FN3142 Quantitative Finance
Question 1
The probability density function of the normal distribution is given by
where µ is the mean and σ
2
is the variance of the distribution.
a [7 marks] Assuming µ = 0 derive the maximum likelihood estimate of σ
2 given the sample of i.i.d data (x1, x2, . . . , xT ).
b [8 marks] Now assume that xt
is conditionally normally distributed as N(0, σt
2
) where
Write down the likelihood function for this model given a sample of data (x1, x2, . . . , xT ).
c [10 marks] Describe how we can obtain estimates for {ω, α, β} for the GARCH(1,1) model and discuss estimation difficulties.
Question 2
Consider the time series process xt that follows
xt = φxt−1 + σ∈t
where ∈ t ∼ N(0, 1) and φ < 1.
a [5 marks] What is the unconditional mean of xt?
b [5 marks] What is the unconditional variance of xt?
c [5 marks] What is the first-order autocorrelation of xt?
d [5 marks] What is the second-order autocorrelation of xt?
e [5 marks] Given a sample of data (x1, x2, . . . , xT ) you estimate the parameters of this process via OLS. Compute an analytical expression for the R2
in this regression and give an interpretation.
Question 3
a [5 marks] Given a loss function L, an optimal forecast is obtained by minimising the conditional expectation of the future loss:
(1)
Given the quadratic loss function
L(y, yˆ) = (y − yˆ)
2 (2)
show that the optimal forecast is the conditional mean.
b [5 marks] Describe how one can test forecast optimality with Mincer-Zarnowitz re-gression.
c [5 marks] Consider a forecast Yˆ
τ
∗ of a variable Yτ . You have 100 observations of Yˆ*τ and Yτ and run the following regression
and obtain the following results:
what null hypothesis should you set up in order to test for forecast optimality? Can this test be conducted with the information given?
d [10 marks] What can be inferred from the results table in part (c)?
Question 4
a [5 marks] What is meant by serial correlation? Give an example of a process with zero serial correlation and an example of a process with positive serial correlation.
b [10 marks] Malkiel (1992) stated that a capital market is efficient if it fully and correctly reflects all relevant information in determining securities prices. Thus, mar-ket efficiency is defined with respect to some information set Ωt
. Describe the three commonly employed definitions of market efficiency that depend on the size of Ωt
.
c [10 marks] Which of the following observations could provide evidence against semi-strong form. market efficiency? In the case of observations that could go against market efficiency, explain what additional information would be needed to conduct a rigorous test.
– Mutual fund managers do not on average make superior returns than the market.
– In a particular year hedge fund managers make superior returns than the market.
– On average hedge fund managers make superior returns than the market.
– Low book-to-market stocks tend to have higher returns than high book-to-market stocks.
– forming a portfolio that goes long stocks that have had large positive returns over the previous year and goes short stocks that have had large negative returns over the previous year generates superior returns than the market.