THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT2309 THE STATISTICS OF INVESTMENT RISK
STAT3609 THE STATISTICS OF INVESTMENT RISK
December 19, 2015 Time: 2:30 p.m. – 4:30 p.m.
Only approved calculators as announced by the Examinations Secretary can be used in
this examination. It is candidates’ responsibility to ensure that their calculator operates
satisfactorily, and candidates must record the name and type of the calculator used on
the front page of the examination script.
• This examination has a total of 100 points.
• Answer ALL SIX questions.
• The points for each question are indicated at the beginning of the
question.
DO NOT TURN OVER
UNTIL YOU ARE TOLD TO DO SO!
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
1. (Total: 8 points)
The table below lists the autocorrelations from lag 1 to lag 6 for the continuously
compounded monthly return rates on a market index based on 100 return rates
data.
Lag 1 2 3 4 5 6
Autocorrelation 0.324 0.287 0.103 −0.04 −0.076 0.193
Test the random walk hypothesis with regard to the return rates on the market
index at the 5% level of significance using
(a) (4 points) the Portmanteau test;
(b) (4 points) the variance-ratio test.
2. (Total: 10 points)
The trading price Pt, 50-day moving average MAt(50) and 100-day moving
average MAt(100) (all in dollars) of a particular stock for 12 consecutive trading
days are given below:
Day t 1 2 3 4 5 6 7 8 9 10 11 12
Pt 80 78 85 86 90 92 93 85 81 78 72 76
MAt(50) 83 82 82 85 87 88 88 84 80 79 76 77
MAt(100) 86 85 84 84 84 84 85 83 81 81 80 80
(a) (4 points) Test the random walk hypothesis using the runs test at the 5%
significance level.
(b) (3 points) Using the single moving average trading rule with MAt(50),
determine when trades should take place and find the rate of return.
(c) (3 points) Using the double moving average trading rule, determine when
trades should take place and find the rate of return.
P. 2 of 8
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
3. (Total: 15 points)
Investor A has a utility function UA(W ) = lnW for W > 0 and investor B has a
utility function UB(W ) =
√
W for W > 0.
(a) (4 points) Show that both A and B are risk-averse investors.
(b) Suppose that for a certain investment, the wealth amount W takes values
100, 200, 300 and 400 with equal probabilities.
(i) (5 points) Find the certainty equivalent amount of investor A for this
investment.
(ii) (6 points) Find the risk premium of investor B for this investment.
4. (Total: 25 points)
Suppose the single index market model (SIMM) and the capital asset pricing
model (CAPM) are both valid for the following assets.
Expected Variance Beta Residual variance
Asset return of return coefficient in SIMM
i µi σ
2
i βi τ
2
i
A 0.105 a 1.5 0.02
B 0.086 0.0146 b 0.0082
C 0.067 0.0258 0.5 c
D d 0.04 e 0.04
(a) (10 points) Compute the values of a, b, c, d and e in the above table.
(b) (2 points) Write down the equation of the capital market line.
(c) (2 points) Write down the equation of the security market line.
(d) (2 points) Which asset has the largest Sharpe ratio?
(e) (3 points) What is the correlation between assets A and C?
(f) (3 points) Is any of the assets equivalent to the market portfolio? Explain
briefly.
(g) (3 points) Is any of the assets equivalent to the risk-free asset? Explain
briefly.
P. 3 of 8
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
5. (Total: 30 points)
The mean vector and variance-covariance matrix (both in percentage on an
annual basis) of four risky assets A, B, C and D (in the same order) are respectively
µ =

8
10
9
11
 and Σ =

144 0 −78 0
0 196 0 −105
−78 0 169 0
0 −105 0 225
 .
The risk-free rate is given to be 5% per annum. Assume that short selling is
allowed.
(a) (2 points) What is the correlation between assets A and C?
(b) (4 points) Determine the tangency portfolio of assets B and D only.
(c) (6 points) Find the expected return on the minimum variance portfolio of
assets C and D only.
(d) (8 points) Find the volatility of the tangency portfolio of assets A, B and C
only.
(e) (10 points) Determine the minimum variance portfolio of the four assets.
6. (Total: 12 points)
(a) Consider a constant correlation model for n assets with equal variance σ2 > 0
and a common correlation ρ for any pair of assets.
(i) (3 points) Find, in terms of n, σ2 and ρ, the variance of the equally-weighted
portfolio of the n assets.
(ii) (4 points) Determine the range of ρ for the model to be valid.
(b) (5 points) Suppose that 3 risky assets with a common variance have the
following correlation matrix:
1 −0.5 −0.5
−0.5 1 −0.5
−0.5 −0.5 1

Describe and verify how a risk-free portfolio can be constructed with these
3 assets.
P. 4 of 8
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
Formula Sheet
• Relation between simple returns R and continuously-compounded (log) returns r:
r = ln(1 +R)
• Simple linear regression:
(i) Model: yi = α+ βxi + ei, i = 1, 2, . . . , n, where ei ∼ N(0, σ2)
(ii) OLS estimates:
αˆ = y¯ − βˆx¯ and βˆ =
∑n
i=1(xi − x¯)(yi − y¯)∑n
i=1(xi − x¯)2
=
Sxy
Sxx
(iii) Unbiased estimate of σ2:
s2 =
1
n− 2
n∑
i=1
(yi − yˆi)2 = 1
n− 2
n∑
i=1
(yi − αˆ− βˆxi)2 = 1
n− 2
(
Syy −
S2xy
Sxx
)
(iv) Standard error of OLS estimates:
SE(αˆ) =
√
V̂ar(αˆ) =
√
s2
Sxx
∑n
i=1 x
2
i
n
and SE(βˆ) =
√
V̂ar(βˆ) =
√
s2
Sxx
• The CAPM equation:
E(Ri) = Rf + βi[E(RM )−Rf ] where βi = Cov(Ri, RM )
Var(RM )
• Black-Litterman model:
Market view: m ∼ N(µCAPM , τΣCAPM )
Subjective view: v|m ∼ N(Pm,Ω)
Conditional view: E(m|v) = µCAPM + τΣCAPMPT(τPΣCAPMPT + Ω)−1(v −PµCAPM )
When Ω = cPΣCAPMP
T: E(m|v) = µCAPM + ττ+cΣCAPMPT(PΣCAPMPT)−1(v −PµCAPM )
• Beta adjustment:
(i) Modified Blume’s method (Merrill Lynch): βAdji =
2
3
βi +
1
3
(ii) Vasicek’s method:
Prior view on βi: βi ∼ N(β0, s20)
Conditional view on βi: β
Adj
i =
s21β0 + s
2
0βˆi
s20 + s
2
1
where βˆi is the regression estimate of βi and s1 is its standard error
(iii) Dimson’s method:
Fit the regression: Ri,t = αi +
N∑
k=−N
βikRM,t−k + ei,t
Adjusted βi: β
Adj
i =
N∑
k=−N
βˆik
P. 5 of 8
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
• The APT equation:
E(Ri) = λ0 + λ1βi1 + λ2βi2 + · · ·+ λLβiL
• EWMA estimators:
(i) µi,t = (1− λ)ri,t−1 + λµi,t−1 or
µi,t =
1−λ
1−λM
∑M
m=1 λ
m−1ri,t−m
(ii) σ2i,t = (1− λ)(ri,t−1 − µi,t−1)2 + λσ2i,t−1 or
σ2i,t =
1−λ
1−λM
∑M
m=1 λ
m−1(ri,t−m − µi,t−m)2
(iii) σij,t = (1− λ)(ri,t−1 − µi,t−1)(rj,t−1 − µj,t−1) + λσij,t−1 or
σij,t =
1−λ
1−λM
∑M
m=1 λ
m−1(ri,t−m − µi,t−m)(rj,t−m − µj,t−m)
• Portfolio performance measures:
(i) Sharpe ratio: SRp =
E(Rp)−Rf
σp
(ii) Treynor ratio: TRp =
E(Rp)−Rf
βp
(iii) Jensen’s alpha: αp = E(Rp)− {Rf + βp[E(RM )−Rf ]} (CAPM)
(iv) Modigliani and Modigliani’s M2: M2p = Rf +
[
E(Rp)−Rf
σp
]
σM − E(RM )
(v) Information ratio: IRp =
E(Rp −Rb)
σ(Rp −Rb) or IRp =
αp
τp
(CAPM)
(vi) Fama performance measure:
Selectivity = Ra − E[R(βa)] and Portfolio risk = E[R(βa)]−Rf
Net selectivity = Ra − E[R(σa)] and Diversification = E[R(σa)]− E[R(βa)]
• Attribution analysis:
Asset allocation =
∑N
i=1(wpi − wbi)Rbi
Security selection =
∑N
i=1wpi(Rpi −Rbi)
• Market timing model:
Treynor and Mazuy model:
Rp,t −Rf,t = αp + bp(RM,t −Rf,t) + cp(RM,t −Rf,t)2 + ep,t
Henriksson and Merton model:
Rp,t −Rf,t = αp + bp(RM,t −Rf,t) + cpDt(RM,t −Rf,t) + ep,t
where Dt = 1 if RM,t > Rf,t; 0 otherwise
P. 6 of 8
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
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e
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ar
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it
z
M
o
d
el
S
o
lv
e
µ
i
−
R
f
=
z i
σ
2 i
+
∑ j6=iz
j
σ
ij
.
S
ol
ve
µ
i
−
R
f
=
z i
σ
2 i
+
∑ j6=iz
j
σ
ij
−
u
i,
(N
o
S
tr
u
ct
u
re
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V
ar
(R
i,
t
)
=
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2 i
T
h
en
w
i
=
z i
z 1
+
··
·+
z N
,
i
=
1,
..
.,
N
.
u
iz
i
=
0,
C
ov
(R
i,
t
,R
j,
t
)
=
σ
ij
u
i
≥
0
,
z i
≥
0,
i
=
1,
..
.,
N
.
T
h
en
w
i
=
z i
z 1
+
··
·+
z N
,
i
=
1,
..
.,
N
.
S
in
gl
e
In
d
ex
F
in
d
z i
=
β
i
τ
2 i
[ µ i−
R
f
β
i
−
A
] ,
(1
)
R
an
k
r i
=
µ
i
−
R
f
β
i
su
ch
th
a
t
r 1
≥
r 2
≥
··
·≥
r N
.
M
ar
ke
t
M
o
d
el
(S
IM
M
)
A
=
σ
2 M
N ∑ i=1β
i(
µ
i
−
R
f
)
τ
2 i
1
+
σ
2 M
N ∑ i=1β
2 i
τ
2 i
.
(2
)
A
b
=
σ
2 M
b ∑ i=1β
i(
µ
i
−
R
f
)
τ
2 i
1
+
σ
2 M
b ∑ i=1β
2 i
τ
2 i
,
b
=
1,
..
.,
N
.
R
i,
t
=
α
i
+
β
iR
M
,t
+
e i
,t
(3
)
F
in
d
b∗
su
ch
th
at
µ
i
=
α
i
+
β
iµ
M
r i
>
A
b
∗
,
i
=
1,
..
.,
b∗
;
σ
ij
=
β
2 i
σ
2 M
+
τ
2 i
(f
or
i
=
j)
r i
≤
A
b
∗
,
i
=
b∗
+
1
,.
..
,N
.
=
β
iβ
j
σ
2 M
(f
or
i
6=
j)
T
h
en
w
i
=
z i
z 1
+
··
·+
z N
,
i
=
1,
..
.,
N
.
(4
)
z i
=
β
i
τ
2 i
[ µ i−
R
f
β
i
−
A
b
∗
] ,
i
=
1,
..
.,
b∗
;
z i
=
0
,
i
=
b∗
+
1
,.
..
,N
.
(5
)
T
h
en
w
i
=
z i
z 1
+
··
·+
z N
,
i
=
1,
..
.,
N
.
C
on
st
an
t
F
in
d
z i
=
1
(1
−
ρ
)σ
i
[ µ i−
R
f
σ
i
−
C
] ,
(1
)
R
an
k
r i
=
µ
i
−
R
f
σ
i
su
ch
th
a
t
r 1
≥
r 2
≥
··
·≥
r N
.
C
or
re
la
ti
on
C
=
ρ
1
−
ρ
+
N
ρ
N ∑ i=1µ
i
−
R
f
σ
i
.
(2
)
C
b
=
ρ
1
−
ρ
+
bρ
b ∑ i=1µ
i
−
R
f
σ
i
,
b
=
1,
..
.,
N
.
(3
)
F
in
d
b∗
su
ch
th
at
M
o
d
el
r i
>
C
b
∗
,
i
=
1,
..
.,
b∗
;
σ
ij
=
ρ
σ
iσ
j
(f
or
i
6=
j)
r i
≤
C
b
∗
,
i
=
b∗
+
1
,.
..
,N
.
T
h
en
w
i
=
z i
z 1
+
··
·+
z N
,
i
=
1,
..
.,
N
.
(4
)
z i
=
1
(1
−
ρ
)σ
i
[ µ i−
R
f
σ
i
−
C
b
∗
] ,
i
=
1,
..
.,
b∗
;
z i
=
0
,
i
=
b∗
+
1
,.
..
,N
.
(5
)
T
h
en
w
i
=
z i
z 1
+
··
·+
z N
,
i
=
1,
..
.,
N
.
P. 7 of 8
S&AS: STAT2309/STAT3609 The Statistics of Investment Risk
T
e
st
s
fo
r
R
a
n
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o
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W
a
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y
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R
u
le
H
0
:
r t
=
ln
P
t
−
ln
P
t−
1
∼
i.
i.
d
.
R
u
n
s
⇒
T
h
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se
q
u
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of
ri
se
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d
d
ro
p
of
p
ri
ce
T
=
#
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f
u
p
w
ar
d
s
an
d
T
→
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( µ=
2n
1
n
2
n
+
1
,
σ
2
=
2
n
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n
2
(2
n
1
n
2
−
n
)
n
2
(n
−
1
)
)
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es
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se
ri
es
is
ra
n
d
om
d
ow
n
w
a
rd
s
tr
en
d
s
as
n
→
∞
Z
=
T
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µ
σ
=
#
of
ru
n
s
n
1
=
#
of
P
t
>
P
t−
1
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#
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f
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q
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os
it
iv
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d
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2
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of
P
t
<
P
t−
1
=
#
o
f
r t
<
0
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ej
ec
t
H
0
if
n
eg
at
iv
e
re
tu
rn
s
is
ra
n
d
om
n
=
n
1
+
n
2
|Z
|>
z
α 2
H
0
:
r t
=
µ
+
e t
,
e t
∼
i.
i.
d
.
N
(0
,σ
2
)
G
iv
en
r 1
,.
..
,r
n
.
P
or
tm
an
-
⇒
ρ
1
=
ρ
2
=
··
·=
ρ
M
=
0
Q
M
=
n
(n
+
2)
M ∑ k=1
ρˆ
2 k
n
−
k
te
au
T
es
t
w
h
er
e
ρ
k
=
C
or
r(
r t
,r
t+
k
)
ρˆ
k
=
n
−
k ∑ t=1(r
t
−
r¯)
(r
t+
k
−
r¯)
n ∑ t=1(r
t
−
r¯)
2
Q
M
→
χ
2
(M
)
a
s
n
→
∞
R
ej
ec
t
H
0
if
r¯
=
1 n
n ∑ t=1r
t
Q
M
>
χ
2 α
(M
)
H
0
:
r t
=
µ
+
e t
,
e t
∼
i.
i.
d
.
N
(0
,σ
2
)
G
iv
en
r 1
,.
..
,r
n
.
V
ar
ia
n
ce
-
⇒
V
R
(q
)
=
V
ar
(r
t
+
r t
−
1
+
··
·+
r t
−
q
+
1
)
qV
ar
(r
t
)
V
R
(q
)
=
1
+
2
q
−
1 ∑ k=1(
1
−
k q
) ρˆ k
R
at
io
T
es
t
=
1
+
2
q
−
1 ∑ k=1(
1
−
k q
) ρ k
Z
=
√ n
(V
R
(q
)
−
1)
√ 2(2
q
−
1
)(
q
−
1
)
3
q
→
N
(0
,1
)
a
s
n
→
∞
R
ej
ec
t
H
0
if
=
1
fo
r
al
l
q
|Z
|>
z
α 2
T
ra
d
in
g
R
u
le
s
F
il
te
r
R
u
le
S
in
g
le
M
o
v
in
g
A
v
e
ra
g
e
D
o
u
b
le
M
o
v
in
g
A
v
e
ra
g
e
–
If
st
o
ck
p
ri
ce
in
c
re
a
se
s
(d
ec
re
a
se
s)
x
%
,
b
u
y
(s
el
l
sh
or
t)
th
e
st
o
ck
;
–
b
u
y
if
P
t
>
M
A
t
(n
)
–
b
u
y
if
M
A
t
(n
1
)
>
M
A
t
(n
2
)
–
If
w
e
a
re
h
o
ld
in
g
(s
el
li
n
g
sh
o
rt
)
st
o
ck
a
n
d
th
e
st
o
ck
p
ri
ce
d
e
c
re
a
se
s
–
se
ll
if
P
t
<
M
A
t
(n
)
–
se
ll
if
M
A
t
(n
1
)
<
M
A
t
(n
2
)
(i
n
cr
ea
se
s)
x
%
fr
om
th
e
h
ig
h
e
st
(l
ow
es
t)
p
ri
ce
,
se
ll
(b
u
y
)
th
e
st
o
ck
w
h
er
e
n
1
<
n
2
an
d
g
o
sh
o
rt
(h
o
ld
)
it
.
*
*
*
*
*
*
*
*
*
*
E
N
D
O
F
P
A
P
E
R
*
*
*
*
*
*
*
*
*
*
P. 8 of 8