeds the value ratio. Similarly, there is a downward bias in Paasche: P PQP Z
M11 M11 . M 00 M 00
To prove the inequality in expression (7), write (8)
Z=
M11 M 00 p1q1 p0q 0 p1q1 p0q 0 , M10 M 01 p1q 0 p0q1 p0q1 p1q 0
with p t p1t , , p nt and q t q1t , , q nt the price and quantity vectors in period t ( t 0,1 ).
Consider the first ratio on the far right-hand side of equation (8), p1q1 p0q1 . The numerator is the observed total cost in period 1, viz. the consumption basket of period 1, q1 , evaluated at period 1 prices, p1. The denominator, p0q1 , is the cost of the same basket at period 0 prices, p 0 . For the consumer to minimise cost, the former is less than the latter, so the ratio is less than unity. As the second ratio on the far right of equation (8) is also less than unity, for similar reasons, it follows that Z5% 8
1. Alumina 2. AMCOR 3. AMP 100 0 72 1 0 0 4. ANZ Banking Group 81 96 0 38 44 33 5. BHP Billiton 100 100 100 100 100 100 6. Brambles 57 100 100 7. Coles Group 0 0 32 1 4 100 100 100 8. Commonwealth Bank of Australia 100 100 100 58 0 83 0 19 9. CSL 49 10. Foster's Group 0 100 72 0 0 100 1 0 11. Macquarie Group 99 98 0 1 100 12. Macquarie Infrastructure Group 81 26 13. National Australia Bank 100 100 100 100 68 2 82 70 14. Newcrest Mining 15. News Corporation 100 100 100 16. News Corporation Preference Shares 100 100 100 17. Origin Energy 18. QBE Insurance Group 2 0 56 71 100 12 19. Rinker Group 100 100 86 100 20. Rio Tinto 0 11 0 2 100 9 100 100 21. St. George Bank 0 0 0 0 0 0 96 0 22. Stockland 23. Suncorp-Metway 0 0 7 100 0 24. Sydney Roads Group 0 25. Telstra Corporation 99 100 100 100 100 98 100 89 26. Wesfarmers 88 0 0 3 0 11 0 0 27. Wesfarmers PPS 28. Westfield Holdings 0 0 29. Westfield Group 100 100 100 99 100 77 30. Westfield Trust 0 0 31. Westpac Banking Corporation 100 100 91 100 2 2 71 7 32. Woodside Petroleum 98 98 100 100 100 100 93 58 33. Woolworths 26 0 0 100 26 71 99 88 Total number of companies in sub-period 20 20 20 20 20 21 20 20 65 65 60 55 60 71 75 75 Percent of redundant stocks Notes: 1. See footnote 6 for information rergarding sub-periods. 2. Blanks indicate that the stock in question is excluded by Standard and Poor’s in the S&P20 index. 17
9
10
11
12
0 73 100 25
0 97 100 0
0 21 100 0
0 98 100 0
55 47 1 100
100 98 0 100
94 7 0 9
100 60 0 5
100
100
100 83
38 49
19
45
3
0 70
100 0 8 0
100 26 0 0
100 0
100
28
0
100 1 0
100 10
100 27
100 1
100
69
100
67
100 100 66 21 71
73 100 98 20 75
35 20 1 20 70
100 52 1 20 60
(Percentage) 25 0 33 83 100 57 50 83 100 25 67 100 92 100 100 100 0 70 100 75 18 50 33 0 100 33 0 0 100 0 83 100 67 20 0
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Hendry, D.F.
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21
