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Efficiency of the silver futures market
Journal of Banking and Finance 11 (1987) 49--64, North-Holland
EFFICIENCY OF THE SILVER FUTURES MARKET
An Empirical Study Using Daily D a t a Raj A G G A R W A L and P.S. S U N D A R A R A G H A V A N * The University of Toledo, Toledo, Ohio 43606, USA Received July 1984, final version received May 1986 This paper investigates whether the silver futures market is efficient with respect to the information contained in the time series of daily price changes. An analysis of the serial correlation of returns on silver futures supports the hypothesis that successive price changes are independent. However, a series of first and second order Markov chain models built using the direction as well as the magnitude of price change, reveals some short-term dependence. This result regarding the non-independence of successive price changes is reinforced by an analysis of upward and downward cycles, and by the extraordinary profits generated by using mechanical filter rules. The conclusion of this study is that the silver futures market does not seem to be efficient even in the weak form and that astute traders and investors can make modest excess risk-adjusted returns by using appropriate trading s~fategies.
I. Introduction This paper considers silver as a financial a:set and investigates the futures price of silver from the point of view of efficient market hypothesis of F a m a (1970) and others. In its weak form, this hypothesis implies that successive price changes are not correlated and that it should not be possible to forecast future price changes based only on the pattern of past price changes. In the semi-strong form the hypothesis proposes that all publicly available information is impounded in security prices so as to leave no opportunity to make extraordinary risk-adjusted returns based ori publicly held information. M a n y researchers have conducted empirical investigations of the applicability of the efficient market hypothesis proposed by F a m a (1970) and others. A review of the related literature indicates mixed findings with regard to market efficiency in the precious metals market, in particular, Tschoegl (1980) has studied the gold market using sequences of daily price changes during the period 1975-1977 by building Markov chain models and lagged correlations analysis and concluded that the market seemed to be efficient at *The authors are grateful to the anonymous referees of this journal for useful comments and to Ted Ko!!ia~ for programming assistance. The authors are re:ponsible for any remaining errors. Both authors have contributed equally. 0378-4206/87/$3.50 ~!) 1987, Elsevier Science Publishers B.V. (North.Holland)
50
R. Aggarwal and P.& Sundararaghavan, EO/~ciencyof the silver futures market
least in the weak form. Abken (1980) also could not find any inefficiencies in the monthly gold price series for the period 1973-1979. Further, Ball et al. (1985) found that the gold options market was efficient for the period 19811982. On the other hand, Booth et al. (1982) found persistent dependence in daily gold prices for the period 1968-1980 and Koutsoyiannis (1983) using daily prices for the period January 1980 to March 1981 also found shortterm time-series dependence in gold returns. In addition, Booth and Kaen (1979) found that extraordinary profits could be made using mecb:mical filter rules in the gold and silver markets while using daily data for the period 1969-1977. However, Solt and Swanson (1981) studied the markets for gold and silver using weekly prices during 1971-1979 and concluded that while there was positive dependence in the price change data, investors cannot easily use this positive dependence to generate superior returns. Thus, as this brief review of the literature indicates, there seems to be some controversy regarding the efficiency of the markets for precious metals such as silver. In order to ~hed some light on this apparent controversy, in this paper we have chosen the silver futures market as the basis for our study. We examine daily silver futures prices for the recent and previously unexamined period of 1980-1984. This period is especially interesting because the October 1979 change in the Federal Reserve operating procedures led to extraordinary fluctuations in interest rates and other financial variables. Before proceeding further we would like to provide another motivation for choosing the futures market for this study. Silver futures contracts are sold on the COMEX in New York and in the CBOT in Chicago in 5000oz. contracts and (on the CBOT in Chicago) in 1000oz. minisilver contracts. Since the introduction of the minisiiver contract in 1981, the barrier to entry has fallen significantly, resulting in an increase in the open interest as well as a significant expansion in the investor composition to include medium income investors. These recent changes make the silver futures market particularly attractive to study from the point of view of market efficiency. First, the transaction cost is very low ($15.00 to $50.00 for a 1000oz. contract). Second, execution of a long or a short contract is very quick as there is a large open interest facilitating an active market. Third, the margin is also relatively low. 1 Thus, an advantage of using the futures market is the relatively low transaction costs (0.25 to 0.50% of the value of the contract) v.'hich, together ~vith the relative ease and the speed with which a contract can be purchased or liquidated, makes it easy for an investor of even modest means to exploit market inefficiencies. One final advantage is that in the futures market any perceived inefficiency can be exploited either by going long or going short, or by other strategies that may not be possible or convenient in the spot market. lit is $1,200.00 for l,O00oz contract and about $3,500.00 for ~ne 5,000oz contract.
R. Aggarwal and P.S. Sundararaghavan, E.~ciency of the silver futures market
51
In this paper, we analyze a series of successive price changes for silver futures contracts as traded in the CBOT and the COMEX to study whether successive price changes are stochastically independent. Furthermore, we test various filter trading rules to determine if in fact extraordinary profits can be generated in the silver futures market. In section 2, we discuss the data used in this study and we justify the ~paration of 1980 from the 1981-1983 period. We also report means, standard 0eviations and third and fourth mot,~ents of various rates of returns, and first and second differences of daii)~ prices for silver futures contracts. In section 3, the serial correlations of first difference of prices and rates of returns on investments for various lagged periods are analyzed and their implicattons for the efficiency of the silver market are discussed. Section 4 examines various Markov chains for absolute price movements as well as for directional price movements. The results of the tests for stationarity and independence and their significance for trading strategies are also reported. In section 5, the price trends as exhibited by upward and downward cycles, are reported. The standard runsup and runsdown test was used to test the hypothesis that the price move~lents are random. The implications of extraordinary runs on trading strategies are also discussed. Section 6 examines if an investor can profit by using filter-based trading rules. Section 7 gives some conclusions and suggestions for future research.
2. Data selection and analysis of data for the study Daily open (O(0), high (H(t)), low (L(t)) and dosing prices (C(t)) of the silver futures contract for the next month as reported in the Wall Street Journal for the period 1980-1984 were used in this study. 2 During late 1979 to early 1980, silver prices expe,ienced extraordinary volatility and moved over very wide ranges mainly because of the attempts of the Hunt brothers and others to corner the market. 3 Consequently, we first examine if our analysis can be conducted for the entire period or if the year 1980 should be examined separately. The mean and standard deviation for 1989 closing price series was 19.29 and 6.76, whereas for years 1981, 1982 and 19•3, the means were 10.5, 7.9 and 12.1, and standard deviations were 2.0, 1.5 and 1.0, implying significant difference between 1980 and other years. In this paper we are more interested 2The term of the futures contract used in this study normally expired within 15-45 days. This choice of using prices on short-term futures contracts helped maintain the parity between the price levels of the various contracts included in the study. Further, the spot silver prices are very highly correlated with these futures contract prices and the spot price for a period could be approximated closely by subtracting a small constant from this futures contract price 3See Barnhill and Powell (1981) for another study indicating the special nature of price changes in 1980.
52
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
in price changes than in the absolute price levels. Therefore, an initial Markov chair, model was constructed to see if the mechanism governing price changes was significantly different in 1980 as compared to other years. A Markov chain model for the price change behavior for 1980 was constructed. 4 The amount of price change determined whether the system was in Down (D), Neutral (N), or Up (U) state. Specifically, if the amount of price change was chosen as 0.10, then if C ( t ) - C ( t - 1 ) < - 0 . 1 0 the system is in state D~ If -0.10__
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
53
Table 1 Statistical distribution of daily silver futures prices 1981-1983.
Variable ~ct Rcl Rm
Series~
Mean
C ( t ) - C ( t - 1)
c(t+ l)-C(t) C(t)
H(t+ l)-H(t)
H(t)
-0.0063650 b
Annualized ¢ rate of Standard return deviation d
Skewness Kurtosis
0.28766
-- 1.235
6.213
-0.0000438
( - 1.6)
0.02496
-0.371
5.011
0.0086860
(317.0)
0.01318
- 1.824
6.723
--0.0088175
(--321.8)
0.01417
--1.329
4.713
-0.0130000
d
0.39900
-- 1.157
6.970
L(t+ l ) - L ( t ) RL~ 6c2
L~t) C(t+2)-C(t)
"C(t), H(t) and L(0 represents the closing, high and low prices respectively for time period t. b6c~ is expected to have a mean very close to zero. CThe figures in parentheses are annualized percentage returns. The signs are to be properly interpreted, because potentially one can sell short and achieve positive returns. However, only some of these returns are possible on an ex-ante basis. dThe rates of return for these series are not defined.
passive strategy for daily trading exists that can generate extraordinary returns. However, these statistics also reveal that if one can successfully trade at the daily high (low), there is some possibility for excess risk adjustment returns by buying long (or selling short). 7 The two-day difference series once again confirms the volatile nature of these markets. 8
3. Serial eorrelation~ Serial correlations provide another way to study the relationships between elements of a time series. Further, the efficient market hypothesis has definite implications concerning serial correlations. Consider the model
Z(t)=
P(t)-P(t-l) P ( t - l!
't =r(:)+e(),
7Once again, this strategy cannot be implemented directly on an ex ante basis. However, it may be possible to exploit this information indirectly. Filter tests in a later section of this paper provide some evidence of excess risk-adjusted returns in the silver market. sit should be noted that all of the price change series exhibit third and fourth moments that indicate deviations from normality. In addition, on 48 of the 673 days covered in these series, limit moves occurred, indicating market disequilibrium. Therefore, the summary statistics for the serie~ may be somewhat biased and any statistical tests assuming normality of the price difference series may be misleading.
54
R. Aggarwaland P.S. Sundararaghavan, Efficiencyof the silverfutures market Table 2 Auto correlation coefficients for one and two day returns." Lag
Seriesb
1
2
3
4
5
10
15
Qd
Q*
c(t+l)-C(O c(0
-0.011
0.052
0.026
0.010 -0.033
-0.056
0.026
9.13
23.68
C(t+ l)-C(t)
-0.035
0.061
0.021
0.009 -0.032
-0.054
0.023
9.81
25.00
H(t+l)-H(t)
0.113 0.041
0.029
0.001
0.011
0.038
16.88 25.00
H(t+2]i-H(t)
c
0.106
0.052
0.020-0.008-0.001
-0.009
15.88 23.68
H(t)-L(t)
0.377 0.269
0.250
0.294
0.235
0.173
C(t+2)-C(t)
'
0.063
0.006-0.035
-0.055
0.060
0.007
0.123 269.00 0.030
23.68
13.35 23.68
"All auto ccrrelations for lags 1-15 were used in calculating Q, even though some are not reported above. bC(t), H(t), L(t) represent respectively the closing, opening, high, and low prices for the time period. The study of the nature of the series may contribute to our understanding of the stochastic information generation process, underlying price movements in the silvers futures market. This insight may help us in devising appropriate filter trading strategies. For results of the filter trading strategies, see tables 6 and 7. CFor the 2-d~y difference, lagging by 1 period is not meaningful, since those are expected to be significantly positively correlated. ~Q=~.]=t aiN, where N is the number of observations and ai is the sample auto-correlation of lag i= 1,2 ..... 15. Q follows Z2 distribution with t degrees of freedom. Q* is the critical value at 5% significance level.
where Z(t) is the percentage return of silver over the period t - 1 to t, and P(t) may represent opening or closing price series, r(t) is the normal return on assets of similar risk category and e'(t) is the random component of return. Efficient ~arket hypothesis implies that
E(e'(t)) = 0 ,
cov
(e'(t),e'(t-
k)) = 0
for k = 1, 2 , . . . ,
and
k < t.
Since r(t) for successive periods may be almost equal, though not identical, a reasonable surrogate measure for cov (e'(t), e'(t-k)) will be the various lagged serial correlation coefficients listed in table 2. None of these correlations were significant. Another method to examine the independence of successive price changes would be to study the serial correlation of the time series of first differences (one day) (P(t+ 1)-P(t)) as well as two day (P(t+2)-P(t)) price change time series where P represents a price series. Table 2 also summarizes the ser~,? coefficients of these difference series. The difference series for daily high prices seemed to be significantly positively correlated for a lag of one day for the one day difference series and
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
55
for a lag of two days for the two day difference series. This indicates dependence of successive daily highs which could be exploited in developing trading strategies? The range series H(t)-L(t) is positively correlated as expected. The strong positive correlation for the one-day lag in this series can be interpreted as an evidence that days with high ranges are likely to be followed by days with high ranges and days with narrow ranges are likely to be followed by narrow range days. Finally, Q-statistics show that the auto correlelograms for all the series except the high to high and the range were not significant at the 5% level. Only the range exhibited a high degree of auto correlation, reinforcing the earlier point that days with high ranges are more likely to be followed by such days and vice versa.
4. Markovian models A number of authors [for example, Tschoegl (1980), Fielitz (1975), Fielitz and Bhargava (1973)] have used Markov chain models to study the behavior of successive price changes. Markov chain models assume stationary transition behavior and that transition probabilities are dependent only on the current state occupied by the system and not on its history. In this section we develop and test a number of Markov chain models in order to examine the behavior of price changes in the silver futures market. There are basically two sets of Markovian models that are relevant in this case, the first set uses the direction of the price movement to define the states of the system, whereas in the second set, the magnitude of the price movement is used to define the states of the system. Table 3 presents second order Markov chain transition matrices, t° This matrix is based on the direction of movement of closing prices. The first concern in the Markov chain models is to test if the matrices are stationary over time. The stationarity property assures us that the Markov chain models built using past data can be used for modeling future price movements. The price series were divided into three periods (1981, 1982, 1983) and for each period Markovian transition matrices were developed. The null hypothesis (Ho) was that the matrices were stationary over time. The alternate hypothesis (H1) was that the transition matrix of price change is not 9Naturally, this strategy cannot be implemented directly on an ex ante basis. However, it may be possible to exploit this information indirectly. Filter tests in a later section of this paper provide some evidence of excess risk-adjusted returns in the silver market. t°While we developed and tested a number of first order and second order Markov transition matrices, due to space limitations, we present only a representative sample in tables 3 and 4.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
56
Table 3 A second order Markov chain closing price changes of silver futures contracts during 1981-1983. (.4 = $o.o5.)"
DD DD
46
DN
I
I
I
t
1I 20
I 140
I DN
0.431 I
0.191 i
i
I
I t
I I I
I I 1 I I I
I
t
I DU 19 ND
NN
NU
UD
UN
uu
DU
i
t
I I l
I I '
I NN I I
[
I
I
I NU t
I I
I
I
I UD I I
I UN I I Ii I I
I
IUU
I
I
I I 106 I I J I 58 I I I I 118
0.38l I I I I I I I 119 113 I 26 ! I I I I I ! I 0.331 0.221 0.451 I [ I I I I I 153 121 44 I -I I I I 0.37 I I I I 0.45 I 0.18
113 ] 13 i I I 0.421 0.291 0.291 i i I I
I
I ND I I
I
I I I
"'
I
III
I I ~ l I i I I I I I I _l i I i 42 125 165 I i I 0.321 0.191 0.491 i I I, 115
7
I10
0.391 I
I 0.251 i
I
I
I I I I I I i
I I I I l I I
I 8
120
I
I 0,361 I i 30 I i I I I
I10 0.54
I
J
i iI i i
I .... 1 i
It ~
I
I
I
I
!
I
I
I
I
I
I
I
!
!
! i 0 " 3 5 1 0.191 i I i I
I I I I 28 I I
I I
I
I
I I 45
I I I
I 0.461
I
148
112 0.491
116
I
I 56
I
!
0.181
0.28 I
I
I
I
I
I I
1
132
I
I 43
i I i
I I I
0.121
0.38 I
137 i
I
97
I
"The entries in the t(~p left-hand corner are actual frequencies while the entries in the bottom right-hand corner are transition probabilities. The states of this Markov chain capture 2 days' history. The system is in state D if P(t)- P ( t - 1)< - J , in state N if -_~ < P ( t ) - P ( t - 1)<,4, and in state U if .4 < P ( t ) - P(t - 1), where P is the price series used. The system is in state D D if it is in state D for two successive price transitions. Similarly, other states are defined. That is, the states contain built-b, history of two successive price changes. Hence, from state DD (the system was in state D in two successive price changes), the system can only go to DD or D N or DU but not to any other state, First order (3 x 3) matrices corresponding to the definition of states here can be derived by samming each column of this table. For e ~ m p l e , frequency correspond~ ing to transition from D to D in one day is obtained by summing column DD of this table. Similarly, frequency from N to D is obtained by summing column N D in this table.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
57
stationary. Using the Anderson-Goodman (1957) stationarity test, the null hypothesis was accepted which implies that one could use these Markov chains based on past data for modeling future price changes. We now examine the transition matrix in table 3 for any evidence of serial dependence in the price series. Serial independence of price changes requires that the rows of the Markov chains be identical. The next test addresses this issue. The null hypothes:~ is that the rows are identical and the alternate hypothesis is that the rows a~e not identical. We apply the chi-square test to the 3 x 3 matrix derived from table 3 where the neutral state is based on a minimum change of $0.05. The observed frequencies of the 3 x 3 matrix (by rows) 107, 58, 118; 45, 28, 56; 131, 43, 97; are compared to the expected frequencies of 117, 54, 112; 54, 24, 51; 112, 51, 108. The chi-square value is 9.635 compared to a critical value of 9.49 rejecting the null hypothesis at the 95% confidence level,tt The results are interesting and the relatively low transaction costs in these markets may permit an investor to exploit the inefficiencies exhibited by the price dependence shown by the transition matrix. ~2 Having examined Markov chains based on di~~ctional price movements we now examine similar transition matrices based on the magnitudes of the price change. Table 4 presents one of these results. We use a chi-square test to examine the price series f~r serial dependence using the Markov matrix in table 4. This test also shows that there is significant serial dependence. ~3 An examination of the transition probabilities in table 4 indicates that price changes seemed random (since appropriate transition probabilities are nearly equal) in 'normal' trading days (which correspond to transitions between states 2, 3, 4 and 5). However, when there is a strong directional trend like an upward price movement or downward price movement, the transition probabilities revealed a significant tendency for either reinforcing the trend or reversing it. To explore the nature of this process further we conduct trend and cycle analysis in section 5 below. 11However, if we define the states based on a minimum change of $0.10, the actual frequencies (by rows) are 65, 83, 78; 79, 77, 77; 83, 72, 69; comps,red to the expected frequencies of 75, 77, 74; 77, 7o,.."1"7;,.7~,. ~ 76, -la. ~. "rh,-...chi.square ~,~h,,~...,.~.~;~.,v,a ~ ,.v...e,-,~,-a tO a critical value of 9.49 (the matrix is doubly stochastic and hence the degree of freedom is 4). '2Another set of chi-square tests using the 9 x 9 matrices tested the null hypothesis that the transition matrix based on one-day price history (3x3 matrix) is not different from the transition matrix based on two-day price history. The null hypothesis was rejected under both definitions of neutral state implying that one-day history is sufficient in a statistical sense. Details of these and other tables not presented here for space reasons are available from the authors. '3The null hypothesis is that the rows are identical. The expected frequency for each cell is given by multiplying the row tota! by the average proportion of the !oral sample that is in the state indicated by the row in question. For example, for cells (1,1) to {6,1) the expected frequencies are 4.5, 7.0, 15.5, 15.5, 8.1 and 4.6. The chi-square value is 52.35 !critical value of 37.65 for degrees of freedom of 25 since the matrix is doubly stochastic). Thus, we reject the null at 95% confidence level. Results of the filter tests in section 6 suggest some approaches for exploiting this serial dependency.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
58
Table 4 Markov transition matrix using closing prices of silver futures contracts during 1981-~.~83. (k = 1.)~ State
-
I
I
'
I
I
I 1
! 2
I 3
I4
I 5
I
I6
I
I
I
I
I
I
I 11
I 9
I 11
i11
I 7
I 6
1 I
0.201
0.161
0.20 !
0.20 1
0.131
I
i
1
I
I
I
0.11
I 5
I 5
I 33
I 29
I 8
I 6
2 [
I 1 13
3
4
5
0.06 I
0.06 ]
i
51
122
I
0.07 I
I
i
i 8
131
0.12
t
0.041
I
I
I11
l l5
I
0.111
i
!
I 6
i
59 0.16 26 0.16
3
I
6
0.38 ~
13
0.34 1
I 58
0.09 I
I 34
I 13
0.27 [
0.30 1
0.18 I
I
I
I
! 54
I 31
! 8
0.31 I
0.281
0.16 I
I
I
I
I
25
I
10
I
0.07
0.04 13
I
0.26 [
0.25 I
0.10 I
0.13 I
I
I
I
I
I 14 I
I 11
I 10
I
I
I
0.10 ~ 0.05 0.23' 0.25 I I I I ....... aThe states of the Markov Chain model of this table are given below: State State State State State State
0.07
0.19 ]
0.18 I I
1: P ( t ) - P(t - k) < - $0.50 2: - $0.50 < P ( t ) - P(t - k) < - $0.25 3 : - $ 0 . 2 5 < P(t) - P(t - k) < 0 4 : 0 < P(t) - P ( t - k) < $0.25 5:$0.25 < P(t) - P(t - k) < $0.50 6:$0.50 < P(t) - P ( t - k),
where P represents the price series used and k denotes the number of days. The top left-hand corner entries are frequencies and the other entries are probabilities.
5. Trend
and cycle analysis
In this section we examine the number of upward and downward cycles (runs) and present our findings and implications for market efficiency. The runs test counts tile number of upward and downward runs and tests it against the expected distribution of runs if the numbers were truly random. If the price changes for the silver futures contracts are truly random, there should b ,~ very few exl, . . . .raoirUllliJ, . . . J: . . . l.y. . .l]Jl:~. ... Table 5 presents some selected statistics about upward and downward cycles using the closing price series. We apply the runs-up-and-down test on these cycles. The null hypothesis is that there are no more and no fewer runs
R. Aggarwal and P.I;. Sundararaghavan, Efficiency of the silver futures market
II
o
0~
E
Y. o
E
=={)
I.i
I
=
.= "0
*d ot,I
~
•- ~ . ~
.~
~,o
u
~
t~
o"
"N
.=.
- ~ . !
~
~
I
e~ "o
eL
~ ~; ~.~ ~ o L
...~.~,~.~.
E
b_
o
E_
J.B.F.- C
59
60
R. Aggarwal and P.S. Sundararaghavan, EO~ciencyof the silverfutures market
in the price series than might be expected as a result of random variation. The alternative hypothesis is that there are significantly more or fewer runs than might be attributed to chance. 14 The number of data points was 673 and the mean and standard deviation of the expected number of runs (both up and down) were calculated as 447 and 10.93. The actual number of runs in this data were 376, which is significantly lower than the expected number of runs, resulting in a Z-score of -6.496, which enables us to reject the null hypothesis confidently. This very strongly suggests some possibility of nonrandomness in the price movements. Next we examine the issue of length of the cycles. Given the number of cycles and assuming a random walk for the direction of price change~ we compared the expected and observed number of cycles of various lengths in the downward and upward cycles. There were more cycles of length one than would be expected under the random walk hypothesis. This implies higher than normal short-term volatility. Calculated chi-square values for the distribution of downward and upward cycles were 10.268 and 8.59 respectively compared to a critical value at the 95~ confidence level of 11.07. Thus, we conclude that given the number of cycles, the number of cycles in each length category were not significantly different from the number expected. Finally, we exami:te the cycles from a trader's point of view. For example, if a trader has observed that the price has gone down for the last two trading days: Can he {she) exploit this information? For this, one should look at the conditional probability of a downward (upward) cycle of length (n+ 1) given that a downward (upward) cycle of length n has been observed. From the conditional probabilities given in table 7, it is clear that it takes some time to obtain a trend. However, if a trend is in the making, then downward trends tend to reinforce themselves, whereas upward tend to reverse. 15 Here again, there is informational inefficiency that an astute trader may be able to exploit. 6. Filter tests
In this section we examine various mechanical filter trading rules to explore whether extraordinary risk-adjusted returns could be made in the silver futures market. We examine various strategies that may be able to exploit the perceived short-term dependence of the silver futures market. 16 a4For details of this test see Soloman (1983). aSThe expected value of all these conditional probabilities is 0.50. However, we are not able to apply the chi-square test since the observed frequency under the conditional class will not be equal to the total expected frequency under the conditional class. t61n this study, the return on a futures contract is defined as the ratio of liquidation price less purchase price to the average daily capital required to maintain the interest including the margin payments.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
61
With that in mind, several mechanical trading rules (commonly known as filter rules) were developed and tested using past data. First, a brief discussion of the rules is in order. For every trading day, there is a target purchase price, which is the previous day's closing price plus the price change for a particular filter. On each trading day, at most one contract is purchased if the price reaches the target price. If the opening price is less than the target price, a contract is bought at the opening. The average investment is the dollar amount required to maintain ownership of outstanding contracts after accounting for margin calls were calculated. 17 The results of the trading rules are presented in table 6. It is clear from table 6 that extraordinary risk-adjusted returns before and after transaction costs were possible using some of the trading rules. It is interesting to note that before transaction costs most of the trading rules resulted in extraordinary profits without large amounts of average investment. In fact, the average investment is negative in quite a few cases, which implies that the profits acquired in the earlier periods are more than sufficient to maintain the interest in the outstanding contracts. On an overall basis, the results of these filter tests show that it is possibi~: to exploit price dependence in the silver futures markets to generate extraordinary returns by using appropriate mechanical trading rules. Thus, the results presented here confirm the serial dependencies found in earlier tests and there seems to be significant evidence of inefficiencies in the silver futures market. The same filters were applied to data for the next thirteen months (October 1983 through September 1984) following the study period, and the results are presented in table 7. It is interesting to note that there are many similarities between tables 6 and 7. The net profit figures differe6 in sign in only 4 out of the 20 filters. Similarly, average investment differed in sign in only 6 out of the 20 filters. This showed remarkable corroboration of the nature of the results expected by using specific filters, thus showing that the filters are robust, Finally, a number of filters continued to generate extraordinary profits providing additional evidence of inefficiency in the silver futures market. 7. Conclusions
In this paper we have tested the efficiency of the silver futures market with ~TThe filter rule for this study taken from Stevenson and Bear (1970) is: Starting from the previous day's closing prke, wait for the price to move up or down by a given amount (the filter size) and then establish a long position. Orders for a stop loss and liquidation at a particular price for a given level of profit are placed at the time of the purchase. These orders are executed when and if the trading price reaches one or the other target price. A small number of contracts (less than 10 for any trading rule) could not be liquidated, because on the same trading day, both liquidation prices (at a profit as well as at a loss) were reached and, therefo[e, without a complete price vs. time chart for each day, it cannot be determined which target was reached first. These trades were omitted from the reports of the filter tests.
62
R. Aggarwal and P.S. Sundararaghavan, Efficiency of P hilverfutures market
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R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
Z ~ : u
63
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64
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
respect to daily prices of silver futures contracts traded on the CBOT and COMEX exchanges. First, we show that the behavior of silver prices in 1980 (during the 'Hunt' episode) was significantly di~erent from their behavior in the period 1981-1983. Second, the dependence of successive price changes was tested using serial correlation methods, Markov chain models, and trend and cycle analysis. The results of these statistical tests indicated some serial dependence in prices as well as strong cycles in price movement.% , t6rd, we conducted filter tests that sbow that it was possible to generate excess returns in the silver futures markets. Finally, the filter tests were applied for one more year of new data, 1983 (October) to 1984 (September) and the results corroborated the findings for the 1981-1983 data. Thus, our results provide support for weak-form inefficiency in the market for silver futures for the period 1981-1983 and there was significant evidence presented in this study to suggest that, with a suitably designed and disciplined trading strategy, traders could make excess risk adjusted returns.
References Abken, Peter A., 1980, The economics of gold price movements, Economic Review (FRB of Richmond), March-April, 3-13. Anderson, T.W. and L.A. Goodman, 1957, Statistical inference about Markov chains, Annals of Mathematical Statistics 28, 80-110. Ball, Clifford, W.N. Torous and A.E. Tschoegl, 1985, An empirical investigation of the EOE gold options market, Journal of Banking and Finance 9, 101-113. Barnhill, T.M. and J.A. Powell, 1981, Silver price volatility: A perspective on the July 1979-April 1980 period, Journal of Futures Markets 1, no. 4, 619-647. Booth, G.G. and F.R. Kaen, 1979, Gold and silver spot prices and market information efficiency, Financial Review, Spring, 21-26. Booth, G.G., F.R. Kaen and P.E. Koveos, 1982, Persistent dependence in gc,~d prices, The Journal of Financial Research, Spring, 85-93. Dryden, M., 1969, Share price movements" A Markovian approach, Journal of Finance 24, 4960. Fama, E., 1970, Efficient capital marke~: A review of theory and empirical work, Journal of Finance 25, 283-417. Fiditz, B.D., 1975, On the stationarity of transition probability matrices of common stocks, Journal of Financial apd Quantitative Analysis 10, 327-339. l=!=iitz, B. and T.N. Bhargava, 1973, The behaviour of stock price relatives - A Markovian analysis, Operations Research 21, 1183-1189. Garman, M.B. and M.J. Klass, 1980, On the estimation of security price volatilities from historical data, Journal of Business 53, no. 1, 67-78. Koutsoyiannis, A., 1983, A short-run pricing model for a speculative asset, tested with data from the gold bullion market, Applied Economics 15, 563-581. Ryan, T.M., 1973, Security prices as Markov processes, Journal of Financial and Quantitative Analysis 8, 17-36. Soloman, S.L., 1983, Simulation o~ waiting line systems ~Prentice-Hal! g.nglewood Cliffs, NJJ. Solt, M.E. and P.J. Swanson, 1981, On the efficiency of the markets for gold and silver, Journal of Business 54, 453-477. Stevenson, R.A. and R.M. Bear, 1970, Commodity futures: Trends or random walks?, Journal of Finance 25, March, 65-81. Tschoegl, A.E., 1980, Efficieacy of the gold market, Journal of Banking and Finance 4, 331-379.
EFFICIENCY OF THE SILVER FUTURES MARKET
An Empirical Study Using Daily D a t a Raj A G G A R W A L and P.S. S U N D A R A R A G H A V A N * The University of Toledo, Toledo, Ohio 43606, USA Received July 1984, final version received May 1986 This paper investigates whether the silver futures market is efficient with respect to the information contained in the time series of daily price changes. An analysis of the serial correlation of returns on silver futures supports the hypothesis that successive price changes are independent. However, a series of first and second order Markov chain models built using the direction as well as the magnitude of price change, reveals some short-term dependence. This result regarding the non-independence of successive price changes is reinforced by an analysis of upward and downward cycles, and by the extraordinary profits generated by using mechanical filter rules. The conclusion of this study is that the silver futures market does not seem to be efficient even in the weak form and that astute traders and investors can make modest excess risk-adjusted returns by using appropriate trading s~fategies.
I. Introduction This paper considers silver as a financial a:set and investigates the futures price of silver from the point of view of efficient market hypothesis of F a m a (1970) and others. In its weak form, this hypothesis implies that successive price changes are not correlated and that it should not be possible to forecast future price changes based only on the pattern of past price changes. In the semi-strong form the hypothesis proposes that all publicly available information is impounded in security prices so as to leave no opportunity to make extraordinary risk-adjusted returns based ori publicly held information. M a n y researchers have conducted empirical investigations of the applicability of the efficient market hypothesis proposed by F a m a (1970) and others. A review of the related literature indicates mixed findings with regard to market efficiency in the precious metals market, in particular, Tschoegl (1980) has studied the gold market using sequences of daily price changes during the period 1975-1977 by building Markov chain models and lagged correlations analysis and concluded that the market seemed to be efficient at *The authors are grateful to the anonymous referees of this journal for useful comments and to Ted Ko!!ia~ for programming assistance. The authors are re:ponsible for any remaining errors. Both authors have contributed equally. 0378-4206/87/$3.50 ~!) 1987, Elsevier Science Publishers B.V. (North.Holland)
50
R. Aggarwal and P.& Sundararaghavan, EO/~ciencyof the silver futures market
least in the weak form. Abken (1980) also could not find any inefficiencies in the monthly gold price series for the period 1973-1979. Further, Ball et al. (1985) found that the gold options market was efficient for the period 19811982. On the other hand, Booth et al. (1982) found persistent dependence in daily gold prices for the period 1968-1980 and Koutsoyiannis (1983) using daily prices for the period January 1980 to March 1981 also found shortterm time-series dependence in gold returns. In addition, Booth and Kaen (1979) found that extraordinary profits could be made using mecb:mical filter rules in the gold and silver markets while using daily data for the period 1969-1977. However, Solt and Swanson (1981) studied the markets for gold and silver using weekly prices during 1971-1979 and concluded that while there was positive dependence in the price change data, investors cannot easily use this positive dependence to generate superior returns. Thus, as this brief review of the literature indicates, there seems to be some controversy regarding the efficiency of the markets for precious metals such as silver. In order to ~hed some light on this apparent controversy, in this paper we have chosen the silver futures market as the basis for our study. We examine daily silver futures prices for the recent and previously unexamined period of 1980-1984. This period is especially interesting because the October 1979 change in the Federal Reserve operating procedures led to extraordinary fluctuations in interest rates and other financial variables. Before proceeding further we would like to provide another motivation for choosing the futures market for this study. Silver futures contracts are sold on the COMEX in New York and in the CBOT in Chicago in 5000oz. contracts and (on the CBOT in Chicago) in 1000oz. minisilver contracts. Since the introduction of the minisiiver contract in 1981, the barrier to entry has fallen significantly, resulting in an increase in the open interest as well as a significant expansion in the investor composition to include medium income investors. These recent changes make the silver futures market particularly attractive to study from the point of view of market efficiency. First, the transaction cost is very low ($15.00 to $50.00 for a 1000oz. contract). Second, execution of a long or a short contract is very quick as there is a large open interest facilitating an active market. Third, the margin is also relatively low. 1 Thus, an advantage of using the futures market is the relatively low transaction costs (0.25 to 0.50% of the value of the contract) v.'hich, together ~vith the relative ease and the speed with which a contract can be purchased or liquidated, makes it easy for an investor of even modest means to exploit market inefficiencies. One final advantage is that in the futures market any perceived inefficiency can be exploited either by going long or going short, or by other strategies that may not be possible or convenient in the spot market. lit is $1,200.00 for l,O00oz contract and about $3,500.00 for ~ne 5,000oz contract.
R. Aggarwal and P.S. Sundararaghavan, E.~ciency of the silver futures market
51
In this paper, we analyze a series of successive price changes for silver futures contracts as traded in the CBOT and the COMEX to study whether successive price changes are stochastically independent. Furthermore, we test various filter trading rules to determine if in fact extraordinary profits can be generated in the silver futures market. In section 2, we discuss the data used in this study and we justify the ~paration of 1980 from the 1981-1983 period. We also report means, standard 0eviations and third and fourth mot,~ents of various rates of returns, and first and second differences of daii)~ prices for silver futures contracts. In section 3, the serial correlations of first difference of prices and rates of returns on investments for various lagged periods are analyzed and their implicattons for the efficiency of the silver market are discussed. Section 4 examines various Markov chains for absolute price movements as well as for directional price movements. The results of the tests for stationarity and independence and their significance for trading strategies are also reported. In section 5, the price trends as exhibited by upward and downward cycles, are reported. The standard runsup and runsdown test was used to test the hypothesis that the price move~lents are random. The implications of extraordinary runs on trading strategies are also discussed. Section 6 examines if an investor can profit by using filter-based trading rules. Section 7 gives some conclusions and suggestions for future research.
2. Data selection and analysis of data for the study Daily open (O(0), high (H(t)), low (L(t)) and dosing prices (C(t)) of the silver futures contract for the next month as reported in the Wall Street Journal for the period 1980-1984 were used in this study. 2 During late 1979 to early 1980, silver prices expe,ienced extraordinary volatility and moved over very wide ranges mainly because of the attempts of the Hunt brothers and others to corner the market. 3 Consequently, we first examine if our analysis can be conducted for the entire period or if the year 1980 should be examined separately. The mean and standard deviation for 1989 closing price series was 19.29 and 6.76, whereas for years 1981, 1982 and 19•3, the means were 10.5, 7.9 and 12.1, and standard deviations were 2.0, 1.5 and 1.0, implying significant difference between 1980 and other years. In this paper we are more interested 2The term of the futures contract used in this study normally expired within 15-45 days. This choice of using prices on short-term futures contracts helped maintain the parity between the price levels of the various contracts included in the study. Further, the spot silver prices are very highly correlated with these futures contract prices and the spot price for a period could be approximated closely by subtracting a small constant from this futures contract price 3See Barnhill and Powell (1981) for another study indicating the special nature of price changes in 1980.
52
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
in price changes than in the absolute price levels. Therefore, an initial Markov chair, model was constructed to see if the mechanism governing price changes was significantly different in 1980 as compared to other years. A Markov chain model for the price change behavior for 1980 was constructed. 4 The amount of price change determined whether the system was in Down (D), Neutral (N), or Up (U) state. Specifically, if the amount of price change was chosen as 0.10, then if C ( t ) - C ( t - 1 ) < - 0 . 1 0 the system is in state D~ If -0.10__
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
53
Table 1 Statistical distribution of daily silver futures prices 1981-1983.
Variable ~ct Rcl Rm
Series~
Mean
C ( t ) - C ( t - 1)
c(t+ l)-C(t) C(t)
H(t+ l)-H(t)
H(t)
-0.0063650 b
Annualized ¢ rate of Standard return deviation d
Skewness Kurtosis
0.28766
-- 1.235
6.213
-0.0000438
( - 1.6)
0.02496
-0.371
5.011
0.0086860
(317.0)
0.01318
- 1.824
6.723
--0.0088175
(--321.8)
0.01417
--1.329
4.713
-0.0130000
d
0.39900
-- 1.157
6.970
L(t+ l ) - L ( t ) RL~ 6c2
L~t) C(t+2)-C(t)
"C(t), H(t) and L(0 represents the closing, high and low prices respectively for time period t. b6c~ is expected to have a mean very close to zero. CThe figures in parentheses are annualized percentage returns. The signs are to be properly interpreted, because potentially one can sell short and achieve positive returns. However, only some of these returns are possible on an ex-ante basis. dThe rates of return for these series are not defined.
passive strategy for daily trading exists that can generate extraordinary returns. However, these statistics also reveal that if one can successfully trade at the daily high (low), there is some possibility for excess risk adjustment returns by buying long (or selling short). 7 The two-day difference series once again confirms the volatile nature of these markets. 8
3. Serial eorrelation~ Serial correlations provide another way to study the relationships between elements of a time series. Further, the efficient market hypothesis has definite implications concerning serial correlations. Consider the model
Z(t)=
P(t)-P(t-l) P ( t - l!
't =r(:)+e(),
7Once again, this strategy cannot be implemented directly on an ex ante basis. However, it may be possible to exploit this information indirectly. Filter tests in a later section of this paper provide some evidence of excess risk-adjusted returns in the silver market. sit should be noted that all of the price change series exhibit third and fourth moments that indicate deviations from normality. In addition, on 48 of the 673 days covered in these series, limit moves occurred, indicating market disequilibrium. Therefore, the summary statistics for the serie~ may be somewhat biased and any statistical tests assuming normality of the price difference series may be misleading.
54
R. Aggarwaland P.S. Sundararaghavan, Efficiencyof the silverfutures market Table 2 Auto correlation coefficients for one and two day returns." Lag
Seriesb
1
2
3
4
5
10
15
Qd
Q*
c(t+l)-C(O c(0
-0.011
0.052
0.026
0.010 -0.033
-0.056
0.026
9.13
23.68
C(t+ l)-C(t)
-0.035
0.061
0.021
0.009 -0.032
-0.054
0.023
9.81
25.00
H(t+l)-H(t)
0.113 0.041
0.029
0.001
0.011
0.038
16.88 25.00
H(t+2]i-H(t)
c
0.106
0.052
0.020-0.008-0.001
-0.009
15.88 23.68
H(t)-L(t)
0.377 0.269
0.250
0.294
0.235
0.173
C(t+2)-C(t)
'
0.063
0.006-0.035
-0.055
0.060
0.007
0.123 269.00 0.030
23.68
13.35 23.68
"All auto ccrrelations for lags 1-15 were used in calculating Q, even though some are not reported above. bC(t), H(t), L(t) represent respectively the closing, opening, high, and low prices for the time period. The study of the nature of the series may contribute to our understanding of the stochastic information generation process, underlying price movements in the silvers futures market. This insight may help us in devising appropriate filter trading strategies. For results of the filter trading strategies, see tables 6 and 7. CFor the 2-d~y difference, lagging by 1 period is not meaningful, since those are expected to be significantly positively correlated. ~Q=~.]=t aiN, where N is the number of observations and ai is the sample auto-correlation of lag i= 1,2 ..... 15. Q follows Z2 distribution with t degrees of freedom. Q* is the critical value at 5% significance level.
where Z(t) is the percentage return of silver over the period t - 1 to t, and P(t) may represent opening or closing price series, r(t) is the normal return on assets of similar risk category and e'(t) is the random component of return. Efficient ~arket hypothesis implies that
E(e'(t)) = 0 ,
cov
(e'(t),e'(t-
k)) = 0
for k = 1, 2 , . . . ,
and
k < t.
Since r(t) for successive periods may be almost equal, though not identical, a reasonable surrogate measure for cov (e'(t), e'(t-k)) will be the various lagged serial correlation coefficients listed in table 2. None of these correlations were significant. Another method to examine the independence of successive price changes would be to study the serial correlation of the time series of first differences (one day) (P(t+ 1)-P(t)) as well as two day (P(t+2)-P(t)) price change time series where P represents a price series. Table 2 also summarizes the ser~,? coefficients of these difference series. The difference series for daily high prices seemed to be significantly positively correlated for a lag of one day for the one day difference series and
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
55
for a lag of two days for the two day difference series. This indicates dependence of successive daily highs which could be exploited in developing trading strategies? The range series H(t)-L(t) is positively correlated as expected. The strong positive correlation for the one-day lag in this series can be interpreted as an evidence that days with high ranges are likely to be followed by days with high ranges and days with narrow ranges are likely to be followed by narrow range days. Finally, Q-statistics show that the auto correlelograms for all the series except the high to high and the range were not significant at the 5% level. Only the range exhibited a high degree of auto correlation, reinforcing the earlier point that days with high ranges are more likely to be followed by such days and vice versa.
4. Markovian models A number of authors [for example, Tschoegl (1980), Fielitz (1975), Fielitz and Bhargava (1973)] have used Markov chain models to study the behavior of successive price changes. Markov chain models assume stationary transition behavior and that transition probabilities are dependent only on the current state occupied by the system and not on its history. In this section we develop and test a number of Markov chain models in order to examine the behavior of price changes in the silver futures market. There are basically two sets of Markovian models that are relevant in this case, the first set uses the direction of the price movement to define the states of the system, whereas in the second set, the magnitude of the price movement is used to define the states of the system. Table 3 presents second order Markov chain transition matrices, t° This matrix is based on the direction of movement of closing prices. The first concern in the Markov chain models is to test if the matrices are stationary over time. The stationarity property assures us that the Markov chain models built using past data can be used for modeling future price movements. The price series were divided into three periods (1981, 1982, 1983) and for each period Markovian transition matrices were developed. The null hypothesis (Ho) was that the matrices were stationary over time. The alternate hypothesis (H1) was that the transition matrix of price change is not 9Naturally, this strategy cannot be implemented directly on an ex ante basis. However, it may be possible to exploit this information indirectly. Filter tests in a later section of this paper provide some evidence of excess risk-adjusted returns in the silver market. t°While we developed and tested a number of first order and second order Markov transition matrices, due to space limitations, we present only a representative sample in tables 3 and 4.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
56
Table 3 A second order Markov chain closing price changes of silver futures contracts during 1981-1983. (.4 = $o.o5.)"
DD DD
46
DN
I
I
I
t
1I 20
I 140
I DN
0.431 I
0.191 i
i
I
I t
I I I
I I 1 I I I
I
t
I DU 19 ND
NN
NU
UD
UN
uu
DU
i
t
I I l
I I '
I NN I I
[
I
I
I NU t
I I
I
I
I UD I I
I UN I I Ii I I
I
IUU
I
I
I I 106 I I J I 58 I I I I 118
0.38l I I I I I I I 119 113 I 26 ! I I I I I ! I 0.331 0.221 0.451 I [ I I I I I 153 121 44 I -I I I I 0.37 I I I I 0.45 I 0.18
113 ] 13 i I I 0.421 0.291 0.291 i i I I
I
I ND I I
I
I I I
"'
I
III
I I ~ l I i I I I I I I _l i I i 42 125 165 I i I 0.321 0.191 0.491 i I I, 115
7
I10
0.391 I
I 0.251 i
I
I
I I I I I I i
I I I I l I I
I 8
120
I
I 0,361 I i 30 I i I I I
I10 0.54
I
J
i iI i i
I .... 1 i
It ~
I
I
I
I
!
I
I
I
I
I
I
I
!
!
! i 0 " 3 5 1 0.191 i I i I
I I I I 28 I I
I I
I
I
I I 45
I I I
I 0.461
I
148
112 0.491
116
I
I 56
I
!
0.181
0.28 I
I
I
I
I
I I
1
132
I
I 43
i I i
I I I
0.121
0.38 I
137 i
I
97
I
"The entries in the t(~p left-hand corner are actual frequencies while the entries in the bottom right-hand corner are transition probabilities. The states of this Markov chain capture 2 days' history. The system is in state D if P(t)- P ( t - 1)< - J , in state N if -_~ < P ( t ) - P ( t - 1)<,4, and in state U if .4 < P ( t ) - P(t - 1), where P is the price series used. The system is in state D D if it is in state D for two successive price transitions. Similarly, other states are defined. That is, the states contain built-b, history of two successive price changes. Hence, from state DD (the system was in state D in two successive price changes), the system can only go to DD or D N or DU but not to any other state, First order (3 x 3) matrices corresponding to the definition of states here can be derived by samming each column of this table. For e ~ m p l e , frequency correspond~ ing to transition from D to D in one day is obtained by summing column DD of this table. Similarly, frequency from N to D is obtained by summing column N D in this table.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
57
stationary. Using the Anderson-Goodman (1957) stationarity test, the null hypothesis was accepted which implies that one could use these Markov chains based on past data for modeling future price changes. We now examine the transition matrix in table 3 for any evidence of serial dependence in the price series. Serial independence of price changes requires that the rows of the Markov chains be identical. The next test addresses this issue. The null hypothes:~ is that the rows are identical and the alternate hypothesis is that the rows a~e not identical. We apply the chi-square test to the 3 x 3 matrix derived from table 3 where the neutral state is based on a minimum change of $0.05. The observed frequencies of the 3 x 3 matrix (by rows) 107, 58, 118; 45, 28, 56; 131, 43, 97; are compared to the expected frequencies of 117, 54, 112; 54, 24, 51; 112, 51, 108. The chi-square value is 9.635 compared to a critical value of 9.49 rejecting the null hypothesis at the 95% confidence level,tt The results are interesting and the relatively low transaction costs in these markets may permit an investor to exploit the inefficiencies exhibited by the price dependence shown by the transition matrix. ~2 Having examined Markov chains based on di~~ctional price movements we now examine similar transition matrices based on the magnitudes of the price change. Table 4 presents one of these results. We use a chi-square test to examine the price series f~r serial dependence using the Markov matrix in table 4. This test also shows that there is significant serial dependence. ~3 An examination of the transition probabilities in table 4 indicates that price changes seemed random (since appropriate transition probabilities are nearly equal) in 'normal' trading days (which correspond to transitions between states 2, 3, 4 and 5). However, when there is a strong directional trend like an upward price movement or downward price movement, the transition probabilities revealed a significant tendency for either reinforcing the trend or reversing it. To explore the nature of this process further we conduct trend and cycle analysis in section 5 below. 11However, if we define the states based on a minimum change of $0.10, the actual frequencies (by rows) are 65, 83, 78; 79, 77, 77; 83, 72, 69; comps,red to the expected frequencies of 75, 77, 74; 77, 7o,.."1"7;,.7~,. ~ 76, -la. ~. "rh,-...chi.square ~,~h,,~...,.~.~;~.,v,a ~ ,.v...e,-,~,-a tO a critical value of 9.49 (the matrix is doubly stochastic and hence the degree of freedom is 4). '2Another set of chi-square tests using the 9 x 9 matrices tested the null hypothesis that the transition matrix based on one-day price history (3x3 matrix) is not different from the transition matrix based on two-day price history. The null hypothesis was rejected under both definitions of neutral state implying that one-day history is sufficient in a statistical sense. Details of these and other tables not presented here for space reasons are available from the authors. '3The null hypothesis is that the rows are identical. The expected frequency for each cell is given by multiplying the row tota! by the average proportion of the !oral sample that is in the state indicated by the row in question. For example, for cells (1,1) to {6,1) the expected frequencies are 4.5, 7.0, 15.5, 15.5, 8.1 and 4.6. The chi-square value is 52.35 !critical value of 37.65 for degrees of freedom of 25 since the matrix is doubly stochastic). Thus, we reject the null at 95% confidence level. Results of the filter tests in section 6 suggest some approaches for exploiting this serial dependency.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silverfutures market
58
Table 4 Markov transition matrix using closing prices of silver futures contracts during 1981-~.~83. (k = 1.)~ State
-
I
I
'
I
I
I 1
! 2
I 3
I4
I 5
I
I6
I
I
I
I
I
I
I 11
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5. Trend
and cycle analysis
In this section we examine the number of upward and downward cycles (runs) and present our findings and implications for market efficiency. The runs test counts tile number of upward and downward runs and tests it against the expected distribution of runs if the numbers were truly random. If the price changes for the silver futures contracts are truly random, there should b ,~ very few exl, . . . .raoirUllliJ, . . . J: . . . l.y. . .l]Jl:~. ... Table 5 presents some selected statistics about upward and downward cycles using the closing price series. We apply the runs-up-and-down test on these cycles. The null hypothesis is that there are no more and no fewer runs
R. Aggarwal and P.I;. Sundararaghavan, Efficiency of the silver futures market
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R. Aggarwal and P.S. Sundararaghavan, EO~ciencyof the silverfutures market
in the price series than might be expected as a result of random variation. The alternative hypothesis is that there are significantly more or fewer runs than might be attributed to chance. 14 The number of data points was 673 and the mean and standard deviation of the expected number of runs (both up and down) were calculated as 447 and 10.93. The actual number of runs in this data were 376, which is significantly lower than the expected number of runs, resulting in a Z-score of -6.496, which enables us to reject the null hypothesis confidently. This very strongly suggests some possibility of nonrandomness in the price movements. Next we examine the issue of length of the cycles. Given the number of cycles and assuming a random walk for the direction of price change~ we compared the expected and observed number of cycles of various lengths in the downward and upward cycles. There were more cycles of length one than would be expected under the random walk hypothesis. This implies higher than normal short-term volatility. Calculated chi-square values for the distribution of downward and upward cycles were 10.268 and 8.59 respectively compared to a critical value at the 95~ confidence level of 11.07. Thus, we conclude that given the number of cycles, the number of cycles in each length category were not significantly different from the number expected. Finally, we exami:te the cycles from a trader's point of view. For example, if a trader has observed that the price has gone down for the last two trading days: Can he {she) exploit this information? For this, one should look at the conditional probability of a downward (upward) cycle of length (n+ 1) given that a downward (upward) cycle of length n has been observed. From the conditional probabilities given in table 7, it is clear that it takes some time to obtain a trend. However, if a trend is in the making, then downward trends tend to reinforce themselves, whereas upward tend to reverse. 15 Here again, there is informational inefficiency that an astute trader may be able to exploit. 6. Filter tests
In this section we examine various mechanical filter trading rules to explore whether extraordinary risk-adjusted returns could be made in the silver futures market. We examine various strategies that may be able to exploit the perceived short-term dependence of the silver futures market. 16 a4For details of this test see Soloman (1983). aSThe expected value of all these conditional probabilities is 0.50. However, we are not able to apply the chi-square test since the observed frequency under the conditional class will not be equal to the total expected frequency under the conditional class. t61n this study, the return on a futures contract is defined as the ratio of liquidation price less purchase price to the average daily capital required to maintain the interest including the margin payments.
R. Aggarwal and P.S. Sundararaghavan, Efficiency of the silver futures market
61
With that in mind, several mechanical trading rules (commonly known as filter rules) were developed and tested using past data. First, a brief discussion of the rules is in order. For every trading day, there is a target purchase price, which is the previous day's closing price plus the price change for a particular filter. On each trading day, at most one contract is purchased if the price reaches the target price. If the opening price is less than the target price, a contract is bought at the opening. The average investment is the dollar amount required to maintain ownership of outstanding contracts after accounting for margin calls were calculated. 17 The results of the trading rules are presented in table 6. It is clear from table 6 that extraordinary risk-adjusted returns before and after transaction costs were possible using some of the trading rules. It is interesting to note that before transaction costs most of the trading rules resulted in extraordinary profits without large amounts of average investment. In fact, the average investment is negative in quite a few cases, which implies that the profits acquired in the earlier periods are more than sufficient to maintain the interest in the outstanding contracts. On an overall basis, the results of these filter tests show that it is possibi~: to exploit price dependence in the silver futures markets to generate extraordinary returns by using appropriate mechanical trading rules. Thus, the results presented here confirm the serial dependencies found in earlier tests and there seems to be significant evidence of inefficiencies in the silver futures market. The same filters were applied to data for the next thirteen months (October 1983 through September 1984) following the study period, and the results are presented in table 7. It is interesting to note that there are many similarities between tables 6 and 7. The net profit figures differe6 in sign in only 4 out of the 20 filters. Similarly, average investment differed in sign in only 6 out of the 20 filters. This showed remarkable corroboration of the nature of the results expected by using specific filters, thus showing that the filters are robust, Finally, a number of filters continued to generate extraordinary profits providing additional evidence of inefficiency in the silver futures market. 7. Conclusions
In this paper we have tested the efficiency of the silver futures market with ~TThe filter rule for this study taken from Stevenson and Bear (1970) is: Starting from the previous day's closing prke, wait for the price to move up or down by a given amount (the filter size) and then establish a long position. Orders for a stop loss and liquidation at a particular price for a given level of profit are placed at the time of the purchase. These orders are executed when and if the trading price reaches one or the other target price. A small number of contracts (less than 10 for any trading rule) could not be liquidated, because on the same trading day, both liquidation prices (at a profit as well as at a loss) were reached and, therefo[e, without a complete price vs. time chart for each day, it cannot be determined which target was reached first. These trades were omitted from the reports of the filter tests.
62
R. Aggarwal and P.S. Sundararaghavan, Efficiency of P hilverfutures market
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respect to daily prices of silver futures contracts traded on the CBOT and COMEX exchanges. First, we show that the behavior of silver prices in 1980 (during the 'Hunt' episode) was significantly di~erent from their behavior in the period 1981-1983. Second, the dependence of successive price changes was tested using serial correlation methods, Markov chain models, and trend and cycle analysis. The results of these statistical tests indicated some serial dependence in prices as well as strong cycles in price movement.% , t6rd, we conducted filter tests that sbow that it was possible to generate excess returns in the silver futures markets. Finally, the filter tests were applied for one more year of new data, 1983 (October) to 1984 (September) and the results corroborated the findings for the 1981-1983 data. Thus, our results provide support for weak-form inefficiency in the market for silver futures for the period 1981-1983 and there was significant evidence presented in this study to suggest that, with a suitably designed and disciplined trading strategy, traders could make excess risk adjusted returns.
References Abken, Peter A., 1980, The economics of gold price movements, Economic Review (FRB of Richmond), March-April, 3-13. Anderson, T.W. and L.A. Goodman, 1957, Statistical inference about Markov chains, Annals of Mathematical Statistics 28, 80-110. Ball, Clifford, W.N. Torous and A.E. Tschoegl, 1985, An empirical investigation of the EOE gold options market, Journal of Banking and Finance 9, 101-113. Barnhill, T.M. and J.A. Powell, 1981, Silver price volatility: A perspective on the July 1979-April 1980 period, Journal of Futures Markets 1, no. 4, 619-647. Booth, G.G. and F.R. Kaen, 1979, Gold and silver spot prices and market information efficiency, Financial Review, Spring, 21-26. Booth, G.G., F.R. Kaen and P.E. Koveos, 1982, Persistent dependence in gc,~d prices, The Journal of Financial Research, Spring, 85-93. Dryden, M., 1969, Share price movements" A Markovian approach, Journal of Finance 24, 4960. Fama, E., 1970, Efficient capital marke~: A review of theory and empirical work, Journal of Finance 25, 283-417. Fiditz, B.D., 1975, On the stationarity of transition probability matrices of common stocks, Journal of Financial apd Quantitative Analysis 10, 327-339. l=!=iitz, B. and T.N. Bhargava, 1973, The behaviour of stock price relatives - A Markovian analysis, Operations Research 21, 1183-1189. Garman, M.B. and M.J. Klass, 1980, On the estimation of security price volatilities from historical data, Journal of Business 53, no. 1, 67-78. Koutsoyiannis, A., 1983, A short-run pricing model for a speculative asset, tested with data from the gold bullion market, Applied Economics 15, 563-581. Ryan, T.M., 1973, Security prices as Markov processes, Journal of Financial and Quantitative Analysis 8, 17-36. Soloman, S.L., 1983, Simulation o~ waiting line systems ~Prentice-Hal! g.nglewood Cliffs, NJJ. Solt, M.E. and P.J. Swanson, 1981, On the efficiency of the markets for gold and silver, Journal of Business 54, 453-477. Stevenson, R.A. and R.M. Bear, 1970, Commodity futures: Trends or random walks?, Journal of Finance 25, March, 65-81. Tschoegl, A.E., 1980, Efficieacy of the gold market, Journal of Banking and Finance 4, 331-379.