In the past, most puzzles for market efficiency came from empirical data and statistical methods. The puzzle presented in Section 3 is a purely theoretical, mathematical one.
Additionally, in Section 4 some empirical evidence is given. I proved the following:
• In discrete time, the expected gain of the SLS strategy is positive when (¯µh;t)t≥ 0 ∀t and (¯µh;t)t > 0 for at least two time points, or when (¯µh;t)t ≤ 0 ∀t and
(¯µh;t)t<0 for at least two points in time .
• In continuous time, the expected gain of the SLS strategy is positive when ¯µ(t)6= 0.
• The expected gain of the standard SLS rule surpasses the expected gain of a simple buy-and-hold strategy for all t > 0 with ¯µδ(t) 6∈ [0, Beg(K,µ)] if¯ K > 1, with Beg(K) → 0 for K → ∞, and all ¯µδ(t) 6∈ R+0 when K ≤ 1 (in continuous time).
• The risk-adjusted return of the standard SLS rule is positive for all K >0, −1<
¯
µ6= 0, and σ2 >0.
• The risk-adjusted return of the standard SLS rule exceeds the risk-adjusted return of a simple buy-and-hold strategy for all −1 < µ <¯ 0 and when K ≤1 for some 0<µ.¯
• When K and I0∗ are chosen large enough, a trader could have realized excess returns in the years 2016 and 2017 by SLS trading the DAX assets, although bid-ask-spreads, trading fees, and interest rates were taken into account.
• In discrete time, the expected gain of the discounted SLSδstrategy for all discount-ing factorsδ∈(0,1] is positive when (¯µh;t)t≥0∀tand (¯µh;t)t>0 for at least two points in time, or when (¯µh;t)t ≤ 0 ∀t and (¯µh;t)t < 0 for at least two points in time.
• In continuous time, the expected gain of the discounted SLSδ strategy for all dis-counting factorsδ∈(0,1] is positive when ¯µδ(t)6= 0.
The findings of Section 3 mean that an SLS trader can expect positive gain (even in discrete time) on all arbitrary small intervals where the trend is not changing its sign.
Only for points in time when the trend changes its sign will the SLS trader face negative expected gains. Note that the price path itself can arbitrarily often change its slope.
When using the SLS rule, the trader does not have to predict the direction of the price or its turning point. Put differently, even if the price has a lot of turning points, the robust positive expectation property still holds. Only if the trend—not the return—changes its sign, the trader expects a loss. When the trend path is to some extent smooth, and trading frequency is increased, the points in time where the trend changes its sign do carry less (or, when going to continuous time, even no) weight.
Clearly, there are some assumptions to discuss, e.g., continuous time trading. How-ever, since the results of this work do not rely on any price path but only on the trend
process, and there are high frequency trading possibilities, only a very hard non-trending assumption could invalidate these results. For example, one must assume that for every point in time with a positive (negative) price trend, for every arbitrary small interval after that point in time, there must be another point in time where the price trend is negative (positive). This would also imply that there are absolutely no identifiable trends in fundamental values. Adequate resources, perfect liquidity, the possibility of short selling, and approximately also the price-taker property can be seen as justified on modern stock exchanges when both the trader and the traded assets are big enough and I0∗ and K are chosen small enough (cf. Section 2.3).
When a person asks me to solve the puzzle, the only—more or less—satisfying answer I can give is that the risk measure is inappropriate (maybeskewnessis better). But there are two problems: First, this idea only works when market efficiency is defined via risk-adjusted returns only (not when it is defined via expected gain). And second, I would run into a problem very similar to the joint hypotheses problem: I conjecture that for nearly every trading strategy, one can find two risk measures: one indicating that risk-adjusted returns are high, and one indicating that risk-risk-adjusted returns are low. And the other way around, I also conjecture that for nearly all risk measures one can find two trading strategy: one that beats the market, and one that is beaten by it. No one can say whether or not the risk measure or the market efficiency hypothesis is wrong.
Thus, I rely on a standard definition of risk-adjustment.
To sum up, there are three ways to solve the puzzle whether a technical trading rule can have a positive expected gain in an efficient market. The three possibilities of how to solve the problem whether or not the SLS rule is beating the market (for a big, rich trader that trades small amounts of highly liquid stocks of big underlying firms) are explained next: First, were I to assume that all assets are risk-neutral—and not only the market as a whole—the results would not hold. However, that would mean that at every point in time, the trend of every single stock is exactly the trend of the bond, no matter how volatile the stock. (Note: it is reasonable to assume that a high-volatile stock is riskier, and hence should have a higher trend.) Second, and a little bit weaker than the first argument, when the trends (the trends and not only the price paths) of the stocks jump in every (infinitesimally small) interval from positive to negative or vice versa, again the results will not hold. That means, the SLS trading rule is challenging the efficient market hypothesis because one had to assume that all charts of all assets are without any trend (or with only infinitesimally short trends) compared to the num´eraire to disprove the SLS rule’s robust positive expectation property. Usually, the difficulty for traders is not that there are no price trends, the difficulties are that the trends are
unknown, i.e., the the price movements are not unpredictable. However, the SLS rule works well even for fully unknown price trends.
Third, and finally, we cannot adequately measure risk. This leads to a risk including joint hypotheses problem, because there is not one risk measure that everyone relies on.
No one can say whether or not the used risk measure or the efficient market hypothesis is wrong. The last point is the most satisfying answer I can give.
Why do not all traders use the SLS rule if it really works well? Before discussing this, I mention that if all traders would use this rule, the market requirements would not be fulfilled anymore, and liquidity especially would be an issue. When all traders in a market follow the same rule (or even a very similar rule), trading volume would decrease (to zero) since all agents want to buy (sell) at the same time, but no one is selling (buying). I guess that again, risk is the answer: When most traders fear the high risk related to SLS trading, and only a very small fraction of traders use this rule, the non-SLS traders will be exploited by the SLS traders, since all market requirements are fulfilled.
At the very end of this work, I mention that the robust positive expectation property is not an arbitrage possibility. The gain is not sure, it is only in expectation. And it needs potentially a very high number of experiments, i.e., of trading processes, to realize, on average, a positive expected gain.
Acknowledgment
“Is Fama wrong?” This question was asked during my presentation at the Interna-tional Symposium on Interdisciplinarity at the Universit`a di Corsica Pasquale Paoli in Corte/Corti, France (Baumann et al., 2017). This question was the motivation for the work at hand. Therefore, I thank the audience of the track Interdisciplinarity in Eco-nomics.
Furthermore, I thank Lars Gr¨une (Applied Mathematics, University of Bayreuth, Germany), Bernhard Herz (International Economics and Finance, University of Bayreuth, Germany), and Michaela Baumann (Analytics Competence Center, N ¨URNBERGER Versicherung, Germany) for their support. I thank Louisa Chen (University of Sussex), Dmitry Shevchenko (Southern Federal University Rostov-on-Don), and Jakob Krause (Martin-Luther-Universit¨at Halle-Wittenberg) for their helpful comments. Additionally, I thank Jane Lael for her professional editing work.
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1 ADIDAS D:ADS
2 ALLIANZ D:ALV
3 BASF D:BAS
4 BAYER D:BAYN
5 BEIERSDORF D:BEI
6 BMW D:BMW
7 COMMERZBANK D:CBK
8 CONTINENTAL D:CON
9 COVESTRO D:1COV
10 DAIMLER D:DAI
11 DEUTSCHE BANK D:DBK
12 DEUTSCHE BOERSE D:DB1
13 DEUTSCHE LUFTHANSA D:LHA
14 DEUTSCHE POST D:DPW
15 DEUTSCHE TELEKOM D:DTE
16 E ON N D:EOAN
17 FRESENIUS D:FRE
18 FRESENIUS MED.CARE D:FME
19 HEIDELBERGCEMENT D:HEI
20 HENKEL D:HEN
21 INFINEON TECHNOLOGIES D:IFX
22 LINDE D:LIN
23 MERCK KGAA D:MRK
24 MUENCHENER RUCK D:MUV2
25 RWE D:RWE
26 SAP D:SAP
27 SIEMENS D:SIE
28 THYSSENKRUPP D:TKA
29 VOLKSWAGEN D:VOW
30 VONOVIA D:VNA
Table 1: List of the 30 assets indexed in the DAX.
×
×
×
× dgL
gL IL
dgS
gS
-IS
-dp p
K I0∗
gSLS
Figure 1: Flow diagram for the standard SLS controller with input (or disturbance) variable return on investment dpp, i.e., price, and output variable gain gSLS. The SLS traders’ parameters areK >0 and I0∗ >0.
p0= 10 I0L= 100 I0S =−100 g0 = 0 B0 = 0
p1 = 12 I1L= 140 I1S =−60 g1= 0 B1 =−80
p1 = 9 I1L= 80 I1S =−120 g1 = 0 B1 = 40
p2 = 13.2 I2L= 168 I2S =−48 g1= 8 B1 =−112
p2 = 10.8 I2L= 112 I2S =−72 g1=−8 B1 =−48
p2 = 9.9 I2L= 96 I2S =−96 g1=−4 B1 =−4
p2 = 8.1 I2L= 64 I2S =−144 g1= 4 B1 =−76 50%
50%
80%
20%
80%
20%
t= 0 1 2
Figure 2: Small SLS trading example in a binomial tree model with three periods and trends: µ1 = 5% andµ2 = 6% and trading parameters: I0∗ = 100 andK= 2.
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
01020304050
expected SLS gain
mu_bar
K = 16, 8, 4, 2, 1, 0.5, 0.25 (top to bottom)
E(g): SLS (solid)
Figure 3: Expected gain of different SLS strategies with I0∗ = 10 and K = 16,8,4,2,1,12,14 . The average trend is ¯µ∈(−1,2].
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0.00.20.40.60.81.0
expected SLS gain
mu_bar
K = 16, 8, 4, 2, 1, 0.5, 0.25 (top to bottom)
E(g): SLS (solid)
Figure 4: Expected gain of different SLS strategies with I0∗ = 10 and K = 16,8,4,2,1,12,14 . The average trend is ¯µ∈[−0.1,0.2].
expected SLS gain
mu_bar
K
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0246810
Figure 5: Contour plot of the expected gain of the SLS strategy for K ∈ (0,10] and µ∈[−0.1,0.2]. The expected gain is positive for all (K, µ) with ¯µ6= 0.
expected SLS gain
mu_bar
K
-1 0 1 2 3 4 5
0246810
Figure 6: Contour plot of the expected gain of the SLS strategy for K ∈ (0,10] and µ∈(−1,5]. The expected gain is positive for all (K, µ) with ¯µ6= 0.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.01
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: SLS (solid)
Figure 7: Risk-adjusted return of different SLS strategies with I0∗ = 10 and K =
1
16,18,14,12,1,2,4 (from top to bottom). All returns are adjusted with the respective standard deviation. The average trend is ¯µ ∈ (−1,5], and the average volatility is σ2 = 1%.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.02
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: SLS (solid)
Figure 8: Risk-adjusted return of different SLS strategies with I0∗ = 10 and K =
1
16,18,14,12,1,2,4 (from top to bottom). All returns are adjusted with the respective standard deviation. The average trend is ¯µ ∈ (−1,5], and the average volatility is σ2 = 2%.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.05
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: SLS (solid)
Figure 9: Risk-adjusted return of different SLS strategies with I0∗ = 10 and K =
1
16,18,14,12,1,2,4 (from top to bottom). All returns are adjusted with the respective standard deviation. The average trend is ¯µ ∈ (−1,5], and the average volatility is σ2 = 5%.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.1
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: SLS (solid)
Figure 10: Risk-adjusted return of different SLS strategies with I0∗ = 10 and K =
1
16,18,14,12,1,2,4 (from top to bottom). All returns are adjusted with the respective standard deviation. The average trend is ¯µ ∈ (−1,5], and the average volatility is σ2 = 10%.
risk-adjusted return (SLS)
mu_bar
K
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0246810
Figure 11: Contour plot of the risk-adjusted return of the SLS strategy for K ∈(0,10]
andµ∈[−0.1,0.2]. For risk adjustment, I use the standard deviation. The risk-adjusted return is positive for all (K, µ) with ¯µ6= 0. The average volatility isσ2 = 1%.
risk-adjusted return (SLS)
mu_bar
K
-1 0 1 2 3 4 5
0246810
Figure 12: Contour plot of the risk-adjusted return of the SLS strategy for K ∈(0,10]
and µ ∈ (−1,5]. For risk adjustment, I use the standard deviation. The risk-adjusted return is positive for all (K, µ) with ¯µ6= 0. The average volatility isσ2 = 1%.
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
01020304050
expected SLS gain
mu_bar
K = 16, 8, 4, 2, 1, 0.5, 0.25 (top to bottom)
E(g): bnh (dashed), SLS (solid)
Figure 13: Expected gain of different SLS strategies (solid lines) with I0∗ = 10 and K = 16,8,4,2,1,12,14 (from top to bottom) compared to the expected gain of a simple buy-and-hold strategy (dashed line) with initial investment 10. The average trend is
¯
µ∈(−1,2].
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0.00.20.40.60.81.0
expected SLS gain
mu_bar
K = 16, 8, 4, 2, 1, 0.5, 0.25 (top to bottom)
E(g): bnh (dashed), SLS (solid)
Figure 14: Expected gain of different SLS strategies (solid lines) with I0∗ = 10 and K = 16,8,4,2,1,12,14 (from top to bottom) compared to the expected gain of a simple buy-and-hold strategy (dashed line) with initial investment 10. The average trend is
¯
µ∈[−0.1,0.2].
expected SLS gain - expected bnh gain
mu_bar
K
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0246810
Figure 15: Contour plot of the expected difference of the gain of the SLS strategy and the bnh rule for K ∈(0,10] and µ∈[−0.1,0.2]. The expected difference is positive for all (K, µ) in the left as well as in the upper-right area.
expected SLS gain - expected bnh gain
mu_bar
K
-1 0 1 2 3 4 5
0246810
Figure 16: Contour plot of the expected difference of the gain of the SLS strategy and the bnh rule for K ∈(0,10] and µ∈(−1,5]. The expected difference is positive for all (K, µ) in the left as well as in the upper-right area.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.01
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: bnh (dashed), SLS (solid)
Figure 17: Risk-adjusted return of different SLS strategies (solid lines) withI0∗= 10 and K = 161,18,14,12,1,2,4 (from top to bottom) compared to the risk-adjusted return of a simple buy-and-hold strategy (dashed line) with initial investment 10. All returns are adjusted with the respective standard deviation. The average trend is ¯µ∈(−1,5], and the average volatility isσ2 = 1%.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.02
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: bnh (dashed), SLS (solid)
Figure 18: Risk-adjusted return of different SLS strategies (solid lines) withI0∗= 10 and K = 161,18,14,12,1,2,4 (from top to bottom) compared to the risk-adjusted return of a simple buy-and-hold strategy (dashed line) with initial investment 10. All returns are adjusted with the respective standard deviation. The average trend is ¯µ∈(−1,5], and the average volatility isσ2 = 2%.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.05
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: bnh (dashed), SLS (solid)
Figure 19: Risk-adjusted return of different SLS strategies (solid lines) withI0∗= 10 and K = 161,18,14,12,1,2,4 (from top to bottom) compared to the risk-adjusted return of a simple buy-and-hold strategy (dashed line) with initial investment 10. All returns are adjusted with the respective standard deviation. The average trend is ¯µ∈(−1,5], and the average volatility isσ2 = 5%.
-1 0 1 2 3 4 5
0246810
risk-adjusted return sigma^2_bar=
0.1
mu_bar
K = 0.0625, 0.125, 0.25, 0.5, 1, 2, 4 (top to bottom)
rar: bnh (dashed), SLS (solid)
Figure 20: Risk-adjusted return of different SLS strategies (solid lines) withI0∗= 10 and K = 161,18,14,12,1,2,4 (from top to bottom) compared to the risk-adjusted return of a simple buy-and-hold strategy (dashed line) with initial investment 10. All returns are adjusted with the respective standard deviation. The average trend is ¯µ∈(−1,5], and the average volatility isσ2 = 10%.
risk-adjusted return (SLS - bnh)
mu_bar
K
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0246810
Figure 21: Contour plot of the difference of the risk-adjusted returns of the SLS rule and of a bnh rule. The average volatility is 1%. The SLS rule is dominant on the left side.
risk-adjusted return (SLS - bnh)
mu_bar
K
-1 0 1 2 3 4 5
0246810
Figure 22: Contour plot of the difference of the risk-adjusted returns of the SLS rule and of a bnh rule. The average volatility is 1%. The SLS rule is dominant in the left and in the lower-right area.
×
×
×
×
×
×
× dgLδ
gLδ ILδ
dgSδ
gSδ
-ISδ
-dp p
K I0∗ δ−1
δ−t
gSLSδ fLδ
fSδ
Figure 23: Flow diagram for the discounted SLS controller with input (or disturbance) variable return on investment dpp, i.e., price, and output variable gain gSLSδ. The SLSδ traders’ parameters areK >0,I0∗>0, andδ∈(0,1].
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0
01020304050
expected discounted SLS_delta gain
mu_bar_delta
K = 16, 8, 4, 2, 1, 0.5, 0.25 (top to bottom)
E(f): SLS_delta (solid)
Figure 24: Expected discounted gain of different SLSδ strategies with I0∗ = 10 and K = 16,8,4,2,1,12,14 (from top to bottom). The average trend is ¯µδ∈(−1,2].
-0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0.00.20.40.60.81.0
expected discounted SLS_delta gain
mu_bar_delta
mu_bar_delta