FN3142

QUANTITAIVE FINANCE

PRELIMINARY EXAM 2019

Question 1

Consider a forecast of a variable, Yt. You have 100 observations of and Yt and

you run the following regression:

The following results are obtained:

a)   What, if anything, can we infer from this output? 60 marks

b)   The picture in the Figure 1 is based on daily returns on General Electric over the period 1990-1999. What does this picture  reveal about the conditional variance of General Electric returns? What volatility model do you suggest for this asset? 40 marks

Figure 1: Figure for question 1b

x-axis represents return on day t; y-axis represents the squared return on day t+1.

Question 2

a)   Define the Efficient Market Hypothesis (EMH) and distinguish between its weak- form, semi-strong and strong form. 15 marks

b)   Discuss Fisher Black's interpretation of the EMH and explain why it maybe difficult to test empirically. 10 marks

c)   What is the role of forecasting models  and search technologies in refining the information set? Define market efficiency according to Granger and Timmerman (2003) 25 marks

d)   After extensive empirical  research, someone finds evidence of predictability in stock returns that cannot be possibly explained as compensation for risk bearing. Is it necessarily a violation of the EMH? 25 marks

e)   Briefly state the random walk model for asset prices and discuss whether the EMH implies the random walk model. 25 marks

Question 3

Consider using a historical simulation method (HS) and a GARCH method for forecasting volatility. After building the so called hit variables:

the following regressions are run, with standard errors in parenthesis corresponding to the parameter estimates:

Hitt(H)s  = 0.0814 +  μt                   (0.0132)

Hitt(GA)RCH  = 0.0207 +  μt          (0.0123)

a)   Describe how the above regression output can be used to test the accuracy of the VaR forecasts from these two models and decide whether the VaR forecasts are accurate or not? 40 marks

b)    Describe how you can determine the best convex combination of the  two forecasts in the sense of mean squared error minimisation. 60 marks

Question 4

a)   Show that a stationary GARCH(1,1) model can be re-written as a function of the unconditional variance and the deviations of the lagged conditional variance and lagged squared residual from the unconditional variance 20 marks

b)   Derive the two-step ahead predicted variance for a GARCH(1,1) as a function of the parameters of the model and the one-step forecast. 40 marks

c)   Derive the three-step ahead predicted variance for a GARCH(1,1) and infer the general expression for a h-step ahead forecast. Give an example of a financial application that may require using a h-step ahead forecast? 40 marks