This section consists of three questions and you are required to answer only two out of three

questions. {50 Marks}.

A1.   (a)  Describe the GARCH(1,1) process.   What stylised features of Önancial data could

be modelled using a GARCH(1,1) process?  Then, derive the h-step ahead forecast conditional on the information set at T, and comment on your Öndings. {10 Marks}

(b)  Describe the IGARCH(1,1) process and comment on its main statistical implication.

{5 Marks}

(c)  Describe two  important  extensions to the  GARCH  model  by describing the  GJR GARCH-M model in detail.  What additional characteristics of Önancial data might

they be able to capture? {10 Marks}

A2.  Consider the CAPM regression model:

yt  = a + pxt +et . t = 1. eee. T.                                         (1)

where yt  is a dependent variable (excess return on an individual portfolio), xt  is the regressor (excess return on the market index), a and p are the intercept and slope parameter, and et ~ iidN(0. a2 ) is the idiosyncratic error. All excess returns are expressed in % per annum.

(a) Some US studies Önd that Önancially distressed Örms have low, not high, average returns, suggesting that the equity market has not properly priced distress risk. This Önding is called ëthe distress anomalyí. To investigate this issue in the UK, we employ the data for UK public companies trading on the London Stock Exchange over the period January 1998- December 2007 (a total of 120 observations), run the CAPM regression for the excess returns on the long-short portfolios that go long in portfolio 1 (constructed as the 10% of stocks with the lowest default risk) and short-sell portfolio 2 (constructed as the 10% of stocks with the highest default risk), and obtain the OLS estimation results:

yt  =6e11 -0e78xt  with R2 = e108

(13.4)     (.198)

where the Ögures in (.) are standard errors and R2  is the adjusted multiple correlation coe¢ cient. Discuss the Önancial implications of these estimation results, and evaluate whether or not the UK data provide evidence in favour of the distress anomaly.  {8 Marks}

(b)  Describe what is meant by  "the value premium. " Then, discuss and compare the alternative explanations of the value premium provided by the fundamental-based and the sentiment-based theories.         {8 Marks}

(c) There has been a large anomaly literature where Örm-speciÖc characteristics such as past returns, book-to-market ratios and size help explain cross sectional returns, which contradicts the prediction of CAPM. In this regard, derive the Fama and French

(1993) three-factor model. Describe how to test its validity using the Fama-MacBeth (1973, FM) two-pass regression approach.  Then, discuss the weakness of both the three-factor model and the FM approach. {9 Marks}

A3.   (a) We have Ötted a RiskMetrics model to the log daily returns for IBM over the period

July 1962 - September 1997 as follows:

rt = at . at = atet . t = 1..... 9190

a t(2)= 0.94at(2)━1 + (1 - 0.94)at(2)━1

From the Ötted model, we have r9190 = 0.05 and 9190(2)= 0.002.  At the 1% and 5% quantiles, evaluate the forecasts of the daily value-at-risk (VaR) on a £1,000,000 long position at 1 and 10 day horizons, where the one-sided 1% and 5% quantile of a standard normal distribution is -2.33 and -1.65.  Then, discuss the weakness of this approach. {8 Marks}

(b)  Consider the daily IBM log returns over July 1962ñSeptember 1997 with a total of 9190 observations.  We use a parametric volatility model to calculate VaR of 1-day horizon at t = 9190 for a long position of $10m.  From the data we have r9189  = -0.002, r9190  = -0.013 and a 9190(2)  = 0.000335. The Ötted AR(1)-GARCH(1,1) model is obtained by

rt = 0.0007 + 0.02rt━1+ut ;  ut = atet

a t(2)  = 0.000004 + 0.08ut(2)━1+0 91a.t(2)━1

In the case that et  follows (i) the standard normal distribution and (ii) a t-distribution with 5 dof, derive the respective VaRs at 1% quantile.  Notice that the 1% quantiles of the standard normal distribution is -2.3262 and the 1% quantiles of t-distribution with 5 dof is -3.3649.  Comparing the results, which approach would you prefer and why? {8 Marks}

(c)  Using any numerical example, show that VaR fails the sub-additivity condition such that disaggregated risk management does not work using this measure, whereas the expected shortfall (ES) satisÖes the sub-additivity condition. {9 Marks}

SECTION B:

This section consists of two questions and you are required to answer BOTH questions. {50

Marks}.

B1.   (a)  Discuss why the parametric GARCH and the nonparametric quantile models tend to

underestimate VaR. Then, describe an alternative approach which can improve the VaR evaluation in terms of backtesting. {13 Marks}

(b)  Describe how to estimate the time-varying optimal hedge ratio by combining the bi-

variate error correction model and the dynamic conditional correlation GARCH model. Explain why this joint approach is more likely to be e§ective in reducing the variance of the hedging portfolio relative to the naive hedging. {12 Marks}

B2. To analyse the determinants of bilateral trade áows amongst the 14 EU member countries

(Austria, Belgium-Luxemburg, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Portugal, Spain, Sweden, United Kingdom), using the annual data over 1960- 2001 (42 years), Serlenga and Shin (2007) consider the panel data gravity model:

yit  = p x1,it  + p x2,it  + 9 z1i  + 9 z2i  + eit .  i = 1. eee. N. t = 1. eee. T         (2)

with two-way error components:

eit  = ai  + θt +uit                                                                              (3)

where yit  is a scalar dependent variable (the bilateral trade áows), x1,it  and x2,it  are k1 x1 and k2 x1 vectors of time-varying regressors, z1i   and z2i   are ,1 x1 and ,2 x1 vectors of time-invariant regressors. The data contains the following variables:

trade: the bilateral trade áows, the sum of logged exports and imports;

gdp: the sum of the logged real GDPs for both trading countries;

sim: the measure of similarity between two trading countries;

rlf: the measure of relative factor endowments;

rer: the logged bilateral real exchange rate;

cee: dummy that is equal to 1 when both belong to European Community;

emu: dummy that is equal to 1 when both adopt the common currency;

dis: geographical distance between capital cities;

bor: dummy that is equal to 1 when the trading partners share a border;

lan: dummy that is equal to 1 when both speak the same language

We assume that unobserved individual e§ects, ai , unobserved time e§ects, θt  and idiosyn-cratic disturbances, uit  follow:

ai ~ iid( . a a(2)); θt ~ iid(0. a 9(2)); uit  ~ iid(0. a u(2)); E(ai uit ) = 0 E(θt uit ) = 0 E(ai θt ) = 0 for all i. te (4)

We further assume that

E(ai Ix1,it ) = 0; E(ai Ix2,it )  0; E(ai Iz1i ) = 0; E(ai Iz2i )  0   (5)

E(θt Ix1,it ) = 0; E(θt Ix2,it )  0; E(θt Iz1i ) = 0; E(θt Iz2i )  0    (6)

where x1,it = {rer. gdp. rlf}. x2,it = {sim. cee. emu}. z1i = {dis. bor} and z2i = {lan}e De- scribe the implications of the assumptions in (5) and (6). Then, discuss an estimation  procedure which can consistently estimate p1  and p2  as well as 91  and 92 . {8 Marks}

(b) Anderson and van Wincoop (2003) propose to include multilateral resistance terms that capture the fact that bilateral trade áows depend on bilateral barriers as well as trade barriers across all trading partners.  To address this important issue, we now allow the error components to follow the multi-factor structure:

eit  = ai  + 一 9 t   + uit                                                                   (7)

where θt  is an r x 1 vector of unobserved common factors, which are correlated with the regressors, and 一i is the r x 1 vector of heterogeneous loading coe¢ cients. Discuss the implications of (7).   Given large T and large  N, describe how to consistently

estimate p1  and p2  as well as 91  and 92  in the model (2) with (7). {8 Marks} (c)  Consider the panel data model:

yit = pxit +ai(*)+uit ; i = 1. eee. N; t = 1. eee. T                            (8)

where yit  is a dependent variable and xit  a scalar regressor. We assume that

a i(*)= 入x-i +ai ; ai ~ N(0. a a(2))  and  uit ~ N(0. au(2))

where x-i = T━1 x xit . We assume that ai ís distributed independently of uit  and xit . Substituting ai(*)  into (8), we obtain:

yit = pxit +入x-i +ai + uit ; i = 1. eee. N; t = 1. eee. T                        (9)

(p . 入) in (9) can be estimated by GLS (i.e. the random e§ects estimator). Then, show that the random e§ects estimator is equivalent to the Öxed e§ects estimator.  Does this imply that there is no real distinction between the FE and the RE approaches? {9 Marks}