daily vs. monthly returns - fields institute · daily vs. monthly returns 1(14) daily vs. monthly...

Daily vs. Monthly returns 1(14)

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 2(14)

Outline

I The problem

I The CTA industry

I The data set

I Daily vs. Monthly

I Pricing of fund-linked products

Daily vs. Monthly returns The problem 3(14)

Problem description

Hedge funds market themselves through monthly data but in amanaged account it is possible to follow a hedge fund investmentevery day.

How will the daily risk and quantitative properties experienced bythe investor differ from what they expect from the monthly figures?

Daily vs. Monthly returns The CTA industry 4(14)

Commodity Trading Advisors

CTA - Managed futures industry

I A 30-year-old asset class

I Chartists and trend followers

I Business legend - Turtle traders

Daily vs. Monthly returns The CTA industry 5(14)

Industry figures

BarclayHedge CTA database collectsmonthly data for CTA programs.Figures from 2010 Q1 shows:

I 1058 funds totally over 20 years

I Annual return 11.6%, Sharperatio 0.41

I 553 active funds managing$217.2B

I Systematic programs constitutethe main part, $169.31B

Daily vs. Monthly returns The dataset 6(14)

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 trendfollowing

Daily vs. Monthly returns The dataset 7(14)

Common hedge fund return biases

I Instant history/Back-fillI Start many funds, keep only the profitable, do not report until

good live performance and use back-fill possibilities.

I Selection biasI Database reporting is voluntary, causing a self-selection bias

I Survivorship biasI Only the fittest survives, blow-ups are rarely reported

Daily vs. Monthly returns The dataset 7(14)

Common hedge fund return biases

I Instant history/Back-fillI Start many funds, keep only the profitable, do not report until

good live performance and use back-fill possibilities.

I Selection biasI Database reporting is voluntary, causing a self-selection bias

I Survivorship biasI Only the fittest survives, blow-ups are rarely reported

Daily vs. Monthly returns The dataset 7(14)

Common hedge fund return biases

I Instant history/Back-fillI Start many funds, keep only the profitable, do not report until

good live performance and use back-fill possibilities.

I Selection biasI Database reporting is voluntary, causing a self-selection bias

I Survivorship biasI Only the fittest survives, blow-ups are rarely reported

Daily vs. Monthly returns Daily vs Monthly figures 8(14)

Moments

Min Mean Median Max

DailyMean return -0.000178 0.000566 0.000546 0.001818St. deviation 0.00242 0.010767 0.00938 0.02550Skewness -1.235 -0.1447 -0.1424 2.3446Kurtosis 3.7552 9.4731 7.093 58.3304

MonthlyMean return -0.0038 0.0123 0.0118 0.0752St. deviation 0.0109 0.0501 0.0443 0.1642Skewness -0.9147 0.2686 0.1447 2.0355Kurtosis 1.8328 4.0179 3.3589 12.3661

Table: Properties of the first four moments for all managers as a group.

Daily vs. Monthly returns Daily vs Monthly figures 9(14)

Non-normality

−0.06 −0.04 −0.02 0 0.02 0.04

0.0010.0030.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99 0.9970.999

Data

Pro

babi

lity

Normal Probability Plot

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

10

20

30

40

50

60

Fitted empirical distribution

Fitted normal distribution

Figure: Left: Normal probability plot of returns clearly showing theoccurrence of fat tails. Right: Empirical distribution (blue) using aEpanechnikov kernel, together with a fitted normal distribution (red)

Daily vs. Monthly returns Daily vs Monthly figures 10(14)

Statistical tests

Daily Monthly

Jacque-Berra test 100% 20%

Lilliefors test 99% 14%

Lags 1 10 1 10

Ljung-Box on returns 50% 50% 12% 26%

Ljung-Box on absolute returns 100% 100% 25% 22%

ARCH-test 90% 97.5% 17% 39%

Table: Percentage of funds rejecting the null hypothesis for statisticaltests on daily and monthly figures.

Daily vs. Monthly returns Daily vs Monthly figures 11(14)

Autocorrelation

0 5 10 15 20 25 30 35 40

0

0.2

0.4

0.6

0.8

1

Autocorrelation

0 5 10 15 20 25 30 35 40

0

0.2

0.4

0.6

0.8

1

Autocorrelation on absolute values

Figure: Left: Autocorrelation function for log-returns. Right:Autocorrelation for the absolute value of the log-returns. Absolute valuesshow an irrefutable correlation, pointing towards the existence ofvolatility clustering.

Daily vs. Monthly returns Pricing of fund-linked products 12(14)

Pricing of fund-linked products

To illustrate the effect of non-normal higher moments we constructsimulation examples using a Normal inverse Gaussian distribution:

dSt = (µ + βσ2(t))St + dt + σ(t)St dBt , S0 = s > 0,

dσ2t = −λσ2

t dt + dLλt , σ20 = y > 0,

Bt - Brownian motion, Lλt - pure jump subordinator.

Daily vs. Monthly returns Pricing of fund-linked products 13(14)

Fixed threshold products

75 80 85 90 950

5

10

15

20

Barrier

% in

crea

sed

risk

of h

ittin

g th

e th

resh

old

75 80 85 90 950

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Barrier

Def

ault

rate

DailyMonthly

Figure: Fixed threshold product over a five years horizon. Left: Rate ofthe fund investment hitting the barrier for a range of barrier values.Right: Percentage increased risk of hitting the barrier when using aNIG-distribution instead of a normal distribution.

Daily vs. Monthly returns Pricing of fund-linked products 14(14)

CPPI

80 82 84 86 88 90 92 94 96 98 100110

120

130

140

150

160

170

180

190

200

210

Barrier

Mea

n of

exp

iry in

dex

DailyMonthly

Figure: Constant proportion portfolio insurance product simulated of overfive years for different insurance levels and leverage factor 4. Theexpected result at expiry is clearly lower for a product simulated usinghigh order moments.