Model Construction & Forecast Based Portfolio Allocation:

Transcription

1 QBUS6830 Financial Time Series and Forecasting Model Construction & Forecast Based Portfolio Allocation: Is Quantitative Method Worth It? Members: Bowei Li (303083) Wenjian Xu ( ) Xiaoyun Lu ( ) 0

2 Executive Summary The aim of this report is to compare four forecasting models: ARMA-ARCH, ARMA-GARCH, ARMA-EGARCH and Historical Simulation of last 00 days (HS-00) 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.

3 Table&of&Contents&. Introduction Exploratory Data Analysis Models for Forecasting Motivations of Model Selection Model Selection ARMA-ARCH-t Model ARMA-GARCH-t Model ARMA-EGARCH-t Historical Simulation Model Construction for the Other Industries Forecast and Accuracy Measures Return Forecast and Accuracy One-Step-Ahead Return Forecast Multi-period Return Forecast Volatility Forecast and Accuracy One-Step-Ahead Volatility Forecast Multi-period Volatility Forecast VaR Forecast and Accuracy One-step ahead forecast of VaR Multi-step ahead forecast of VaR Optimal Portfolio Allocation Portfolio Allocation Methods Fixed Weight Portfolio Construction Dynamic Weight Portfolio Construction Utility Score for Portfolios Conclusion List of References

4 . 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 (202). The daily returns of the sector portfolios are employed in this report with a time span of 0 years, constituting a sample size of The in sample period is from June 2002 to 3 December 2009 with a size of 880, 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 (200). The log return has the attractive attribute of additive. The Figure 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 370th 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 -2.6%, 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. 3

5 Table : Summary statistics for each asset Figure : Portfolio Returns (In-sample period) Mean Median Std Min Max Skewness Kurtosis Energy Financials Telecomm Consumer 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.3%. Energy had the highest volatility of.6, which has been observed in the plot above. The Consumer industry has the lowest volatility of.04%. Overall, the daily returns range between % and 9.2%, 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

6 3. Models for Forecasting 3. 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 ARMA-ARCH-t Model As for the mean equation, the AIC find the appropriate orders of AMRA (3, 3) while SIC favours ARMA (, ). We will trust SIC and choose ARMA (, ). Again, AIC and SIC are used to find the appropriate orders for ARCH (p): 5

7 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 (, )-ARCH (9) -t model: r = r a + a a = σε ε ~ t (0,) * t t t t t t t t 8.04 (s.e) (0.027) (0.48) (0.48) σ = a a a a a t t t 2 t 3 t 4 t 5 (s.e) (0.029) (0.039) (0.033) (0.028) (0.033) (0.028) a a + 0.2a + 0.3a t 6 t 7 t 8 t 9 (0.029) (0.030) (0.038) (0.037) The absolute value of AR parameter is less than, so the AR () is stationary. Moreover, the absolute value of MA parameter is less than as well; therefore, the MA () 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 5 th parameter in the ARCH. The volatility persistence is estimated 0.866, which is less than, so the stationarity requirement is met. The low degree of freedom estimates (8.04) indicates much fatter tails than a Gaussian. The standardized residuals aˆ t little remaining auto-correlation or obvious heteroskedasticity. / ˆ σ, are plotted below (Figure 4). According to the ACF, it shows t 6

8 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 (, ). The ARMA actually helps here, though their parameter estimates are insignificant. LB tests: H0: ρ = ρ2 =... = ρm = 0 HA:at least one of ρi 0 ( i=,2... m) Table 2: LB test results (residuals) for ARMA-ARCH-t m=7, d.f=5 m=22, d.f=0 m=27,d.f=5 p-value The histogram of these transformed standardized residuals (see Figure 5) appears slightly more fattailed than a normal distribution, with two large outliers around 3.5 and 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: H0 :skewness=0 AND kurtosis=3 H A :the distribution is not Normal 7

9 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 and kurtosis of 2.99, which seem very close to the Gaussian distribution (0 and 3). The ACF for the squared standardised residuals aˆ / ˆ σ are plotted below in Figure 6. It displays 2 2 t t some significant correlations at the 8 th lag and 6 th 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=7, d.f=5 m=22, d.f=0 m=27,d.f=5 p-value The p-value of the LB test indicates strongly significant remaining ARCH effects in the squared standardized residuals. Therefore, the ARMA (, )-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 (, )-ARCH (6)-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=0 m=34,d.f=5 p-value 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 (, )-ARCH (6)-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 (, )-ARCH (9)-t as chosen by SIC. 8

10 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 (, )-GARCH (, ) with t-distribution as suggested by SIC. r = r a + a a = σε ε ~ t (0,) * t t t t t t t t 8.89 (s.e)(0.026) (0.4) (0.4) σ = a σ t t t (s.e) (0.0026) (0.02) (0.0) The absolute value of AR parameter is less than, so the AR () is stationary. Moreover, the absolute value of MA parameter is less than as well; therefore, the MA () 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 = 0.995, very high and close to, 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). The standardized residuals aˆ Figure 7: Returns and its standard deviation remaining auto-correlation or obvious heteroskedasticity. t / ˆ σ, are plotted below (See Figure 8). The plots appear to show little t 9

11 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 (, ) even though their parameter estimates are insignificant. Table 5: LB test results (residuals) for ARMA-GARCH-t LB test: residuals m=0, d.f=5 m=5, d.f=0 p-value 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: and 3.00, 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 0

12 reject the null hypothesis of Gaussian residuals. The model fits the data very well in the tails of the distribution. Figure 0: ACF for squared standardised residuals The squared transformed standardised residuals seem to display significant autocorrelation at the 3 th and the 6 th 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(,)-GARCH(,)-t model captures the volatility dynamics reasonably well. Table 6: LB test results (squared residuals) for ARMA-GARCH-t m=0, d.f=5 m=5, d.f=0 p-value In summary, the ARMA (, )-GARCH (, )-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 ARMA-EGARCH-t In terms of the EGARCH model, we choose to fit the ARMA (, )-EGARCH (, )-t: r = r + 0. a + a a = σε ε ~ t (0,) * t t t t t t t t 9.40 (s.e)(0.023) (0.402) (0.406) log( σ ) = log( σ ) + 0.5( ε E( t )) 0.05ε 2 2 * t t t t- (s.e) (0.003) (0.004) (0.023) (0.04) The absolute value of AR parameter is less than, so the AR () is stationary. Moreover, the absolute value of MA parameter is less than as well; therefore, the MA () 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 <, the stationarity requirement has

13 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. The level of asymmetry is estimated as On average, the negative shocks of around 2 standard deviations have a 22.73% higher volatility than positive shocks of the same size. Figure : New information curve Figure 2: 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 20 th to the 600 th observation). During the high volatility period (around the 600 th 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 600 th observation), as its volatility persistence is the highest among the three models. 2

14 3.2.4 Historical Simulation The final model applied is a non-parametric model, which is Historical Simulation with data from the last 00 days (HS-00), the equations are shown below: r r s r r ( ) t = t i σ t = (00) = t i (00) 00 i= 00 i= The sample mean of last 00 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 3 and Figure Return HS-00 return Figure 3: Compare daily return and 00-day mean return HS Figure 4: 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. 3

15 Energy: ARMA(,)-ARCH()-t r = r 0.94 a + a a = σε ε ~ t (0,) * t t t t t t t t 8.54 (s.e)(0.0064) (0.055) (0.048) σ = a a + 0.6a a a t t t 2 t 3 t 4 t 5 (s.e) (0.074) ( 0.030) (0.029) (0.040) (0.034) (0.029) a a a a + 0.2a a t 6 t 7 t 8 t 9 t 0 t (0.038) (0.033) (0.025) (0.030) (0.038) (0.032) Energy: ARMA(,)-GARCH(,)-t r = r a + a a = σε ε ~ t (0,) * t t t t t t t t 9.0 (s.e)(0.072) (0.47) (0.48) σ = a σ t t t (s.e) (0.005) (0.009) (0.0092) Energy: ARMA(,)-EGARCH(,0)-t r = r a + a a = σε ε ~ t (0,) * t t t t t t t t 9.3 (s.e)(0.009) (0.093) (0.085) log( σ ) = log( σ ) + 0.2( ε E( t )) 0.02ε 2 2 * t t t- 9.3 t- (s.e) (0.003) (0.003) (0.02) (0.0) Financials: ARMA(2,)-ARCH(8)-t r = r 0.0r 0.76 a + a a = σε ε ~ t (0,) * t t t 2 t t t t t t 8.3 (s.e) (0.028) (0.54) (0.026) (0.54) Financials: ARMA(2,)-GARCH(,)-t r σ = a r+ 0.3a r+ 0.3a + a0.aa a a + 0.3at + 0.2a + 0.3a (s.e)(0.026) (0.025) (0.033) (0.47) (0.036) (0.027)(0.035) (0.47)(0.035) (0.033) (0.04) (0.039) (0.038) * 2 2 t t = t t 0.02 t 2 t t 3 t + t t4 t= σε tt 5t εt ~ t (0,) t 7 t 8 σ = a σ t t t (s.e) (0.002) (0.0) (0.02) 4

16 Financials: ARMA(2,)-EGARCH(,0)-t r = r r a + a a = σε ε ~ t (0,) * t t t 2 t t t t t t 0.8 (s.e)(0.028) (0.607) (0.026) (0.607) * log( σ ) = log( σ ) ( ε E( t )) 0.44ε 2 2 t t t- 0.8 t- (s.e) (0.005) (0.005) (0.026) (0.06) Telecomm: ARMA(2,)-ARCH(7)-t r = r 0.039r a + a a = σε ε ~ t (0,) * t t t 2 t t t t t t 6.2 (s.e)(0.06) (0.392) (0.027) (0.392) σ = a + 0.4a a a + 0.a t t t 2 t 3 t a a t 5 t 6 t 7 (s.e) (0.068) (0.047) (0.039) (0.039) (0.03) (0.038) (0.032) (0.036) Telecomm: ARMA(2,)-GARCH(,)-t r = r 0.047r a + a a = σε ε ~ t (0,) * t t t 2 t t t t t t 6.38 (s.e)(0.07) (0.45) (0.023) (0.45) σ = a σ t t t (s.e) (0.006) (0.0) (0.0) Telecomm: ARMA(2,)-EGARCH(,0)-t r = r 0.043r a + a a = σε ε ~ t (0,) * t t t 2 t t t t t t 6.65 (s.e)(0.05) (0.44) (0.023) (0.44) log( σ ) = log( σ ) ( ε E( t )) 0.026ε 2 2 * t t t t- (s.e) (0.003) (0.004) (0.022) (0.04) 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. So, the mean equations modelled in ARMA are stationary and invertible. In the volatility models, there are several insignificant parameters, such as the 5 th ARCH parameter for Energy. And, the absolute values of the all the parameters in each volatility equation are less than, which means the ARCH and GARCH models are stationary. The EGARCH models are also stationary, as the β are less than. On the other hand, some models volatility persistence can be 5

17 very close to. 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: H0: ρ = ρ2 =... = ρm = 0 HA:at least one of ρi 0 ( i=,2... m) JB tests: H0 :skewness=0 AND kurtosis=3 H A :the distribution is not Normal 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(,)-ARCH()-t for Energy ARMA(2,)-ARCH(8)- t for Financials ARMA(2,)-ARCH(7)- t for Telecomm p-value of LB test: 0.07 (m=9, d.f=5) 0.08 (m=7, d.f=5) 0.06 (m=6, d.f=5) residuals 0.08 (m=24, d.f=0) 0.00 (m=22, d.f=0) 0.70 (m=2, d.f=0) p-value of JB test p-value of LB test: squared residuals (m=9, d.f=5) (m=7, d.f=5) (m=6, d.f=5) (m=24, d.f=0) (m=22, d.f=0) (m=2, d.f=0) 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 (, )-ARCH ()-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. 6

18 Table 8: Diagnostic Test results after Re-fitting ARMA-ARCH-t model ARMA(5,5)-ARCH(6)- t for Energy ARMA(2,)-ARCH(6)- t for Financials ARMA(2,)-ARCH(6)- t for Telecomm p-value of LB (m=32, d.f=5) (m=25, d.f=5) (m=25, d.f=5) test: residuals 0.00 (m=37, d.f=0) (m=30, d.f=0) (m=30, d.f=0) p-value of JB test p-value of LB (m=32, d.f=5) (m=25, d.f=5)) 0.00 (m=25, d.f=5) test: squared residuals 0.002(m=37, d.f=0) (m=30, d.f=0) (m=30, d.f=0) 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(,)- ARMA(2,)- GARCH(,)-t for GARCH(,)-t for Energy Financials ARMA(2,)- GARCH(,)-t for Telecomm p-value of LB test: (m=0, d.f=5) (m=, d.f=5) (m=, d.f=5) residuals (m=5, d.f=0) (m=6, d.f=0) (m=6, d.f=0) p-value of JB test p-value of LB test: squared residuals (m=0, d.f=5) (m=, d.f=5) (m=, d.f=5) 0.07 (m=5, d.f=0) (m=6, d.f=0) (m=6, d.f=0) 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. 7

19 Therefore, we may try refining the models of Energy and Telecomm with higher orders in GARCH to capture adequate lagged volatilities. Table 0: Diagnostic Test results after Re-fitting ARMA-GARCH-t model ARMA(,)-GARCH(5,5)-t ARMA(2,)-ARCH(5,5)-t for Energy for Telecomm p-value of LB test: residuals (m=8, d.f=5) (m=9, d.f=5) 0.20 (m=23, d.f=0) 0.80 (m=24, d.f=0) p-value of JB test p-value of LB test: (m=8, d.f=5) (m=9, d.f=5) squared residuals 0.04(m=23, d.f=0) (m=24, d.f=0) 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: Energy: r r s Et ; = Et ; i σ Et ; = E(00) 00 i= Financials: r r s Ft ; = Ft ; i σ Ft ; = F(00) 00 i= Telecomm: r r s Tt ; = Tt ; i σtt ; = T(00) 00 i= 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. 8

20 4. Return Forecast and Accuracy 4.. 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 Energy Forecast Period Return ARMA-ARCH ARMA-GARCH ARMA-EGARCH HS(00) Figure 5 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 6: First 25 one-step-ahead Forecasted Energy returns under four models versus actual returns Figure 6 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. 9

21 Table: Accuracy Measures for one-step-ahead Forecasted daily Energy returns under four models ARMA(,)- ARMA(,)- ARMA(,)- HS-00 ARCH()-t GARCH(,)-t EGARCH(,)-t RMSE MAD The units of RMSE and MAD are the same units as percentage returns. The typical errors made are between.06% and.4% 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-00 ranks last under MAD while ARMA-GARCH-t ranks last under RMSE. As RMSE is sensitive to outliers, MAD is more trustworthy. 6 Energy 6 Financials Telecomms 3 Cons Staples Figure 7: 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-00 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.4% 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-00 ranks last under both MAD and RMSE for all portfolios. 20

22 Table 2: Accuracy Measures for one-step-ahead Forecasted daily Portfolio returns under four models ARMA- ARMA- ARMA- HS-00 ARCH GARCH EGARCH Energy RMSE MAD Financials RMSE MAD Telecomm RMSE MAD Consumer RMSE MAD 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. 0.5 Multi-period Forecasts under 4 Models Percentage Return ARMA-ARCH ARMA-GARCH ARMA-EGARCH HS(00) Forecast Period Figure 8: 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.4% in terms of percentage returns. These errors are large and similar to the errors made from one-step-ahead forecasts. 2

23 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-00 method. For Consumer, the best method is ARMA-EGARCH-t and the worst is HS method. Table 3: 607-step-ahead Forecasted portfolio returns under four models ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS(00) Energy RMSE MAD Financials RMSE MAD Telecomm RMSE MAD Consumer RMSE MAD Volatility Forecast and Accuracy 4.2. 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 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 is depicted in figure9. The Proxy 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 for most of times. Therefore, the true volatility process should also sit on top of the absolute return shocks. The HS-00 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 () recovers after exactly days. The GARCH and EGARCH are on top of each other on most days. 22

24 6 5 4 Energy Proxy volatility ARMR-ARCH ARMR-GARCH ARMR-EARCH HS(00) Figure 9: Forecasted Portfolios volatilities under four models and Proxy 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. 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 20, 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 Energy Proxy 2 volatility ARMA-ARCH ARMA-GARCH ARMA-EGARCH HS(200) 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. Figure2 shows Proxy 3 versus volatility forecasts is presented below. 23

25 Energy Proxy 3 Volatility ARMA-ARCH ARMA-GARCH ARMA-EGARCH HS(00) Figure 2: 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 4: RMSE for one-step-ahead forecasted daily Energy volatilities under four models ARMA(,)-ARCH()-t ARMA(,)-GARCH(,)-t ARMA(,)-EGARCH(,0)-t HS-00 Proxy Proxy Proxy Table 5: MAD for one-step-ahead forecasted daily Energy volatilities under four models ARMA(,)-ARCH()-t ARMA(,)-GARCH(,)-t ARMA(,)-EGARCH(,0)-t HS-00 Proxy Proxy Proxy 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, the best method is the ARMA-EGARCH model. For Proxy 2 and 3, the best method is the ARMA-GARCH. The HS-00 ranks last for all proxies. Table 6: RMSE for one-step-ahead forecasted daily Portfolio volatilities under four models ARMA- ARMA- ARMA- HS(00) ARCH-t GARCH-t EGARCH-t Energy Proxy Proxy Proxy Financials Proxy Proxy Proxy Telecomm Proxy Proxy Proxy Consumer Proxy Proxy Proxy

26 Table 7: MAD for one-step-ahead forecasted daily Portfolio volatilities under four models ARMA- ARMA- ARMA- HS(00) ARCH-t GARCH-t EGARCH-t Energy Proxy Proxy Proxy Financials Proxy Proxy Proxy Telecomm Proxy Proxy Proxy Consumer Proxy Proxy Proxy 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 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 Multi-period Volatility Forecasts under 4 Models ARMA-ARCH ARMA-GARCH ARMA-EGARCH HS(00) Figure 22: 607-step-ahead Energy Volatility Forecasts under four models 25

27 Table 8: RMSE for 607-step-ahead forecasted daily Portfolio volatilities under four models ARMA- ARMA- ARMA- HS-00 ARCH-t GARCH-t EGARCH-t Energy Proxy Proxy Proxy Financials Proxy Proxy Proxy Telecomm Proxy Proxy Proxy Consumer Proxy Proxy Proxy Table 9: MAD for 607-step-ahead forecasted daily Portfolio volatilities under four models ARMA- ARMA- ARMA- HS-00 ARCH-t GARCH-t EGARCH-t Energy Proxy Proxy Proxy Financials Proxy Proxy Proxy Telecomm Proxy Proxy Proxy Consumer Proxy Proxy Proxy All portfolios RMSE and MAD for volatility forecast are presented in the above tables. The errors made are ranged from 0.26% to.8% 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. One-step ahead forecast of VaR 26

28 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-00 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 (-step) Energy ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value

29 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 In addition, the ARCH is the only model over-estimates the risk level, with less violations and violation rates significantly less than 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 2: Accuracy Test for Financials (-step) Financials ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value Table 22: Accuracy Test for Telecomm (-step) Telecomm ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value Table 23: Accuracy Test for Consumer (-step) Consumer ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value

30 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 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 29

31 According to Figure 23 and 24, the ARMA-EGARCH-t model and HS-00 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-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value 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-00 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-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value Table 26: Accuracy Test for Telecomm (multi-step) Telecomm ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value

32 Table 27: Accuracy Test for Consumer (multi-step) Consumer ARMA-ARCH-t ARMA-GARCH-t ARMA-EGARCH-t HS-00 Number of Violations ! !/! Confidence Interval (0.0327,0.0673) Independence Test DQ Test Loss Function Value 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-00 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. 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 200). Under this strategy, higher portfolio weights are allocated on asset with lower forecasted VaR which is one 3