Question 1 (Visualisation of time plots)

In the dataset ”Q1 2.csv”, there is one variable named as ”rt”.  Based on the variable ”rt”, complete the following questions,

(a) Obtain a line plot of rt , and comment on the plot;

(b) Obtain a line plot of rt(2), and comment on the plot;

(c) Obtain a correlogram of rt , and provide a comment on the ACF of the plot;

(d) Obtain a correlogram of rt(2), and provide a comment on the ACF of the plot;

(e) What test can be used as an alternative to check the results in  (c)? Use R to obtain the results, and provide the H0  and HA  and test statistic and conclusion.

(f) What test can be used as an alternative to check the results in  (d)? Use R to obtain the results, and provide the H0  and HA  and test statistic and conclusion.

Question 2 (Correlogram)

Based on your plot obtained in Question 1 (c), answer the following:

(a) Can we fit an ARMA-type model (eg, AR, MA or ARMA model) to rt ? Briefly explain;

(b) Is it reasonable to fit an AR(1) model? Briefly explain;

(c) Is it reasonable to fit an AR(2) model? Briefly explain;

(d) Is it reasonable to fit an MA(1) model? Briefly explain;

(e) Is it reasonable to fit an MA(2) model? Briefly explain;

(f) Is it reasonable to fit an ARMA(4,4) model? Briefly explain;

Question 3 (ADF test and (G)ARCH models)

(a) Suppose we have a time series yt  for t = 1, 2, ...,T, we wish to check whether yt is stationary. The R output for ADF test (unit root test) is provided below.  Please conduct an ADF test to answer whether yt  is stationary (make sure your answer contains proper H0   and HA , DF regression, test statistic, p-value and decision).

Suppose that we have the two (G)ARCH models as follows,

ARCH(2) :  σt(2)      =   ω + α 1 et(2)− 1 + α2 et(2)−2 ; GARCH(1, 1) :  σt(2)      =   ω + α 1 et(2)− 1 + β1 σt(2)− 1 ;

(1)

(2)

We have obtained and plotted 100-step ahead forecasts of σt  in the graph based on the two models;

(b) Which model is more likely to produce the plots of forecast for σt(2)  in the graph on the (a) (left panel)? Explain briefly.

(c) Which model is more likely to produce the plots of forecast for σt(2)  in the graph on the (b) (right panel)? Explain briefly.

Suppose that we have the two (G)ARCH models as follows,

GJR − GARCH(1, 1) :  σt(2)      =   ω + α 1 et(2)− 1 + β1 σt(2)− 1 + γet(2)− 1 It − 1 ;     (3) ARCH(3) :  σt(2)      =   ω + α 1 et(2)− 1 + α2 et(2)−2 + α2 et(2)−3 ;         (4)

We have obtained and plotted news impact curve (NIC) in the graph based on the two models;

(d) Which model is more likely to produce the plot on the left panel? Explain briefly.

(e) Which model is more likely to produce the plot on the right panel? Explain briefly.

Question 4 (Mean and variance)

Let rt  denotes the continuously compounded returns of a financial asset at time t. If rt  follows an ARMA(1,2) model,

rt  = ϕ0 + ϕ1 rt − 1 + et + θ1 et − 1 + θ2 et −2 ,

(a) Derive the unconditional mean of rt , E(rt ) (show all necessary steps and conditions).

(b) For given information available at time t, derive the 1-step, 2-step and 3-step ahead forecasts of rt  (show all necessary steps and conditions).

(c) If we estimate the ARMA(1,2) model and obtain ϕ0  = 0.2, ϕ 1  = 0.4, θ 1  = −0.3 and θ2  = 0.45, compute the E(rt=3|It=2), E(rt=4|It=2) and E(rt=5|It=2) based on the information provided in the Table 1.

Table 1: Monthly returns

t           rt         et

1         3.0    0.4

2        -0.9    0.6

3            ?       -

4            ?       -

5            ?       -

Consider the following ARMA(2,1)-ARCH(2) model,

rt      =   ϕ0 + ϕ1 rt − 1 + ϕ2 rt −2 + et + θ1 et − 1 ,

et      =   σt zt ,

σt(2)      =   ω + α 1 et(2)− 1 + α2 et(2)−2 .

(d) For given information available at time t, derive the 1-step, 2-step and 3- step ahead forecasts of variance of rt  (show all necessary steps and conditions).

(e) If we estimate the model, we obtain ϕ0   = 0.2, ϕ 1   = 0.3, ϕ2   = −0.2, θ 1  = 0.25, ω = 0.15, α 1  = 0.43 and α2  = 0.18.  Compute the var(rt=4|It=3), var(rt=5|It=3) and var(rt=6|It=3) based on the information provided in the Table 2.

Table 2: Monthly returns

Consider the following GARCH(2,2) model,

rt      =   µ + et ,

et      =   σt zt ,

σt(2)      =   ω + α 1 et(2)− 1 + α2 et(2)−2 + β1 σt(2)− 1 + β2 σt(2)−2 ,

(f) For given information available at time t, derive the 1-step, 2-step and 3- step ahead forecasts of variance of rt  (show all necessary steps and conditions).

(g) If we estimate the model, we obtain µ  = 0.35, ω  = 0.15, α 1   = 0.2, α2  = 0.1, β 1  = 0.55 and β2  = 0.25. Compute the var(rt=4|It=3), var(rt=5|It=3) and var(rt=6|It=3) based on the information provided in the Table 3.

(h) Comment on the estimated ARCH(2) model in (b) and GARCH(1,2) model in (d), is there any issue with the two estimated models? Briefly explain.

Table 3: Monthly returns

Question 5 (Value at Risk)

The dataset ’Q5.csv’contains adjusted closing price of Nasdaq index.  You are employed as an analyst by an investment bank in Melbourne. Assume that you invested a certain amount of Nasdaq index with US$800,000 for the bank.

(a) (Historical approach) According to daily continuously compounded re- turns of the index, what is the value at risk (VaR) for the bank’s holding of the index during the next 24 hours at the 95% level of confidence?

(b) (Parametric approach) What is the one-day VaR for the bank’s holding of the index at the 95% level of confidence? Assume that the daily return follows the normal distribution.

(c) (Parametric approach) What is the one-week (5 trading days) VaR for the investor’s holding of the index at the 99% level of confidence? Assume that the daily return follows the Student’s t distribution.

We fit a GARCH(1,1) model,

rt      =   µ + et ,

σt(2)      =   ω + α 1 et(2)− 1 + β1 σt(2)− 1 ,

where zt  are independent and identically distributed as a Student’s t distri- bution.

(e) (Conditional VaR) If we estimate ω = 0.00008, α 1  = 0.00098 and β 1  = 0.97.  Given et   = 0.0078 and σt   = 0.0038, calculate the one-day conditional value-at-risk for the bank’s holding of the index at the 99% confidence level (You are not required to use R to estimate the model, you can calculate the 1-step ahead forecast σt+1  by hand based on the available information).

Assume that the daily return of the index follows the Student’s t distri- bution. Given the RiskMetrics model,

rt      =   µ + et ,

et      =   σt zt ,

σt(2)      =   (1 − λ)rt(2)− 1 + λσt(2)− 1 ,

where λ = 0.94; and zt  are independent and identically distributed as the Stu- dent t distribution with 6 degrees of freedom.

(f)  (Riskmetrics)  Calculate the one-day  conditional value-at-risk for the bank’s holding of the index at the 99% confidence level.

(g) Is the estimation of VaR based on the GARCH(1,1) and Riskmetrics models reasonable? Briefly explain;