CITY UNIVERSITY OF HONG KONG
Course code & title
:
MS6217
Statistical Modelling in Finance
Session
:
Semester B,
2004-2005
Time allowed
:
3 hours
This paper has 13 pages (including this page)
Instructions to candidates:
1.
2.
Answer ALL FOUR questions
Show sufficient work for each question
Materials, aids and instruments permitted during examination:
Approved calculator
1
Question 1 (25 marks)
Consider the following transfer function model,
Yt    v( B) X t  N t
(1.1)
where v( B)  vo  v1 B  v 2 B 2  ....... , and B is the backward shift operator.
a)
An equivalent way of representing (1.1) is
Yt    w( B) X t b /  ( B)   q ( B) Q ( B L )at /  p ( B) P ( B L )
.
(1.2)
Explain the purposes of w(B),  (B) and b in (1.2). Explain also why (1.2) is an
equivalent way of representing (1.1).
(5 marks)
b)
To examine the relationship between the monthly forward rates and spot rates of
the Australian dollar from May 1991 to April 1997, a transfer function model is
specified with { f t }, the first difference of the 90-day forward rate measured as
U.S. cents per Australian dollar, as the input series; and {st }, the first difference
of the spot rate, as the output series. Both { f t } and {st } are stationary.
i)
Describe step by step how you would “prewhiten” { f t } and {st } in the
transfer function model. Are the prewhitened series necessarily white
noise?
(5 marks)
ii)
Suppose that the sample cross-correlations between the prewhitened
{ f t } and prewhitened {st } are given by,
r fs (5)  0.001 ;
rfs(-4) = -0.102;
rfs(-3) = 0.004;
rfs(-2) = 0.000;
rfs(-1) = 0.024;
rfs(0) = -0.113;
r fs (1)  0.012 ;
rfs(2) = 0.130;
rfs(3) = 0.516;
rfs(4) = 0.202;
rfs(5) = 0.193;
rfs(6) = 0.101;
The standard deviation of r fs (i ) and rsf (i ) is approximately (n  i) 1 / 2 ,
where n is the number of observations in the sample.
Discuss how you would identify a tentative transfer function model on the
basis of the above results.
(5 marks)
2
iii)
Suppose that the model identified in ii) is estimated and the Ljung-BoxPierce test for the significance of the residuals are:
Q * (6)  31.2
Q (12)  39.5
*
Q * (18)  42.4
with p-value = 0.0
with p-value = 0.0
with p-value = 0.0
So what do you conclude? Discuss how you would proceed to specify a
final transfer function model.
(3 marks)
iv)
In practice, what other diagnostic tests are necessary to examine the
validity of the model?
(5 marks)
v)
Discuss how you would test if there exists significant arbitraging
opportunities in the Australian foreign exchange market using the
estimated transfer function model.
(2 marks)
3
Question 2 (15 marks)
a)
Briefly discuss the motivation behind log transformation of seasonal time series.
Figures 2a and b show, respectively, the earnings per share and log earnings per
share of a listed company in the NYSE from 1981Q1 to 2001Q4. Discuss if the
objectives of log transformations have been achieved.
(2 marks)
Fig.2a: Earnings per share
18
16
14
earnings
12
10
8
6
4
2
0
0
10
20
30
40
50
60
70
80
t
Fig 2b. Log earnings per share
3
2.5
2
log earnings
1.5
1
0.5
0
0
10
20
30
40
50
60
70
80
-0.5
-1
t
b)
Denote the log earnings by xt . Upon examining the ACF’s and PACF’s of xt
and its first regular and seasonal differences, the investigator estimates the
models:
i)
(1  B 4 )(1  B) xt  (1  B)at ;
4
ii) (1  B 4 )(1  B) xt  (1  B)(1  B 4 )at ;
where at ~ i.i.d .(0,  2 ) . Give expressions of these models in terms of the
notation ARIMA(p,d,q)(P,D,Q)L.
(2 marks)
c)
Show that for model ii), the autocorrelation coefficient of the differenced series at


lag 3 is given by  3   / (1   2 )(1   2 ) , while for model i),  3  0.
(3 marks)
d)
Results on the estimation of the two models are given as follows. Compare the
performance of the models using the in-sample diagnostics.
(5 marks)
Estimation Results of Model i)
Maximum Likelihood Estimation
Parameter
Estimate
Standard
Error
Approx
t Value
Pr > |t|
Lag
MA1,1
0.76654
0.07655
10.01
<.0001
1
Variance Estimate
Std Error Estimate
AIC
SBC
Number of Residuals
0.008836
0.093999
-147.514
-145.145
79
Autocorrelation Check of Residuals
To
Lag
6
12
18
24
ChiSquare
10.80
19.14
22.02
24.79
DF
Pr >
ChiSq
--------------------Autocorrelations--------------------
5
11
17
23
0.0555
0.0587
0.1838
0.3614
0.169 0.148 -0.119 -0.251 -0.031 -0.014
0.254 0.050 -0.030 0.018 -0.135 0.076
-0.075 0.095 -0.060 -0.065 -0.001 -0.080
0.081 -0.008 0.050 -0.028 -0.119 -0.023
Estimation Results of Model ii)
Maximum Likelihood Estimation
Parameter
Estimate
Standard
Error
t Value
Approx
Pr > |t|
Lag
MA1,1
MA2,1
0.68089
0.31449
0.09049
0.11227
7.52
2.80
<.0001
0.0051
1
4
5
Variance Estimate
Std Error Estimate
AIC
SBC
Number of Residuals
0.008137
0.090203
-152.753
-148.014
79
Autocorrelation Check of Residuals
To
Lag
6
12
18
24
e)
ChiSquare
3.38
10.17
13.70
16.39
DF
Pr >
ChiSq
--------------------Autocorrelations--------------------
4
10
16
22
0.4960
0.4256
0.6212
0.7961
0.042 0.150 -0.114
0.187 0.070 -0.089
-0.105 0.114 -0.084
0.025 -0.008 0.070
0.007 -0.053
0.050 -0.121
-0.032 -0.000
0.004 -0.132
0.021
0.097
-0.057
0.027
Obtain forecast of earnings per share for period 85 (i.e., 2002Q1) using the
preferred model chosen in part d) and the following information:
Obs.
80
81
82
83
84
xt
2.3016
2.7850
2.6858
2.7738
2.4519
x̂t (predicted value of xt )
2.4816
2.7855
2.8240
2.7707
2.6465
xt  xˆt
-0.18
-0.0005
-0.1382
0.0031
-0.1946
(3 marks)
6
Question 3 (37 marks)
a)
Your financial advisor has suggested to you a protective put strategy on your
investment: buy shares in a market stock fund and purchase put options on those
shares with three month maturity and exercise price of $1040. The stock fund is
currently at $1200. However, your professor at university has suggested instead
that you buy a three-month call option on the index fund with exercise price of
$1120 and buy three-month T-bills with face value of $1120.
i)
ii)
iii)
In a table and on a graph, illustrate the payoffs to each of these strategies as a
function of the stock fund value in three months.
(6 marks)
Which portfolio would you think must require a greater initial outlay to
establish?
(2 marks)
Suppose the market prices of the securities are as follows:
Stock fund
T-bill (face value $1120)
Call (exercise price $1120)
Put (exercise price $1040)
b)
$1200
$1080
$160
$8
iv)
Make a table of profits realized for each portfolio for the following values of
the stock price in 3 months:
ST = 0, 1040, 1120, 1200 and 1280. Graph
the profits to each portfolio as a function of ST on a single graph.
(9 marks)
Which strategy is riskier?
(2 marks)
i)
Use the Black-Scholes formula ( C  So e  T P( Z  d1 )  Xe  rT P ( Z  d 2 ) ),
where d1 
ln( So / X )  (r     2 / 2)T
 T
and d2  d1   T to find the value
of a European style call option on the following stock:
Stock price ( S o = 100);
Annual interest rate (r = 0.10);
Time to expiry (T = 3 months);
ii)
c)
Exercise price (X = 95);
Dividend yield (  = 0);
Standard deviation (   0.5 )
(4 marks)
Suppose that the standard deviation on the stock increases. Will the option be
worth more or less with the higher volatility?
(2 marks)
In order to better understand the dependence of a security on volatility, an
investigator estimates a GARCH(1,1) model for the monthly excess returns of the
S&P 500 index ( y t ) from January 1926 to December 1991. Results of estimation
using SAS are given as follows:
GARCH Estimates
7
SSE
MSE
Log Likelihood
SBC
Normality Test
2.70454693
0.00341
1269.46195
-2512.2257
95.0061
Observations
Uncond Var
Total R-Square
AIC
Pr > ChiSq
Variable
DF
Estimate
Standard
Error
Intercept
ARCH0
ARCH1
GARCH1
1
1
1
0.007453
0.0000818
0.1203
0.8545
0.001547
0.0000238
0.0197
0.0189
1
792
???
.
-2530.9239
<.0001
t Value
4.82
3.44
6.12
45.15
Approx
Pr > |t|
<.0001
0.0006
<.0001
<.0001
i)
Write down the estimated GARCH process. What assumptions on the
parameters are necessary in order for the process to be valid?
(5 marks)
ii)
Obtain the unconditional variance of the process.
iii)
What other diagnostic tests would you use to examine the adequacy of the
model?
(3 marks)
8
(4 marks)
Question 4 (23 marks)
To examine the international transmission of bond market movements, an investigator
considers 960 daily close-of-trade observations from April 1986 to December 1989 on
the yields of government bonds with less than five years to maturity for the bond markets
of the U.K. and U.S.
a)
Using the Augmented Dickey Fuller (ADF) test in conjunction with the Dolado’s
sequential testing procedure discussed in class, test for the unit root hypothesis for
each of the bond yield series at the 10% level of significance. Some of the
following information may be useful:
Test Statistic

U

U
UK
-1.14
-1.01
US
-1.21
-1.73
4.01
6.17
3.92
2.22
where  is the test statistic for testing H o :   0 vs. H1 :   0 in models without
a linear trend;
 U is the test statistic for testing H o :   0 vs. H1 :   0 in models with a linear
trend;
 is the test statistic for testing H o :     0 vs. H1 :   0,  0 in models
without a linear trend;
U is the test statistic for testing H o :     0 vs. H1 :   0,   0 in models
with a linear trend; and
the regression used for testing is:
p
xt    t  xt 1 
  i xt i   t
(8 marks)
i 1
b)
Discuss (without having to perform the testing) how the investigator could test if
the bond “returns” (as opposed to bond yields) are stationary?
(2 marks)
c)
Suppose that each of the bond yield series is I(1). The investigator then tests if
there is a long run relationship between the series using the bivariate
Cointegrating Regression Augmented Dickey Fuller (CRADF) test. The results
are summarized as follows:
CRADF
UK/US
US/UK
p = augmentation
level
0
0
-4.27
-3.98
9
Conduct the test at the 10% level of significance. What do you conclude?
(2 marks)
d)
The investigator then constructs and estimates the following Error Correction
model:
UKt   01  1 zt 1 
USt   02   2 zt*1 
q
m
i UKt i   j USt  j  1t
i 1
j 1
m*
q*
i*UKt i   *j USt  j   2t
i 1
i)
ii)
iii)
e)
(5.1)
(5.2)
j 1
What is the purpose of including the error correction terms in (5.1) and
(5.2)?
(2 marks)
What are the expected signs of the coefficients 1 and  2 ? Carefully
explain your answers.
(2 marks)
Suggest how the investigator could determine the appropriate number of
lagged terms in the Error Correction model.
(2 marks)
One interesting hypothesis to examine is whether movements in one bond market
have a tendency to “Granger cause” movements in the other. Discuss how the test
of Ganger causality may be conducted. Suppose the test statistic for testing
UKt  USt is 9.563 and q  q*  m  m*  6 . Conduct the test at the 5%
level of significance.
(3 marks)
f)
Given your answers to parts c) and e), discuss if USt is expected to “Granger
cause” UK t .
(2 marks)
10