FN3142 ZA

BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas in Economics and Social Sciences

Quantitative Finance

Monday, 22 May 2017 : 10:00 to 13:00

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

σt(2) = ω + βσt(2)-1 + αxt(2)-1

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 di伍culties.

Question 2

Consider the time series process xt  that follows

xt  = φxt-1 + σct

where ct  ~ 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:

Given the quadratic loss function

L(y, ˆ(y)) = (y -ˆ(y))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 Y(ˆ)τ*   of a variable Yτ .  You have 100 observations of Y(ˆ)τ* and Yτ  and run the following regression

Yτ = Q + βY(ˆ)τ* + ετ

and obtain the following results:

            Estimate       Std Error

α         -0.10           0.02

β          1.51           0.30

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 e伍cient if it fully and correctly re丑ects all relevant information in determining securities prices. Thus, mar- ket e伍ciency is defined with respect to some information set Ωt. Describe the three commonly employed definitions of market e伍ciency that depend on the size of Ωt.

c [10 marks] Which of the following observations could provide evidence against semi- strong form. market e伍ciency? In the case of observations that could go against market e伍ciency, 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.

– Mutual fund managers do not on average 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.

FN3142 ZB

BSc degrees and Diplomas for Graduates in Economics, Management, Finance and the Social Sciences, the Diplomas in Economics and Social Sciences

Quantitative Finance

Monday, 22 May 2017 : 10:00 to 13:00

Question 1

Imagine the following gamble.  First, flip a fair coin to determine the amount of your bet: if heads, you bet $1, if tails you bet $2.  Second, flip again:  if heads, you win the amount of your bet, if tails, you lose it.  For example, if you flip heads and then tails, you lose  $1; if you flip tails and then heads you win $2.) Let X be the amount you bet, and let Y be your net winnings (negative if you lost).

a [10  Marks] Show that the covariance between X and Y is zero. b [15 Marks] Show that X  and Y are not independent.

Question 2

Consider the zero-mean MA(1) process Xt  :

Xt  = ut + δut-1     where ut  i~.i.d  N(0, σu(2))

a [5 Marks] Find E[Xt], Et [Xt+1], Et [Xt+2]

b [5 Marks] Find √0  = Var[Xt]

c [15  Marks] Derive  the  autocorrelation  function  (ACF).  Now,  imagine you  have a parameter estimate of δ = 0.70.  Plot the autocorrelation function as a function of the number of lags.

Question 3

a [5  Marks] Assume daily returns that are normally distributed with constant mean and variance, i.e., they are given by

Rt+1  = σνt+1

νt+1 i~.i.d N(0, 1)

where the time increment t + 1 is 1-day.  Derive the following formula for the Value-at- Risk at the α% (VaR) critical level and 1-day horizon.

where Φ is the standard normal cumulative density function.

b [10 Marks] Describe the ‘historical simulation’ and RiskMetrics approaches to mea- suring Value-at-Risk.

c [10 Marks] The expected shortfall ESt α +1 at the critical level α% and 1-day horizon can be defined as

Using the VaR formula from part  (a) derive the following formula for the 1-day ex- pected shortfall at critical level α

where φ is the standard normal probability density function.  Hint: From the properties of the normal distribution we know that

if z is normally distributed.

Question 4

a [5 Marks] Roberts (1967) defines three types of information set available to investors:

(i) weak form eflciency; (ii) semi-strong form eflciency; (iii) strong form eflciency. Report a definition for each of these.

b [5 Marks] To which information set, if any, do the following variables belong?

– Stock prices today.

– The risk free rate today.

– Next year’s production figures just approved by a company’s board of directors.

– The nominal size of the short position George Soros has in European equity.

– Stock prices tomorrow.

c [5  Marks] What is the eflcient market hypothesis statement according to Malkiel (1992)?

d [10 Marks] Black (1986) gives an alternative definition of market eflciency. What is it and why is Black’s definition diflcult to test?