Daily vs. Monthly returns
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Daily vs. Monthly returns
Empirical evidence from Commodity Trading Advisors
Martin Groth
martin.groth@brummer.se
Bachelier World Congress
Toronto, Canada, June 2010
Daily vs. Monthly returns
Outline
I
The problem
I
The CTA industry
I
The data set
I
Daily vs. Monthly
I
Pricing of fund-linked products
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Daily vs. Monthly returns
The problem
Problem description
Hedge funds market themselves through monthly data but in a
managed account it is possible to follow a hedge fund investment
every day.
How will the daily risk and quantitative properties experienced by
the investor differ from what they expect from the monthly figures?
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Daily vs. Monthly returns
Commodity Trading Advisors
CTA - Managed futures industry
I
A 30-year-old asset class
I
Chartists and trend followers
I
Business legend - Turtle traders
The CTA industry
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Daily vs. Monthly returns
Industry figures
BarclayHedge CTA database collects
monthly data for CTA programs.
Figures from 2010 Q1 shows:
I
1058 funds totally over 20 years
I
Annual return 11.6%, Sharpe
ratio 0.41
I
553 active funds managing
$217.2B
I
Systematic programs constitute
the main part, $169.31B
The CTA industry
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Daily vs. Monthly returns
The dataset
The data set
I
Daily return series from 77 CTA funds of which 65 were active
I
No proforma, only live trading
I
At least 2 years track record
I
Mainly classic CTA strategies, mid- to -long term trend
following
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Daily vs. Monthly returns
The dataset
Common hedge fund return biases
I
Instant history/Back-fill
I
Start many funds, keep only the profitable, do not report until
good live performance and use back-fill possibilities.
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Daily vs. Monthly returns
The dataset
Common hedge fund return biases
I
Instant history/Back-fill
I
I
Start many funds, keep only the profitable, do not report until
good live performance and use back-fill possibilities.
Selection bias
I
Database reporting is voluntary, causing a self-selection bias
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Daily vs. Monthly returns
The dataset
Common hedge fund return biases
I
Instant history/Back-fill
I
I
Selection bias
I
I
Start many funds, keep only the profitable, do not report until
good live performance and use back-fill possibilities.
Database reporting is voluntary, causing a self-selection bias
Survivorship bias
I
Only the fittest survives, blow-ups are rarely reported
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Daily vs. Monthly returns
Daily vs Monthly figures
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Moments
Min
Mean
Median
Max
Mean return
St. deviation
Skewness
Kurtosis
-0.000178
0.00242
-1.235
3.7552
0.000566
0.010767
-0.1447
9.4731
0.000546
0.00938
-0.1424
7.093
0.001818
0.02550
2.3446
58.3304
Mean return
St. deviation
Skewness
Kurtosis
-0.0038
0.0109
-0.9147
1.8328
0.0123
0.0501
0.2686
4.0179
0.0118
0.0443
0.1447
3.3589
0.0752
0.1642
2.0355
12.3661
Daily
Monthly
Table: Properties of the first four moments for all managers as a group.
Daily vs. Monthly returns
Daily vs Monthly figures
Non-normality
Normal Probability Plot
60
Fitted empirical distribution
Probability
0.999
0.997
0.99
0.98
0.95
0.90
Fitted normal distribution
50
40
0.75
30
0.50
0.25
20
0.10
0.05
0.02
0.01
0.003
0.001
10
−0.06
−0.04
−0.02
Data
0
0.02
0.04
0
−0.06
−0.04
−0.02
0
0.02
0.04
Figure: Left: Normal probability plot of returns clearly showing the
occurrence of fat tails. Right: Empirical distribution (blue) using a
Epanechnikov kernel, together with a fitted normal distribution (red)
0.06
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Daily vs. Monthly returns
Daily vs Monthly figures
Statistical tests
Jacque-Berra test
Lilliefors test
Lags
Ljung-Box on returns
Ljung-Box on absolute returns
ARCH-test
Daily
100%
99%
1
50%
100%
90%
10
50%
100%
97.5%
Monthly
20%
14%
1
12%
25%
17%
10
26%
22%
39%
Table: Percentage of funds rejecting the null hypothesis for statistical
tests on daily and monthly figures.
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Daily vs. Monthly returns
Daily vs Monthly figures
Autocorrelation
Autocorrelation
Autocorrelation on absolute values
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
35
40
Figure: Left: Autocorrelation function for log-returns. Right:
Autocorrelation for the absolute value of the log-returns. Absolute values
show an irrefutable correlation, pointing towards the existence of
volatility clustering.
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Daily vs. Monthly returns
Pricing of fund-linked products
Pricing of fund-linked products
To illustrate the effect of non-normal higher moments we construct
simulation examples using a Normal inverse Gaussian distribution:
dSt
= (µ + βσ 2 (t))St + dt + σ(t)St dBt , S0 = s > 0,
dσt2 = −λσt2 dt + dLλt , σ02 = y > 0,
Bt - Brownian motion, Lλt - pure jump subordinator.
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Daily vs. Monthly returns
Pricing of fund-linked products
Fixed threshold products
0.6
% increased risk of hitting the threshold
0.7
Daily
Monthly
Default rate
0.5
0.4
0.3
0.2
0.1
0
75
80
85
Barrier
90
95
20
15
10
5
0
75
80
85
Barrier
90
95
Figure: Fixed threshold product over a five years horizon. Left: Rate of
the fund investment hitting the barrier for a range of barrier values.
Right: Percentage increased risk of hitting the barrier when using a
NIG-distribution instead of a normal distribution.
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Daily vs. Monthly returns
Pricing of fund-linked products
CPPI
210
Daily
Monthly
200
Mean of expiry index
190
180
170
160
150
140
130
120
110
80
82
84
86
88
90
Barrier
92
94
96
98
100
Figure: Constant proportion portfolio insurance product simulated of over
five years for different insurance levels and leverage factor 4. The
expected result at expiry is clearly lower for a product simulated using
high order moments.
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