ńņš. 15 |

16-Dec-98

16-Nov-99

16-Dec-99

16-Nov-00

16-Dec-00

16-May-98

16-Jul-98

16-May-99

16-Jul-99

16-May-00

16-Jul-00

16-May-01

16-Jul-01

16-Aug-98

16-Sep-98

16-Aug-99

16-Sep-99

16-Aug-00

16-Sep-00

Figure 10.4 Annualised RiskMetrics volatility with varying decay factors

A comparison of different decay factors can be found in the sheet named āDifferentDe-

cayChartā 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

=Ļ+ +

Ļt2 2 2

Ī±i Īµtā’i Ī²j Ļtā’j (10.8)

j =1

i=1

max(p,q)

(Ī±i + Ī²i ) < 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 ļ¬‚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 ļ¬‚at functions of

their parameters, which makes it difļ¬cult for optimisers).

We will restrict our analysis to the ļ¬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 ļ¬‚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: Ļt2 = Ļ + Ī±Īµtā’1 + Ī²Ļtā’1 . If we replace Ļ by 0 and Ī± by

2 2

7

1 ā’ Ī², 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):

Ļt2 (1) = E(Ļt+1/t ) = Ļ + Ī±Īµt2 + Ī²Ļt2

2

Ļt2 (2) = E(Ļ + Ī±Īµt+1 + Ī²Ļt2 (1)) = Ļ + Ī±E(Īµt+1 ) + Ī²Ļt2 (1)

2 2

= Ļ + Ī±E(Ļt2 (1)) + Ī²Ļt2 (1)

= Ļ + (Ī± + Ī²)Ļt2 (1)

Ļt2 (3) = Ļ + (Ī± + Ī²)Ļt2 (2) = Ļ(1 + (Ī± + Ī²)) + (Ī± + Ī²)2 Ļt2 (1)

Hence for a forecasting horizon h, we have:

1 ā’ (Ī± + Ī²)hā’1

Ļt2 (h) = Ļ + (Ī± + Ī²)hā’1 Ļt2 (1) (10.9)

1 ā’ (Ī± + Ī²)

When h tends to inļ¬nity, we have the following long-term variance of the GARCH(1,1)

process:

Ļ

lim Ļt2 (h) = Ļ 2 = (10.10)

1ā’Ī±ā’Ī²

hā’ā

where Ļ is the unconditional variance.

The spreadsheet-based application (āDataGARCHā 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 Ļ 2 using historical data and the

standard time series approach:

N

1 1

Ļ= (rt ā’ r)2 =

2

E(RR ) (10.11)

or

N T

t=1

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

Ļ = Ļ 2 (1 ā’ Ī± ā’ Ī²) (10.12)

Ļ 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 (āDataGARCH VTā 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 Īµt > 0

rt = Ļt Īµt Ļt2 = Ļ + Ī±Īµtā’1 + Ī²Ļtā’1 + Ī³ Stā’1 Īµtā’1 St =

2 2 2

(10.13)

1 if Īµt < 0

with Ī± > 0, Ī² > 0, Ī³ > 0 and Ī± + Ī² + 1 Ī³ < 1. When returns are positive, Ī³ Stā’1 Īµtā’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 Ī³ , a

positive number, meaning that the impact of this particular negative return is bigger on

the estimation of volatility, as shown below:

Ļt2 = Ļ + (Ī± + Ī³ )Īµtā’1 + Ī²Ļtā’1

2 2

if Īµt < 0,

We can forecast the volatility term structure:

hā’1

1 ā’ Ī± + Ī² + 1Ī³ hā’1

=Ļ + Ī± + Ī² + 1Ī³

2 2 2

ĻGJR,t (h) ĻGJR,t (1) (10.14)

1ā’ Ī±+Ī² + 2

1

Ī³

2

with ĻGJR,t (1) = Ļ + Ī±Īµt2 + Ī²Ļt2 + Ī³ St Īµt2 /2. We can derive the long-term variance pro-

2

cess from this model as:

Ļ

lim ĻGJR,t (h) = Ļ 2 =

2

(10.15)

1 ā’ Ī± ā’ Ī² ā’ Ī³ /2

hā’ā

The spreadsheet-based application (Figure 10.7) is located in āDataGARCHGJR VTā

(Bourgoin001.xls).

Figure 10.7 The GJR model with variance targeting

322 Applied Quantitative Methods for Trading and Investment

Remarks:

ā¢ Here we use again the variance targeting process (note that $I$4, omega is not greyed

out) in order to facilitate the optimisation process.

ā¢ 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+0.5ā— gamma is strictly less than 1).

ā¢ 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 weā™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:

ā¢ 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.

ā¢ 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%

16-May-98

16-Jul-98

16-Nov-98

16-Mar-99

16-May-99

16-Jul-99

16-Nov-99

16-Mar-00

16-May-00

16-Jul-00

16-Nov-00

16-Mar-01

16-May-01

16-Jul-01

16-Apr-98

16-Sep-98

16-Sep-00

16-Dec-00

16-Jan-01

16-Feb-01

16-Oct-98

16-Dec-98

16-Jan-99

16-Feb-99

16-Apr-99

16-Sep-99

16-Oct-99

16-Dec-99

16-Jan-00

16-Feb-00

16-Apr-00

16-Oct-00

16-Apr-01

16-Jun-98

16-Aug-98

16-Jun-99

16-Aug-99

16-Jun-00

16-Aug-00

16-Jun-01

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 Ī± + Ī² = 1 = Ī» compared to Ī± + Ī² < 1 for the GARCH model.

ā¢ 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 weā™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 ļ¬‚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, weā™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 ļ¬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%

7-Aug-01

12-Aug-01

17-Aug-01

22-Aug-01

27-Aug-01

1-Sep-01

6-Sep-01

11-Sep-01

16-Sep-01

21-Sep-01

26-Sep-01

1-Oct-01

6-Oct-01

11-Oct-01

16-Oct-01

21-Oct-01

26-Oct-01

31-Oct-01

5-Nov-01

10-Nov-01

15-Nov-01

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 speciļ¬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:

Ļ1,t = (1 ā’ Ī»)Īµ1,tā’1 + Ī»Ļ1,tā’1

2 2 2

Ļ2,t = (1 ā’ Ī»)Īµ2,tā’1 + Ī»Ļ2,tā’1

2 2 2

(10.16)

Ļ12,t = (1 ā’ Ī»)Īµ1,tā’1 Īµ2,tā’1 + Ī»Ļ12,tā’1

2 2

The initialisation of Ļ1,0 , Ļ2,0 , Ļ12,0 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 deļ¬nite

by construction as long as Ī» is the same everywhere in the covariance matrix.9 But the

drawbacks are still the same in a multivariate framework: ļ¬‚at term structure of forecast,

too much persistence of shocks and too low volatilities during calm periods.

The spreadsheet of Figure 10.10 (āRiskMetricsā 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:

Ļi,j = Ļi,j /(Ļi Ļj )

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 deļ¬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 deļ¬ne the probability distribution, here

we will use the multivariate normal distribution:

ā’1

1 ā’1

f (x, Āµ, ) = (x ā’ Āµ) (x ā’ Āµ) (10.17)

exp

(2Ļ )n/2 | |1/2 2

where Āµ is the vector of the mean and the covariance matrix for two assets with zero

mean, the multivariate normal distribution looks like this:

ā’1 2

x2

x1 2Ļ12 x1 x2

1

f (x1 , x2 , Ļ1 , Ļ2 , Ļ12 ) = ā’ +2

exp

2ĻĻ1 Ļ2 (1 ā’ Ļ12 ) 2(1 ā’ Ļ12 ) Ļ1

2 2 2 2

Ļ1 Ļ2 Ļ2

(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

= ā’ ln(2Ļ ) ā’ ln(Ļ1,t ) ā’ ln(Ļ2,t ) ā’ ln(1 ā’ Ļ12,t )

2

2

t=1

2 2

x1,t x2,t

2Ļ12,t x1,t x2,t

1

ā’ ā’ +2 (10.19)

2(1 ā’ Ļ12,t )

2 2 Ļ1,t Ļ2,t

Ļ1,t Ļ2,t

The optimisation routine will be to maximise the log-likelihood function with respect to

the decay factor:

max ln L(Ī») with 0 < Ī» ā¤ 1

326 Applied Quantitative Methods for Trading and Investment

Here the covariance speciļ¬cation is:

Ė2 Ė2

Ļ1,t = (1 ā’ Ī»)Īµ1,tā’1 + Ī»Ļ1,tā’1

2

Ė2 Ė2

Ļ2,t = (1 ā’ Ī»)Īµ2,tā’1 + Ī»Ļ2,tā’1

2

(10.20)

Ė Ė2

Ļ12,t = (1 ā’ Ī»)Īµ1,tā’1 Īµ2,tā’1 + Ī»Ļ12,tā’1

2

Ė

where Ī» is the optimal decay factor.

The spreadsheet of Figure 10.11 (āOptimalDecayā 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 ļ¬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:

Ļ1,t = Ļ11 + Ī±11 Īµ1,tā’1 + Ī²11 Ļ1,tā’1

2 2 2

Ļ2,t = Ļ22 + Ī±22 Īµ2,tā’1 + Ī²22 Ļ2,tā’1

2 2 2

(10.21)

Ļ12,t = Ļ12 + Ī±12 Īµ1,tā’1 Īµ2,tā’1 + Ī²12 Ļ12,tā’1

2 2

The spreadsheet of Figure 10.12 (āFull Diagonal GARCHā in Bourgoin002.xls) is more

complex because we have to deal with multiple constraints:

Ī±ij + Ī²ij < 1, Ī±ij > 0, Ī²ij > 0 and Ļii > 0

Applied Volatility and Correlation Modelling 327

Figure 10.12 The diagonal GARCH model

Note that Ļij 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:

Ļij = Ļ 2 (1 ā’ Ī±ij ā’ Ī²ij ) (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 ļ¬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 (āFull Diagonal GARCH VTā 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 simpliļ¬ed model of the ātraditionalā type. We consider that all assets under

analysis have the same parameters, i.e., Ī±ij = Ī± and Ī²ij = Ī², 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 (Ī± and Ī²):

Ļ1,t = Ļ 2 (1 ā’ Ī± ā’ Ī²) + Ī±Īµ1,tā’1 + Ī²Ļ1,tā’1

2 2 2

1

Ļ2,t = Ļ 2 (1 ā’ Ī± ā’ Ī²) + Ī±Īµ2,tā’1 + Ī²Ļ2,tā’1

2 2 2

(10.23)

2

Ļ12,t = Ļ 2 (1 ā’ Ī± ā’ Ī²) + Ī±Īµ1,tā’1 Īµ2,tā’1 + Ī²Ļ12,tā’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 (āScalar GARCH VTā 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 (ļ¬ve).

We can note a clear drop in the log-likelihood function to ā’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 (ā’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 ļ¬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 Ć— 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, Ī± and Ī².

Here we follow the idea of Athayde (2001) on the estimation of these parameters. If a

ānaturalā index exists for the data, we can calculate a univariate GARCH model and apply

13

The optimisation routine took only 7 seconds to ļ¬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 ānaturalā 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 ādownā to the covariance

matrix, like Athayde (2001).

So the model looks like this. From either a synthetic or a real index, we have:

Ė2

Ļindex,t = ĻIndex + Ī±ĪµIndex,tā’1 + Ī²Ļindex,tā’1

Ė2

2

(10.24)

Ė

Ė

We then plug in the persistence parameters (Ī± and Ī²) ādownā into the multivariate

equation:

Ė Ė2

Ļ1,t = Ļ 2 (1 ā’ Ī± ā’ Ī²) + Ī±Īµ1,tā’1 + Ī²Ļ1,tā’1

Ė Ė2

2

1

Ė Ė2

Ļ2,t = Ļ 2 (1 ā’ Ī± ā’ Ī²) + Ī±Īµ2,tā’1 + Ī²Ļ2,tā’1

Ė Ė2

2

(10.25)

2

Ė Ė2

Ļ12,t = Ļ 2 (1 ā’ Ī± ā’ Ī²) + Ī±Īµ1,tā’1 Īµ2,tā’1 + Ī²Ļ12,tā’1

Ė Ė

2

12

The result is an optimisation procedure that runs on a single univariate GARCH(1,1)

model. Asymmetric speciļ¬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 (āFast GARCHā in Bourgoin002.xls). The ļ¬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

ā’0.20 Over-reaction of the fast GARCH

and RiskMetrics

ā’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 (āGraphicalResultsā 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

Weā™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). Weā™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, weā™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 ļ¬‚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), āOrthogonal GARCH: An Empirical Validation in

Equities, Foreign Exchange and Interest Ratesā, School of Mathematical Sciences Discussion

Paper, Sussex University.

Athayde, G. (2001), āForecasting Relationship between Indexes of Different Countries: A New

Approach to the Multivariate GARCHā, Forecasting Financial Markets Conference, London,

May 2001.

Black, F. (1976), āStudies of Stock Market Volatility Changesā, Proceedings of the American Sta-

tistical Association, Business and Economic Statistics Edition, 177ā“181.

Bourgoin, F. (2000), āLarge Scale Problem in Conditional Correlation Estimationā, in Advances in

Quantitative Asset Management, C. Dunis (ed.), Kluwer Academic, Dordrecht.

Bourgoin, F. (2002), āFast Calculation of GARCH Correlationā, Forecasting Financial Markets

Conference, London, May 2001.

Diebold, F. X., A. Hickman, A. Inoue and T. Schuermann (1997), āConverting 1-Day Volatility to

h-Day Volatility: Scaling by Sqrt(h) is Worse than You Thinkā, Working Paper, Department of

Economics, University of Pennsylvania.

Ding, X. and R. F. Engle (1994), āLarge Scale Conditional Covariance Matrix: Modelling, Estima-

tion and Testingā, UCSD Discussion Paper.

Engle, R. F. and J. Mezrich (1995), āGrappling with GARCHā, Risk Magazine, 8 (9), 112ā“117.

Engle, R. F. and J. Mezrich (1996), āGARCH for Groupsā, Risk Magazine, 9 (8), 36ā“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), āRelationship between the Expected Value

and the Volatility of the Nominal Excess Returns on Stocksā, Journal of Finance, 48 (5),

1779ā“1801.

J. P. Morgan (1995), RiskMetricsļ™ Technical Document, Version 4.0.

J. P. Morgan (1997), RiskMetricsļ™ Monitor Fourth Quarter, pp. 3ā“11.

Ledoit, O. (1999), āImproved Estimation of the Covariance Matrix of Stock Returns with an Appli-

cation to Portfolio Selectionā, UCLA Working Paper.

Ledoit, O., P. Santa-Clara and M. Wolf (2001), āFlexible Multivariate GARCH Modeling with an

Application to International Stock Marketsā, UCLA Working Paper.

Pourahmadi, M. (1999), āJoint Meanā“Covariance Models with Applications to Longitudinal Data:

Unconstrained Parametrizationā, Biometrika, 86, 677ā“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

meanā“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 efļ¬ciency (LeBaron, 1991, 1992; Levich and Thomas,

1993; Schulmeister, 1988; Taylor, 1980, 1986, 1990a,b, 1992, 1994), traders have had

more proļ¬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, diversiļ¬cation, risk and also return one might expect under various price

trend model hypotheses. This chapter ļ¬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 meanā“variance trading rule allocation model.

11.2 DATA

To illustrate the main ļ¬ndings on the statistical properties of moving averages and

also our meanā“variance trading rules allocation model we use a sample of ļ¬ve cur-

rency pairs USDā“JPY, EURā“USD, GBPā“USD, USDā“CAD and AUDā“USD. The data

Applied Quantitative Methods for Trading and Investment. Edited by C.L. Dunis, J. Laws and P. NaĀØm

Ä±

ļ™ 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

USDā“JPY EURā“USD GBPā“USD USDā“CAD AUDā“USD

ā’0.03% ā’0.02%

Mean 0.03% 0.00% 0.01%

Stdev 0.75% 0.59% 0.46% 0.32% 0.65%

ā’5.65% ā’2.28% ā’2.15% ā’1.54% ā’2.65%

Min

Max 3.22% 2.64% 2.04% 1.61% 4.69%

ā’0.84 ā’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

ā’0.0120 ā’0.0316 ā’0.0254 ā’0.0282

a(2) 0.0187

ā’0.0402 ā’0.0098 ā’0.0449 ā’0.0059

a(3) 0.0277

ā’0.0205 ā’0.0162 ā’0.0297

a(4) 0.0009 0.0053

ā’0.0181 ā’0.0684 ā’0.0122

a(5) 0.0079 0.0397

ā’0.0236 ā’0.0094

a(6) 0.0013 0.0195 0.0148

ā’0.0131 ā’0.0148 ā’0.0212 ā’0.0377 ā’0.0014

a(7)

ā’0.0060 ā’0.0355 ā’0.0248

a(8) 0.0390 0.0208

ā’0.0472 ā’0.0027

a(9) 0.0021 0.0746 0.0097

ā’0.0034

a(10) 0.0746 0.0335 0.0013 0.0097

a

a(n) means the autocorrelation coefļ¬cient of order n.

60%

40%

USD-JPY

20%

Cumulative spot returns

USD-CAD

0%

GBP-USD

ā’20%

ā’40% AUD-USD

EUR-USD

ā’60%

USD-JPY

EUR-USD

GBP-USD

USD-CAD

AUD-USD

ā’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 ļ¬nancial research, we use daily spot returns,

deļ¬ned as ln(St /Stā’1 ). Each of the exchange rates is sampled at 16:00 GMT. We use a

hypothetical transaction cost of ā’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 āDataā

of Lequeux001.xls, shows the summary statistics of the daily logarithmic returns for the

ļ¬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 ļ¬nancial time series (Taylor, 1986).

It is also worth noting that the time series have had a signiļ¬cant mean over the period

and that they all exhibit positive serial dependencies of ļ¬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 USDā“JPY, EURā“USD and AUDā“USD, as shown in the

worksheet āCumulative Spot Returnsā and reproduced in Figure 11.1.

11.3 MOVING AVERAGES AND THEIR

STATISTICAL PROPERTIES

Market efļ¬ciency depends upon rational, proļ¬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 participantsā™

activity and their motivation. Central banks attempt to dampen the volatility of their

currency because stability assists trade ļ¬‚ows and facilitates the control of inļ¬‚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 proļ¬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 inefļ¬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 = Btā’1 Xt where Xt = ln(Pt /Ptā’1 ) is the underlying logarithmic return, Pt the asset

price at time t, and Btā’1 the signal triggered by the trading rule at time t ā’ 1. Btā’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 USDā“JPY and moving average trading rule of 117 days

ļ£± ļ£¼

deļ¬ned as:

m

ļ£“ ļ£“

1

ļ£“B = 1 (long position) ļ£“

ļ£“ tā’1 ļ£“

ļ£“ ļ£“

if Ptā’1 > Ptā’i

ļ£² ļ£½

m i=1

ļ£“ ļ£“

m

ļ£“ ļ£“

ļ£“ Btā’1 = ā’1 if Ptā’1 < 1 (short position) ļ£“

ļ£“ ļ£“

Ptā’i

ļ£³ ļ£¾

m i=1

11.4 TRADING RULE EQUIVALENCE

In this chapter we consider a slightly modiļ¬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

mā’1 mā’2

ln(Ct ) < aj ln(Ctā’j ) dj Xtā’j < 0

Linear rules

j =0 j =0

mā„2 aj = 1/m dj = (m ā’ j ā’ 1)

Simple MA

Table 11.3 Numerical example of signal generation when using prices or logarithmic price returns

mā’2 mā’1 mā’2 mā’1

Xt = dj = ln(Ct ) <

Date Spot

(m ā’ j ā’ 1)

ln(Pt /Ptā’1 )

price j =0 j =0 j =0 j =0

dj Xtā’j dj Xtā’j < 0 aj ln(Ctā’j )

1

ln(Ctā’j )

m

30/01/2002 13 298

31/01/2002 13 378 0.005998 1

01/02/2002 13 387 0.000673 2

ā’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

ā’0.00864

11/02/2002 13 365 8

ā’0.00563 ā’0.0006 ā’1 ā’1

12/02/2002 13 290 9 13 355

ā’0.00517 ā’1 ā’1

13/02/2002 13 309 0.001429 13 356

ā’0.00588 ā’0.01172 ā’1 ā’1

14/02/2002 13 231 13 341

ā’0.00226 ā’1 ā’1

15/02/2002 13 266 0.002642 13 329

ā’0.00053 ā’0.00556 ā’1 ā’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 USDā“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

āResultsā worksheet of the ļ¬le āLequeux001.xlsā.

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 āMAā in

Lequeux001.xls. The user should ļ¬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

ńņš. 15 |