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Figure 10.4 Annualised RiskMetrics volatility with varying decay factors

A comparison of different decay factors can be found in the sheet named “DifferentDe-
cayChart” in Bourgoin001.xls (Figure 10.4).
As we will see in the next section, the RiskMetrics and optimal decay models are, in
fact, simpler versions of the GARCH model.


10.3.3 The GARCH model

The general form of the GARCH(p, q) model is:

p q
=ω+ +
σt2 2 2
±i µt’i βj σt’j (10.8)
j =1
i=1

max(p,q)
(±i + βi ) < 1, this
Where all the parameters are positive or equal to zero and i=1
implies that the unconditional volatility of the process exists (we will show this later on)
and provides the necessary stationary condition for the stochastic process.
The GARCH model allows more ¬‚exibility to be built into the volatility process than
RiskMetrics but the extra complexity comes at a price, in that the optimisation is more
complex (maximum likelihood functions are usually known to be very ¬‚at functions of
their parameters, which makes it dif¬cult for optimisers).
We will restrict our analysis to the ¬nancial industry standard (Engle and Mezrich, 1995),
the GARCH(1,1) with just one lag for the residual and one lag for the conditional variance.
As we can see from the above equation, the RiskMetrics equation is embedded in it.7 We
have shown previously that the term structure of volatility forecast is ¬‚at for RiskMetrics,
here however the GARCH(1,1) allows for a mean-reversion process to take place for the

In a GARCH(1,1), the volatility equation becomes: σt2 = ω + ±µt’1 + βσt’1 . If we replace ω by 0 and ± by
2 2
7

1 ’ β, then the model falls back to the RiskMetrics equation.
Applied Volatility and Correlation Modelling 319

volatility. That is, the volatility mean reverts to a long-term average, the unconditional
volatility of the process.
To forecast the volatility, we have to iterate equation (10.8) forward (with p = 1 and
q = 1):

σt2 (1) = E(σt+1/t ) = ω + ±µt2 + βσt2
2


σt2 (2) = E(ω + ±µt+1 + βσt2 (1)) = ω + ±E(µt+1 ) + βσt2 (1)
2 2


= ω + ±E(σt2 (1)) + βσt2 (1)
= ω + (± + β)σt2 (1)
σt2 (3) = ω + (± + β)σt2 (2) = ω(1 + (± + β)) + (± + β)2 σt2 (1)

Hence for a forecasting horizon h, we have:

1 ’ (± + β)h’1
σt2 (h) = ω + (± + β)h’1 σt2 (1) (10.9)
1 ’ (± + β)

When h tends to in¬nity, we have the following long-term variance of the GARCH(1,1)
process:
ω
lim σt2 (h) = σ 2 = (10.10)
1’±’β
h’∞


where σ is the unconditional variance.
The spreadsheet-based application (“DataGARCH” sheet in Bourgoin001.xls) is not
very different from the optimal decay sheet; only now omega and alpha are part of the
optimisation routine (Figure 10.5).




Figure 10.5 The GARCH model
320 Applied Quantitative Methods for Trading and Investment

However because we know the long-term volatility σ 2 using historical data and the
standard time series approach:

N
1 1
σ= (rt ’ r)2 =
2
E(RR ) (10.11)
or
N T
t=1


We can replace ω by using equations (10.10) and (10.11) together:

ω = σ 2 (1 ’ ± ’ β) (10.12)

ω is then NOT required in the optimisation process! The technique has been pioneered
by Engle and Mezrich (1995) and is called the variance targeting technique, so we can
make the optimisation routine simpler by reducing the number of parameters from three
to two. One should note that on the spreadsheet in Figure 10.6 (“DataGARCH VT” in
Bourgoin001.xls) only alpha and beta are greyed out.
As we can see the likelihood function $I$8 is identical to the normal GARCH model
in the previous example, so there seems to be no loss of likelihood in the model and we
were able to reduce the number of parameters at the same time.




Figure 10.6 The GARCH model with variance targeting


10.3.4 THE GJR model

Since Black (1976) it is well accepted that stocks have an asymmetric response to news.
In order to account for these asymmetries encountered in the market, Glosten, Jagannathan
Applied Volatility and Correlation Modelling 321

and Runkle (GJR; Glosten et al., 1993) extended the framework of the GARCH(1,1) to
take this into account. The augmented model can be written as:

0 if µt > 0
rt = σt µt σt2 = ω + ±µt’1 + βσt’1 + γ St’1 µt’1 St =
2 2 2
(10.13)
1 if µt < 0

with ± > 0, β > 0, γ > 0 and ± + β + 1 γ < 1. When returns are positive, γ St’1 µt’1 2
2
is equal to zero and the volatility equation collapses to a GARCH(1,1) equation; on
the contrary, when returns are negative, the volatility equation is augmented by γ , a
positive number, meaning that the impact of this particular negative return is bigger on
the estimation of volatility, as shown below:

σt2 = ω + (± + γ )µt’1 + βσt’1
2 2
if µt < 0,

We can forecast the volatility term structure:
h’1
1 ’ ± + β + 1γ h’1
=ω + ± + β + 1γ
2 2 2
σGJR,t (h) σGJR,t (1) (10.14)
1’ ±+β + 2
1
γ
2

with σGJR,t (1) = ω + ±µt2 + βσt2 + γ St µt2 /2. We can derive the long-term variance pro-
2

cess from this model as:
ω
lim σGJR,t (h) = σ 2 =
2
(10.15)
1 ’ ± ’ β ’ γ /2
h’∞

The spreadsheet-based application (Figure 10.7) is located in “DataGARCHGJR VT”
(Bourgoin001.xls).




Figure 10.7 The GJR model with variance targeting
322 Applied Quantitative Methods for Trading and Investment

Remarks:

• Here we use again the variance targeting process (note that $I$4, omega is not greyed
out) in order to facilitate the optimisation process.
• One good way to ensure that the optimisation has good starting values is to give alpha
and beta the values of a GARCH(1,1) process with gamma equal to a very low positive
value (say 0.01 for example as long as alpha+beta+0.5— gamma is strictly less than 1).
• One should also note that the likelihood function is slightly higher in GJR(1,1) with
2548.37 against 2543.64 for the GARCH(1,1). The likelihood ratio test8 performed
in $I$10 allows us to conclude that this model is capturing some of the unaccounted
volatility dynamics of the simple GARCH model.

10.3.5 Model comparison
Since we™ve calculated several types of volatility model, we can now look at the one-
step-ahead volatility forecast over time (Figure 10.8).
Several remarks can be made:

• The GJR model shows clearly that it is more sensitive to the big downward movements
of the S&P500. For example, in August 1998, it shows a substantial increase in volatility
compared to the GARCH model or any of the weighted-decay models.
• There are clearly two distinct types of model here: the weighted-decay on the one side
and the GARCH-type model on the other. This can be shown from the high degree
of persistence of the volatility with the weighted-decay models (after a shock their

LTCM melt-down generates 1600
45.00%
substantially more volatility
with an asymmetric model 1500
40.00%
1400
35.00%
1300 S&P500 Index

30.00%
Volatility




1200

1100
25.00%
1000
20.00%
900
15.00%
800
Weighted-decay models
under-estimate volatility
700
10.00%
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GARCH Volatility VT Volatility RiskMetrics GJR Ann. Volatility VT Optimal Decay Ann. Vol
S&P500 Index

Figure 10.8 Comparison between different volatility models

8
For the likelihood ratio test, refer to Gallant (1997, p. 181).
Applied Volatility and Correlation Modelling 323

volatility is consistently higher than the GARCH-type models), because the persistence
parameters ± + β = 1 = » compared to ± + β < 1 for the GARCH model.
• We can also see that during calm periods, the weighted-decay volatilities tend to
decrease much more than the GARCH models, this is due to the fact that there is
no mean reversion embedded in these models. If we take the extreme case, the volatil-
ity will progressively tend to zero if the asset returns are zero in the near future; this
has been one of the main criticisms of the RiskMetrics model.


The difference in the volatility estimation not only differs historically, but also when we
try to forecast the term structure of volatility in the future (Figure 10.9). As we™ve shown
earlier, the weighted-decay methodology does not provide any insight into the future
behaviour of the term structure of volatility since there is no information in its term
structure (the term structure of forecast is ¬‚at and equal to the one-step-ahead volatility
forecast). On the other hand, the GARCH-type model shows us the mean-reversion process
to the long-term volatility (the unconditional volatility) happening over time. Here, we
can see that the GARCH model mean reverts slightly quicker than the GJR model, but
both models show that the volatility is likely to increase in the next 5 months (horizon
of forecasting).
In this section, we™ve seen that it is possible to estimate and forecast volatility using
complex volatility models within the Excel framework. However if one wants, for example,
to do complicated statistical tests for diagnostic checking, Excel will not be a suitable
framework for a proper econometric analysis of ¬nancial time series. But for common
day-to-day analysis, the practicality and user friendliness is unrivalled.
The next section will introduce the modelling of conditional correlation using the same
framework utilised above.


LT Volatility
22.00%


20.00%


GJR-VT(1,1)
18.00%
GARCH(1,1)

16.00%

RiskMetrics & optimal decay models
14.00%


12.00%


10.00%
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12-Aug-01
17-Aug-01
22-Aug-01
27-Aug-01
1-Sep-01
6-Sep-01
11-Sep-01
16-Sep-01
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1-Oct-01
6-Oct-01
11-Oct-01
16-Oct-01
21-Oct-01
26-Oct-01
31-Oct-01
5-Nov-01
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20-Nov-01
25-Nov-01
30-Nov-01
5-Dec-01
10-Dec-01
15-Dec-01
20-Dec-01
25-Dec-01
30-Dec-01




Figure 10.9 S&P500 term structure of volatility forecast for different models
324 Applied Quantitative Methods for Trading and Investment

10.4 MULTIVARIATE MODELS
In this section, we will show how to perform multivariate models to calculate conditional
correlation estimation and forecast the term structure using Excel. Several models will
be considered, the J.P. Morgan RiskMetrics, the optimal decay model and three GARCH
models: the full diagonal GARCH, and its simpler derivative with variance targeting, and
the superfast GARCH model (Bourgoin, 2002). For convenience purposes and simplicity,
we will consider only a two-variable system, more can be added but with additional
complexity in the spreadsheet set-up. Each calculation devoted to a speci¬c asset or
cross-term (volatility for asset 1, asset 2 and the covariance) will have its own background
colour in the workbook Bourgoin002.xls. In this way it is easier to read and understand
the spreadsheet.


10.4.1 The RiskMetrics model

Let us consider a two-asset model, where both volatilities and the covariance follow the
RiskMetrics (RiskMetrics Technical Document, p. 82) equation:

σ1,t = (1 ’ »)µ1,t’1 + »σ1,t’1
2 2 2


σ2,t = (1 ’ »)µ2,t’1 + »σ2,t’1
2 2 2
(10.16)
σ12,t = (1 ’ »)µ1,t’1 µ2,t’1 + »σ12,t’1
2 2


The initialisation of σ1,0 , σ2,0 , σ12,0 is set to the unconditional volatilities and covariance.
The model has very appealing attributes in a multivariate framework: it is very quick to
calculate for a large size covariance matrix and the covariance is always positive de¬nite
by construction as long as » is the same everywhere in the covariance matrix.9 But the
drawbacks are still the same in a multivariate framework: ¬‚at term structure of forecast,
too much persistence of shocks and too low volatilities during calm periods.
The spreadsheet of Figure 10.10 (“RiskMetrics” in Bourgoin002.xls) shows the imple-
mentation of RiskMetrics for volatilities and correlations for two stock indexes, the French
CAC40 and the German DAX30. The volatilities are calculated in exactly the same
way as in Bourgoin001.xls, but here we add the cross-term to calculate the covariance.
Equation (10.16) is calculated in columns F, G and H respectively. Column I is the
resulting correlation by applying the standard textbook formula:

ρi,j = σi,j /(σi σj )


10.4.2 The optimal decay model

The optimal decay model is a generalisation of the RiskMetrics model where the optimal
decay is not predetermined at 0.94. The objective function is the likelihood function. In

9
A matrix is positive de¬nite as long as its minimum eigenvalue is strictly positive. This is crucial for risk
management purposes because otherwise a linear combination of assets in the portfolio can yield to a negative
portfolio variance. For a more detailed explanation, refer to J.P. Morgan (1997).
Applied Volatility and Correlation Modelling 325




Figure 10.10 The RiskMetrics model bivariate



order to write the likelihood function, we need to de¬ne the probability distribution, here
we will use the multivariate normal distribution:

’1
1 ’1
f (x, µ, ) = (x ’ µ) (x ’ µ) (10.17)
exp
(2π )n/2 | |1/2 2

where µ is the vector of the mean and the covariance matrix for two assets with zero
mean, the multivariate normal distribution looks like this:

’1 2
x2
x1 2ρ12 x1 x2
1
f (x1 , x2 , σ1 , σ2 , ρ12 ) = ’ +2
exp
2πσ1 σ2 (1 ’ ρ12 ) 2(1 ’ ρ12 ) σ1
2 2 2 2
σ1 σ2 σ2
(10.18)
The likelihood function for a time-varying covariance matrix is the following:

T T
ln L = ln f (x1,i , x2,i , ) = ln(f (x1,i , x2,i , ))
i=1 i=1
T
1
= ’ ln(2π ) ’ ln(σ1,t ) ’ ln(σ2,t ) ’ ln(1 ’ ρ12,t )
2
2
t=1
2 2
x1,t x2,t
2ρ12,t x1,t x2,t
1
’ ’ +2 (10.19)
2(1 ’ ρ12,t )
2 2 σ1,t σ2,t
σ1,t σ2,t

The optimisation routine will be to maximise the log-likelihood function with respect to
the decay factor:
max ln L(») with 0 < » ¤ 1
326 Applied Quantitative Methods for Trading and Investment

Here the covariance speci¬cation is:

ˆ2 ˆ2
σ1,t = (1 ’ »)µ1,t’1 + »σ1,t’1
2


ˆ2 ˆ2
σ2,t = (1 ’ »)µ2,t’1 + »σ2,t’1
2
(10.20)
ˆ ˆ2
σ12,t = (1 ’ »)µ1,t’1 µ2,t’1 + »σ12,t’1
2


ˆ
where » is the optimal decay factor.
The spreadsheet of Figure 10.11 (“OptimalDecay” in Bourgoin002.xls) is identical to
the RiskMetrics sheet up to column I. The only major difference here is that we calculate
the log-likelihood function (column J) at every time step (equation (10.18)). The sum
of the log-likelihood functions (equation (10.19)) is performed in cell M9. Since we are
trying to ¬nd the optimal lambda, we need to maximise this cell (see Solver settings). The
optimal decay factor is obtained after using the Solver and optimising the spreadsheet in
cell M8.




Figure 10.11 The bivariate optimal decay model


10.4.3 The diagonal GARCH model
In the diagonal GARCH model, each variance and covariance term has its own dynamics
(set of parameters), so the functional form for the model is:

σ1,t = ω11 + ±11 µ1,t’1 + β11 σ1,t’1
2 2 2


σ2,t = ω22 + ±22 µ2,t’1 + β22 σ2,t’1
2 2 2
(10.21)
σ12,t = ω12 + ±12 µ1,t’1 µ2,t’1 + β12 σ12,t’1
2 2


The spreadsheet of Figure 10.12 (“Full Diagonal GARCH” in Bourgoin002.xls) is more
complex because we have to deal with multiple constraints:

±ij + βij < 1, ±ij > 0, βij > 0 and ωii > 0
Applied Volatility and Correlation Modelling 327




Figure 10.12 The diagonal GARCH model

Note that ωij is allowed to be negative because otherwise you constrain the long-term
correlation to be positive between the two assets.10
The spreadsheet is identical to the optimal decay spreadsheet with regard to the log-
likelihood function. The optimisation, on the other hand, has multiple constraints (17 in
total!) and nine parameters as indicated in the Solver settings. Each variance (1 and 2)
and the covariance have their own parameters (alphas, betas and omegas).
Because the optimisation procedure is quite complicated, it might require a long time
to run.11

10.4.4 The diagonal GARCH model with variance targeting
For this model, we apply the variance targeting technique mentioned in Section 10.3.3 to
the multivariate context:
ωij = σ 2 (1 ’ ±ij ’ βij ) (10.22)
ij

Because most of the time the alphas and betas tend to have very similar optimised values
between each equation, the complexity arises from the ¬t of the omegas, using the variance

10
This can be derived easily from the long-term correlation implied by the diagonal model.
11
It took 35 seconds to run on a PIII-600 MHz with 256 Mb RAM and Windows 2000.
328 Applied Quantitative Methods for Trading and Investment




Figure 10.13 The diagonal GARCH model with variance targeting

targeting technique accelerates the optimisation procedure dramatically.12 In practice, the
model appeal is greatly enhanced.
In the spreadsheet of Figure 10.13 (“Full Diagonal GARCH VT” in Bourgoin002.xls),
we can see that the spreadsheet is almost identical to the diagonal GARCH model, the
only subtle difference is that Omega1, Omega2 and Omega12 are not part of the opti-
misation procedure (check cells M41 and M42), they have been replaced by formulas
(equation (10.22)) as shown in cell M12. The number of constraints is reduced to 15
from 17 and most importantly the number of parameters is now only six instead of nine.

10.4.5 The scalar GARCH model with variance targeting
This is the most simpli¬ed model of the “traditional” type. We consider that all assets under
analysis have the same parameters, i.e., ±ij = ± and βij = β, and we use the variance
targeting technique in order to eliminate the omegas from the estimation procedure, so
the number of parameters to optimise becomes completely independent from the number
of variables in the model and equal to two (± and β):

σ1,t = σ 2 (1 ’ ± ’ β) + ±µ1,t’1 + βσ1,t’1
2 2 2
1

σ2,t = σ 2 (1 ’ ± ’ β) + ±µ2,t’1 + βσ2,t’1
2 2 2
(10.23)
2

σ12,t = σ 2 (1 ’ ± ’ β) + ±µ1,t’1 µ2,t’1 + βσ12,t’1
2 2
12

12
Instead of 35 seconds, the optimisation routine took only 17 seconds, half the time required by the general
model with hardly any reduction in the likelihood function.
Applied Volatility and Correlation Modelling 329




Figure 10.14 The scalar GARCH model with variance targeting

The worksheet of Figure 10.14 (“Scalar GARCH VT” in Bourgoin002.xls) shows how
this is done. Only MVDGAlpha1 and MVDGBeta1 are optimised, the omegas are derived
from the unconditional variances and the covariance calculated in F3:H3. Here the number
of constraints is a lot smaller (¬ve).
We can note a clear drop in the log-likelihood function to ’10 489.96, whereas before
we had hardly any changes of likelihood at all. Although quite restrictive, the structural
constraints of the model (alphas and betas identical across all equations) provide a very
appealing model where the number of parameters is always two, so the optimiser will be
very quick.13 In this particular case, we can see that the log-likelihood function is lower
than the optimal decay (’10 481) but the model guarantees the mean-reversion process,
avoiding the underestimation of the volatilities during calm periods.

10.4.6 The fast GARCH model
The purpose of this model is to enable us to calculate conditional correlation in a very
fast and easy manner. It was ¬rst presented by Bourgoin (2002), and follows in the
footsteps of Ding and Engle (1994), Ledoit (1999), Pourahmadi (1999), Bourgoin (2000)
and Athayde (2001) to deal with a very large number of assets. For the purpose of clarity,
we will only present the model in a 2 — 2 setting but the application is straightforward
to generalise to a very large number of assets.
As we have seen in the previous paragraph it is possible to use the variance targeting
process on a scalar GARCH in order to obtain a multivariate model with just two para-
meters regardless of the size of the covariance matrix. This is very appealing because the
optimisation routine will converge quickly to the results. However the problem would be
much simpler if we could avoid the estimation of the levels of persistence, ± and β.
Here we follow the idea of Athayde (2001) on the estimation of these parameters. If a
“natural” index exists for the data, we can calculate a univariate GARCH model and apply

13
The optimisation routine took only 7 seconds to ¬nd the solution.
330 Applied Quantitative Methods for Trading and Investment

the parameters of persistence to the multivariate scalar GARCH model. On the other hand
when there is not a “natural” index for the data, we construct a synthetic index return
from an equal-weighted portfolio of all the assets under analysis and perform a univariate
GARCH on this synthetic return series. Bourgoin (2002) showed that within a portfolio
of equities, bonds or FX, there is not much difference between the persistence parameters
(alpha and beta) of each component within the portfolio; hence it is practically equivalent
to a scalar GARCH model.
So, as an extension to the previous model, we calculate a single univariate GARCH
model for the synthetic index (here a GARCH(1,1) but any model discussed in the volatil-
ity section can be used) and apply its persistence parameters “down” to the covariance
matrix, like Athayde (2001).
So the model looks like this. From either a synthetic or a real index, we have:

ˆ2
σindex,t = ωIndex + ±µIndex,t’1 + βσindex,t’1
ˆ2
2
(10.24)

ˆ
ˆ
We then plug in the persistence parameters (± and β) “down” into the multivariate
equation:
ˆ ˆ2
σ1,t = σ 2 (1 ’ ± ’ β) + ±µ1,t’1 + βσ1,t’1
ˆ ˆ2
2
1

ˆ ˆ2
σ2,t = σ 2 (1 ’ ± ’ β) + ±µ2,t’1 + βσ2,t’1
ˆ ˆ2
2
(10.25)
2

ˆ ˆ2
σ12,t = σ 2 (1 ’ ± ’ β) + ±µ1,t’1 µ2,t’1 + βσ12,t’1
ˆ ˆ
2
12

The result is an optimisation procedure that runs on a single univariate GARCH(1,1)
model. Asymmetric speci¬cations can be used as well, as long as we can specify the
long-term covariance terms like in the GJR model.
Now let us look at the set-up required to perform this new model in the spreadsheet
example of Figure 10.15 (“Fast GARCH” in Bourgoin002.xls). The ¬rst task required is to




Figure 10.15 The fast GARCH model
Applied Volatility and Correlation Modelling 331

1.00


0.80


0.60


0.40


0.20


0.00


’0.20 Over-reaction of the fast GARCH
and RiskMetrics

’0.40
01-Jan-00
15-Mar-00
28-May-00
10-Aug-00
23-Oct-00
05-Jan-01
20-Mar-01
02-Jun-01
15-Aug-01
28-Oct-01
10-Jan-02
25-Mar-02
07-Jun-02
20-Aug-02
02-Nov-02
15-Jan-03
30-Mar-03
12-Jun-03
25-Aug-03
07-Nov-03
20-Jan-04
03-Apr-04
16-Jun-04
29-Aug-04
11-Nov-04
24-Jan-05
08-Apr-05
21-Jun-05
03-Sep-05
16-Nov-05
29-Jan-06
13-Apr-06
26-Jun-06
08-Sep-06
21-Nov-06
03-Feb-07
18-Apr-07
01-Jul-07
13-Sep-07
26-Nov-07
08-Feb-08
22-Apr-08
05-Jul-08
17-Sep-08
30-Nov-08
12-Feb-09
27-Apr-09
10-Jul-09
22-Sep-09
Excel Full Diag(1,1) RiskMetrics
Fast Garch Correlation

Figure 10.16 Multivariate GARCH correlation model comparison

calculate the returns on an equal-weighted portfolio (column L) and perform a univariate
GARCH(1,1) on it (columns M and N for the variance and the log-likelihood function
respectively); this is done in the grey cells. The cells Q25:Q27 contain the variance
targeting omegas derived from the unconditional covariance matrix and the alpha and
beta calculated for the univariate GARCH model. The rest follows through in the same
way as the other spreadsheets.

10.4.7 Model comparison
We can plot the time-varying correlation resulting from the various models as in
Figure 10.16 (“GraphicalResults” in Bourgoin002.xls). We can see that despite the added
complexity and difference in the models, they provide the same patterns over time.
We can notice that the fast GARCH and RiskMetrics tend to slightly overestimate and
underestimate the correlation during stress times.

10.5 CONCLUSION
We™ve seen from this chapter that it is possible to calculate GARCH models in Excel,
from the most simple one in a univariate setting (RiskMetrics) to a fairly complicated
model in a multivariate framework (diagonal GARCH). We™ve shown that quite quickly
when we increase the complexity of the model, the number of parameters and constraints
increases dramatically, as well as the time required for the optimisation. In order to
deal with the high dimensionality problem, we™ve shown a new technique called the
fast GARCH that is easily implemented in Excel and can provide a solution when the
number of assets under analysis becomes large. The spreadsheets show how to perform
the statistical analysis, build the maximum likelihood function required for the Solver in
order to obtain the parameters for each model and several comparison charts have also
332 Applied Quantitative Methods for Trading and Investment

been produced. Although it is not recommended to use Excel as an advanced statistical
package, the ¬‚exibility and insight gained by the spreadsheet-based approach should
outweigh the drawbacks, at least in the beginning.

REFERENCES
Alexander, C. and A. M. Chibumba (1997), “Orthogonal GARCH: An Empirical Validation in
Equities, Foreign Exchange and Interest Rates”, School of Mathematical Sciences Discussion
Paper, Sussex University.
Athayde, G. (2001), “Forecasting Relationship between Indexes of Different Countries: A New
Approach to the Multivariate GARCH”, Forecasting Financial Markets Conference, London,
May 2001.
Black, F. (1976), “Studies of Stock Market Volatility Changes”, Proceedings of the American Sta-
tistical Association, Business and Economic Statistics Edition, 177“181.
Bourgoin, F. (2000), “Large Scale Problem in Conditional Correlation Estimation”, in Advances in
Quantitative Asset Management, C. Dunis (ed.), Kluwer Academic, Dordrecht.
Bourgoin, F. (2002), “Fast Calculation of GARCH Correlation”, Forecasting Financial Markets
Conference, London, May 2001.
Diebold, F. X., A. Hickman, A. Inoue and T. Schuermann (1997), “Converting 1-Day Volatility to
h-Day Volatility: Scaling by Sqrt(h) is Worse than You Think”, Working Paper, Department of
Economics, University of Pennsylvania.
Ding, X. and R. F. Engle (1994), “Large Scale Conditional Covariance Matrix: Modelling, Estima-
tion and Testing”, UCSD Discussion Paper.
Engle, R. F. and J. Mezrich (1995), “Grappling with GARCH”, Risk Magazine, 8 (9), 112“117.
Engle, R. F. and J. Mezrich (1996), “GARCH for Groups”, Risk Magazine, 9 (8), 36“40.
Gallant, A. R. (1997), An Introduction to Econometric Theory, Princeton University Press, Prince-
ton, NJ.
Glosten, L. R., R. Jagannathan and D. Runkle (1993), “Relationship between the Expected Value
and the Volatility of the Nominal Excess Returns on Stocks”, Journal of Finance, 48 (5),
1779“1801.
J. P. Morgan (1995), RiskMetrics™ Technical Document, Version 4.0.
J. P. Morgan (1997), RiskMetrics™ Monitor Fourth Quarter, pp. 3“11.
Ledoit, O. (1999), “Improved Estimation of the Covariance Matrix of Stock Returns with an Appli-
cation to Portfolio Selection”, UCLA Working Paper.
Ledoit, O., P. Santa-Clara and M. Wolf (2001), “Flexible Multivariate GARCH Modeling with an
Application to International Stock Markets”, UCLA Working Paper.
Pourahmadi, M. (1999), “Joint Mean“Covariance Models with Applications to Longitudinal Data:
Unconstrained Parametrization”, Biometrika, 86, 677“690.
Tsay, S. T. (2002), Analysis of Financial Time Series, Wiley InterScience, New York.
11
Optimal Allocation of Trend-Following
Rules: An Application Case of Theoretical
Results

PIERRE LEQUEUX



ABSTRACT
This chapter builds upon previously published results on the statistical properties of trading
rules to propose an allocation model for trading rules based on simple moving averages.
This model could easily be extended to cover a larger universe of linear trading rules and
is presented here purely as a proof of concept. We use theoretical results on volatility, cor-
relation, transactions costs and expected returns of moving averages trading rules within a
mean“variance framework to determine what should be the optimal weighting of trading
rules to maximise the information ratio, thereby presenting an unbiased methodology of
selecting an ex-ante optimal basket of trading rules.

11.1 INTRODUCTION
Moving averages have now been used for many years by both academics and market
participants. Whereas the former have usually been using moving averages trading rules
as a means of testing for market ef¬ciency (LeBaron, 1991, 1992; Levich and Thomas,
1993; Schulmeister, 1988; Taylor, 1980, 1986, 1990a,b, 1992, 1994), traders have had
more pro¬t-motivated goals in mind. It is therefore quite surprising that only a relatively
small section of literature has been devoted to the statistical properties of trend-following
trading rules. This is clearly one of the most important issues for a market practitioner.
Statistical properties of trading rules may give clearer insights in regard of the level of
transactions cost, diversi¬cation, risk and also return one might expect under various price
trend model hypotheses. This chapter ¬rst presents some of the mainstream results that
have been published by Acar and Lequeux (1995, 1996), Acar et al. (1994) and Acar
(1994). In the latter part of this chapter we demonstrate how these theoretical results
could be used within a mean“variance trading rule allocation model.

11.2 DATA
To illustrate the main ¬ndings on the statistical properties of moving averages and
also our mean“variance trading rules allocation model we use a sample of ¬ve cur-
rency pairs USD“JPY, EUR“USD, GBP“USD, USD“CAD and AUD“USD. The data

Applied Quantitative Methods for Trading and Investment. Edited by C.L. Dunis, J. Laws and P. Na¨m
±
™ 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
334 Applied Quantitative Methods for Trading and Investment
Table 11.1 Summary statistics of daily logarithmic returns of exchange rates

USD“JPY EUR“USD GBP“USD USD“CAD AUD“USD

’0.03% ’0.02%
Mean 0.03% 0.00% 0.01%
Stdev 0.75% 0.59% 0.46% 0.32% 0.65%
’5.65% ’2.28% ’2.15% ’1.54% ’2.65%
Min
Max 3.22% 2.64% 2.04% 1.61% 4.69%
’0.84 ’0.25
Skew 0.27 0.08 0.24
Kurtosis 5.08 1.21 1.17 2.41 2.95
a(1)a 0.0459 0.0249 0.0526 0.0218 0.0210
’0.0120 ’0.0316 ’0.0254 ’0.0282
a(2) 0.0187
’0.0402 ’0.0098 ’0.0449 ’0.0059
a(3) 0.0277
’0.0205 ’0.0162 ’0.0297
a(4) 0.0009 0.0053
’0.0181 ’0.0684 ’0.0122
a(5) 0.0079 0.0397
’0.0236 ’0.0094
a(6) 0.0013 0.0195 0.0148
’0.0131 ’0.0148 ’0.0212 ’0.0377 ’0.0014
a(7)
’0.0060 ’0.0355 ’0.0248
a(8) 0.0390 0.0208
’0.0472 ’0.0027
a(9) 0.0021 0.0746 0.0097
’0.0034
a(10) 0.0746 0.0335 0.0013 0.0097
a
a(n) means the autocorrelation coef¬cient of order n.


60%



40%

USD-JPY
20%
Cumulative spot returns




USD-CAD

0%
GBP-USD

’20%



’40% AUD-USD


EUR-USD
’60%
USD-JPY
EUR-USD
GBP-USD
USD-CAD
AUD-USD
’80%
15/02/1996
15/04/1996
15/06/1996
15/08/1996
15/10/1996
15/12/1996
15/02/1997
15/04/1997
15/06/1997
15/08/1997
15/10/1997
15/12/1997
15/02/1998
15/04/1998
15/06/1998
15/08/1998
15/10/1998
15/12/1998
15/02/1999
15/04/1999
15/06/1999
15/08/1999
15/10/1999
15/12/1999
15/02/2000
15/04/2000
15/06/2000
15/08/2000
15/10/2000
15/12/2000
15/02/2001
15/04/2001
15/06/2001
15/08/2001
15/10/2001
15/12/2001
15/02/2002




Figure 11.1 Cumulative return of spot exchange rates

covers the period 15/02/1996 to 12/03/2002, a total of 1582 daily observations. For each
exchange rate we have the spot foreign exchange rate and the relevant 1-month inter-
est rate series. As is usual in empirical ¬nancial research, we use daily spot returns,
de¬ned as ln(St /St’1 ). Each of the exchange rates is sampled at 16:00 GMT. We use a
hypothetical transaction cost of ’0.03%, which generally represents the operating cost
Optimal Allocation of Trend-Following Rules 335

in the interbank market. Table 11.1, taken from cells O1:X17 of the worksheet “Data”
of Lequeux001.xls, shows the summary statistics of the daily logarithmic returns for the
¬ve exchange rates sampled. As can be seen in Table 11.1, all the series under review are
clearly non-normal. This is to be expected for most ¬nancial time series (Taylor, 1986).
It is also worth noting that the time series have had a signi¬cant mean over the period
and that they all exhibit positive serial dependencies of ¬rst order. Overall the spot
exchange rates studied here have exhibited a trend over the period 1996 to 2002. This
is more particularly true for USD“JPY, EUR“USD and AUD“USD, as shown in the
worksheet “Cumulative Spot Returns” and reproduced in Figure 11.1.



11.3 MOVING AVERAGES AND THEIR
STATISTICAL PROPERTIES

Market ef¬ciency depends upon rational, pro¬t-motivated investors (Arnott and Pham,
1993). Serial dependencies or in more simple terms, trends, have been commonly observed
in currency markets (Kritzman, 1989; Silber, 1994) and are linked to market participants™
activity and their motivation. Central banks attempt to dampen the volatility of their
currency because stability assists trade ¬‚ows and facilitates the control of in¬‚ation level.
A central bank might intervene directly in the currency market in times of high volatility,
such as the ERM crisis in 1992, or act as a discreet price-smoothing agent by adjusting
their domestic interest rate levels. Other participants in the foreign exchange markets,
such as international corporations, try to hedge their currency risk and therefore have no
direct pro¬t motivation. Arguably they may have the same effect as a central bank on
exchange rate levels. The success of momentum-based trading rules in foreign exchange
markets is most certainly linked to the heterogeneity of market agents and their different
rationales, which render the FX markets inef¬cient. Autoregressive models such as moving
averages are indeed particularly well suited to trade foreign exchange markets, where it
is estimated that around 70% of strategies are implemented on the back of some kind
of momentum strategies (Billingsley and Chance, 1996). Lequeux and Acar (1998) have
demonstrated the added value of using trend-following methods by proposing a transparent
active currency benchmark that relies on simple moving averages to time the currency
market. The AFX benchmark they proposed had a return over risk ratio exceeding 0.641
for the period January 1984 to November 2002, and this compares well to other traditional
asset classes. Moving average trading rules are an attractive proposition as a decision rule
for market participants because of their simplicity of use and implementation. The daily
simple moving average trading rules work as follows: when the rate penetrates from below
(above) a moving average of a given length m, a buy (sell ) signal is generated. If the
current price is above the m-moving average, then it is left long for the next 24 hours,
otherwise it is held short. Figure 11.2 illustrates the working of such a trading rule.
The rate of return generated by a simple moving average of order m is simply calculated
as: Rt = Bt’1 Xt where Xt = ln(Pt /Pt’1 ) is the underlying logarithmic return, Pt the asset
price at time t, and Bt’1 the signal triggered by the trading rule at time t ’ 1. Bt’1 is


1
See http://www.quant.cyberspot.co.uk/AFX.htm for more information on the AFX.
336 Applied Quantitative Methods for Trading and Investment
15 000
If Pt < MA then Sell
USD Buy JPY
14 500


14 000


13 500


13 000
MA 117 DAYS


12 500


12 000


11 500

SPOT USD-JPY
11 000 If Pt > MA then Buy
USD Sell JPY

10 500


10 000
Feb-96
Apr-96
Jun-96
Aug-96
Oct-96
Dec-96
Feb-97
Apr-97
Jun-97
Aug-97
Oct-97
Dec-97
Feb-98
Apr-98
Jun-98
Aug-98
Oct-98
Dec-98
Feb-99
Apr-99
Jun-99
Aug-99
Oct-99
Dec-99
Feb-00
Apr-00
Jun-00
Aug-00
Oct-00
Dec-00
Feb-01
Apr-01
Jun-01
Aug-01
Oct-01
Dec-01
Feb-02
Figure 11.2 Spot USD“JPY and moving average trading rule of 117 days


± 
de¬ned as:
m
 
1
B = 1 (long position) 
 t’1 
 
if Pt’1 > Pt’i
 
m i=1
 
m
 
 Bt’1 = ’1 if Pt’1 < 1 (short position) 
 
Pt’i
 
m i=1



11.4 TRADING RULE EQUIVALENCE
In this chapter we consider a slightly modi¬ed version of moving averages. Instead of
using spot prices the moving average signals generated are based on logarithmic prices.
Simple moving averages by construction are autoregressive models with constrained
parameters. Therefore the major difference between linear predictors such as simple and
weighted moving averages and momentum trading rules resides in the weighting scheme
used. Simple moving averages give decreasing weights to past underlying returns whereas
momentum rules give an equal weight. Table 11.2 shows the equivalence for the simple
moving average. Results for other linear predictors such as weighted, exponential or
double moving averages crossover rules can be found in Acar and Lequeux (1998).
As an illustration Table 11.3 shows how trading signals would be generated for a simple
moving average trading rule of order 10.
When not strictly identical (as in the case of the momentum method), signals generated
by trading rules based on logarithmic prices are extremely similar to the ones generated
by rules based on prices. Acar (1994) shows that using Monte Carlo simulations the
signals are identical in at least 97% of the cases. Figure 11.3 clearly depicts this by
showing the returns generated by simple moving averages trading rules of order 2 to 117
Optimal Allocation of Trend-Following Rules 337
Table 11.2 Return/price signals equivalence

Rule Parameter(s) Price sell signals Equivalent return sell signals

m’1 m’2
ln(Ct ) < aj ln(Ct’j ) dj Xt’j < 0
Linear rules
j =0 j =0

m≥2 aj = 1/m dj = (m ’ j ’ 1)
Simple MA

Table 11.3 Numerical example of signal generation when using prices or logarithmic price returns
m’2 m’1 m’2 m’1
Xt = dj = ln(Ct ) <
Date Spot
(m ’ j ’ 1)
ln(Pt /Pt’1 )
price j =0 j =0 j =0 j =0
dj Xt’j dj Xt’j < 0 aj ln(Ct’j )
1
ln(Ct’j )
m

30/01/2002 13 298
31/01/2002 13 378 0.005998 1
01/02/2002 13 387 0.000673 2
’0.00682
04/02/2002 13 296 3
05/02/2002 13 333 0.002777 4
06/02/2002 13 357 0.0018 5
07/02/2002 13 362 0.000374 6
08/02/2002 13 481 0.008866 7
’0.00864
11/02/2002 13 365 8
’0.00563 ’0.0006 ’1 ’1
12/02/2002 13 290 9 13 355
’0.00517 ’1 ’1
13/02/2002 13 309 0.001429 13 356
’0.00588 ’0.01172 ’1 ’1
14/02/2002 13 231 13 341
’0.00226 ’1 ’1
15/02/2002 13 266 0.002642 13 329
’0.00053 ’0.00556 ’1 ’1
18/02/2002 13 259 13 325
19/02/2002 13 361 0.007663 0.000299 13 328 1 1


14.0%

y = 0.9796x + 0.0019
12.0% 2
R = 0.9774



10.0%
MA log returns




8.0%


6.0%


4.0%


2.0%


0.0%
0% 2% 4% 6% 8% 10% 12% 14%
MA price returns

Figure 11.3 Comparison of returns generated by the moving averages using log returns or prices
338 Applied Quantitative Methods for Trading and Investment

Press to run simulation
0.98
Select statistic to plot
0.96
as X and Y
0.94

0.92

0.90

0.88

0.86

Select currency, MA order
0.84
and transaction cost
0.82
0 20 40 60 80 100 120 140



Figure 11.4 Correlation of signals as a function of the moving average order

based both on prices and logarithmic returns for USD“JPY over the period 26/07/1996
to 12/03/2002. The near equivalence is quite obvious. Note that the reader can recreate
Figure 11.3 by drawing an XY plot of the data contained in (B3:B118, N3:N118) of the
“Results” worksheet of the ¬le “Lequeux001.xls”.
Figure 11.4 further illustrates the relationship between logarithmic returns and price-
based trading rule signals. The reader can furthermore investigate the various performance
statistics of simple moving averages of order 2 to 117 by using the worksheet “MA” in
Lequeux001.xls. The user should ¬rst select a currency pair and then press the simulation
button. All statistics will then be calculated for a moving average of order 2 to 117. The
results can then be visualised as an XY plot by selecting the data for X and Y as shown in
Figure 11.4. Alternatively the user can select a currency pair and specify the moving aver-
age order to compare the differences between logarithmic and price based trading rules.

11.5 EXPECTED TRANSACTIONS COST UNDER ASSUMPTION
OF RANDOM WALK
When evaluating the performance of a trading rule taking into account transaction costs is
crucial. Whereas this can be done empirically, doing so implies making some assumptions
in regard of the underlying dependencies. It may therefore induce a bias in the rule
selection process. Acar and Lequeux (1995) demonstrated that over a period of T days,
there are an expected number, N , of round turns2 and therefore a total trading cost (TC), c,
equal to cN. The number N of round turns is a stochastic variable, which depends mainly
on the individual forecaster Ft rather than on the underlying process. Assuming that the
underlying time series, Xt , follows a centred identically and independently distributed
normal law, the expected number of round turns generated by a moving average of order
m, supposing that a position is opened at the beginning of the period and that the last
position is closed at the end day of the period, is given by:

1 1

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