FN3142 Quantitative Finance
Question 1
Assume daily returns that are normally distributed with constant mean and variance, i.e., they are given by
where the time increment t + 1 is 1-day.
(a) [25 marks] Derive the following formula for the Value-at-Risk at the α% (V aR) crit-ical level and 1-day horizon.
where Φ is the standard normal cumulative density function.
(b) [25 marks] The expected shortfall at the critical level α% and 1-day horizon can be defined as
Using the V aR formula from part (a) derive the following formula for the 1-day expected 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.
(c) [50 marks] Prove that the relative difference between the 1-day expected shortfall and 1-day Value-at-Risk, as a proportion of the 1-day Value-at-Risk converges to zero when α goes to zero, i.e., show that
Question 2
(a) [40 marks] Forecast optimality is judged by comparing properties of a given forecast with those that we know are true. An optimal forecast generates forecast errors which, given a loss function, must obey some properties. Under a mean-square-error loss function what three properties must the optimal forecast error et+h|t ≡ Yt+h −Yˆ
t
∗
+h
for a horizon h possess?
(b) [30 marks] Can an economic forecast be ‘optimal’ but have poor forecasting power (in terms of a low R2
from a regression of the forecasted variable on a constant and the forecast)? If so, give an example or else explain why not.
(c) [30 marks] Can an economic forecast fail to be ‘optimal’ and still forecast well (in terms of a high R2
from a regression of the forecasted variable on a constant and the fore-cast)? If so, give an example or else explain why not.
Question 3
Consider the following moving average process
(a) [25 marks] Find E[Yt
], Et
[Yt+1], Et
[Yt+3]
(b) [25 marks] Find γ0 = V ar[Yt
]
(c) [50 marks] Find γ1 = Cov[Yt
, Yt−1], γ2 = Cov[Yt
, Yt−2], γj = Cov[Yt
, Yt−j
] for j ≥ 3.
Question 4
Denote a series of daily closing prices for IBM as S = S1, S2, S3, . . . , Sn with corresponding gross returns given by
(a) [10 marks] What is meant by serial correlation? Explain.
(b) [10 marks] Give an example of a process with zero serial correlation and an example of a process with positive serial correlation? Explain.
(c) [10 marks] Does the series S have positive, negative, or zero serial correlation accord-ing to (i) the random walk hypothesis; and (ii) the efficient market hypothesis ? Explain.
(d) [10 marks] Does the series R have positive, negative, or zero serial correlation accord-ing to (i) the random walk hypothesis; and (ii) the efficient market hypothesis ? Explain.
(e) [10 marks] Explain the difference between pure and statistical arbitrage. Give exam-ples for each.
(f) [10 marks] Does weak-form. market efficiency imply strong-form. market efficiency? What about the reverse? Explain your reasoning.
(g) [20 marks] Suppose Yahoo announced this morning that its profits from last quarter have dropped by 17% compared to the previous quarter. If Yahoo’s closing price today was up 3% from yesterday, can you conclude that this is evidence against the efficient market hypothesis?
(h) [20 marks] Give TWO examples of frictions such that even if the current market price is different from what it ‘should’ be (on the basis of forecasted future returns and dividends), an arbitrage opportunity may not exist.