Post on 19-Dec-2015
transcript
Outline
Moving average rulesChannel rulesFilter rulesRule evaluation
Statistical significance and risk Breakeven transaction costs
Monte-carlo and bootstrap tests
Moving Average Trading Rules (Simplest)
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at ,L =1L
pt− jj=0
L−1
∑
pt
≥ at ,L : Buy
< at ,L : Sell ⎧ ⎨ ⎩
Multiple Averages
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at ,L =1L
pt− jj=0
L−1
∑
at ,S =1S
pt− jj=0
S−1
∑
at ,S
≥ at ,L : Buy
< at ,L : Sell ⎧ ⎨ ⎩
Bands
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at ,L =1L
pt− jj=0
L−1
∑
at ,S =1S
pt− jj=0
S−1
∑
at ,S
≥ (1+ B)at ,L : Buy
< (1− B)at ,L : Sell ⎧ ⎨ ⎩
Channel Rule
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mt−1 = min(pt−L ,K pt−1)
M t−1 = max(pt−L ,K pt−1)
t = Buy
pt
< (1− B)mt−1 : Sell
≥ (1+ B)mt−1 : Buy
Otherwise : neutral
⎧
⎨ ⎪
⎩ ⎪
t = Sell
pt
≤ (1− B)M t−1 : Sell
≥ (1+ B)M t−1 : Buy
Otherwise : neutral
⎧
⎨ ⎪
⎩ ⎪
Filter Rule
Buy period to sell Price falls by f fraction from recent price
maxSell period to buy
Price rises by f fraction from recent price min
Significance Test I: Buy-Sell
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r B =1nB
rtt∈B∑ r S =
1nS
rtt∈S∑
sB
2 = var(rB ) sS
2 = var(rS )
z =r B − r S
(sB
2 /nB + sS
2 /nS )(1/2)
Significance Test II: Dynamic strategy, genmatrule.m
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st =1: Buy
−1: Sell ⎧ ⎨ ⎩
xt = strt
x =1n
xtt=1
n
∑ , s2 = var(x)
z =x
(s2 /n)(1/2)
Probability of a Price Rise
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p1 = Pr(rt+1 > 0 | t is buy)
p2 = Pr(rt+1 > 0 | t is sell)
p = Pr(rt+1 > 0)
sp1
2 = var(p1) =p(1− p)
nB
sp2
2 = var(p2 ) =p(1− p)
nS
z =p1 − p2
(sp1
2 + sp2
2 )(1/2)z'=
p1 − 0.5(0.25 /nB )(1/2)
Results From Equity Markets
Brock, Lakonishok, LeBaron(1992) Dow data (daily/100 years) Standard MA rules (5, 50, 150, 200 day) Stat sig predictability Volatility forecasts
Sullivan, Timmermann and White(1999), LeBaron(2000) Results drop in 90’s
Day and Wang(2002) Nonsynchronous prices
Global Equity Markets
Bessembinder and Chen(1995) Repeat results for Asia
Hudson, Dempsey, and Keasey(1996) Long range results form UK
Consistent predictability over many years, many countries
Predictability falling over time
FX Markets
Generally stronger predictability than equity markets Levich and Thomas(1993) LeBaron(1992)
Some connections with intervention LeBaron(1999)
Transaction Costs
Costs of trading: ImportantOften assume proportionalDepends on strategyFirst strategy:
Simple (Long/short) futures Long in buy periods Short in sell periods
Transaction Costs: I. Simple long/short futures
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PT = (1+ r1' )(1+ r2
' )(1− c)(1− c)(1− r3' )(1− r4
' ) −1
log(1− c) ≈ −c
log(1+ PT ) = r1 + r2 − c − c − r3 − r4
ER = log(1+ PT ) = strtt=1
T
∑ − 2Ntradec
Daily Sharpe ratio =(1/T )ERσ (strt )
Annual Sharpe =(250 /T )(ER)
250σ (strt )
Breakeven Transaction Costs:
Simple long/short futures
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PT = (1+ r1
' )(1+ r2
' )(1− c)(1− c)(1− r3
' )(1− r4
' ) −1
Breakeven : PT = 0 log(1− c) ≈ −c
log(1) = r1 + r2 − c − c − r3 − r4
0 = strtt=1
T
∑ − 2N tradec
c =strt
t=1
T
∑2N trade
Transaction Costs: Simple Equity Strategy
Equity strategy: Sell: Hold risk free Buy: Leverage position
Invest own $1, borrow additional $1 Designed to replicate risk on buy and hold
Transaction costs: Equity strategy
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WT = (1 + 2r1' − r ' )(1 + 2r2
' − r ' )K
(1 − c)(1 + r ' )(1 + r ' )
log(WT ) = 2r1 − r + 2r2 − r − c + r + r
Adjust for oportunity cost(excess return)
ER = 2r1 − r + 2r2 − r − c + r + r − 4r
ER = (2rt − 2r)t=1
2
∑ − c
ER = It,B 2(rt
t=1
T
∑ − r) − Ntradec
Sharpe =(1/T )ERσ (ER)
Annual Sharpe =250(1/T )(ER)
250σ (ER)
Breakeven Transaction Costs: Equity portfolio
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0 = 2r1 − r + 2r2 − r − c + r + r − 4r
0 = (2rt − 2r)t=1
2
∑ − c
0 = It,B 2(rt
t=1
T
∑ − r) − Ntradec
c =
It,B 2(rt − r)t=1
T
∑Ntrade
Results
US equity(Dow): 0.22% for recent periods Smaller than most T-cost estimates
Older periods (up to 1%) Currencies: large returns for 0.2% transaction
levels (6-10%) (Sharpe ratios near 1) All near zero beta
Recent Results
All trading rule returns falling in the 1990’s
Increased efficiency?LeBaron(1999): FX interventions
Removing intervention period removes most fx predictability
Few US interventions in the 90’s
Evidence Summary
Generally large statistical significanceMarginal break even results
Except FX Big returns after T-costs Good risk tradeoff
Careful: All results getting smaller over all recent periods!!!!
Bootstrap Tests
Brock, Lakonishok, and LeBaron(1999) Scrambled returns series
(Monte-carlo: simulated normal returns) Destroy patterns Evaluate rules on scrambled series Compare with original Matlab:
bsmarule.m