t 10 g sg j B0  0; . . . ; BT < 0PT Ee fi t1 Rt sg j Z0  1; . . . ; ZT  0 Ee i "1 "T 1 T s ( T 1sF s if T  1F"F sif T  0
Because T is a random variable, we have to compute the unconditionalcharacteristic function of the realized return as follows.FRRT s 
1XT0
PNEe fi t1 Rt IfBt 10 g sg j N  T Pr N  T
1XF "T 1 sF
sPp T 1 1
T1
 1
PF
s  P 1
p
1
p  1
PF
s
F spF" s
Note that the function form of FRRT s is a special case of Proposition 1 ofAcar and Satchell (1995).Proposition 2If we condition the result in Equation (5.10) on the event fR0  0g, we haveFRRT s 
1 pF s1 pF" s
5:11
161
Some exact results for moving-average trading rulesProofSubstitute P  1 into Equation (5.10) and the result follows.Proposition 3If the asset return, Rt , is independently identically distributed, we haveFRRT s 
s1  p pF" s1 pF" s
qF
5:12
ProofThe assumption that the asset return is independently identically distributedimplies that p and q in Equation (5.8) satisfy the condition p  q  1, which inturn impliesP
12
qp
q
1
qp
Substitute this condition into Equation (5.10), rearrange it and the resultfollows.
Proposition 4For the realized return from the geometric moving-average 2; 1 rule, we cancompute the expected realized return, E RRT , asE RRT  
11
p
fPp"
1
p g
5:13
where " and  are the means of the unexpected shocks, "t and t ,respectively.
ProofTo derive the expected realized return, we ®rst have to dierentiate thecharacteristic function of the realized return.F 0RRT s 
1
PF 0
SinceE RRT  
iF 0RRT 0
s  P 1
p

pF sF 0" s1 pF" s 2
F 0 s1 pF" s

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Advanced Trading Rules
we havepF 0F 0" 0F 0 0iP 1 p1 pF" 0 2 1 pF" 0p"P  P 1 p1 p 2 1 p
E RRT   i 1
111

PF 0 0
p
Pp"
1

p 
From Equation (5.13), we know that for the expected realized return to bepositive, the inequality" 
1 pPp 
must be satis®ed. One sucient condition for that inequality to be satis®ed isthat:" 5 This condition basically states that if the unexpected positive shocks in themarket generally have much more impact on the asset return than theunexpected negative shocks, the sequential trading strategy based on thesignals from the geometric moving-average 2; 1 rule will on average createpositive realized returns. Furthermore, the inequality can be satis®ed too aslong as:p if "  
Pp 5 1
This second sucient condition implies that the unexpected positive shocksare highly persistent and much more persistent than the unexpected negativeones even when their means are not much dierent from each other. To seethis, we substitute:P
12
qp
q
into the coecient of the RHS of the inequality and rearrange it as:11 p11 p1 qObviously, we know from this coecient that in order to have a positiveexpected realized return, state one in which the asset return is positive must bevery persistent, say, p  0.90, and much more persistent than state zero inwhich the asset return is negative, i.e., p 5 q. For example, suppose that p isequal to 0.90, and that q is equal to 0.99, which means that state zero is even
Some exact results for moving-average trading rules
163
more persistent than state one. Then, in order to have positive expectedrealized return, the inequality"  1:22should be satis®ed. In this case, it will be easily violated if the mean ofnegative shocks is more or less equal to that of positive shocks. However, ifstate zero is much less persistent than state one, e.g., q  0.5, the inequalitybecomes:"  0:13and can be more easily satis®ed than the previous one.Proposition 5If we condition the result in Equation (5.13) on the event fR0  0g, we have:E RRT  
1
1
p
 p"
1
p 
5:14
ProofSubstitute P  1 into Equation (5.13) and the result follows.Proposition 6Assume that the asset return, Rt is independently identically distributed, theexpected realized return in Equation (5.13) becomes:1E RRT   f p 2 "q
q g
5:15
ProofSubstitute p  q  1 and P  p  1 q into Equation (5.13) and the resultfollows.Basically, Equations (5.14) and (5.15) present similar results. Both equations have the same ®rst sucient condition for getting a positive expectedrealized return as Equation (5.13). However, the second condition ofEquation (5.14) says that as long as state one is highly persistent, an investor,who already owns the target asset when he or she decides to adopt thesequential trading strategy, can make pro®ts on average. In this case, whetherstate zero is persistent or not is irrelevant. In contrast to Equation (5.14),Equation (5.15) even has the same second condition as Equation (5.13), butthe degree of persistence of state one relative to state zero required by theformer is much higher than the latter. For example, we also assume thatp  0.9 and q  0.5.
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Advanced Trading Rules
The inequality:q"  2 pbecomes"  0:31This inequality is apparently much more restricted than:"  0:135.4 THE MOVING-AVERAGE (1, 1) RULEFor the moving-average 1; 1 rule, Equation (5.4) becomes:Xn 2nj  1Rt jRt  limn!1n 1j1
5:16
In order to make the moving-average 1; 1 rule work, the RHS ofEquation (5.16) has to converge to a random process or a scalar. Thisrequires the sequence of coecients:nj  1n 1to be absolutely summable. Obviously, it is not because the summationequals:2
n2
In other words, the RHS of Equation (5.16) does not converge in probabilityto a random process or a scalar even though the asset return is assumed tofollow a stationary process. Therefore, we cannot extend the results of thegeometric moving-average 2; 1 rule to the geometric moving-average 1; 1rule based on Equation (5.16). However, if we take the natural logarithm onboth sides of Equation (5.3), we can have:ln Pt 
n 1Xj0
ln Pt 1 =n
5:17
Assuming the natural logarithm of asset price, ln Pt , to follow a stationaryprocess, we can assert that the RHS of Equation (5.17), according to the Lawof Large Numbers, will converge in probability to its long-term mean level,denoted by 1 . Hence, Equation (5.17) becomes:ln Pt  1
5:18
Some exact results for moving-average trading rules
165
Similarly, provided that the asset price follows a stationary stochasticprocess, the arithmetic moving-average rules described by Equation (5.2) canbe simpli®ed to be:Pt  2
5:19
where 2 is the long-term mean of Pt .As usual, our purpose is to derive the distribution of the realized returnfrom the sequential trading strategy based on the moving-average 1; 1 rule.In order to work on asset returns, we choose to focus on the case of thenatural logarithm of asset prices as described by Equation (5.18). However,the following derivation can be easily duplicated for the case of asset pricelevels. Accordingly, the geometric moving-average 1; 1 rule will generate abuying signal when the natural logarithm of the asset price is greater than orequal to its long-term mean and a selling signal when the opposite happens.Based on this characteristic, we assume that the natural logarithm of the assetprice is generated by the general Knight±Satchell±Tran process:ln Pt  1  Zt "t
1
Zt t
5:20
where the random variables "t , t and Zt have the same speci®cations asdescribed in Section 5.3 and are independent to one another for 8t.Equation (5.20) states that the natural logarithm of the asset price¯uctuates around its long-term mean and whether it is greater than thelong-term mean or not depends on the outcome of the Markovian randomvariable Zt . It can be easily shown that ln Pt is stationary. Moreover, we canpresent the asset return, denoted by:as
Rt  ln Pt
ln Pt 1
Rt  Zt "t
Zt 1 "t 1
1
Zt t  1
Zt 1 t 1
5:21
It can be shown that the asset return is a ®rst-order moving-average processwhich satis®es:E Rt   0Var Rt   2P  2"   2"   1 P  2   2 P  2"   2"   1 P  2   2 E Rt i Rt j  0
if jiif ji
jj  1jj > 1
Proposition 7Assume that the asset return is generated by the ®rst-order moving-averageprocess as described by Equation (5.21). The probability distribution of therealized return resulting from the geometric moving-average 1; 1 rule can
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Advanced Trading Rules
be described by the characteristic function:FRRT s  1
PqF s  F
s  1
p1
PF"
s  F
s5:22
ProofFrom Equation (5.18) we see that the geometric moving-average 1; 1 rulecan be expressed as:Bt  0; iff ln Pt  1 ;
Bt < 0; otherwise
Moreover, we can also see from Equation (5.20) that:ln Pt  0; iF Zt  1;
ln Pt < 0; iF Zt  0
Therefore, the information contained in the sequence of trading signals,fB0 ; . . . ; Bt g, is identical to that contained in the sequence of Markovianrandom variables, fZ0 ; . . . ; Zt g.To derive the distribution of the realized return, we have to compute theprobability for an investor to hold the target asset until the end of period Tconditional on the event that a buying signal is generated by the geometricmoving-average 1; 1 rule at the end of period zero.Pr N  T  Pr BT < 0; BT 1  0; . . . ; B1  0; B0  0 Pr ZT  0; ZT 1  1; . . . ; Z1  1; Z0  1 Pr ZT  0; ZT 1  1; . . . ; Z1  1 j Z0  1 Pr Z0  1(Pp T 1 1 p if T  11 Pif T  0where P is the stationary distribution of state one of the Markov process inEquation (5.8).Now we can derive the conditional characteristic function of the realizedreturn for T periods from the sequential trading strategy based on thegeometric moving-average 1; 1 rule as follows:PNFRRT jNT s  Ee fi t1 Rt IfBt 1 0g sg j N  TPT Ee fi t1 Rt IfBt 1 0g sg j B0  0; . . . ; BT < 0PT Ee fi t1 Rt sg j Z0  1; . . . ; ZT  0 Ee i "1 "0  "2 "1  "T 1 "T 2  t "T 1 s (F" s  F sqF s  F s  1 pF" s F
if T  1s
if T  0
Some exact results for moving-average trading rules
167
Finally, the unconditional characteristic function becomes:1PNXEe fi t1 Rt IfBt 1 0g sg j N  T Pr N  TFRRT s T0
P 1
1X
s  F
s
 1
PfqF s  F
s
 1
pF"
s  F
sg
 1
pF"
PqF s  F
 1
p1
PF"
pT 1
T1
ss  F
s
Proposition 8If we condition the result in Equation (5.22) on the event:fln P0  1 gwe have:FRRT s  F"
s  F
5:23
s
ProofSubstitute P  1 into Equation (5.22) and the result follows.Proposition 9If the natural logarithm of asset price, ln Pt , is independently identicallydistributed, we have:FRRT s  q 2 F s  F
s  1 pqF"
s  F
s
5:24
ProofThe assumption that the natural logarithm of asset price is independentlyidentically distributed implies that p and q in Equation (5.8) satisfy thecondition p  q  1, which in turn implies:P
1
2
qp
q
1
qp
Substitute this condition into Equation (5.22), rearrange it and the resultfollows.Proposition 10For the realized return from the geometric moving-average 1; 1 rule as inEquation (5.22), we can compute the expected realized return, E RRT ,
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Advanced Trading Rules
as follows:E RRT  
1
p1
P"   
5:25
where " and  are the means of the unexpected shocks, "t and t ,respectively.ProofTo derive the expected realized return, we have to ®rst dierentiate thecharacteristic function of the realized return.F 0RRT s  1
PqF 0 s
F 0
s
1
p1
PF 0"
s  F 0
s
Since:E RRT  
iF 0RRT 0
we have:E RRT  
i 1
PqF 0 0

1
p1
F 0 0  i1
p1
PF 0" 0  F 0 0
P"   
Proposition 11If we condition the result in Equation (5.25) on the event:fln P0  1 gwe have:E RRT  
"   
5:26
ProofSubstitute P  1 into Equation (5.25) and the result follows.Proposition 12Assume that the natural logarithm of asset price, ln Pt , is independentlyidentically distributed, the expected realized return in Equation (5.25)becomes:E RRT  
1
pq"   
5:27
ProofSubstitute p  q  1 and P  p  1 q into Equation (5.25) and the resultfollows.Equation (5.25), Equation (5.26) and Equation (5.27) all suggest that aslong as the asset price follows the general Knight±Satchell±Tran process,investors will lose money on average by taking the sequential trading strategy
Some exact results for moving-average trading rules
169
that exploits the information contained in the whole past history of assetprices. In addition, if an investor already owns the target asset when he beginsto follow the strategy, he has to take the risk of losing more money than hehas to if he does not own it in the ®rst place. This is because p 1 P isalways positive. Interestingly, if the natural logarithm of the asset price isactually independently identically distributed and p  q  1, the sequentialtrading strategy based on the geometric moving-average 1; 1 rule does theleast damage to investors' terminal wealth. Furthermore, if it actually followsa random walk, 8 the possible loss from the trading strategy can be reduced toa minimum.
5.5 APPLICATIONS TO UK STOCK AND FUTURES MARKETSIn order to illustrate the implications of the theoretical results in Section 5.3,we use the empirical results of Knight, Satchell and Tran to actually calculatethe expected realized returns of UK FT100 index and futures from thesequential trading strategy based on the geometric moving-average 2; 1rule. Knight, Satchell and Tran assume that the unexpected shocks, "t and t ,follow two dierent scale gamma distributions with parameters 1 ; 1  and2 ; 2 , as described by Equation (5.6) and Equation (5.7) respectively. Table5.1 presents the maximum likelihood estimates of the parameters. These MLestimates suggest that in both markets neither state one nor state zero is verypersistent. On the contrary, they are quite transitory. In other words, once theasset return enters one of the two states today, it has about 50 per cent chanceto switch to the other state tomorrow. Besides, state zero is relativelypersistent to state one in both markets. The fact that the sum of the estimatesof p and q is very close to one, i.e., p^  q^  1, implies that asset return isalmost independently identically distributed in both markets.Before we calculate the expected realized return, E RRT , we ®rst have tocalculate the expected values of "t and t . We know that the characteristicTable 5.1 Parameter estimates ofFT100 futures and FT100 index
^1^1^2^2p^q^
Futures
Index
1.3835171.380.749798.7500.48050.4859
1.3640204.400.7499214.040.48160.5080
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Advanced Trading Rules
function of a scale gamma distribution with parameters is
;  is:
Hence, from the characteristic functions of "t and t , we can calculate theirexpected values by:1
" 
1
 
2
and2
respectively. According to the ML estimates of parameters for the FT100futures, we can compute "  0.0081 and   0.0076. Note that there is notmuch dierence between " and  . Therefore, we can expect that thegeometric moving-average 2; 1 rule cannot create much pro®t for investorsin FT100 futures market. We calculate the expected realized return inEquation (5.13) and Equation (5.15) as follows:1:0 1:0336 0:4805 0:0081 0:5195 0:0076  0:014%0:51951:0E RRT   0:4805 2 0:00810:4859 0:0076  0:38%0:4859
E RRT  
As we expect, the geometric moving-average 2; 1 rule does not work verywell in FT100 futures market. On average, it produces only 0.014 per cent perday. On the other hand, if futures return is very close to independentlyidentically distributed, it will make investors in this market lose 0.38 per centper day on average.In the case of FT100 index, we can calculate "  0.0067 and   0.0035.Note that the expected value of positive shocks in UK stock market is muchgreater than the expected value of negative shocks. In fact, the former isalmost a double of the latter. This leads us to expect that the geometricmoving-average 2; 1 rule would work well in the stock market or at leastmuch better than it does in FT100 futures market. We compute the expectedrealized return in Equation (5.13) and Equation (5.15) for FT100 index asfollows:1:0 1:0104 0:4816 0:0067 0:5184 0:0035  0:28%0:51841:0E RRT   0:4816 2 0:00670:5080 0:0035  0:04%0:5080
E RRT  
Some exact results for moving-average trading rules
171
Interestingly, although the geometric moving-average 2; 1 rule still cannotbring investors in the stock market pro®t if the stock return is independentlyidentically distributed, it does confer investors positive average return up toaround 0.3 per cent per day.5.6 CONCLUSIONSWe have analysed theoretically the probability characteristics of the realizedreturns from two naive moving-average trading rules. Our results say thateven a naive geometric moving-average 2; 1 rule, which generates tradingsignals according to the information contained in the asset return at presenttime, can bring investors pro®ts as long as the positive shocks to Markovianasset returns in ®nancial markets have on average much more in¯uence thanthe negative shocks. The other sucient condition to make such a movingaverage rule pro®table is that the positive shocks must be very persistentthemselves and much more persistent than the negative ones. For thegeometric moving-average 1; 1 rule, we prove that it always deliversnegative realized returns to investors provided that the asset price followsthe general Knight±Satchell±Tran process. However, this negative return canbe reduced to a minimum if the asset price is in fact a random walk.Although we apply the results for the geometric moving-average 2; 1 ruleto UK FT100 index and futures markets by using the empirical results ofKnight, Satchell and Tran and ®nd that it can let investors make pro®ts inboth markets up to on average 0.3 per cent and 6.0 per cent per monthrespectively if asset returns are not independently identically distributed, wedo not statistically test whether these numbers are signi®cant or not. Therefore, the empirical test of the theoretical results in this article will be the topicof our future research.REFERENCESAcar, E. (1993) Economic evaluation of ®nancial forecasting, PhD Thesis, CityUniversity, London.Acar, E. and Satchell, S.E. (1997) A theoretical analysis of trading rules: anapplication to the moving average case with Markovian returns, AppliedMathematical Finance, 4, 1±16.Akgiray, V. (1989) Conditional heteroscedasticity in time series of stock returns:Evidence and forecasts, Journal of Business, 62, 55±80.Blattberg, R.C. and Gonedes, R.C. (1974) A comparison of the stable and studentdistributions as statistical models for stock prices, Journal of Business, 47, 244±80.Brock, W., Lakonishok, J. and LeBaron, B. (1992) Simple technical rules and thestochastic properties of stock returns, Journal of Finance, 47, 1731±64.
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Clark, P.K. (1973) A subordinated stochastic process model with ®nite variancefor speculative prices, Econometrica, 41, 135±55.Fama, E. (1965) The behaviour of stock prices, Journal of Business, 47, 244±80.Knight, J.L., Satchell, S.E. and Tran, K.C. (1995) Statistical modelling ofasymmetric risk in asset returns, Applied Mathematical Finance, 2, 155±72.Kon, S.J. (1984) Models of stock returns: a comparison, Journal of Finance, 39, 147±65.LeBaron, B. (1991) Technical trading rules and regime shifts in foreign exchange,Working Paper, Department of Economics, University of Wisconsin-Madison.Levich, R.M. and Thomas, L.R. (1993) The signi®cance of technical trading-rulepro®ts in the foreign exchange market: a bootstrap approach, Journal ofInternational Money and Finance, 12, 451±74.Mandelbrot, B. (1963) The variations of certain speculative prices, Journal ofBusiness, 36, 394±419.Merton, R. (1976) Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 125±44.Neftci, S.N. (1991) Naive trading rules in ®nancial markets and Wiener-Kolmogorov prediction theory: a study of `technical analysis', Journal of Business, 64,549±71.Tucker, A.L. (1992) A re-examination of ®nite- and in®nite-variance distributionsas models of daily stock returns, Journal of Business and Economic Statistics, 10,73±81.
NOTES1. A random variable B is a Markov time if the eventAt  fB < tgis Ot -measurable ± that is, whether B is less than t or not can be decided giventhe information available at time t, Ot . See Neftci (1991) for an example thatdescribes the dierence between Markov times and non-Markov times.2. The models he uses are the GARCH 1; 1 and the Markov switching modeldeveloped by Hamilton (1989).3. As is standard in this literature, we assume the following approximation Xj 1Pt j 1Pt jPtPt lnRt i1 lnPtPt jPt 1Pt ji04. For example,