model construction & forecast based portfolio allocation · qbus 6830 assignment 2 310113083;...

QBUS 6830 Assignment 2 310113083; 308077237; 311295347

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QBUS6830

Financial Time Series and Forecasting

Model Construction &

Forecast Based Portfolio Allocation:

Is Quantitative Method Worth It?

Members: Bowei Li (310113083)

Wenjian Xu (308077237)

Xiaoyun Lu (311295347)

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Executive Summary

The aim of this report is to compare four forecasting models: ARMA-ARCH, ARMA-GARCH,

ARMA-EGARCH and Historical Simulation of last 100 days (HS-100) for four different industries

in Australia, which are Energy, Financial, Telecomm and Consumer. The forecasts generated for

each industry under each model are utilised for allocating portfolio weights on the basis of three

allocation strategies: return, standard deviation and Value at Risk (VaR). Finally, the models are

evaluated based on the investment outcome.

It was found that the GARCH type models did better in terms of forecast accuracy and investment

outcomes in general. As for the return forecast accuracy, the ARCH and the EGARCH perform

better overall. However, these two models do not necessarily generate higher returns. As for forecast

of volatility, the GARCH and the EGARCH performed better in terms of accuracy, however, the

ARCH generate the best investment outcomes under the Volatility Strategy. It provides the highest

return and lowest standard deviation across different forecasting frequencies. As for VaR, the

forecasts generated by the EGARCH are accurate in general. Moreover, the model also performs

better in terms of investment outcome.

By comparing across different allocating strategies, the VaR Strategy, which is also the most

conservative strategy, generated the best investment outcomes either in the context of return or risk.

It also has the highest utility score among all the strategies in our analysis. The Volatility Strategy

ranked the second, and the Return Strategy performed the worst. It could be more truthful when

doing investment after the GFC period. In this report, we also compared our outcomes with the

simple equally weighted portfolio. The equally weighted portfolio ranked before the portfolio

outcome on the basis of Return Strategy. However, Volatility Strategy and VaR Strategy performed

better than the simple equally weighted portfolio in terms of investment outcome. Therefore, it is

reasonable to conclude that all our quantitative effort is worth doing.

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Table&of&Contents&1. Introduction ........................................................................................................................................ 3"

2. Exploratory Data Analysis ................................................................................................................. 3"

3. Models for Forecasting ...................................................................................................................... 5"

3.1 Motivations of Model Selection ................................................................................................... 5"

3.2 Model Selection ........................................................................................................................... 5"

3.2.1 ARMA-ARCH-t Model ........................................................................................................ 5"

3.2.2 ARMA-GARCH-t Model ..................................................................................................... 9"

3.2.3 ARMA-EGARCH-t ............................................................................................................ 11"

3.2.4 Historical Simulation .......................................................................................................... 13"

3.3 Model Construction for the Other Industries ............................................................................. 13"

4. Forecast and Accuracy Measures ..................................................................................................... 18"

4.1 Return Forecast and Accuracy ................................................................................................... 19"

4.1.1 One-Step-Ahead Return Forecast ....................................................................................... 19"

4.1.2 Multi-period Return Forecast .............................................................................................. 21"

4.2 Volatility Forecast and Accuracy ............................................................................................... 22"

4.2.1 One-Step-Ahead Volatility Forecast ................................................................................... 22"

4.2.2 Multi-period Volatility Forecast ......................................................................................... 25"

4.3 VaR Forecast and Accuracy ........................................................................................................... 26"

4.3.1 One-step ahead forecast of VaR ......................................................................................... 26"

4.3.2 Multi-step ahead forecast of VaR ....................................................................................... 29"

5. Optimal Portfolio Allocation ........................................................................................................... 31"

5.1 Portfolio Allocation Methods .................................................................................................... 31"

5.2" Fixed Weight Portfolio Construction .................................................................................... 34"

5.3" Dynamic Weight Portfolio Construction ............................................................................... 34"

5.5" Utility Score for Portfolios .................................................................................................... 36"

6. Conclusion ....................................................................................................................................... 39"

7. List of References ............................................................................................................................ 40"

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1. Introduction The aim of this report is to evaluate and compare different models and portfolio allocation methods

in the context of four sector portfolios in Australia. The ARMA-ARCH, ARMA-GARCH and

ARMA-EGARCH models are utilised as parametric models while the historical simulation method

as the non-parametric model in this report. In addition, the four portfolios under evaluation are:

Energy, Financials, Telecommunication and Consumer Staples, data of which are from Yahoo

Finance (2012). The daily returns of the sector portfolios are employed in this report with a time span

of 10 years, constituting a sample size of 2487. The in sample period is from 1 June 2002 to 31

December 2009 with a size of 1880, and therefore a forecasting period of 607.

The report will first present an Exploratory Data Analysis and model construction of the in-sample

period data. Next, one-step-ahead and multi-step-ahead forecasts will be conducted to generate return,

volatility and VaR forecasts during the forecast period by using the models identified above.

Forecast accuracy will also be evaluated and analysed. Afterwards, the forecasts will be used to

generate weights for portfolio allocation under three strategies which are Return Strategy, Volatility

Strategy and VaR Strategy. The final part of this report will discusses and assesses the investment

outcomes under different models and different allocation methods by comparing the mean, standard

deviation and utility score of each portfolio. As a result, we can evaluate whether investments are

better off by using the quantitative methods.

2. Exploratory Data Analysis The log returns are calculated and used within the entire report as suggested by Tsay (2010). The log

return has the attractive attribute of additive. The Figure 1 below plots each portfolio log returns over

time. As can be seen from the plot, all portfolios tend to have a daily mean return around zero. The

portfolio returns have a considerably higher volatility during the GFC period (after the 1370th data

point) as separated by the blue line. Overall, the Energy portfolio shows the highest return close to

9.2% and the lowest return around -12.6%, which are pointed out with red arrow in the plot.

Moreover, the Energy sector exhibits high volatility over time, even during the pre-crisis period with

its extreme returns circled in magenta. In addition, the Financials industry presents remarkably

higher volatility during the GFC period in comparison with the non-crisis period, which is circled in

green. The consumer portfolio exhibited the lowest volatility among the four industries.

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Figure 1: Portfolio Returns 2002-2009 (In-sample period)

Table 1: Summary statistics for each asset

Mean Median Std Min Max Skewness Kurtosis Energy 0.069 0.140 1.605 -12.576 9.206 -0.506 8.737

Financials 0.004 0.020 1.328 -8.990 8.812 0.010 9.417 Telecomm -0.018 -0.003 1.277 -10.845 7.180 -0.551 7.840 Consumer 0.030 0.023 1.044 -10.561 6.812 -0.450 11.873

As can be seen from the table above, all assets had positive average and median daily returns around

zero during the in sample period. In addition, Financials and Telecomm have standard deviation

around 1.3%. Energy had the highest volatility of 1.61, which has been observed in the plot above.

The Consumer industry has the lowest volatility of 1.04%. Overall, the daily returns range between -

12.58% and 9.21%, which is also the highest value and lowest value of Energy industry.

A clear overview of the skewness and kurtosis can be obtained by combining the summary statistics

with their histograms (Figure 2). All portfolios exhibit negative skewness, except for Telecomm.

Moreover, the histogram suggests that the skewness of each portfolio is influenced by their extreme

values, which are a number of extreme negative values outweigh the positive ones for the portfolios

except for Telecomm. The kurtosis of each portfolio returns are way above 3, which indicates the

existence of outliers and fat-tails in return distribution, therefore, the forecast models below used the

t-distribution instead of the Gaussian distribution.

Figure 2: Histogram for each portfolio 2002-2009

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3. Models for Forecasting 3.1 Motivations of Model Selection

The ARMA model is used as quantitative method for modelling the mean equation. We expect the

financial time series data to be analysed have autocorrelation effects. Therefore, ARMA model are

used to capture the autocorrelation effects and patterns by including the lagged return series and

including the lagged error series in the mean equations.

As for the variance equations, ARCH, GARCH and EGARCH are employed for modelling the

volatilities as the financial time series data may have the issue of heteroskedasticity. The ARCH

model is represented as the basic volatility model, which is expected to characterize the time series

data by including lagged innovation terms, while GARCH model is a more generalised model by

including lagged variance terms. The EGARCH model is chosen as it has fewer restrictions on its

parameters in the equations because of the log form variance equation. In addition, the model is able

to measure the leverage effect.

Moreover, due the issue of fat-tailed behaviour discussed above, the error terms will be in standard

Student-t distribution, which allows for higher kurtosis and fatter tails to capture outliers. Therefore,

the quantitative models for asset returns are ARMA-ARCH, ARMA-GARCH and ARMA-EGARCH

with t distribution.

3.2 Model Selection

The section below will discuss each model for each asset in details. Firstly, the orders for each model

are selected based on the result of AIC and SIC. After that, several tests are conducted against the

assumptions for each model. The LB test will be conducted to test for any autocorrelation among

standardised residual and the ARCH effect among the squared residuals. Further, the JB test is

conducted for test the normality of the standardised residuals. Finally, models will be refined based

on the results for each test. The Consumer industry will be evaluated in detail for illustration.

3.2.1 ARMA-ARCH-t Model

As for the mean equation, the AIC find the appropriate orders of AMRA (3, 3) while SIC favours

ARMA (1, 1). We will trust SIC and choose ARMA (1, 1). Again, AIC and SIC are used to find the

appropriate orders for ARCH (p):

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Figure 3: AIC & SIC for choosing ARCH orders

The Figure 3 shows that both AIC and SIC prefer ARCH (9). So, we will fit ARMA (1, 1)-ARCH (9)

-t model:

*1 1 t t 8.04

2 2 2 2 2 21 2 3 4 5

0.047 0.16 0.12 = ~ (0,1)(s.e) (0.027) (0.48) (0.48)

0.17 0.18 0.097 0.066 0.091 0.052(s.e) (0.029) (0.039) (0.033) (0.0

t t t t t t

t t t t t t

r r a a a t

a a a a a

σ ε ε

σ

− −

− − − − −

= − + +

= + + + + +

2 2 2 26 7 8 9

28) (0.033) (0.028) 0.069 0.061 0.12 0.13 (0.029) (0.030) (0.038) (0.037)

t t t ta a a a− − − −+ + + +

The absolute value of AR parameter is less than 1, so the AR (1) is stationary. Moreover, the

absolute value of MA parameter is less than 1 as well; therefore, the MA (1) is invertible. The

parameter estimates for AR and MA are not significant. However, we still leave them in the mean

equation at this stage. All the parameter estimates are positive and significantly different to zero,

expect for the 5th parameter in the ARCH. The volatility persistence is estimated 0.866, which is less

than 1, so the stationarity requirement is met. The low degree of freedom estimates (8.04) indicates

much fatter tails than a Gaussian.

The standardized residuals ˆ ˆ/t ta σ , are plotted below (Figure 4). According to the ACF, it shows

little remaining auto-correlation or obvious heteroskedasticity.

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Figure 4: Standardised rediduals and its ACF

To confirm that, the LB test is conducted with outcomes summarized in the table below. The p-value

from the LB test shows that we do not reject the null hypothesis of no remaining autocorrelation, at 5%

significance level. So, it seems the mean equation is reasonably modelled well by ARMA (1, 1). The

ARMA actually helps here, though their parameter estimates are insignificant.

LB tests: 0 1 2: ... 0 : at least one of 0 ( 1,2... )m A iH H i mρ ρ ρ ρ= = = = ≠ = Table 2: LB test results (residuals) for ARMA-ARCH-t

m=17, d.f=5 m=22, d.f=10 m=27,d.f=15 p-value 0.177 0.415 0.674

The histogram of these transformed standardized residuals (see Figure 5) appears slightly more fat-

tailed than a normal distribution, with two large outliers around 3.5 and -3.5. However, the QQ-plot

does not depart much at all from normality in either upper or lower tail, which may seem that it is

very close to a standard Gaussian.

Figure 5: Histogram for standardised residual and its QQ plot

As for the normality of the standardized residuals, the JB test can be conducted with:

0 : skewness=0 AND kurtosis=3 : the distribution is not NormalAH H

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The result shows a p-value of 0.5, indicating the Gaussianity of standardized residuals. Furthermore,

we cannot reject that the residuals come from a Student-t distribution. The sample skewness and

kurtosis are also calculated to confirm the test: it has a skewness of -0.018 and kurtosis of 2.99,

which seem very close to the Gaussian distribution (0 and 3).

The ACF for the squared standardised residuals 2 2ˆ ˆ/t ta σ are plotted below in Figure 6. It displays

some significant correlations at the 8th lag and 16th lag, which means the volatility equation might not

well specified by an ARCH (9).

Figure 6: ACF for squared standardised residuals

Table 3: LB test results (squared residuals) for ARMA-ARCH-t m=17, d.f=5 m=22, d.f=10 m=27,d.f=15

p-value 0.0018 0.0004 0.0000

The p-value of the LB test indicates strongly significant remaining ARCH effects in the squared

standardized residuals. Therefore, the ARMA (1, 1)-ARCH (9)-t may have not capture adequate

ARCH effects.

To refine the model, we will choose a higher order ARCH model. For example, we choose to re-fit

ARMA (1, 1)-ARCH (16)-t, and again, conduct LB test for the squared residuals. Table 4: LB test results (squared residuals) after refine

m=24, d.f=5 m=29, d.f=10 m=34,d.f=15 p-value 0.0001199 0.0000989 0.0001131

As can be seen from the table, the p-values are still very small and thereby indicating strongly

significant remaining ARCH effects in the standardized residuals in the ARMA (1, 1)-ARCH (16)-t

model. This might be explained the properties of the ARCH model that it does not include any

lagged variance in the volatility equation. Therefore, the model cannot be simply improved by

increase ARCH’s order. We will still use the ARMA (1, 1)-ARCH (9)-t as chosen by SIC.

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3.2.2 ARMA-GARCH-t Model

The ARMA-GARCH model follows the same step as the ARMA-ARCH model. We will fit the

ARMA (1, 1)-GARCH (1, 1) with t-distribution as suggested by SIC.

*1 1 t t 8.89

2 2 21 1

0.050 0.14 0.080 = ~ (0,1)(s.e)(0.026) (0.41) (0.41)

0.0059 0.071 0.924(s.e) (0.0026) (0.012) (0.011)

t t t t t t

t t t

r r a a a t

a

σ ε ε

σ σ

− −

− −

= − + +

= + +


absolute value of MA parameter is less than 1 as well; therefore, the MA (1) is invertible. Again, all

the coefficients for the AR and MA in the mean equation are insignificant. However, we still leave

them in the mean equation at this stage. All parameter estimates in the GARCH model are positive

with volatility persistence estimated as 0.071 + 0.924 = 0.995, very high and close to 1, indicating

strong persistence in volatility and slow mean reversion. Low degrees of freedom (8.89) estimate

indicating much fatter tails than a Gaussian.

The standard deviation process seems nice and smooth, and sits nicely “on the shoulder” of the

returns data below (shown in Figure 7).

Figure 7: Returns and its standard deviation

The standardized residuals ˆ ˆ/t ta σ , are plotted below (See Figure 8). The plots appear to show little

remaining auto-correlation or obvious heteroskedasticity.

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Figure 8: Standardised rediduals and its ACF

To confirm the observation in ACF plot, the LB test is conducted with outcomes summarized in the

table below. The high p-value from the LB test shows that there is no remaining autocorrelation, at 5%

significance level. So, it seems the mean equation is reasonably modelled well by ARMA (1, 1) even

though their parameter estimates are insignificant.

Table 5: LB test results (residuals) for ARMA-GARCH-t LB test: residuals m=10, d.f=5 m=15, d.f=10

p-value 0.7810 0.8542

The histogram of these transformed standardized residuals is shown in Figure 9. The plot appears

slightly more fat-tailed than a normal distribution, with one large negative outlier around -4.

However, the QQ-plot does not depart much at all from normality in either upper or lower tail, which

may seem that it is very close to a standard Gaussian.

Figure 9: Histogram for standardised residual and its QQ plot

The sample skewness and kurtosis are: -0.019 and 3.001, which seem very close to the Gaussian

distribution with 0 and 3, respectively. In addition, the JB test has the p-value of 0.5, thus cannot

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reject the null hypothesis of Gaussian residuals. The model fits the data very well in the tails of the

distribution.

Figure 10: ACF for squared standardised residuals

The squared transformed standardised residuals seem to display significant autocorrelation at the 13th

and the 16th lag. We can further conduct the LB test to test the ARCH effect. The results for LB test

are listed in the table below. The high p-value indicates that there are no clearly significant

remaining ARCH effects in the data. It seems that the ARMA(1,1)-GARCH(1,1)-t model captures

the volatility dynamics reasonably well. Table 6: LB test results (squared residuals) for ARMA-GARCH-t

m=10, d.f=5 m=15, d.f=10 p-value 0.1132 0.2263

In summary, the ARMA (1, 1)-GARCH (1, 1)-t model have reasonably well captured the mean,

volatility and distribution processes of the Consumer returns. It cannot be clearly rejected on any of

these three criteria or aspects.

3.2.3 ARMA-EGARCH-t

In terms of the EGARCH model, we choose to fit the ARMA (1, 1)-EGARCH (1, 1)-t:*

1 1 t t 9.40

2 2 *1 t-1 9.40 t-1

0.041 0.167 0.11 = ~ (0,1)(s.e)(0.023) (0.402) (0.406)

log( ) 0.004 0.988log( ) 0.15(| | (| |)) 0.051 (s.e) (0.003) (0.004)

t t t t t t

t t

r r a a a t

E t

σ ε ε

σ σ ε ε

− −

−

= − + +

= − + + − −

(0.023) (0.014)


absolute value of MA parameter is less than 1 as well; therefore, the MA (1) is invertible. Again, all

the coefficients for the AR and MA in the mean equation are insignificant. However, we still leave

them in the mean equation at this stage. Moreover, |β|=0.988 < 1, the stationarity requirement has

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been fulfilled in the equation. Low degrees of freedom (9.4) estimate indicates much fatter tails than

a Gaussian. The coefficients in the volatility equation are significant, particular with the significant

leverage term.

The NIC is plotted in Figure 11. The level of asymmetry is estimated as 1.2273. On average, the

negative shocks of around 2 standard deviations have a 22.73% higher volatility than positive shocks

of the same size.

Figure 11: New information curve

Figure 12: Volatility for ARCH, GARCH and EGARCH model

As can be seen from the plot, the ARCH estimates are quite noisy and less smooth, compared to the

other series. It gives higher volatility estimates than the other two models over time. On the contrary,

the GARCH volatilities seem smoother than the ARCH estimated volatilities. The EARCH estimates

provide the lowest volatility during the low volatility period (from the 210th to the 600th observation).

During the high volatility period (around the 1600th observation), the GARCH estimates are higher

than EARCH but lower than ARCH. The GARCH and EGARCH are almost on top of each other on

most days. The GARCH volatilities “recover” the slowest from the GFC from 2008 (around the

1600th observation), as its volatility persistence is the highest among the three models.

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3.2.4 Historical Simulation

The final model applied is a non-parametric model, which is Historical Simulation with data from the

last 100 days (HS-100), the equations are shown below:

( )2100 100

2 2(100) (100)

1 1

1 1 100 100 1t t i t t i

i ir r s r rσ− −

= =

= = = −−∑ ∑

The sample mean of last 100 days are used when estimating the mean and standard deviation. The

return and the standard deviation are expected to be smoother than other models, shown in Figure 13

and Figure 14.

Figure 13: Compare daily return and 100-day mean return

Figure 14: Compare volatility among each model

3.3 Model Construction for the Other Industries Similar to fitting Consumer, we are using AIC and SIC to find the appropriate orders for AMRA and

then to find the orders for ARCH and GARCH, and thereby to fit EGARCH. We are still using the

Student-t distribution for characterizing the error terms. The equations of the fitted models for the

other three industries are below.

0 200 400 600 800 1000 1200 1400 1600 1800-12

-10

-8

-6

-4

-2

0

2

4

6

8

ReturnHS-100 return

0 200 400 600 800 1000 1200 1400 1600 18000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

HS-100

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*1 1 t t 8.54

2 2 2 2 2 21 2 3 4 5

Energy: ARMA(1,1)-ARCH(11)-t

0.0083 0.92 0.94 = ~ (0,1)(s.e)(0.0064) (0.055) (0.048)

0.42 0.087 0.061 0.16 0.068 0.027(s.e) (0.074) (

t t t t t t

t t t t t t

r r a a a t

a a a a a

σ ε ε

σ

− −

− − − − −

= + − +

= + + + + +

2 2 2 2 2 26 7 8 9 10 11

0.030) (0.029) (0.040) (0.034) (0.029)

0.098 0.084 0.034 0.064 0.12 0.063 (0.038) (0.033) (0.025) (0.030) (0.038

t t t t t ta a a a a a− − − − − −+ + + + + +

) (0.032)

*1 1 t t 9.01

2 2 21 1

Energy: ARMA(1,1)-GARCH(1,1)-t

0.2 0.84 0.83 = ~ (0,1)(s.e)(0.072) (0.47) (0.48)

0.0093 0.052 0.94(s.e) (0.0051) (0.009) (0.0092)

t t t t t t

t t t

r r a a a t

a

σ ε ε

σ σ

− −

− −

= − + +

= + +

*1 1 t t 9.13

2 2 *1 t-1 9.13 t-1

Energy: ARMA(1,1)-EGARCH(1,0)-t

0.009 0.912 0.927 = ~ (0,1)(s.e)(0.009) (0.093) (0.085)

log( ) 0.005 0.991log( ) 0.121(| | (| |)) 0.021

t t t t t t

t t

r r a a a t

E t

σ ε ε

σ σ ε ε

− −

−

= + − +

= + + − −

(s.e) (0.003) (0.003) (0.021) (0.011)

*1 2 1 t t 8.13

2 2 2 2 21 2 3 4

Financials: ARMA(2,1)-ARCH(8)-t

0.013 0.75 0.011 0.76 = ~ (0,1)(s.e) (0.028) (0.54) (0.026) (0.54)

0.15 0.12 0.13 0.13 0.1 0.

t t t t t t t

t t t t t

r r r a a a t

a a a a

σ ε ε

σ

− − −

− − − −

= + − − +

= + + + + + 2 2 2 25 6 7 8089 0.13 0.12 0.13

(s.e) (0.025) (0.033) (0.036) (0.035) (0.035) (0.033) (0.04) (0.039) (0.038) t t t ta a a a− − − −+ + +*

1 2 1 t t 8.20

2 2 21 1

Financials: ARMA(2,1)-GARCH(1,1)-t

0.014 0.74 0.012 0.744 = ~ (0,1)(s.e)(0.026) (0.47) (0.027) (0.47)

0.004 0.927 0.072(s.e) (0.00

t t t t t t t

t t t

r r r a a a t

a

σ ε ε

σ σ

− − −

− −

= + − − +

= + +

2) (0.011) (0.012)

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*1 2 1 t t 10.81

2 21 t-1

Financials: ARMA(2,1)-EGARCH(1,0)-t

0.037 0.259 0.039 0.293 = ~ (0,1)(s.e)(0.028) (0.607) (0.026) (0.607)

log( ) 0.018 0.9741log( ) 0.222(| |

t t t t t t t

t t

r r r a a a tσ ε ε

σ σ ε

− − −

−

= − + + +

= − + + *10.81 t-1(| |)) 0.144

(s.e) (0.005) (0.005) (0.026) (0.016) E t ε− −

*1 2 1 t t 6.12

2 2 2 2 21 2 3 4

Telecomm: ARMA(2,1)-ARCH(7)-t

0.014 0.435 0.039 0.439 = ~ (0,1)(s.e)(0.016) (0.392) (0.027) (0.392)

0.54 0.22 0.14 0.07 0.02 0.1

t t t t t t t

t t t t t

r r r a a a t

a a a a a

σ ε ε

σ

− − −

− − − −

= + − − +

= + + + + + 2 2 25 6 70.064 0.103

(s.e) (0.068) (0.047) (0.039) (0.039) (0.03) (0.038) (0.032) (0.036) t t ta a− − −+ +

*1 2 1 t t 6.38

2 2 21 1

Telecomm: ARMA(2,1)-GARCH(1,1)-t

0.01 0.295 0.047 0.289 = ~ (0,1)(s.e)(0.017) (0.45) (0.023) (0.45)

0.013 0.059 0.934(s.e) (0.006) (0.011

t t t t t t t

t t t

r r r a a a t

a

σ ε ε

σ σ

− − −

− −

= + − − +

= + +

) (0.011)

*1 2 1 t t 6.65

2 21 t-1 6

Telecomm: ARMA(2,1)-EGARCH(1,0)-t

0.005 0.367 0.043 0.363 = ~ (0,1)(s.e)(0.015) (0.44) (0.023) (0.44)

log( ) 0.002 0.989log( ) 0.124(| | (|

t t t t t t t

t t

r r r a a a t

E t

σ ε ε

σ σ ε

− − −

−

= + − − +

= + + − *.65 t-1|)) 0.026

(s.e) (0.003) (0.004) (0.022) (0.014) ε−

In the mean equations, some of the AR and the MA coefficients are insignificant, but we still leave

them in the mean equation at this stage. Also, all the absolute values of coefficients of the AR and

the MA terms in each model are less than 1. So, the mean equations modelled in ARMA are

stationary and invertible.

In the volatility models, there are several insignificant parameters, such as the 5th ARCH parameter

for Energy. And, the absolute values of the all the parameters in each volatility equation are less

than 1, which means the ARCH and GARCH models are stationary. The EGARCH models are also

stationary, as the 1| |β are less than 1. On the other hand, some models’ volatility persistence can be

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very close to1. Among them, Financials has the highest volatility persistence (0.999) from its

GARCH, while Telecomm has the lowest volatility persistence (0.723) from its ARCH.

The low degrees of freedom estimates from each model indicate the fatter tail than a Gaussian.

We then will test the models by conducting LB and JB tests.

LB tests: 0 1 2: ... 0 : at least one of 0 ( 1,2... )m A iH H i mρ ρ ρ ρ= = = = ≠ =

JB tests: 0 : skewness=0 AND kurtosis=3 : the distribution is not NormalAH H

First, the summary of the tests for ARMA-ARCH-t models in each industries are shown in the

tables below.

Table 7: Diagnostic Test results of the ARMA-ARCH-t model ARMA(1,1)-ARCH(11)-t

for Energy ARMA(2,1)-ARCH(8)-

t for Financials ARMA(2,1)-ARCH(7)-

t for Telecomm p-value of LB test: residuals

0.017 (m=19, d.f=5) 0.018 (m=17, d.f=5) 0.016 (m=16, d.f=5) 0.018 (m=24, d.f=10) 0.100 (m=22, d.f=10) 0.170 (m=21, d.f=10)

p-value of JB test 0.009 0.080 0.029 p-value of LB test: squared residuals


The LB test results suggest that there are significant remaining autocorrelation effects in the mean

equation of Energy, and some significant remaining ARCH effects in the volatility equation of

Energy. Both the mean and volatility equation are not well modeled by the ARMA (1, 1)-ARCH

(11)-t model for Energy.

On the other hand, for Financials and Telecomm, the LB tests for the transformed residuals show that

there may be little significant remaining autocorrelation in the mean equation. So, the mean

equations are reasonably well-modeled by the ARMA for Financials and Telecomm. However, for

Financials and Telecomm, the LB tests for the squared residuals suggest that there are clearly

significant remaining ARCH effects in the data. It seems that the ARMA-ARCH-t models have not

captured the volatility dynamics reasonably well.

Furthermore, the JB tests confirm that the transformed standardized residuals are not a standard

Gaussian, except for Financials.

In summary, all the ARMA-ARCH-t models have not captured adequate ARCH effects. So we will

try refining the models which have higher orders to capture adequate ARCH effects. In particular, for

Energy, we will increase the orders in both ARMA and ARCH.

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Table 8: Diagnostic Test results after Re-fitting ARMA-ARCH-t model ARMA(5,5)-ARCH(16)-

t for Energy ARMA(2,1)-ARCH(16)-

t for Financials ARMA(2,1)-ARCH(16)-

t for Telecomm p-value of LB test: residuals



0.000 (m=32, d.f=5) 0.000 (m=25, d.f=5)) 0.001 (m=25, d.f=5)

0.002(m=37, d.f=10) 0.000 (m=30, d.f=10) 0.009 (m=30, d.f=10)

The above results show that the remaining autocorrelation effects and ARCH effects are still

significant for Energy. For Financials and Telecomm, the models actually are becoming worse, by

showing both remaining significant autocorrelation effects and ARCH effects. Also, the normality is

still rejected for the transformed standardized residuals. We might need to explore a new suitable

distribution that has fatter tails than student-t distribution to characterize to dynamics of the data.

Hence, now it may be difficult to find better models by just adjusting the orders of ARMA and

ARCH.

As a result, we will still use the models as chosen by SIC.

Secondly, the summary of the tests for ARMA-GARCH-t models in each industries are shown in

the table below.

Table 9: Diagnostic Test results of the ARMA-GARCH-t model ARMA(1,1)-

GARCH(1,1)-t for Energy

ARMA(2,1)-GARCH(1,1)-t for

Financials

ARMA(2,1)-GARCH(1,1)-t for

Telecomm p-value of LB test: residuals




The LB tests for the transformed residuals show that there is almost no significant remaining

autocorrelation in the mean equations of all the three industries, which are reasonably well-modeled

by the ARMA.

On the other hand, except for Financials, the LB tests for the squared residuals suggest that there are

significant remaining ARCH effects in the data of Energy and Telecomm. It seems that the ARMA-

GARCH-t models have not captured the volatility dynamics reasonably well.

In addition, the JB tests suggest that the transformed standardized residuals for all the three industries

are still not a standard Gaussian.

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Therefore, we may try refining the models of Energy and Telecomm with higher orders in GARCH

to capture adequate lagged volatilities. Table 10: Diagnostic Test results after Re-fitting ARMA-GARCH-t model ARMA(1,1)-GARCH(5,5)-t

for Energy ARMA(2,1)-ARCH(5,5)-t

for Telecomm p-value of LB test: residuals

0.090 (m=18, d.f=5) 0.049 (m=19, d.f=5) 0.210 (m=23, d.f=10) 0.180 (m=24, d.f=10)

p-value of JB test 0.003 0.033 p-value of LB test: squared residuals

0.004 (m=18, d.f=5) 0.025 (m=19, d.f=5) 0.014(m=23, d.f=10) 0.037 (m=24, d.f=10)

The above results show that there are still remaining significant ARCH effects in the volatility

equations. Also, the normality is rejected for the transformed standardized residuals. It may be

difficult to find better models by just adjusting the orders of GARCH.

As a result, we will still use the models as chosen by SIC.

The Historical Simulation method for the other three industries:

1002 2

; ; ; (100)1

1002 2

; ; ; (100)1

1002 2

; ; ; (100)1

1Energy:

1001

Financials: 100

1Telecomm:

100

E t E t i E t Ei

F t F t i F t Fi

T t T t i T t Ti

r r s

r r s

r r s

σ

σ

σ

−=

−=

−=

= =

= =

= =

∑

∑

∑

4. Forecast and Accuracy Measures

In Section 3, we have discussed several asset return models. In this part, we are going to forecast

asset returns and risks using these different models/methods. Firstly, we will forecast with fixed

horizon and moving origin. In-sample size will increase by one and models will be re-estimated for

every period we move forward. In addition, we will assess the forecasting accuracy measures, for

forecasted returns and volatilities of the four sectors throughout the forecasting period. Secondly,

multi-period (607-step-ahead) forecasts with fixed origin will be calculated and evaluated in order to

construct portfolios in next section. Besides, we also generated five-step-ahead forecasts in order to

construct sectors in next section, but we will not focus on analysis of these forecasts here. One

industry will be analysed in details, and results of other industries will be presented.

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4.1 Return Forecast and Accuracy

4.1.1 One-Step-Ahead Return Forecast

We will focus on Energy to analyse of one-step-ahead forecasted returns and accuracy measures.

The following figure summarizes the dynamics of forecasted returns under the four models.

Figure 15 One-step-Forecasted Energy returns under four models versus actual returns

As we can see from the plot, none of the forecasts seem to “follow” the directions or magnitudes of

the actual Energy returns. This pattern is repeated for all the other industries, too.

Figure 16: First 25 one-step-ahead Forecasted Energy returns under four models versus actual returns

Figure 16 shows more close characteristics of forecasted returns. The ARMA-ARCH forecasts (in

“+”), the ARMA-GARCH forecasts (in “*”) and the ARMA-EGARCH forecasts (in diamonds) are

on top of each other, while historical simulation (in triangle) forecasts are different from the ARMA-

ARCH type models’ forecasts in directions in some occasions. However, none of the forecasts

follows the magnitude or directions of actual data. To assess these forecasts numerically, we can

calculate the RMSE and MAD of these forecasts, as shown in the following table.

0 100 200 300 400 500 600 700-6

-4

-2

0

2

4

6Energy

Forecast Period ReturnARMA-ARCHARMA-GARCHARMA-EGARCHHS(100)

0 5 10 15 20 25 30 35 40 45 50-4

-3

-2

-1

0

1

2

3

4

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Table11: Accuracy Measures for one-step-ahead Forecasted daily Energy returns under four models

ARMA(1,1)-ARCH(11)-t

ARMA(1,1)-GARCH(1,1)-t

ARMA(1,1)-EGARCH(1,1)-t

HS-100

RMSE 1.4085 1.4093 1.4077 1.4085 MAD 1.0647 1.0655 1.0641 1.0662

The units of RMSE and MAD are the same units as percentage returns. The typical errors made are

between 1.06% and 1.41% in terms of percentage returns. These seem large for daily return, which

numerically explains why our forecasts do not follow the dynamics of real data. The best method,

most accurate under both accuracy measures, is the ARMA-EGARCH-t model, followed by the

ARMA-ARCH-t. The HS-100 ranks last under MAD while ARMA-GARCH-t ranks last under

RMSE. As RMSE is sensitive to outliers, MAD is more trustworthy.

Figure 17: one-step-ahead forecasted Portfolios returns under four models and actual returns

The above figure summarizes the dynamics of forecasted returns of four portfolios under the four

models. These forecasts are “flat” compared to real return data. In addition, forecasts of ARMA-

ARCH type models are close to each other while HS-100 forecast somewhat deviates from them.

However, none of the forecasts seem to follow the directions or magnitudes of the actual portfolio

returns. This pattern is same for all the four portfolios.

All portfolios’ RMSE and MAD are presented in the following table. The typical errors made are

between 0.65% and 1.41% in terms of percentage returns. These errors are really large for daily

returns. For Telecomm and Consumer, the best method is the ARMA-ARCH-t model. For Energy

and Financials, the best method is the ARMA-EGARCH-t model and ARMA-GARCH-t respectively.

The HS-100 ranks last under both MAD and RMSE for all portfolios.

0 100 200 300 400 500 600 700-6

-4

-2

0

2

4

6Energy

0 100 200 300 400 500 600 700-4

-2

0

2

4

6Financials

0 100 200 300 400 500 600 700-10

-5

0

5

10Telecomms

0 100 200 300 400 500 600 700-3

-2

-1

0

1

2

3Cons Staples

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Table 12: Accuracy Measures for one-step-ahead Forecasted daily Portfolio returns under four models

ARMA-ARCH

ARMA-GARCH

ARMA-EGARCH

HS-100

Energy RMSE 1.4085 1.4093 1.4077 1.4085 MAD 1.0647 1.0655 1.0641 1.0662

Financials RMSE 1.1669 1.1656 1.1658 1.1710 MAD 0.8882 0.8874 0.8877 0.8922

Telecomm RMSE 1.1622 1.1632 1.1624 1.1677 MAD 0.8458 0.8473 0.8473 0.8491

Consumer RMSE 0.8410 0.8419 0.8416 0.8442 MAD 0.6567 0.6576 0.6568 0.6587

4.1.2 Multi-period Return Forecast

Multi-period return forecasts can also be calculated and evaluated. The following figure summarizes

the 607 forecasts under four models for Energy. As we can see from the figure, return forecasts

under the HS method are constant over the whole forecast period. For ARCH type models, ARMA-

EGARCH-t forecast recover the quickest to its long run mean (constant coefficient of the mean

equation). ARMA-GARCH-t comes second, and it bounces back and forth before recovery as the

model has negative AR coefficient. ARMA-ARCH-t forecast recover the slowest. Besides, these

multi-period forecasts are less volatile than one-step-ahead forecasts as they all recover to their long

run mean. However, none of the forecasts matches the direction or magnitude of the real return data.

Figure 18: 607-step-ahead Forecasted Energy returns under four models

All portfolios’ RMSE and MAD for 607-step-ahead forecasts are presented in the following table.

They can roughly be regarded as performance to predict long-run mean for each industry. The typical

errors made are between 0.65% and 1.41% in terms of percentage returns. These errors are large and

similar to the errors made from one-step-ahead forecasts.

0 100 200 300 400 500 600 7000.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

Forecast Period

Perce

ntage

Retu

rn

Multi-period Forecasts under 4 Models

ARMA-ARCHARMA-GARCHARMA-EGARCHHS(100)

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In summary, as for Energy, the best method is the HS method and the worst are ARMA-ARCH-t and

ARMA-GARCH-t, indicating the long-run mean for Energy return series is closer to the HS forecast.

For Financials, the best method is ARMA-EGARCH and the worst is HS method. . For Telecomm,

the best method is ARMA-GARCH-t and the worst is HS-100 method. For Consumer, the best

method is ARMA-EGARCH-t and the worst is HS method.

Table 13: 607-step-ahead Forecasted portfolio returns under four models

ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS(100) Energy RMSE 1.4093 1.4093 1.4086 1.4057

MAD 1.0653 1.0653 1.0648 1.0632 Financials RMSE 1.1658 1.1660 1.1646 1.1737

MAD 0.8867 0.8868 0.8861 0.8932 Telecomm RMSE 1.1629 1.1628 1.1628 1.1643

MAD 0.8466 0.8466 0.8467 0.8495 Consumer RMSE 0.8386 0.8388 0.8383 0.8441

MAD 0.6551 0.6554 0.6549 0.6603

4.2 Volatility Forecast and Accuracy

4.2.1 One-Step-Ahead Volatility Forecast

As volatility is an unobserved process, we need volatility proxies to assess volatility forecast

accuracy. These proxies are stated as following. Proxy 1 is the square mean-corrected daily return.

Proxy 2 is the percentage log intra-day range. Proxy 3 is overnight-movement-adjusted log intra-day

range.

We will focus on Energy to analyse forecasted volatility and accuracy measures. Forecasted

volatility for Energy versus Proxy 1 is depicted in figure19. The Proxy 1 volatilities are in green. The

three GARCH type models’ forecasts seem mostly similar, and mostly to “sit on top” or “on the

shoulders” of the absolute return shocks. This is what expected since the theoretical return shock

under these models are less than the standard deviation as error term for expected to be less than 1

for most of times. Therefore, the true volatility process should also sit on top of the absolute return

shocks. The HS-100 takes the longest to recover from extreme returns, and its forecasted volatility is

the smoothest. The ARCH forecasts are quite noisy and less smooth, compared to the other series.

The ARCH (11) recovers after exactly 11 days. The GARCH and EGARCH are on top of each other

on most days.

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Figure 19: Forecasted Portfolios volatilities under four models and Proxy 1 Figure 20 is the forecasted volatility for Energy as well as Proxy 2. As we can see in the figure, this

proxy does not have close-to-zero volatility estimates like Proxy 1. By using intra-day range data, the

efficiency increase with this proxy which is never zero on a trading day. However, this proxy has

completely missed the overnight price movements as intra-day range does not include overnight

returns. Note that on Aug 5 2011, Energy has dropped significantly by almost 6%, and the intra-day

range was even bigger (more than 7%). Therefore, there is a sharp peaked volatility in the middle of

the plot at around 400 days.

Figure 20: Forecasted Portfolios volatilities under four models and Proxy 2 Proxy 3, on the other hand, takes overnight returns into consideration. However, as closing prices

close to opening price for the next day for these portfolio indices. Proxy 3 and Proxy 2 do not make

much difference in this case. Figure21 shows Proxy 3 versus volatility forecasts is presented below.

0 100 200 300 400 500 600 7000

1

2

3

4

5

6Energy

Proxy 1 volatilityARMR-ARCHARMR-GARCHARMR-EARCHHS(100)

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8Energy

Proxy 2 volatilityARMA-ARCHARMA-GARCHARMA-EGARCHHS(200)

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Figure 21: Forecasted Portfolios volatilities under four models and Proxy 3 As for volatility forecast accuracy, the following two tables present RMSE and MAD measures for

Energy under four different models.

Table 14: RMSE for one-step-ahead forecasted daily Energy volatilities under four models ARMA(1,1)-ARCH(11)-t ARMA(1,1)-GARCH(1,1)-t ARMA(1,1)-EGARCH(1,0)-t HS-100

Proxy1 0.9268 0.9316 0.9225 0.9604 Proxy2 0.6483 0.6412 0.6414 0.6885 Proxy3 0.6372 0.6324 0.6288 0.6868

Table 15: MAD for one-step-ahead forecasted daily Energy volatilities under four models ARMA(1,1)-ARCH(11)-t ARMA(1,1)-GARCH(1,1)-t ARMA(1,1)-EGARCH(1,0)-t HS-100

Proxy1 0.7581 0.7537 0.7509 0.7812 Proxy2 0.5282 0.5183 0.5283 0.5469 Proxy3 0.5099 0.5006 0.5075 0.5356

The typical errors made are between 0.5% and 0.94% in terms of percentage returns. They are less

than the errors made in return forecast. For Proxy 1, the best method is the ARMA-EGARCH model.

For Proxy 2 and 3, the best method is the ARMA-GARCH. The HS-100 ranks last for all proxies.

Table 16: RMSE for one-step-ahead forecasted daily Portfolio volatilities under four models ARMA-

ARCH-t ARMA-

GARCH-t ARMA-

EGARCH-t HS(100)

Energy Proxy1 0.9268 0.9316 0.9225 0.9604 Proxy2 0.6483 0.6412 0.6414 0.6885 Proxy3 0.6372 0.6324 0.6288 0.6868

Financials Proxy1 0.7592 0.7472 0.7536 0.8097 Proxy2 0.5514 0.5447 0.5380 0.6444 Proxy3 0.5637 0.5605 0.5483 0.6620

Telecomm Proxy1 0.8818 0.8768 0.8587 0.8849 Proxy2 0.6144 0.5977 0.5736 0.6056 Proxy3 0.6098 0.5974 0.5749 0.6066

Consumer Proxy1 0.5801 0.5663 0.5722 0.5622 Proxy2 0.3446 0.3316 0.3343 0.3409 Proxy3 0.3419 0.3310 0.3296 0.3422

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8

9Energy

Proxy 3 VolatilityARMA-ARCHARMA-GARCHARMA-EGARCHHS(100)

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Table 17: MAD for one-step-ahead forecasted daily Portfolio volatilities under four models

ARMA- ARCH-t

ARMA-GARCH-t

ARMA-EGARCH-t

HS(100)





All industries’ RMSE and MAD for volatility forecast are presented in the above tables. For

Telecomm and Financials, the best method is the ARMA-EGARCH-t model and the worst is HS for

most proxies. For Consumer, the best method is the ARMA-GARCH-t model and the worst is

ARMA-ARCH-t.

4.2.2 Multi-period Volatility Forecast

Multi-period return forecasts can also be calculated and evaluated. The following figure summarizes

the 607 forecasts under four models for Energy. As we can see from the figure, volatility forecast

under HS method is constant over whole forecast period. As for ARMA-ARCH type models.

ARMA-ARCH forecasts recover quickest. ARMA-EGARCH comes second. And ARMA-GARCH

forecasts recover slowest, as the model is the most volatility-persistent for Energy.

Figure 22: 607-step-ahead Energy Volatility Forecasts under four models

0 100 200 300 400 500 600 7000.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8Multi-period Volatility Forecasts under 4 Models

ARMA-ARCHARMA-GARCHARMA-EGARCHHS(100)

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Table 18: RMSE for 607-step-ahead forecasted daily Portfolio volatilities under four models ARMA-

ARCH-t ARMA-

GARCH-t ARMA-

EGARCH-t HS-100





Table 19: MAD for 607-step-ahead forecasted daily Portfolio volatilities under four models

ARMA-ARCH-t

ARMA-GARCH-t

ARMA-EGARCH-t

HS-100





All portfolios’ RMSE and MAD for volatility forecast are presented in the above tables. The errors

made are ranged from 0.26% to 1.18% in terms of percentage returns. They are bigger than errors

made in one-step-ahead forecast. For Consumer, Telecomm and Financials, the best method is the

ARMA-EGARCH-t model and the worst is ARMA-ARCH-t for all proxies. However, for Energy,

the best method is the HS method and the worst is ARMA-ARCH-t. ARMA-ARCH-t model

forecasts deviate from the true volatility series most, while EGARCH and HS forecasts are closer to

true volatility series.

4.3 VaR Forecast and Accuracy

4.3.1 One-step ahead forecast of VaR

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As described above, each model is re-estimated every period with moving origin and fixed horizon.

Figure 23 shows the forecast and accuracy of VaR under four models. The Energy industry will be

evaluated in detail as indication.

Figure 23: VaR at 5% for Energy

The plot above shows the forecasted VaR for Energy sector over the whole forecast period. As can

be seen from the plot, especially the circled area, the HS-100 estimates staying at a low level for 99

days after extreme shocks and located far away from the data in those periods. The ARCH, GARCH

and EGARCH are on top of each other on most days. However, the ARCH moves back closer to the

returns after extreme shocks, as pointed out with purple arrows.

Figure 24: Violation at 5% for Energy The plot above shows the violations from the VaR forecast at 5%. As can be seen from the table,

most of the returns violate all the models forecasts. However, the forecast under ARCH model has a

few more violations as circled in purple. It seems there are more violations under low volatility

period under each model. Table 20: Accuracy Test for Energy (1-step)

Energy ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 23 39 58 56

! 0.0379 0.0643 0.0956 0.0923 !/! 0.76 1.29 1.91 1.85

Confidence Interval (0.0327,0.0673) Independence Test 0.06 0.13 0.05 0.20

DQ Test 0.0067 0.0015 0.0000 0.0000 Loss Function Value 99.68 97.99 102.80 103.25

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The table above shows the violation rates and the tests results for each model. The ARCH and

GARCH have the violation rate within the confidence interval and not significantly different to 0.05.

In addition, the ARCH is the only model over-estimates the risk level, with less violations and

violation rates significantly less than 0.05. All other models are under-estimate the risk level.

In terms of the GARCH, it has the lowest loss function value, which indicates that the model

forecasts are closest to the true VaR levels. Moreover, it has also passed the independence test,

indicating that it has tracked the dynamic risk well. The EGARCH shows the largest number of

violations, and it gives the second largest loss function value as well as a p-value of zero for DQ test.

Therefore, the EGARCH has not tracked the dynamic risk well. As for the Historical Simulation

model, it has passed the independence test, with a p-value of 0.2. However, it might also not track

the dynamic risk well since it has the largest loss function value and a p-value of zero from DQ test.

Actually, no model could pass the DQ test.

Table 21: Accuracy Test for Financials (1-step)

Financials ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 44 42 34 36

! 0.076 0.070 0.056 0.059 !/! 1.45 1.38 1.12 1.19



Table 22: Accuracy Test for Telecomm (1-step)

Telecomm ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 31 26 29 30

! 0.051 0.043 0.048 0.049 !/! 1.02 0.86 0.96 0.99



Table 23: Accuracy Test for Consumer (1-step)

Consumer ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 38 40 32 33

! 0.063 0.066 0.053 0.054 !/! 1.25 1.32 1.05 1.09



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The three tables above show the violation rates and tests results for each industry. As for Financial

industry, the ARMA-EGARCH-t model perform the best, it has the violation rates closest to the 5%

expected level. In addition, it captures the dynamic risk as it passed the independence and DQ test as

well as providing the lowest loss function value. In terms of Telecomm, the ARMA-EGARCH-t

method has the violation rate closest to the expected level; however, the ARMA-ARCH-t model

forecasts are closest to the true VaR levels as it shows the lowest loss function value. As for

Consumer, the ARMA-EGARCH-t model also performs the best as it has the lowest loss function

value as well as providing the violation rates that close to the expected violation rate level.

4.3.2 Multi-step ahead forecast of VaR

As the multi-step ahead forecast is used when allocating fixed portfolio weights, the volatility tends

to its long run mean. In turn, the Value at Risk also tends to smooth over time. The plot below shows

the smoothed VaR with 5% violation rate.

Figure 23: VaR at 5% for Energy

Figure 24: Violation at 5% for Energy

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According to Figure 23 and 24, the ARMA-EGARCH-t model and HS-100 method forecasts have

few more violations over the period, especially during the high volatility period. The ARMA-

ARCH-t model lies under all other models in the first plot, and therefore, it provides the least number

of violations, which can be verified in the table below.

Table 24: Accuracy Test for Energy (multi-step)

Energy ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 23 39 58 56

! 0.038 0.064 0.096 0.092 !/! 0.76 1.29 1.91 1.85



As can be seen from the table above, the ARMA-ARCH model provides the violation rate of 0.038,

which is within the confidence interval. However, it over-estimated the risk level, as it has fewer

amounts of violations than expected. All other models under-estimates the level of risk. The ARMA-

GARCH has the lowest loss function value; therefore, the ARMA-GARCH model forecasts are

closest to the true VaR. As for ARMA-EGARCH and HS-100 model, they perform the worst, as they

provides far more violations than expected and has the highest loss function values than other models.

Other Models:

Table 25: Accuracy Test for Financials (multi-step)

Financial ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCHt- HS-100 Number of Violations 15 24 62 22

! 0.025 0.040 0.102 0.036 !/! 0.49 0.79 2.04 0.72



Table 26: Accuracy Test for Telecomm (multi-step)

Telecomm ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 17 17 30 10

! 0.028 0.028 0.049 0.017 !/! 0.56 0.56 0.99 0.33



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Table 27: Accuracy Test for Consumer (multi-step)

Consumer ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-100 Number of Violations 10 14 34 40

! 0.017 0.023 0.056 0.066 !/! 0.33 0.46 1.12 1.32



Based on the three tables above, the ARMA-GARCH-t model performs the best as it provides the

closest number of violations to the expectation (30.35) and the second lowest loss function value.

The HS-100 method has the lowest loss function value, which means the forecasts are closest to the

true VaR. As for Telecomm and Consumer, the ARMA-EGARCH-t model performs the best as it

provides both the closest amount of violations to expectation and the lowest loss function values.

5. Optimal Portfolio Allocation

In this section, we are trying to find an optimal portfolio allocation method using forecasts, and try to

perform better than equally-weighted portfolio that is often very hard to beat in real data. Three

strategies will be employed and generated forecasts will assist our portfolio allocation. Performance

will be assessed using actual data over the whole forecast period with three criteria, average return of

portfolios, standard deviation of portfolios and Utility Scores of portfolios.

5.1 Portfolio Allocation Methods

In our portfolio allocation, three different rules are applied when choosing the optimal portfolio

weights, which are: Return Strategy, Volatility Strategy and VaR Strategy.

As for Return Strategy, which is the most aggressive rule, weights are allocated based on their

forecasted returns. That is, higher portfolio weights are allocated on asset with higher forecasted

returns as higher return represents higher utility for investors.

In terms of Volatility Strategy, it is more conservative than the Return Strategy as it takes asset

volatility or risk into consideration. Under this strategy, higher portfolio weights are allocated on

asset with lower forecasted volatilities. The rationale behind this allocation method is that investors

are risk averse and prefer lower volatility.

The VaR measures the quantiles of returns, which shows the minimum amount of loss of a portfolio

under normal market condition during a period with a certain probability level (Jorion 2001). Under

this strategy, higher portfolio weights are allocated on asset with lower forecasted VaR which is one

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of the most widely used risk measure by financial practitioners and has also made its way into the

Basel II capital-adequacy framework (McNeil, Fry and Embrechts 2005). It is the most conservative

rule among the three strategies, which assign more weight on asset with the lower VaR. The rationale

behind this allocation method is that investors dislike losses.

5.1.1 Fixed Weight Portfolio Allocation Method

In fixed weight portfolio allocation, we are constructing portfolios using multi-period forecasts. The

weight for each asset is based on the average of performance of these forecasts. The fixed allocation

weight is stated as following in details under each strategy.

Table 28: Fixed weight allocation strategies

Fixed weight Return strategy Volatility strategy VaR strategy

Formula ; 4

1

ˆ

ˆ

iR i

ii

rWr

=

=

∑

; 4

1

1ˆ

1ˆ

iVol i

ii

Wσ

σ=

=

∑ ; 4

1

1

1i

VaR i

ii

VaRW

VaR=

=

∑

;R iW denotes the weight assigned on asset i under Return Strategy and ir refers to the average 607-

period return forecasts for asset i (i=1,2,3 and 4 that denote Energy, Financials, Telecomm and

Consumer respectively, and these notations will be used throughout this section). ;Vol iW denotes the

weight assigned on asset i under Volatility Strategy and ˆiσ refers to the average 607-period volatility

forecasts for asset i. ;VaR iW denotes the weight assigned on asset i under VaR Strategy and iVaR refers

to the average 607-period VaR forecasts for asset i.

We will use these three weight allocation methods to assign fixed weight to each asset throughout the

forecasting period.

5.1.2 Dynamic Portfolio Allocation Method

In dynamic portfolio allocation, we are constructing portfolios whose weights are re-allocated every

period and every five period. Adjusting weight every period, we can make our utmost quantitative

efforts to compare the performance to portfolios with less quantitative efforts, i.e. fixed weight

portfolios and dynamic portfolios re-allocated every five period. Adjusting weight every five period

is common in finance industry. It makes sense because five-day performance generally represents

weekly performance and is less costly than adjusting weight every period. Moreover, model re-

estimations are in accordance with weigh adjustments, that is to say, models are re-estimated every

period and every five period. In this way, weights re-allocation will be precise and reliable. When re-

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estimating models, new realized forecast period data will be employed in model re-estimation, and

the last realized forecast period data will be regarded as new forecast origin.

For re-allocation weight every period, the allocation weight is stated as following in details under

each strategy.

Table 29: 1-day dynamic weight allocation strategies Return strategy Volatility strategy VaR strategy

Formula ; 1|

; , 4

; 1|1

ˆ

ˆ

i t tR i t

i t ti

rW

r

+

+=

=

∑ ; 1|

; , 4

; 1|1

1ˆ

1ˆ

i t tVol i t

i t ti

Wσ

σ

+

+=

=

∑ , 1|

; , 4

, 1|1

1

1i t t

VaR i t

i t ti

VaRW

VaR

+

+=

=

∑

; ,R i tW denotes the weight assigned on asset i under Return Strategy in day t and ; 1|i t tr + refers to the

one-step-ahead forecast from origin t for asset i. ; ,Vol i tW denotes the weight assigned on asset i under

Volatility Strategy in day t and ; 1|ˆ i t tσ + refers to the one-step-ahead volatility forecast from origin t for

asset i. ; ,VaR i tW denotes the weight assigned on asset i under VaR Strategy in day t and , 1|i t tVaR + refers

to the one-step-ahead VaR forecast from origin t for asset i.

For re-allocation weight every five period, the allocation weight is stated as following in details

under each strategy.

Table 30: 5-day dynamic weight allocation strategies Return strategy Volatility strategy VaR strategy

Formula

5

; |1

; , 4 5

; |1 1

ˆ

ˆ

( ,..., 5)

i t j tj

R i T

i t j ti j

rW

r

T t t

+=

+= =

=

= +

∑

∑∑

15

; |1

; , 14 5

; |1 1

1 ˆ5

1 ˆ5

( ..., 5)

i t j tj

Vol i T

i t j ti j

W

T t t

σ

σ

−

+=

−

+= =

# $% &' (=# $% &' (= +

∑

∑ ∑

15

; |1

; , 14 5

; |1 1

15

15

( ..., 5)

i t j tj

VaR i T

i t j ti j

VaRW

VaR

T t t

−

+

=

−

+

= =

" #$ %& '=" #$ %& '= +

∑

∑ ∑

; ,R i TW denotes the weight assigned on asset i under Return Strategy in day T and ; |i t j tr + refers to the j-

step-ahead return forecast from origin t for asset i. ; ,VaR i TW denotes the weight assigned on asset i

under VaR Strategy in day T and ; |i t j tVaR + refers to the j-step-ahead volatility forecast from origin t

for asset i. ; ,R i TW denotes the weight assigned on asset i under Return Strategy in day T and ; |i t j tr +

refers to the j-step-ahead forecast from origin t for asset i.

We will use these weight allocation methods to assign dynamic weight to each asset throughout the

forecasting period.

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5.2 Fixed Weight Portfolio Construction

Table 31: Fixed weight portfolio performance

fixed weighted Return-Strategy Volatility-Strategy VaR-Strategy MEAN STD MEAN STD MEAN STD ARCH -0.02676 1.06685 -0.01446 0.878 -0.01491 0.883 GARCH -0.02777 1.0824 -0.01573 0.894 -0.01621 0.897 EGARCH -0.02931 1.1239 -0.01762 0.909 -0.01811 0.914 HS -0.0307 1.1460 -0.01619 0.894 -0.01661 0.896

For the fixed weighted portfolio, the volatility-Strategy-weighted portfolio through the ARMA-

ARCH-t modelling has the highest return -0.01446% per day, while the second highest return (-

0.1491%) is generated by the VaR-Strategy-weighted portfolio under the ARMA-ARCH-t modelling.

The Return Strategy weighted portfolio ranked the third strategy that has the return (-0.02676% per

day) through the ARMA-ARCH-t modelling.

Also, the fixed weighted portfolio allocated by the volatility-Strategy has the smallest standard

deviation, which is 0.878 under the ARMA-ARCH-t method. The VaR-Strategy-weighted portfolio

has generated the second lowest standard deviation (0.883, very close to the lowest one), under the

ARMA-ARCH-t method. The Return Strategy weighted portfolio still ranked the third highest,

which is 1.0668, through the ARMA-ARCH-t method.

To summarize, the portfolio weighted according to the Volatility Strategy performs the best, and the

VaR-Strategy weights make the portfolio generate the second best, among the fixed weighted

portfolios. Also, the best allocation criteria are based on the ARMA-ARCH-t modelling. The

historical simulation method never contributes to the best performance.

Table 32: Equally weighted portfolio performance

Equally weighted return std

-0.01746 0.917

Comparing with the fixed weighted portfolio as shown above, in terms of the return and standard

deviation, the performance of the equally weighted portfolio is worse than the volatility-Strategy-

weighted and the VaR-Strategy-weighted portfolios but still better than the return-Strategy-weighted

portfolio.

5.3 Dynamic Weight Portfolio Construction

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Table 33: 1-Day dynamic weigh portfolio performance

1-day dynamic Return- Strategy Volatility-Strategy VaR-Strategy

MEAN STD MEAN STD MEAN STD ARCH -0.0299 1.1932 -0.00986 0.8683 -0.00993 0.8695

GARCH -0.02399 1.1960 -0.01006 0.8721 -0.00985 0.8719 EGARCH -0.067 2.7290 -0.01064 0.870131 -0.00955 0.8713

HS -0.1155 5.5315 -0.01569 0.882208 -0.01506 0.878

For the portfolios that are dynamically weighted in every day, the VaR- Strategy-weighted portfolio

through the ARMA-EGARCH-t method has the highest return -0.0096% per day, compared to the

returns from the Return-Strategy-weighted and the Volatility-Strategy-weighted portfolios. The

second highest portfolio weighted dynamically in everyday is generated from the ARMA-ARCH-t

method by the weights according to the Volatility Strategy, which is -0.0099% per day, very close to

the highest one. The Return Strategy weighted portfolio has ranked as the third strategy (-0.024% per

day) under the ARMA-GARCH-t method, among the everyday-dynamic-portfolios.

Furthermore, the standard deviation of the Volatility Strategy weighted portfolio under the ARMA-

ARCH-t method is the lowest (0.8683%), in the everyday-dynamic-portfolios. The daily-dynamic-

portfolio that is weighted in the VaR Strategy has ranked the second, which is 0.8695% through the

ARMA-ARCH-t method. The Return-Strategy has generated the third lowest standard deviation

(1.19%, based on the ARMA-ARCH-t method) for the everyday-dynamic-portfolio, compared with

VaR-Strategy and Volatility-Strategy.

In comparison, the performance of the equally weighted portfolio is worse than the Volatility

Strategy weighted and the VaR Strategy weighted portfolios but still better than the Return Strategy

weighted portfolio.

Table 34: 5-Day dynamic weigh portfolio performance

Return Strategy Volatility Strategy VaR Strategy

MEAN STD MEAN STD MEAN STD ARCH -0.04487 1.1666 -0.00958 0.8669 -0.01022 0.8755

GARCH -0.04323 1.1486 -0.00979 0.8690 -0.01031 0.8738 EGARCH -0.07881 1.4806 -0.0088 0.8687 -0.00982 0.8598

HS -0.76803 11.083 -0.02724 0.9754 -0.01581 0.8780

Comparing the performance of the portfolio weighted in every five days, the volatility-Strategy-

weighted portfolio (under the ARMA-EGARCH-t method) has the highest return (-0.0088%) but the

second lowest standard deviation (0.867% under the ARMA-ARCH-t). The VaR-Strategy-weighted

portfolio’s standard deviation is the lowest (0.86%). The portfolio weighted according to the Return

Strategy ranks the third in performance.

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In summary, the weights according to Volatility Strategy and VaR Strategy can perform well in the

dynamical portfolios. Also, the best allocation criteria are based on the GARCH-type parametric

models. The historical simulation method never contributes to the best performance.

Comparing with the equally weighted portfolio, the Volatility Strategy weighted and the VaR-

Strategy-weighted portfolios can perform better than the equally weighted portfolio, but the return-

weighted portfolio can never beat it.

5.4 Compare Dynamic and Fixed Portfolio Construction

Based on the tables and discussion above, under the Return Strategy, the equally weighted portfolio

generates the highest return and lowest standard deviation. On the contrary, the 5-day dynamic

weight portfolio ranks the last, with the lowest return under EGARCH model. However, the 5-day

dynamic weight portfolio provides the highest return and the lowest level of risk under the Volatility

Strategy. As for the VaR Strategy, the equally weighted portfolio performs the worst with the lowest

return and highest volatility. The 1-day dynamic weight portfolio generates the highest return under

EGARCH model and the 5-day dynamic weight portfolio provides the lowest standard deviation on

the basis of EGARCH model.

5.5 Utility Score for Portfolios

We have discussed the performance of constructed portfolios under each strategy in terms of average

return and average volatility. In order to assess risk and return together, we need to explore other

tools, one of which is the Utility Function. Bodie, Kane and Marcus (2011) presented a Utility Score

Function in terms of return and standard deviation to assess investment performance, which is

commonly used by financial theorists and CFA Institue. The Utility Score Function is presented as

following: 212

U r Aσ= − : r refers to the expected return of an asset and 2σ denotes the variance of

the asset. 12

is a scaling convention. A is risk aversion coefficient, which ranges from 1 to 6. 1

reflects least risk-averse while 6 reflects most risk averse.

Portfolios receive higher Utility Scores for more attractive risk-return profiles, thus, for Utility

Scores the bigger the better. They are calculated using decimal returns rather than percentage returns.

Hereby, we can link returns and risk collectively to performance via assessing the Utility Score for

each portfolio. This measure put different penalty on volatility, and a commonly used one is A=3.

Take A=3 for instance, the following table presents all Utility Scores for fixed weight portfolios

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under different allocation methods and model forecasts. Besides, we consider the Utility Score for

equally weighted portfolio.

Table 35: Utility Scores for fixed weight portfolios ( A =3)

Return Strategy Volatility Strategy VaR Strategy Equally Weighted ARMA-ARCH -0.00044 -0.00026 -0.00027

-0.00030 ARMA-GARCH -0.00045 -0.00028 -0.00028 ARMA-EGARCH -0.00048 -0.00030 -0.00031 HS(100) -0.00050 -0.00028 -0.00029

As we can see from the table, Return Strategy gives the worst performance in terms of Utility Score.

Volatility Strategy and VaR Strategy, which is slightly better than the former, are both better than

Return Strategy with regard to Utility. Regarding different models, ARMA-ARCH gives the best

Utility in general. The worst is HS(100) for Return Strategy, ARMA-EGACH for Volatility and VaR

strategy. Moreover, it is obvious that the Utility Scores of portfolios under Returns Strategy are

worse than that of equally-weighted portfolios, and that portfolios constructed via Volatility and VaR

Strategy are better than equally-weighted portfolios in terms of Utility.

As for dynamic portfolio allocations, we can also evaluate the portfolio utility performance. Again,

take A=3 for instance, the following tables present all the Utility Scores for the dynamic portfolios

under different allocation methods and model forecasts.

Table 36: Utility Scores for dynamic portfolios re-allocated every day ( A =3)



Table 37: Utility Scores for dynamic portfolios re-allocated every five days ( A =3)



For both portfolios re-allocated every day, Return Strategy gives the worst performance in terms of

Utility Score. Volatility Strategy and VaR Strategy, which is slightly better than the former, are both

better than Return Strategy with regard to Utility. Regarding different models, ARMA-ARCH gives

the best Utility while HS(100) is the worst for Return Strategy. As for Volatility and VaR Strategy,

all ARMA-ARCH type models gives similar results while HS(100) comes the worst. Moreover, the

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portfolios under Returns Strategy are worse than equally-weighted portfolio, and portfolios under

Volatility and VaR Strategy are better than equally-weighted portfolios in dynamic construction.

Last but not least, among all portfolios, portfolio under Volatility Strategy with ARMA-EGARCH

forecasts is the best while the portfolio under Return Strategy with HS(100) method is the worst in

terms of Utility.

We can also evaluate the Utility Scores of these three strategies and compare them to equally-

weighted portfolios with different risk-aversion via the following figure. The figure summarized

average Utility Scores of all the portfolios under each strategy. As we can see from the figure, with

different risk aversions, Utility Scores of Volatility Strategy, VaR Strategy and equally-weighted

portfolio are close to each other, while Utility Scores of Return Strategy is far below them. Besides,

if we see then graph closely, it can be found that VaR Strategy is the best, Volatility Strategy comes

the second, and they both are slightly better than the equally-weighted portfolios. Therefore, we can

draw a conclusion that: in our three portfolio strategies, VaR Strategy is the best, Volatility

Strategy comes second and Return Strategy is the worst; quantitative efforts are valuable here

as VaR and Volatility Strategy can beat equally weighted portfolio that is hard to beat in real

financial data.

Figure 25: average Utility Scores of all the portfolios under each strategy

0 1 2 3 4 5 6 7-20

-15

-10

-5

0Utility Score vs Risk Aversion

Risk%Aversion

Utility%Score

3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7

-1.5

-1.45

-1.4

Utility Score vs Risk Aversion

Risk%Aversion

Utility%Score

Equal Weight StrategyReturn StrategyVolatility StrategyVaR Strategy

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6. Conclusion

In conclusion, all the parametric models we have used have met the requirement of stationarity and

inevitibility. Among these parametric models, it seems that the ARMA models perform well to

catch the autocorrelation patterns in the time series data, except for Energy that has the remaining

autocorrelation effects in the residuals. The parametric volatility models have performed moderately

to capture the heteroskedasitc patterns. The ARCH models have not been able to characterize the

ARCH effects in the data of the all four assets. In comparison, GARCH has modelled volatilities

well for Financials and Consumer but could not perform well for Energy and Telecomm. EGARCH,

as an alternative to model volatilities, has been expected to present the leverage effects in the

equations. This could be useful to characterize impacts of the negative shocks on the volatilities,

especially in the period of the Global Financial Crisis. The normality has been met for Financials

and Consumer by using the Student-t distribution, but still not satisfied for Energy and Telecomm. It

has been expected that a fatter tailed distribution could be employed to capture the tail behaviours in

the models.

In terms of the forecast and accuracy measure for return, the ARMA-ARCH-type models work better

than the Historical Simulation in general. The HS-100 ranks the last under either RMSE or MAD

accuracy test method. Moreover, the HS-100 also provides the lowest portfolio return under the

Return Strategy. The ARMA-EGARCH model performs the best among the models for forecasting

the volatility; however, it does not provide the best investment outcome under the Volatility Strategy.

The ARMA-ARCH model has the best investment outcomes. Therefore, the methods that did the

best in terms of accuracy do not necessarily do the best in terms of profit or risk. The GARCH and

EGARCH model performs better in terms of accuracy under VaR, in addition, these two models also

generate higher returns under VaR Strategy. That is, the two models did better in both accuracy and

profit, which is favourable when doing investments. In terms of the portfolio returns and risks under

each strategy, the VaR Strategy performs the best, which is also the most conservative method. In the

context of disappointing market performance after the GFC, the VaR Strategy controls the loss the

best. In the contrast, the Return Strategy performs the worst. However, the Volatility Strategy and

the VaR Strategy perform better than equally weighted method. Therefore, the quantitative efforts

are worth doing.

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7. List of References

Bodie, Z., Kane, A., Marcu, A. J., 2011, Investments 9th ed. McGraw-Hill/Irwin, New York.

Historical prices, 2012, Yahoo Finance, USA, viewed 1 June 2012, <http://finance.yahoo.com.au/>

Jorion, P., 2007, Value at Risk, 3nd ed. McGraw-Hill, US.

McNeil, A., Fry, R. and Embrechts, P., 2005, Quantitative Risk Management:Concepts, Techniques and Tools, Princeton University Press, New Jersey.

Tsay, R. S., 2010, Analysis of Financial Time Series, 3rd ed, John Wiley & Sons, Inc., Hoboken, New Jersey.

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