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Department of Accounting
and Finance
M.Sc.
Finance
M.Sc. International Banking
and Finance
An
d
M.Sc. International Accounting
and Finance
40909: Basic Statistics
for Finance
Wednesday 7 th January 2009 10.30am – 12.00pm (1½
hours)
Instructions for
Candidates
Answer THREE Questions (in the answer
book provided) [Failure to comply will
result in papers not being marked]
Calculators must not be used to store text and/or formulae nor be capable of communication. Invigilators may require calculators to be reset. All answers are to be written in the exam paper provided in ink.^ Please write clearly as illegible writing cannot be marked. Failure to follow these requirements will lead to a deduction of marks.
To Be Issued: Normal
Distribution Tables
Family Name:
Other Name:
Course (please indicate by ticking appropriate box)
Finance Int. Banking & Fin Int. Accounting & Fin
Please Note: This question paper is to be
returned along with your
examination answer booklet, you should
complete the above and slip this paper into your
answer booklet. Under no circumstances is a
copy of this paper to leave the exam room.
SOLUTIO
NS
(8 marks)
X = 250/10 = £25 (million); X med = (25 + 25)/2 = £ (million)
c) The joint probability distribution of two discrete random variables, X and Y, is as follows:
X
0 10 20 30 40
0 0.08 0.08 0.04 0.03 0. 2 Y 1 0
i. Write down P[X = 30, Y = 10]. P(30, 10) = 0.
(2 marks)
ii. Derive the marginal probability distribution of Y.
(2 marks)
Y : 0 10 20
P(y):
iii. Derive the conditional distribution of X, given Y = 10.
X
P(x│Y
= 10) :
0.2 0.16 0.16 0.
0.32 iv. Find the mean
value of X, given Y = 10.
E(X
│Y =
(4 marks)
(3 marks)
P(10 < R < 16) = P( ‐0.5 < Z < 1) = 1 – 0.1587 –
b) The expected returns (%) of the securities in an equally‐weighted three security portfolio, and their respective standard deviations, are given by
E(R 1 ) = 6E
(R 2 ) = 9 E
(R 3 ) = 12
σ 1 = 1^ σ^2 = 2^ σ^3 = 3
There is no correlation between the returns to securities 1 and 3, that between the returns to securities 1 and 2 is – 0.5, whilst that between the returns to securities 2 and 3 is 0.25.
i. Find the expected return to the portfolio and its standard deviation.
(16 marks)
R p = (1/3)[R 1 + R 2 + R 3 ] so that
E(R p) = (1/3)[27] = 9 (%)
ρ 12 = ‐0.5 → σ 12 = ‐0.5(1)(2) = ‐1; ρ 23 = 0.25 → σ 23 = 0.25(2) (3) = 1.
Var(R p) = (1/9)[1^2 +^22 + 3^2 + 2(‐1) + 2(0) + 2(1.5)] = 15/
Hence SD(R p) = 1.29 (%)
ii. On the assumption that the individual returns are normally distributed, what is the probability that the portfolio return is greater than 10%?
(4 1/3 marks)
P(Rp > 10) = P(Z > 0.775) = 0.
(TOTAL 33 1/3 MARKS)
Q3. a)
The matrices to be used in this question are: A = ⎢
⎥ and B = ⎢ ⎥
i. Calculate (BA) (3 marks)
i. Show that z has a maximum value of 20 when x = 2 and y = 4.
(4 marks)
(6 marks)
(6 marks)
(7 marks)
δz/δx = ‐ 2x – 4 + 2y, δz/δy = 2x ‐ 4y + 12; δ 2 z/δx 2 = ‐ 2; δ 2 z/δy 2 = ‐ 4; δ 2 z/δx
= 2.
Solving the first two equations gives x = 2, y = 4.
Second order conditions satisfy requirements for a maximum at z = 20.
ii. Suppose that the constraint is imposed that x = y. What difference does this make to your answer to part (i)?
x = y → z = ‐ x 2 + 8x. dz/dx = ‐ 2x + 8 = 0 → x = 4, y = 4.
(4 1/3 marks)
Second order condition satisfies requirement for maximum at z = 16.
(TOTAL 33 1/3 MARKS)
the relevance of sample
size to the use of
X as an estimator of μ x^.^ (8 marks)
Parts (i), (ii) and (iv) are standard bookwork.
ii. Explain what is meant by the statement that “ X and Xmed are each unbiased estimators of μ x but X is more efficient than X med ”.
Parts (i), (ii) and (iv) are standard bookwork.
(5 marks)
iii. What do you understand by the phrase “95% confidence interval estimate”?
Parts (i), (ii) and (iv) are standard bookwork.
(7 marks)
iv. Suppose that
σ 2 is known to equal 16. A random sample of 100
observations yields
μ x.
X = 10.8. Find a 95% confidence interval estimate for
(3 1/3 marks)
10.8 ± 1.96{√(16/100)}or 10.8 ± 0.
b) The profitability of 16 companies drawn at random from a given industrial sector is measured before a major tax regime alteration: let X i denote a measure of profitability (in %) of the i’th company. An independent sample of 16 companies is drawn at random after the alteration: let Y i denote profitability (in %) of the i’th company in this second sample. Suppose that it may be assumed that the
variability of profitability is not affected by the change so that we may use σ 2 =
2.42 to represent both var (X) and var(Y). The data show that
10.8.
X = 12.5 and Y =
Does it appear that the tax regime change has
affected profitability?
95% confidence interval
