BEO2431 Risk Management Model Semester 2
BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
Victoria University College of Business
BEO2431 Risk Management Model
Semester 2 2014
Case Study 1
Presented by:
ZHIYI ZHOU 3923070
KAIYUAN ZHANG 3920263
Presented to:
Chinthana Hatangala
Tutorial No.3
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BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
A. Compute the returns (RT) of the following as:
RT = ((Pt – Pt-1)/P t-1)
Share Prices (Monthly Data: Nov1987 to June 2014)
US$: Exchange rates: United States Dollar
SP500: Share price indices: United States: S&P 500
ASX: Share market: Share price indices: S&P/ASX 200
TOPIX: Share price indices: Japan: TOPIX
FTSE: Share price indices: United Kingdom: FTSE 100
ST: Share price indices: Singapore: Straits Times:
For each countries share price and exchange rate
(a) Plot each countries return over the time period. Comment on the volatility and
volatility clustering of the returns (Use Excel)
The volatility clustering is caused by the large changes in price. After recalling and
analyzing the phenomenon, several economic mechanisms are discussed depending
on the explaining of the volatility clustering.
1. Returns - US$
Returns-US$
Returns-US
0.15
High
Volatility
Returns-US $
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
Time
Figure A1
As shown in the Figure A1, the return of US$ experienced large fluctuation during
1987 to 1989, 1998 to 2004 and 2007 to 2013. The first volatility happened during
November 1987 to January 1989 changed between -10% and 6%. The second
volatility area (between -8% and 6%) happened during 1998 to 2004. And the most
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BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
significant volatility happened between September 2007 and July 2013, where the
market crash (-16%) happened. The high volatility shows the unstable financial
environment which can be considered as bad news for investment while the low
volatility shows an anticipated announcement. The volatility may be caused by
government’s intervention of finance, as government run a tight monetary policy
during 70s to 80s.
2. Returns – SP500
Returns-SP500
Returns-SP500
Returns-SP500
0.15
High
Volatility
Low
0.1
Volatility
0.05
0
-0.05 87 89 90 91 92 94 95 96 97 99 00 01 02 04 05 06 07 09 10 11 12 14
- - - - - - - - - - - - - - - - - - - - - -0.1
ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb
N
N
N
N
N
N
M
M
M
M
M
-0.15
-0.2
Market
Time
Crash
Figure A2
In Figure A2, there are only two low volatility areas, which are during July 1992 to
January 1996 and during March 2004 to July 2006. The high volatility areas cluster
during November 1987 to May 1991, March 1997 to January 2003 and September
2007 to July 2013. The high volatility area shows the unstable United Stated share
price indices. During the first 4 years of this line chart, the returns fluctuated between
-9% and 10%, where the United States government run a tight monetary policy. The
second high volatility area changed between -15% and 10%, where the financial crisis
happened. The last high volatility was during the subprime crisis and the market
crash, -16%, happened between September 2007 and November 2008. Therefore, the
volatility clustering can show the finance environment correctly and accurately.
5
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BEO2431 Risk Management Model Semester 2 2014
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ZHIYI ZHOU & KAIYUAN ZHANG
3. Returns – ASX
Returns-ASX
Returns-ASX
0.15
0.1
0.05
High
Volatility
Returns-ASX
Low
Volatility
0
-0.05-87 -89 -90 -91 -92 -94 -95 -96 -97 -99 -00 -01 -02 -04 -05 -06 -07 -09 -10 -11 -12 -14
ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb
-0.1
N F M A N F M A N F M A N F M A N F M A N F
-0.15
Market
Time
Crash
Figure A3
Compared with the two markets above, the share price indices fluctuated stronger.
But the scope of the fluctuation is weaker than United States Market. Except for the
time period during November 2002 to November 2007, this market experienced
relatively strong fluctuation in the rest of time during 1987 to 2014. The market crash
(-12%) happened near July 2009, which is probably caused by the subprime crisis
influence from United States. For the most time of Australian financial market, the
returns change a lot. There is not an exactly gap between the low volatility clustering
and high volatility clustering.
4. Returns – TOPIX
Returns-TOPIX
Returns-TOPIX
0.3
Market
Crash
0.2
0.1
High
Volatility
Returns-TOPIX
Low
Volatility
0
-0.1-87 -89 -90 -91 -92 -94 -95 -96 -97 -99 -00 -01 -02 -04 -05 -06 -07 -09 -10 -11 -12 -14
ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb
-0.2
N F M A N F M A N F M A N F M A N F M A N F
-0.3
Time
Figure A4
Compared with United States and Australian financial markets, fluctuation scope of
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BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
the returns of Japanese financial market changes stronger. Different with the first two
countries’ financial markets, the returns of first two years did not change a lot
comparing with the following 7 years. During January 1991 to August 1992, the
Japanese financial market experienced the highest and lowest returns, which are 19%
and -21%. Because of the close relationship between Japanese and United States
financial market, the unstable phenomena in Japan may be affected by the tight
monetary policy of United States government. In the year of 2009, Japan was also
influenced by subprime crisis. Therefore, there is a high volatility area during 2008 to
2010.
5. Returns – FTSE
Returns-FTSE 100
Returns-FTSE 100
Returns-FTSE 100
0.2
0.15
0.1
0.05
High
Volatility
Low
Volatility
0
-0.05 87 89 90 91 92 94 95 96 97 99 00 01 02 04 05 06 07 09 10 11 12 14
- - - - - - - - - - - - - - - - - - - - - ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb
-0.1
N
N
N
N
N
N
M
M
M
M
M
-0.15
Time
Market
Crash
Figure A5
The United Kingdom financial market has two low volatility areas, which are between
July 1994 to November 1997 and July 2004 to November 2007. The first fluctuation
was during July 1989 to July 1994 and the returns reached its highest in 1989. The
second high volatility was during November 1997 to July 2004 which is the period of
global finance crisis. The market crash happened in 2008, which is also during the
subprime crisis. The line chart of returns fully explains the financial phenomenon,
which shows the importance of volatility clustering.
5
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BEO2431 Risk Management Model Semester 2 2014
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ZHIYI ZHOU & KAIYUAN ZHANG
6. Returns - ST
Returns-ST
Returns-ST
High
Volatility
0.4
Returns-ST
0.3
Low
0.2
Volatility
0.1
0
-0.1 87 89 90 91 92 94 95 96 97 99 00 01 02 04 05 06 07 09 10 11 12 14
- - - - - - - - - - - - - - - - - - - - - ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb
-0.2
N
N
N
N
N
N
M
M
M
M
M
-0.3
Market
Crash
Time
Figure A6
Compared with the countries’ financial markets above, the financial market of
Singapore has longer low volatility area and higher returns. The first 10 years from
November 1987 to November 1997 experienced stationary returns. During the global
finance crisis, the returns changed from -20% to 29%, which is the largest gap within
the six markets. From November 2002 to November 2007, the returns kept stable and
flat. And we can tell from the line chart that Singapore was also affected by the
subprime crisis. The returns in Singapore financial market fluctuated large during
these two periods of time.
7. Conclusion
After analyze all of these six markets returns, it can be found that volatility exist in
each return all the time. Commonly, the degree of the volatility of returns is the
response of the real economic activities in the countries.
(b) Perform summary descriptive statistics on the returns (Use Excel) and explain
the use of the following measures in finance:
1. Arithmetic mean, 2. Geometric Mean, 3. Cumulative wealth Index 4. Standard
Deviation (risk), 5. Skewness, 6. Kurtosis and 7. Coefficient of Variation (CV),
8. The probability of obtaining negative return.
5
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BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
Table Ab1
US$
SP500
ASX
TOPIX
FTSE
ST
Arithmetic Mean
0.001431609
0.007625755
0.005147363
0.00041186
0.005446177
0.006474094
Geometric Mean
0.00090801
0.006737345
0.004399867
-0.00119186
0.004559892
0.004327462
Cumulative Wealth Index
1.335791265
8.516499283
4.057142857
0.68356808
4.268656716
3.964905248
Standard Deviation
0.032199969
0.041954946
0.038603833
0.05647056
0.042067213
0.065524483
Kurtosis
2.274434463
1.227791784
0.344470699
0.87443209
0.513573283
3.416980811
-0.562719367
-0.612193408
-0.368007992
-0.1023303
-0.274207174
0.060597263
22.49215927
5.501743278
7.499730498
137.109706
7.724172784
10.12102738
-0.044459938
-0.181760571
-0.133338125
-0.00729343
-0.129463702
-0.098804199
48.40%
42.86%
44.83%
49.60%
44.83%
46.02%
Skewness
CV
Z-Score
Prob of Z-Score
1. Arithmetic Mean
The formula of arithmetic mean is shown below:
n
∑ Ri
AM = i=1n
Arithmetic mean measures the average performance of all of the data and predicts the expected return for next period. As the formula above
shows that arithmetic mean equals to the sum of all the returns divided by the number of years. In the Table Ab1, it is obvious that SP500 has
the highest arithmetic mean (0.007625755) while TOPIX has the lowest arithmetic mean (0.00041186). Therefore, considering the arithmetic
mean aspect, SP500 can be regarded as the best investment as a whole, which means it has the highest investment return in the next period, while
TOPIX has the lowest expected return in the next period.
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BEO2431 Risk Management Model Semester 2 2014
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ZHIYI ZHOU & KAIYUAN ZHANG
2. Geometric mean
The formula of geometric mean is:
GM =√n (RR 1)(RR 2 )(RR ¿ ¿ 3)… (RR ¿ ¿ n)−1 ¿ ¿
Geometric mean is used to illustrate the degree of the growth of the return by taking
the antilog of the product resulting from multiplying a series of return relatives
together. Therefore, geometric mean can be considered as compound rate of return
over time. From Table Ab1, SP500 has the highest geometric mean (0.006737345),
which means SP500 has the greatest variability of the return, while TOPIX has the
weakest variability of return. Therefore, SP500 has the greatest spread, as well as the
compound return. In contract, TOPIX (-0.00119186) has the lowest spread, as well as
the compound return.
3. Cumulative Wealth Index
The formula of cumulative wealth index is:
CWI n=WI 0 ( RR 1 ) ( RR 2 ) … ( RRn )
CWI n: The cumulative wealth index as of the end of the period n
WI 0: The beginning index value, typically $1
Cumulative wealth index is the accumulation of wealth all the time. It measures the
cumulative effect of returns over time, typically on the basis pf a $1 invested. It can
be calculated from the initial wealth, then to estimate the return in the end of year. It
reflects the level of wealth rather than the change in the wealth. As shown in Table
Ab1, SP500 has the largest cumulative index (8.516499283) and TOPIX has the
lowest cumulated index (0.68356808). Hence, SP500 gains the highest accumulation
of wealth while TOPIX has the lowest one, from which we can predict that SP500
will have the highest return in the end of the year.
4. Standard Deviation
The equation of the standard deviation is:
T
N
σ = N −1 ∑ ( X t − X´ )2
t=1
√
The standard deviation is a measure of how spread out numbers are. In other words,
standard deviation is a statistical measurement that sheds light on historical volatility.
In the field of finance, the standard deviation is applied to the annual rate of return of
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BEO2431 Risk Management Model Semester 2 2014
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an investment to measure the investment's volatility. In Table Ab1, ST has the largest
standard deviation (0.065524483), which means ST stock is unstable, while US$ has
the lowest deviation (0.032199969). Hence, people will face more risk when
investing in ST stock whereas it is safest to invest in US$.
5. Skewness
The formula used to calculate skewness is:
T
n
´ )3 /σ 3
S=
( X t− X
∑
(n−1)(n−2) t =1
Skewness describes asymmetry from the normal distribution in a set of statistical data.
Skewness can come in the form of "negative skewness" or "positive skewness",
depending on whether data points are skewed to the left (negative skew) or to the
right (positive skew) of the data average. The third situation is that if there is no
skewness, it means that distributions are symmetric (normal distribution). When
mean return > median return > mode, it can be considered as positive skew. When
mean return < median return < mode, it can be considered as negative skew. Except
for ST, the other 5 returns in the stock markets are negative skewed.
6. Kurtosis
The formula of kurtosis is:
T
nn 1
K
X X
n 1n 2n 3 t 1 t
4
2
3n 1
/
n 2n 3
4
Kurtosis is a statistical measure used to describe the distribution of observed data
around the mean. It is sometimes referred to as the ‘volatility of volatility’. The
kurtosis of distributions is in one of three categories of classification:
Mesokurtic: Mesokurtic is a distribution that is peaked in the same way as any
normal distribution, not just the standard normal distribution. When ρ is close to
1/2 are considered to be mesokurtic.
Leptokurtic: A leptokurtic distribution is one that has kurtosis greater than a
mesokurtic distribution.
Platykurtic: Platykurtic distributions are those that have a peak lower than a
mesokurtic distribution.
According to Table Ab1, we can tell that the kurtosis of FTSE is close to 0.5, so the
kurtosis of FTSE can be considered as mesokurtic. The kurtosis of ASX is lower than
0.5, therefore it is leptokurtic. And the kurtosis of other returns is higher than 0.5,
which can be considered as playkurtic.
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7. Coefficient of Variation (CV)
The equation of CV is shown below:
CV =
σ
X´
The coefficient of variation represents the ratio of the standard deviation to the mean,
and it is a useful statistic for comparing the degree of variation from one data series to
another. In the investing world, the CV can be used to determine how much volatility
(risk) investors are assuming in comparison to the amount of return they can expect
from the investment. In the other words, the lower the ratio of standard deviation to
mean return is, the better risk-return will be traded off. In accordance to Table Ab1,
SP500 has the lowest CV (5.501743278), which means that investing SP500 will face
the least risk. In the meantime, the highest CV (137.109706) from TOPIX illustrate
that investing TOPIX will face more risk.
8. The probability of obtaining negative return
To get the probability of obtaining negative return, the first step is to calculate the zscore. The equation of calculating negative z-score is:
( R− R´ )
0−´x
z= σ (Pr= σ )
Probability of negative return shows the probability of receiving a negative return for
each portfolio over the selected time periods. From Table Ab1, we can tell that it is
the most likely that investing TOPIX market would get the negative returns while
investing SP500 would has least probability to get negative returns.
9. Conclusion
After considering all of these 8 categories, the most typical markets are SP500 and
TOPIX. When investing SP500, the investors would gain more returns in accordance
with the arithmetic means, geometric means and cumulative wealth index and face
less risk according to standard deviation, CV and the probability of obtaining negative
returns. In the other hands, TOPIX gains less returns but high risk. Therefore,
investing SP500 would be better than investing TOPIX.
Victoria University College of Business
BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
(c) Compare risk and return from these investments using the appropriate plot.
Table Ac1
Risk
Returns
(STD)
(Arithmetic Mean)
US$
0.032199969
0.001431609
SP500
0.041954946
0.007625755
ASX
0.038603833
0.005147363
TOPIX
0.05647056
0.000411864
FTSE
0.042067213
0.005446177
ST
0.065524483
0.006474094
Risk and Return
Return
0.01
0.01
0.01
0.01
0.01
0
0
0
0
0
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
Risk
Figure Ac1
Generally speaking, in the share market, high return comes with high risk. They are
positively correlated. Investors choose different projects to satisfy their different
investment needs. The investors are classified into three types:
Risk averse: This kind of investors aims to avoid risk.
Risk neutral: Risk neutral want to maximize returns without regard to the risks.
Risk seeking: The risk seeking investor always abandons some returns to
increase the risks.
As shown in Figure Ac1, US$ (exchange rate) is the typical example for the ‘high
return high risk’ theory. US$ has the lowest standard deviation which ca be regarded
as risk among these six share prices indices and it also has a considerable low return.
In contract, ST has the highest risk to invest in and considerable high return. There is
also exception existing, for example TOPIX has the second highest standard deviation
Victoria University College of Business
BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
but the lowest arithmetic mean. After comparing all of six share price indices, risk
averse investors may be interested in US$ as it has the lowest risk. And risk neutral
investors may pay more attention on SP500 because of its high return. For the risk
seeking investors, they may invest in ST which has the highest risk, because they
would like to sacrifice the returns to increase risks. Throughout six share price
indices, SP500 has the highest return but medium risk, which may be better to invest
in.
(d) Perform a histogram graph of the returns and explain the distribution of the
returns. (Use SPSS/ Excel)
1. US$ Return
US$ Frequency
Frequency
63 60
53 58
60
Frequency
40
20
0
1
1
4
6
7
17
22
15
8
4
0 0 0 0
91 326 236 394 429 463 498 532 567 601 636 267 295 261 227 192 158 ore
2
2 2 4 2 2 2 2 2 2 2 2 6 7 7 7 7 7 M
58 966 347 982 490 998 506 014 522 030 538 004 445 937 429 921 413
4
6 4 1 1 0 8 7 6 4 3 1 0 1 2 4 5 7
.1 .1 -0. 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 -0. 0.0 0.0 0.0 0.0 0.0
-0 -0
US$
Figure D1
As shown in Figure D1, US$ return shows a negative skewness histogram (mean <
median < mode). The mean of US$ is 0.001431609 and the median and the mode of
US$ are same which is 0.014457295. And we can also get it from the skewness
which is -0.562719367. We can tell that US$ return has a long left tail and its
kurtosis is larger than 0.5 which means that it has a greater peak than the normal
distribution. Hence, the US$ return is leptokurtic distribution.
2. SP500 Return
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SP500 Frequency
Frequency
Frequency
60
50
40
30
20
10
0
57 55
42
31
1
2
1
3
49
24
9 10 12
12 8
3
0
0
3 88 03 18 33 47 62 77 92 07 22 64 49 34 19 04 89 re
7
o
1 5 0 4 8 2 6 0 4 9 3 2 8 4 0 6 1
09 773 238 702 166 631 095 560 024 488 953 582 117 653 189 724 260 M
3
9 2 6 9 3 6 0 3 7 0 3 2 9 5 2 8 5
16 .15 .13 .11 .10 .08 .07 .05 .03 .02 .00 .01 .02 .04 .06 .07 .09
.
-0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 0 0 0 0
SP500
Figure D2
Figure D2 shows that SP500 frequency has a long left tail which means it is negative
skewness (mean < median< mode). As it has a greater kurtosis than the normal
distribution (1.227791784 > 0.5). Therefore, SP500 frequency is a leptokurtic
distribution.
3. ASX Return
ASX Frequency
Frequency
60
44 44
Frequency
40
46
36
24 24
20
0
51
1
3
1
12 12
10 10
1
0
0 0
4 84 83 83 82 82 81 81 81 32 42 21 21 22 22 23 23 re
8
o
15 464 813 162 511 860 209 558 907 774 939 045 696 347 998 649 300 M
1
7 5 3 2 0 8 7 5 3 9 4 1 2 4 5 7 9
26 111 096 081 066 050 035 020 005 .00 .02 040 055 070 085 100 115
1
.
.
.
.
.
0 0 0. 0. 0. 0. 0. 0.
.
.
.
.
-0 -0 -0 -0 -0 -0 -0 -0 -0
ASX
Figure D3
From Figure D3, the mean of ASX return is 0.005147363. The median is 0.00977432
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BEO2431 Risk Management Model Semester 2 2014
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and the mode is 0.02493942. So mean < median < mode, which means it is negative
skewness. And its kurtosis is less than 0.5. Therefore, ASX return is platykurtic
distribution, which has a lower peak than normal distribution.
4. TOPIX Return
TOPIX Frequency
Frequency
60
Frequency
40
21
20
0
1
1
1
4
31
40
56 52
47
30
12 9
4
8
1
1
0
8 55 01 47 93 64 86 32 78 75 29 83 37 59 44 98 52 re
0
o
54 192 731 269 807 834 884 423 961 499 961 422 884 134 807 268 730 M
6
5 8 0 3 5 0 0 3 5 1 8 6 3 1 8 6 3
04 181 159 136 113 .09 068 045 022 000 022 045 068 .09 113 136 159
2
.
.
.
.
0 0. 0. 0.
. -0 0. 0. 0. 0. 0. 0. 0.
-0 -0 -0 -0 -0
TOPIX
Figure D3
The arithmetic mean of TOPIX frequency is 0.00041186. The mode and median of
TOPIX are 0.000149975. As the skewness of TOPIX return distribution is negative.
Because kurtosis of TOPIX return is 0.874432085, larger than 0.5, it is a leptokurtic
distribution which means that it has a greater peak than normal distribution.
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5. FTSE Return
FTSE Frequency
Frequency
53 52 50
60
Frequency
40
20
0
1
1
2
6 10 11
21
27
42
17 13
9
2
1
1
1 8 5 3 8 7 5 2 9 4 6 9 2 5 7 9 3 re
25 310 796 282 976 253 739 225 710 803 317 831 346 860 374 588 403 Mo
8
41 431 820 210 659 989 378 768 157 452 063 673 284 894 505 111 726
0
3 1 9 8 0 4 3 1 0 1 3 4 6 7 9 1 2
.1 .1 .0 .0 -0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.1
-0 -0 -0 -0
-
FTSE
Figure D5
Similar with the first four returns above, FTSE return frequency has the negative
skewness. As shown in Figure D5, the distribution of FTSE frequency has a left tail
(mean < median < mode). Considering of the kurtosis, FTSE return distribution is
very close to 0.5, which is called mesokurtic distribution.
6. ST Return
ST Frequency
Frequency
73
41
40
0
4
1
4
4
7
17
88
46
16 7
6
1
1
1
2
0
-0
.2
-0 393
.2 9
4
-0 087 46
.1 1 6
7 4
-0 80 48
.1 3 3
4
-0 473 49
.1 5 9
4
-0 166 51
.0 7 6
4
-0 859 53
.0 9 3
5 4
-0 53 54
.0 1 9
2 4
0. 46 566
00 34
0. 60 582
03 45
0. 67 401
06 25
0. 74 384
09 05
0. 80 368
12 85
0. 87 351
15 65
0. 94 334
19 45
0. 01 318
22 25
0. 08 301
25 05
14 28
85 4
26
8
M
or
e
Frequency
80
ST
Figure D6
Compared with the five return frequencies above, ST return frequency is the only one
which has the positive skweness (mean < median < mode) and it has a long right tail.
Victoria University College of Business
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ZHIYI ZHOU & KAIYUAN ZHANG
Considering of the kurtosis which is 3.416980811, it is higher than 0.5, which means
it has a greater peak. Therefore, it is a leptokurtic distribution.
(e) Perform normality tests on the returns and explain why asset return is nonnormal. (Use Excel)
Table E1
US$
Mean
0.001431609
Median
Kurtosis
0.002485244
Skewness
2.274434
-0.562719367
5
SP500
0.007625755
0.011111111
1.227791
-0.612193408
8
ASX
0.005147363
0.0089982
0.344470
-0.368007992
7
TOPIX
0.000411864
0.000883392
0.874432
-0.102330303
1
FTSE
0.005446177
0.007740687
0.513573
-0.274207174
3
ST
0.006474094
0.010557855
3.416980
0.060597263
8
To test if the distribution is normal or not, mean, median, kurtosis and skewness are
applied to.
Mean & Median
Normal distribution’s mean and median are both equal to zero. As shown in Figure
E1, all of the returns’ mean and median are not equal to each other and do not equal to
zero neither. Therefore, the asset returns of all of these six share price indices are
non-normal in this field.
Kurtosis & Skewness
The normality requires that the asset return distributions’ kurtosis and skewness need
to be equal to zero. We can find out from the Figure E1 that all the kurtosis is larger
than zero. And the first five asset returns’ skewness is all negative and the last one is
positive. Therefore, the asset returns of all of these six share price indices are nonnormal in this field.
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Section II. Variance, Covariance Matrix and EMH Testing
B. Share Price Returns & Exchange Rate Return
(Data: Jan1986 to Jan 2014):
(a) Compute the correlation coefficients matrix for share prices and exchange
rates. (Use SPSS/ Excel) Comment on the between the shares refers to asset
selection for portfolio
Correlation
US$
SP500
ASX
TOPIX
FTSE
ST
US$
SP500
1
0.377035
0.386429
0.28858
0.326469
0.133918
1
0.62947
0.445345
0.788593
0.102732
Table Ba1
ASX
TOPIX
1
0.464591
0.647592
0.047323
1
0.430079
0.097533
FTSE
ST
1
0.058489
1
Correlation Coefficient measures the level of relation between two securities which is
the commonly used to calculate the portfolio risk. The correlation coefficient (r)
which is found by dividing the covariance between the returns on the securities by the
product of the standard deviations of their returns:
ρ
AB
=
Cov ( R
σ ,σ
A
A ,R B)
B
The value of the correlation is always located between -1 and 1. (-1AB 1)
•
=0, there is absolute no relation between the two group of the data.
•
=1, it means the two securities are perfect positive connecting and
positive relationship between the two parts of data so the movements of
one of seize will lead to the other one changes in the same direction
.
= -1, it means the two securities are perfect negative connecting, total
negative relationship between the two parts of data so if the movement
of one of seize happens the other one will change with the former one in
opposite direction.
•
It can be found from the table above that the correlation coefficient between SP500
and FTSE is 0.78859 which is the highest one among all the data. In other words,
these two shares affect each other in a obviously high degree so the investment
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concludes these two shares is a risky choice, However, the correlation coefficient
between ASX and ST is only 0.04732, which is the lowest among all the correlation
coefficient.
If investors invest all funds in a single asset, for example shares. They may face to
higher risk than asset portfolios. In investment portfolio, negative correlation can
help investors reduce risks. Because correlation of two assets is negative means if
one share price depreciates, the other asset share price will appreciate. In this case,
there is no negative correlation, so investors should choose the portfolio of ASX and
ST, because they have smallest correlation.
(b) Compute the variance and covariance matrix for share prices and exchange rate.
(Use SPSS/Excel) Comment on these covariance.
US$
0.001034
0.000508
0.000479
0.000523
0.000441
0.000282
US$
SP500
ASX
TOPIX
FTSE
ST
SP500
0.001755
0.001016
0.001052
0.001387
0.000282
Table Bb1
ASX
TOPIX
0.001486
0.00101
0.001048
0.000119
0.003179
0.001018
0.00036
FTSE
ST
0.001764
0.000161
0.00428
Variance is a measure of variability. In addition, it means the squared deviations from
the mean or expected value. Contrasts to Variance, Covariance are a raw measure of
the degree of association between two variables. Negative covariance implies that an
increase in returns on asset A is associated with a decrease in returns on asset B,
which means that they are expected to move in different direction. Positive covariance
implies move in same direction.
The formula of the covariance COV
n
COV ( Ri , R j )=
−
( Ri ,R j ) :
−
( Ri,t −R i ) ( R j,t −R j )
∑
t=1
The formula of the variance: n−1
n
2
σ 2 =∑ ( R i−E (R )) P i
t =1
It can be seen from the table that all of covariance data is positive which means the
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increase of the return in share A will lead to the increase of return in share B in the
same direction. In addition, the covariance between FTSE and SP 500 is the highest.
Therefore, SP and FTSE have the highest degree move in same direction, which
means there is an increase in returns on SP500, there will be a possibility that the
returns on FTSE will go up.
(c) Is the Share Prices satisfying weak-form efficiency? Each market, compute u t
where ut= pt –pt-1 and test to see if ut display any patterns of autocorrelation and
comment on the results
An efficient market describes the prices of all securities quickly and fully reflects all
available information about the assets. Furthermore, the efficient market hypothesis
divides the efficiency into three types that contain weak-form efficiency, semi-strong
form efficiency and strong-form efficiency. The correct implication of a weak-form
efficient market hypothesis is that the past history of price information is of no value
in assessing future changes in prices. If stock prices are determined in a market that is
weak-form efficient, historical price and volume data should already be reflected in
current prices and should be of no value in predicting future changes.
Statistical test the independence of stock-price changes is one-way to test for weakform efficiency. If statistical tests suggest that price changes are independent, the
implication is that knowing and using the past sequence of price information is of no
value to investors. Autocorrelation tests are measuring the autocorrelation between
prices changes for various lags, such as one day two day so on. Positive or negative
autocorrelation would indicate the possible existence of potentially profitable trading
strategies. Zero correlation is consistent with the random walk hypothesis.
Autocorrelation of Return is that Letting Xt be the return of an asset, the
autocorrelation between Xt and Xt-j is estimated as
T
∑ ( X t −X )( X t− j −X )
r j= t= j+1 T
∑t=1 ( X t −X ) 2
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US
$
US$
(t-1)
SP50
0(t-1)
ASX(
t-1)
SP5
00
AS
X
TO
PI
X
Table Bc1
FT ST
SE
0.0
49
76
0.0
646
45
0.1
198
41
TOP
IX(t1)
FTS
E(t1)
0.0
859
23
0.0
181
7
ST(t1)
0.0
601
34
It can be found from the table there are five prices are positive except FTSE which is 0.01817. In addition, the biggest one is 0.119841 and the smallest one is -001817. In
the table, both of the positive numbers and negative numbers are close to zero.
Therefore, the share prices all satisfy the weak-form efficiency.
Section III. Portfolio Construction
C. Calculate the expected returns and standard deviations on the following
portfolios:
1. Portfolio1: 50% Australian share (ASX200) and 50% UK share (FTSE 100)
Table C1
Share
Return
Weight
Standard
Correlation
Coefficient
0.647592
ASX
0.00515
0.5
Deviation
0.03860
FTSE
0.00545
0.5
0.04207
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Expected Return:
E( R p )=∑ W i ×E( Ri )
E( R )=0 . 5×0 . 00515+0 .5×0 . 00545=0 . 0053=0 .53 %
Standard Deviation:
δ p=√W 2A δ2A+W 2B δ 2B +2 ρ AB W A W B δ A δ B
δ P= √0.52×0.0386 2 +0.52×0.04207 2+2×0.5×0.5×0.0386×0.04207×0.647592
δ P=0.036617=3.66%
2. Portfolio 2: 45% Singapore Share, 25% Australian Share and 30% USA (S&P
500) Share
Share
Return
Table C2
Weight
Standard
ST
0.00647
0.45
Deviation
0.06552
ASX
0.00515
0.25
0.03860
SP
0.00763
0.30
0.04195
Correlation
Coefficient
ρST , ASX=0.04732
ρST ,SP =0.102732
ρ ASX ,SP=0.62947
Expected Return:
E( R p )=∑ W i ×E( Ri )
E(( R P )=0 .00647×0 . 45+0 . 00515×0 .25+0 . 00763×0 .3=0 .6488 %
Standard Deviation:
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δ p =√W 2A δ2A+W 2B δ 2B +W 2C δ2C +2 ρ AB W A W B+2ρ AC W A W C δ A δ C +2 ρ BC W B W C δ B δ C
2 22 22 2
δP=¿√0.45 ×0. 65 2 +0.25 ×0. 386 +0.3 ×0. 4195 +2×0. 4732×0.45×0.25¿ 0. 65 2×0. 386+2×0.102732×0.45×0.3×0. 65 2×0. 4195+2×0.62947׿0. 3860×0. 4195¿ δp=0. 37107=3.71%¿
3. Portfolio 3: 20% Australian share, 20% US share, 20% Singapore Share, 20%
United Kingdom and 20% Japan share
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Return
Share
ASX
SP500
ST
FTSE
Table C3
Weight
Standard
0.005147
0.2
Deviation
0.038604
0.007626
0.2
0.041955
0.006474
0.2
0.065524
0.005446
0.2
0.042067
0.000412
0.2
0.056471
TOPIX
Correlation
Coefficient
ρ ASX ,SP=0.62947
ρ ASX ,ST=0.04732
ρ ASX ,FT =0.64759
ρ ASX ,TO=0.46459
ρSP ,ST =0.102732
ρSP ,FT =0.78859
ρSP ,TO=0.44534
ρST ,FT =0.05848
ρST ,TO=0.09753
ρ FT,TO=0.43008
Expected Return:
E( R p )=∑ W i ×E( Ri )
=W ASX × R ASX +W SP500 × RSP 500 + W ST × R ST +W FTSE × R FTSE +W TOPIX × RTOPIX
=0.2×0.005147+0.2×0.007626+0.2×0.006474+0.2×0.005446+0.2×0.000412
= 0.005021
= 0.5021%
Standard deviations
W ASX2 σ ASX2 +W SP500 2 σ SP 500 2+W ST 2 σ ST 2+ W FTSE 2 σ FTSE2
+W TOPIX2 σ TOPIX 2
+2W ASX W SP 500 ρ ASX SP 500 σ ASX σ SP 500 +2 W ASX W ST ρ ASX ST σ ASX σ ST
+2 W ASX W FTSE ρ ASX FTSE σ ASX σ FTSE
σ P=
+ 2W ASX W TOPIX ρ ASX TOPIX σ ASX σ TOPIX +2 W SP 500 W ST ρ SP500 ST σ SP 500 σ ST
+2 W SP500 W FTSE ρ SP500 FTSE σ SP 500 σ FTSE
+2 W SP 500 W TOPIX ρSP 500 TOPIX σ SP 500 σ TOPIX +2 W ST W FTSE ρST FTSE σ ST σ FTSE
+2 W ST W TOPIX ρ ST TOPIX σ ST σ TOPIX +2 W FTSE W TOPIX ρ FTSE TOPIX σ FTSE σ TOPIX
√
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0.202 ×0.038604 2+ 0.202 × 0.0419552+ 0.202 × 0.0655242
+0.202 ×0.0420672 +0.20 2 ×0.0564712
+ 2× 0.20 ×0.20 ×0.62947 × 0.038604 ×0.041955
+ 2× 0.20× 0.20 ×0.047323 ×0.038604 × 0.065524
+2× 0.20 ×0.20 × 0.647592×0.038604 × 0.042067
= +2 ×0.20 ×0.20 × 0.0 .464591× 0.038604 ×0.056471
+2× 0.20 ×0.20 × 0.102732× 0.041955× 0.065524
+2× 0.20 ×0.20 × 0.788593× 0.041955× 0.042067
+2 ×0.20 ×0.20 × 0.445345× 0.041955 ×0.056471
+ 2× 0.20 ×0.20 ×0.058489 × 0.065524 ×0.042067
+2× 0.20 ×0.20 × 0.097533× 0.065524 ×0.056471
+2 ×0.20 ×0.20 × 0.430079× 0.042067 ×0.056471
√
= 0.0331 =3.31%
4. Comment on the results by using the plot of risk and return
Table C4
Portfolio 1
Portfolio 2
Portfolio 3
Return (Mean)
0.53%
0.6488%
0.5021%
STD (Risk)
3.66%
3.71%
3.31%
0.04
0.04
0.04
0.04
0.03
0.03
0.03
Return
Standard Deviation
0.02
0.02
0.01
0.01
0.01
0.01
0
0.5
1
1.5
2
0.01
2.5
3
3.5
Figure C4
By the analysis of the plot of risk and return, it can be proved that higher return is
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always with higher risk. If the investor intends to get increasingly return, they need to
face higher risk.
If investors want to get the risk averse, they will choose the portfolio 3.
If investors want to get the risk neutral, they will choose the portfolio 1.
If investors want to get the risk seeking, they will choose portfolio 2.
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Victoria University College of Business
BEO2431 Risk Management Model
Semester 2 2014
Case Study 1
Presented by:
ZHIYI ZHOU 3923070
KAIYUAN ZHANG 3920263
Presented to:
Chinthana Hatangala
Tutorial No.3
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A. Compute the returns (RT) of the following as:
RT = ((Pt – Pt-1)/P t-1)
Share Prices (Monthly Data: Nov1987 to June 2014)
US$: Exchange rates: United States Dollar
SP500: Share price indices: United States: S&P 500
ASX: Share market: Share price indices: S&P/ASX 200
TOPIX: Share price indices: Japan: TOPIX
FTSE: Share price indices: United Kingdom: FTSE 100
ST: Share price indices: Singapore: Straits Times:
For each countries share price and exchange rate
(a) Plot each countries return over the time period. Comment on the volatility and
volatility clustering of the returns (Use Excel)
The volatility clustering is caused by the large changes in price. After recalling and
analyzing the phenomenon, several economic mechanisms are discussed depending
on the explaining of the volatility clustering.
1. Returns - US$
Returns-US$
Returns-US
0.15
High
Volatility
Returns-US $
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
Time
Figure A1
As shown in the Figure A1, the return of US$ experienced large fluctuation during
1987 to 1989, 1998 to 2004 and 2007 to 2013. The first volatility happened during
November 1987 to January 1989 changed between -10% and 6%. The second
volatility area (between -8% and 6%) happened during 1998 to 2004. And the most
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significant volatility happened between September 2007 and July 2013, where the
market crash (-16%) happened. The high volatility shows the unstable financial
environment which can be considered as bad news for investment while the low
volatility shows an anticipated announcement. The volatility may be caused by
government’s intervention of finance, as government run a tight monetary policy
during 70s to 80s.
2. Returns – SP500
Returns-SP500
Returns-SP500
Returns-SP500
0.15
High
Volatility
Low
0.1
Volatility
0.05
0
-0.05 87 89 90 91 92 94 95 96 97 99 00 01 02 04 05 06 07 09 10 11 12 14
- - - - - - - - - - - - - - - - - - - - - -0.1
ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb
N
N
N
N
N
N
M
M
M
M
M
-0.15
-0.2
Market
Time
Crash
Figure A2
In Figure A2, there are only two low volatility areas, which are during July 1992 to
January 1996 and during March 2004 to July 2006. The high volatility areas cluster
during November 1987 to May 1991, March 1997 to January 2003 and September
2007 to July 2013. The high volatility area shows the unstable United Stated share
price indices. During the first 4 years of this line chart, the returns fluctuated between
-9% and 10%, where the United States government run a tight monetary policy. The
second high volatility area changed between -15% and 10%, where the financial crisis
happened. The last high volatility was during the subprime crisis and the market
crash, -16%, happened between September 2007 and November 2008. Therefore, the
volatility clustering can show the finance environment correctly and accurately.
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3. Returns – ASX
Returns-ASX
Returns-ASX
0.15
0.1
0.05
High
Volatility
Returns-ASX
Low
Volatility
0
-0.05-87 -89 -90 -91 -92 -94 -95 -96 -97 -99 -00 -01 -02 -04 -05 -06 -07 -09 -10 -11 -12 -14
ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb
-0.1
N F M A N F M A N F M A N F M A N F M A N F
-0.15
Market
Time
Crash
Figure A3
Compared with the two markets above, the share price indices fluctuated stronger.
But the scope of the fluctuation is weaker than United States Market. Except for the
time period during November 2002 to November 2007, this market experienced
relatively strong fluctuation in the rest of time during 1987 to 2014. The market crash
(-12%) happened near July 2009, which is probably caused by the subprime crisis
influence from United States. For the most time of Australian financial market, the
returns change a lot. There is not an exactly gap between the low volatility clustering
and high volatility clustering.
4. Returns – TOPIX
Returns-TOPIX
Returns-TOPIX
0.3
Market
Crash
0.2
0.1
High
Volatility
Returns-TOPIX
Low
Volatility
0
-0.1-87 -89 -90 -91 -92 -94 -95 -96 -97 -99 -00 -01 -02 -04 -05 -06 -07 -09 -10 -11 -12 -14
ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb ay ug ov eb
-0.2
N F M A N F M A N F M A N F M A N F M A N F
-0.3
Time
Figure A4
Compared with United States and Australian financial markets, fluctuation scope of
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the returns of Japanese financial market changes stronger. Different with the first two
countries’ financial markets, the returns of first two years did not change a lot
comparing with the following 7 years. During January 1991 to August 1992, the
Japanese financial market experienced the highest and lowest returns, which are 19%
and -21%. Because of the close relationship between Japanese and United States
financial market, the unstable phenomena in Japan may be affected by the tight
monetary policy of United States government. In the year of 2009, Japan was also
influenced by subprime crisis. Therefore, there is a high volatility area during 2008 to
2010.
5. Returns – FTSE
Returns-FTSE 100
Returns-FTSE 100
Returns-FTSE 100
0.2
0.15
0.1
0.05
High
Volatility
Low
Volatility
0
-0.05 87 89 90 91 92 94 95 96 97 99 00 01 02 04 05 06 07 09 10 11 12 14
- - - - - - - - - - - - - - - - - - - - - ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb
-0.1
N
N
N
N
N
N
M
M
M
M
M
-0.15
Time
Market
Crash
Figure A5
The United Kingdom financial market has two low volatility areas, which are between
July 1994 to November 1997 and July 2004 to November 2007. The first fluctuation
was during July 1989 to July 1994 and the returns reached its highest in 1989. The
second high volatility was during November 1997 to July 2004 which is the period of
global finance crisis. The market crash happened in 2008, which is also during the
subprime crisis. The line chart of returns fully explains the financial phenomenon,
which shows the importance of volatility clustering.
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6. Returns - ST
Returns-ST
Returns-ST
High
Volatility
0.4
Returns-ST
0.3
Low
0.2
Volatility
0.1
0
-0.1 87 89 90 91 92 94 95 96 97 99 00 01 02 04 05 06 07 09 10 11 12 14
- - - - - - - - - - - - - - - - - - - - - ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb ay Aug ov Feb
-0.2
N
N
N
N
N
N
M
M
M
M
M
-0.3
Market
Crash
Time
Figure A6
Compared with the countries’ financial markets above, the financial market of
Singapore has longer low volatility area and higher returns. The first 10 years from
November 1987 to November 1997 experienced stationary returns. During the global
finance crisis, the returns changed from -20% to 29%, which is the largest gap within
the six markets. From November 2002 to November 2007, the returns kept stable and
flat. And we can tell from the line chart that Singapore was also affected by the
subprime crisis. The returns in Singapore financial market fluctuated large during
these two periods of time.
7. Conclusion
After analyze all of these six markets returns, it can be found that volatility exist in
each return all the time. Commonly, the degree of the volatility of returns is the
response of the real economic activities in the countries.
(b) Perform summary descriptive statistics on the returns (Use Excel) and explain
the use of the following measures in finance:
1. Arithmetic mean, 2. Geometric Mean, 3. Cumulative wealth Index 4. Standard
Deviation (risk), 5. Skewness, 6. Kurtosis and 7. Coefficient of Variation (CV),
8. The probability of obtaining negative return.
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Table Ab1
US$
SP500
ASX
TOPIX
FTSE
ST
Arithmetic Mean
0.001431609
0.007625755
0.005147363
0.00041186
0.005446177
0.006474094
Geometric Mean
0.00090801
0.006737345
0.004399867
-0.00119186
0.004559892
0.004327462
Cumulative Wealth Index
1.335791265
8.516499283
4.057142857
0.68356808
4.268656716
3.964905248
Standard Deviation
0.032199969
0.041954946
0.038603833
0.05647056
0.042067213
0.065524483
Kurtosis
2.274434463
1.227791784
0.344470699
0.87443209
0.513573283
3.416980811
-0.562719367
-0.612193408
-0.368007992
-0.1023303
-0.274207174
0.060597263
22.49215927
5.501743278
7.499730498
137.109706
7.724172784
10.12102738
-0.044459938
-0.181760571
-0.133338125
-0.00729343
-0.129463702
-0.098804199
48.40%
42.86%
44.83%
49.60%
44.83%
46.02%
Skewness
CV
Z-Score
Prob of Z-Score
1. Arithmetic Mean
The formula of arithmetic mean is shown below:
n
∑ Ri
AM = i=1n
Arithmetic mean measures the average performance of all of the data and predicts the expected return for next period. As the formula above
shows that arithmetic mean equals to the sum of all the returns divided by the number of years. In the Table Ab1, it is obvious that SP500 has
the highest arithmetic mean (0.007625755) while TOPIX has the lowest arithmetic mean (0.00041186). Therefore, considering the arithmetic
mean aspect, SP500 can be regarded as the best investment as a whole, which means it has the highest investment return in the next period, while
TOPIX has the lowest expected return in the next period.
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2. Geometric mean
The formula of geometric mean is:
GM =√n (RR 1)(RR 2 )(RR ¿ ¿ 3)… (RR ¿ ¿ n)−1 ¿ ¿
Geometric mean is used to illustrate the degree of the growth of the return by taking
the antilog of the product resulting from multiplying a series of return relatives
together. Therefore, geometric mean can be considered as compound rate of return
over time. From Table Ab1, SP500 has the highest geometric mean (0.006737345),
which means SP500 has the greatest variability of the return, while TOPIX has the
weakest variability of return. Therefore, SP500 has the greatest spread, as well as the
compound return. In contract, TOPIX (-0.00119186) has the lowest spread, as well as
the compound return.
3. Cumulative Wealth Index
The formula of cumulative wealth index is:
CWI n=WI 0 ( RR 1 ) ( RR 2 ) … ( RRn )
CWI n: The cumulative wealth index as of the end of the period n
WI 0: The beginning index value, typically $1
Cumulative wealth index is the accumulation of wealth all the time. It measures the
cumulative effect of returns over time, typically on the basis pf a $1 invested. It can
be calculated from the initial wealth, then to estimate the return in the end of year. It
reflects the level of wealth rather than the change in the wealth. As shown in Table
Ab1, SP500 has the largest cumulative index (8.516499283) and TOPIX has the
lowest cumulated index (0.68356808). Hence, SP500 gains the highest accumulation
of wealth while TOPIX has the lowest one, from which we can predict that SP500
will have the highest return in the end of the year.
4. Standard Deviation
The equation of the standard deviation is:
T
N
σ = N −1 ∑ ( X t − X´ )2
t=1
√
The standard deviation is a measure of how spread out numbers are. In other words,
standard deviation is a statistical measurement that sheds light on historical volatility.
In the field of finance, the standard deviation is applied to the annual rate of return of
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an investment to measure the investment's volatility. In Table Ab1, ST has the largest
standard deviation (0.065524483), which means ST stock is unstable, while US$ has
the lowest deviation (0.032199969). Hence, people will face more risk when
investing in ST stock whereas it is safest to invest in US$.
5. Skewness
The formula used to calculate skewness is:
T
n
´ )3 /σ 3
S=
( X t− X
∑
(n−1)(n−2) t =1
Skewness describes asymmetry from the normal distribution in a set of statistical data.
Skewness can come in the form of "negative skewness" or "positive skewness",
depending on whether data points are skewed to the left (negative skew) or to the
right (positive skew) of the data average. The third situation is that if there is no
skewness, it means that distributions are symmetric (normal distribution). When
mean return > median return > mode, it can be considered as positive skew. When
mean return < median return < mode, it can be considered as negative skew. Except
for ST, the other 5 returns in the stock markets are negative skewed.
6. Kurtosis
The formula of kurtosis is:
T
nn 1
K
X X
n 1n 2n 3 t 1 t
4
2
3n 1
/
n 2n 3
4
Kurtosis is a statistical measure used to describe the distribution of observed data
around the mean. It is sometimes referred to as the ‘volatility of volatility’. The
kurtosis of distributions is in one of three categories of classification:
Mesokurtic: Mesokurtic is a distribution that is peaked in the same way as any
normal distribution, not just the standard normal distribution. When ρ is close to
1/2 are considered to be mesokurtic.
Leptokurtic: A leptokurtic distribution is one that has kurtosis greater than a
mesokurtic distribution.
Platykurtic: Platykurtic distributions are those that have a peak lower than a
mesokurtic distribution.
According to Table Ab1, we can tell that the kurtosis of FTSE is close to 0.5, so the
kurtosis of FTSE can be considered as mesokurtic. The kurtosis of ASX is lower than
0.5, therefore it is leptokurtic. And the kurtosis of other returns is higher than 0.5,
which can be considered as playkurtic.
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7. Coefficient of Variation (CV)
The equation of CV is shown below:
CV =
σ
X´
The coefficient of variation represents the ratio of the standard deviation to the mean,
and it is a useful statistic for comparing the degree of variation from one data series to
another. In the investing world, the CV can be used to determine how much volatility
(risk) investors are assuming in comparison to the amount of return they can expect
from the investment. In the other words, the lower the ratio of standard deviation to
mean return is, the better risk-return will be traded off. In accordance to Table Ab1,
SP500 has the lowest CV (5.501743278), which means that investing SP500 will face
the least risk. In the meantime, the highest CV (137.109706) from TOPIX illustrate
that investing TOPIX will face more risk.
8. The probability of obtaining negative return
To get the probability of obtaining negative return, the first step is to calculate the zscore. The equation of calculating negative z-score is:
( R− R´ )
0−´x
z= σ (Pr= σ )
Probability of negative return shows the probability of receiving a negative return for
each portfolio over the selected time periods. From Table Ab1, we can tell that it is
the most likely that investing TOPIX market would get the negative returns while
investing SP500 would has least probability to get negative returns.
9. Conclusion
After considering all of these 8 categories, the most typical markets are SP500 and
TOPIX. When investing SP500, the investors would gain more returns in accordance
with the arithmetic means, geometric means and cumulative wealth index and face
less risk according to standard deviation, CV and the probability of obtaining negative
returns. In the other hands, TOPIX gains less returns but high risk. Therefore,
investing SP500 would be better than investing TOPIX.
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(c) Compare risk and return from these investments using the appropriate plot.
Table Ac1
Risk
Returns
(STD)
(Arithmetic Mean)
US$
0.032199969
0.001431609
SP500
0.041954946
0.007625755
ASX
0.038603833
0.005147363
TOPIX
0.05647056
0.000411864
FTSE
0.042067213
0.005446177
ST
0.065524483
0.006474094
Risk and Return
Return
0.01
0.01
0.01
0.01
0.01
0
0
0
0
0
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
Risk
Figure Ac1
Generally speaking, in the share market, high return comes with high risk. They are
positively correlated. Investors choose different projects to satisfy their different
investment needs. The investors are classified into three types:
Risk averse: This kind of investors aims to avoid risk.
Risk neutral: Risk neutral want to maximize returns without regard to the risks.
Risk seeking: The risk seeking investor always abandons some returns to
increase the risks.
As shown in Figure Ac1, US$ (exchange rate) is the typical example for the ‘high
return high risk’ theory. US$ has the lowest standard deviation which ca be regarded
as risk among these six share prices indices and it also has a considerable low return.
In contract, ST has the highest risk to invest in and considerable high return. There is
also exception existing, for example TOPIX has the second highest standard deviation
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but the lowest arithmetic mean. After comparing all of six share price indices, risk
averse investors may be interested in US$ as it has the lowest risk. And risk neutral
investors may pay more attention on SP500 because of its high return. For the risk
seeking investors, they may invest in ST which has the highest risk, because they
would like to sacrifice the returns to increase risks. Throughout six share price
indices, SP500 has the highest return but medium risk, which may be better to invest
in.
(d) Perform a histogram graph of the returns and explain the distribution of the
returns. (Use SPSS/ Excel)
1. US$ Return
US$ Frequency
Frequency
63 60
53 58
60
Frequency
40
20
0
1
1
4
6
7
17
22
15
8
4
0 0 0 0
91 326 236 394 429 463 498 532 567 601 636 267 295 261 227 192 158 ore
2
2 2 4 2 2 2 2 2 2 2 2 6 7 7 7 7 7 M
58 966 347 982 490 998 506 014 522 030 538 004 445 937 429 921 413
4
6 4 1 1 0 8 7 6 4 3 1 0 1 2 4 5 7
.1 .1 -0. 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.0 -0. 0.0 0.0 0.0 0.0 0.0
-0 -0
US$
Figure D1
As shown in Figure D1, US$ return shows a negative skewness histogram (mean <
median < mode). The mean of US$ is 0.001431609 and the median and the mode of
US$ are same which is 0.014457295. And we can also get it from the skewness
which is -0.562719367. We can tell that US$ return has a long left tail and its
kurtosis is larger than 0.5 which means that it has a greater peak than the normal
distribution. Hence, the US$ return is leptokurtic distribution.
2. SP500 Return
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SP500 Frequency
Frequency
Frequency
60
50
40
30
20
10
0
57 55
42
31
1
2
1
3
49
24
9 10 12
12 8
3
0
0
3 88 03 18 33 47 62 77 92 07 22 64 49 34 19 04 89 re
7
o
1 5 0 4 8 2 6 0 4 9 3 2 8 4 0 6 1
09 773 238 702 166 631 095 560 024 488 953 582 117 653 189 724 260 M
3
9 2 6 9 3 6 0 3 7 0 3 2 9 5 2 8 5
16 .15 .13 .11 .10 .08 .07 .05 .03 .02 .00 .01 .02 .04 .06 .07 .09
.
-0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 0 0 0 0
SP500
Figure D2
Figure D2 shows that SP500 frequency has a long left tail which means it is negative
skewness (mean < median< mode). As it has a greater kurtosis than the normal
distribution (1.227791784 > 0.5). Therefore, SP500 frequency is a leptokurtic
distribution.
3. ASX Return
ASX Frequency
Frequency
60
44 44
Frequency
40
46
36
24 24
20
0
51
1
3
1
12 12
10 10
1
0
0 0
4 84 83 83 82 82 81 81 81 32 42 21 21 22 22 23 23 re
8
o
15 464 813 162 511 860 209 558 907 774 939 045 696 347 998 649 300 M
1
7 5 3 2 0 8 7 5 3 9 4 1 2 4 5 7 9
26 111 096 081 066 050 035 020 005 .00 .02 040 055 070 085 100 115
1
.
.
.
.
.
0 0 0. 0. 0. 0. 0. 0.
.
.
.
.
-0 -0 -0 -0 -0 -0 -0 -0 -0
ASX
Figure D3
From Figure D3, the mean of ASX return is 0.005147363. The median is 0.00977432
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and the mode is 0.02493942. So mean < median < mode, which means it is negative
skewness. And its kurtosis is less than 0.5. Therefore, ASX return is platykurtic
distribution, which has a lower peak than normal distribution.
4. TOPIX Return
TOPIX Frequency
Frequency
60
Frequency
40
21
20
0
1
1
1
4
31
40
56 52
47
30
12 9
4
8
1
1
0
8 55 01 47 93 64 86 32 78 75 29 83 37 59 44 98 52 re
0
o
54 192 731 269 807 834 884 423 961 499 961 422 884 134 807 268 730 M
6
5 8 0 3 5 0 0 3 5 1 8 6 3 1 8 6 3
04 181 159 136 113 .09 068 045 022 000 022 045 068 .09 113 136 159
2
.
.
.
.
0 0. 0. 0.
. -0 0. 0. 0. 0. 0. 0. 0.
-0 -0 -0 -0 -0
TOPIX
Figure D3
The arithmetic mean of TOPIX frequency is 0.00041186. The mode and median of
TOPIX are 0.000149975. As the skewness of TOPIX return distribution is negative.
Because kurtosis of TOPIX return is 0.874432085, larger than 0.5, it is a leptokurtic
distribution which means that it has a greater peak than normal distribution.
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5. FTSE Return
FTSE Frequency
Frequency
53 52 50
60
Frequency
40
20
0
1
1
2
6 10 11
21
27
42
17 13
9
2
1
1
1 8 5 3 8 7 5 2 9 4 6 9 2 5 7 9 3 re
25 310 796 282 976 253 739 225 710 803 317 831 346 860 374 588 403 Mo
8
41 431 820 210 659 989 378 768 157 452 063 673 284 894 505 111 726
0
3 1 9 8 0 4 3 1 0 1 3 4 6 7 9 1 2
.1 .1 .0 .0 -0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0.1
-0 -0 -0 -0
-
FTSE
Figure D5
Similar with the first four returns above, FTSE return frequency has the negative
skewness. As shown in Figure D5, the distribution of FTSE frequency has a left tail
(mean < median < mode). Considering of the kurtosis, FTSE return distribution is
very close to 0.5, which is called mesokurtic distribution.
6. ST Return
ST Frequency
Frequency
73
41
40
0
4
1
4
4
7
17
88
46
16 7
6
1
1
1
2
0
-0
.2
-0 393
.2 9
4
-0 087 46
.1 1 6
7 4
-0 80 48
.1 3 3
4
-0 473 49
.1 5 9
4
-0 166 51
.0 7 6
4
-0 859 53
.0 9 3
5 4
-0 53 54
.0 1 9
2 4
0. 46 566
00 34
0. 60 582
03 45
0. 67 401
06 25
0. 74 384
09 05
0. 80 368
12 85
0. 87 351
15 65
0. 94 334
19 45
0. 01 318
22 25
0. 08 301
25 05
14 28
85 4
26
8
M
or
e
Frequency
80
ST
Figure D6
Compared with the five return frequencies above, ST return frequency is the only one
which has the positive skweness (mean < median < mode) and it has a long right tail.
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Considering of the kurtosis which is 3.416980811, it is higher than 0.5, which means
it has a greater peak. Therefore, it is a leptokurtic distribution.
(e) Perform normality tests on the returns and explain why asset return is nonnormal. (Use Excel)
Table E1
US$
Mean
0.001431609
Median
Kurtosis
0.002485244
Skewness
2.274434
-0.562719367
5
SP500
0.007625755
0.011111111
1.227791
-0.612193408
8
ASX
0.005147363
0.0089982
0.344470
-0.368007992
7
TOPIX
0.000411864
0.000883392
0.874432
-0.102330303
1
FTSE
0.005446177
0.007740687
0.513573
-0.274207174
3
ST
0.006474094
0.010557855
3.416980
0.060597263
8
To test if the distribution is normal or not, mean, median, kurtosis and skewness are
applied to.
Mean & Median
Normal distribution’s mean and median are both equal to zero. As shown in Figure
E1, all of the returns’ mean and median are not equal to each other and do not equal to
zero neither. Therefore, the asset returns of all of these six share price indices are
non-normal in this field.
Kurtosis & Skewness
The normality requires that the asset return distributions’ kurtosis and skewness need
to be equal to zero. We can find out from the Figure E1 that all the kurtosis is larger
than zero. And the first five asset returns’ skewness is all negative and the last one is
positive. Therefore, the asset returns of all of these six share price indices are nonnormal in this field.
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Section II. Variance, Covariance Matrix and EMH Testing
B. Share Price Returns & Exchange Rate Return
(Data: Jan1986 to Jan 2014):
(a) Compute the correlation coefficients matrix for share prices and exchange
rates. (Use SPSS/ Excel) Comment on the between the shares refers to asset
selection for portfolio
Correlation
US$
SP500
ASX
TOPIX
FTSE
ST
US$
SP500
1
0.377035
0.386429
0.28858
0.326469
0.133918
1
0.62947
0.445345
0.788593
0.102732
Table Ba1
ASX
TOPIX
1
0.464591
0.647592
0.047323
1
0.430079
0.097533
FTSE
ST
1
0.058489
1
Correlation Coefficient measures the level of relation between two securities which is
the commonly used to calculate the portfolio risk. The correlation coefficient (r)
which is found by dividing the covariance between the returns on the securities by the
product of the standard deviations of their returns:
ρ
AB
=
Cov ( R
σ ,σ
A
A ,R B)
B
The value of the correlation is always located between -1 and 1. (-1AB 1)
•
=0, there is absolute no relation between the two group of the data.
•
=1, it means the two securities are perfect positive connecting and
positive relationship between the two parts of data so the movements of
one of seize will lead to the other one changes in the same direction
.
= -1, it means the two securities are perfect negative connecting, total
negative relationship between the two parts of data so if the movement
of one of seize happens the other one will change with the former one in
opposite direction.
•
It can be found from the table above that the correlation coefficient between SP500
and FTSE is 0.78859 which is the highest one among all the data. In other words,
these two shares affect each other in a obviously high degree so the investment
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concludes these two shares is a risky choice, However, the correlation coefficient
between ASX and ST is only 0.04732, which is the lowest among all the correlation
coefficient.
If investors invest all funds in a single asset, for example shares. They may face to
higher risk than asset portfolios. In investment portfolio, negative correlation can
help investors reduce risks. Because correlation of two assets is negative means if
one share price depreciates, the other asset share price will appreciate. In this case,
there is no negative correlation, so investors should choose the portfolio of ASX and
ST, because they have smallest correlation.
(b) Compute the variance and covariance matrix for share prices and exchange rate.
(Use SPSS/Excel) Comment on these covariance.
US$
0.001034
0.000508
0.000479
0.000523
0.000441
0.000282
US$
SP500
ASX
TOPIX
FTSE
ST
SP500
0.001755
0.001016
0.001052
0.001387
0.000282
Table Bb1
ASX
TOPIX
0.001486
0.00101
0.001048
0.000119
0.003179
0.001018
0.00036
FTSE
ST
0.001764
0.000161
0.00428
Variance is a measure of variability. In addition, it means the squared deviations from
the mean or expected value. Contrasts to Variance, Covariance are a raw measure of
the degree of association between two variables. Negative covariance implies that an
increase in returns on asset A is associated with a decrease in returns on asset B,
which means that they are expected to move in different direction. Positive covariance
implies move in same direction.
The formula of the covariance COV
n
COV ( Ri , R j )=
−
( Ri ,R j ) :
−
( Ri,t −R i ) ( R j,t −R j )
∑
t=1
The formula of the variance: n−1
n
2
σ 2 =∑ ( R i−E (R )) P i
t =1
It can be seen from the table that all of covariance data is positive which means the
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increase of the return in share A will lead to the increase of return in share B in the
same direction. In addition, the covariance between FTSE and SP 500 is the highest.
Therefore, SP and FTSE have the highest degree move in same direction, which
means there is an increase in returns on SP500, there will be a possibility that the
returns on FTSE will go up.
(c) Is the Share Prices satisfying weak-form efficiency? Each market, compute u t
where ut= pt –pt-1 and test to see if ut display any patterns of autocorrelation and
comment on the results
An efficient market describes the prices of all securities quickly and fully reflects all
available information about the assets. Furthermore, the efficient market hypothesis
divides the efficiency into three types that contain weak-form efficiency, semi-strong
form efficiency and strong-form efficiency. The correct implication of a weak-form
efficient market hypothesis is that the past history of price information is of no value
in assessing future changes in prices. If stock prices are determined in a market that is
weak-form efficient, historical price and volume data should already be reflected in
current prices and should be of no value in predicting future changes.
Statistical test the independence of stock-price changes is one-way to test for weakform efficiency. If statistical tests suggest that price changes are independent, the
implication is that knowing and using the past sequence of price information is of no
value to investors. Autocorrelation tests are measuring the autocorrelation between
prices changes for various lags, such as one day two day so on. Positive or negative
autocorrelation would indicate the possible existence of potentially profitable trading
strategies. Zero correlation is consistent with the random walk hypothesis.
Autocorrelation of Return is that Letting Xt be the return of an asset, the
autocorrelation between Xt and Xt-j is estimated as
T
∑ ( X t −X )( X t− j −X )
r j= t= j+1 T
∑t=1 ( X t −X ) 2
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US
$
US$
(t-1)
SP50
0(t-1)
ASX(
t-1)
SP5
00
AS
X
TO
PI
X
Table Bc1
FT ST
SE
0.0
49
76
0.0
646
45
0.1
198
41
TOP
IX(t1)
FTS
E(t1)
0.0
859
23
0.0
181
7
ST(t1)
0.0
601
34
It can be found from the table there are five prices are positive except FTSE which is 0.01817. In addition, the biggest one is 0.119841 and the smallest one is -001817. In
the table, both of the positive numbers and negative numbers are close to zero.
Therefore, the share prices all satisfy the weak-form efficiency.
Section III. Portfolio Construction
C. Calculate the expected returns and standard deviations on the following
portfolios:
1. Portfolio1: 50% Australian share (ASX200) and 50% UK share (FTSE 100)
Table C1
Share
Return
Weight
Standard
Correlation
Coefficient
0.647592
ASX
0.00515
0.5
Deviation
0.03860
FTSE
0.00545
0.5
0.04207
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Expected Return:
E( R p )=∑ W i ×E( Ri )
E( R )=0 . 5×0 . 00515+0 .5×0 . 00545=0 . 0053=0 .53 %
Standard Deviation:
δ p=√W 2A δ2A+W 2B δ 2B +2 ρ AB W A W B δ A δ B
δ P= √0.52×0.0386 2 +0.52×0.04207 2+2×0.5×0.5×0.0386×0.04207×0.647592
δ P=0.036617=3.66%
2. Portfolio 2: 45% Singapore Share, 25% Australian Share and 30% USA (S&P
500) Share
Share
Return
Table C2
Weight
Standard
ST
0.00647
0.45
Deviation
0.06552
ASX
0.00515
0.25
0.03860
SP
0.00763
0.30
0.04195
Correlation
Coefficient
ρST , ASX=0.04732
ρST ,SP =0.102732
ρ ASX ,SP=0.62947
Expected Return:
E( R p )=∑ W i ×E( Ri )
E(( R P )=0 .00647×0 . 45+0 . 00515×0 .25+0 . 00763×0 .3=0 .6488 %
Standard Deviation:
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δ p =√W 2A δ2A+W 2B δ 2B +W 2C δ2C +2 ρ AB W A W B+2ρ AC W A W C δ A δ C +2 ρ BC W B W C δ B δ C
2 22 22 2
δP=¿√0.45 ×0. 65 2 +0.25 ×0. 386 +0.3 ×0. 4195 +2×0. 4732×0.45×0.25¿ 0. 65 2×0. 386+2×0.102732×0.45×0.3×0. 65 2×0. 4195+2×0.62947׿0. 3860×0. 4195¿ δp=0. 37107=3.71%¿
3. Portfolio 3: 20% Australian share, 20% US share, 20% Singapore Share, 20%
United Kingdom and 20% Japan share
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Return
Share
ASX
SP500
ST
FTSE
Table C3
Weight
Standard
0.005147
0.2
Deviation
0.038604
0.007626
0.2
0.041955
0.006474
0.2
0.065524
0.005446
0.2
0.042067
0.000412
0.2
0.056471
TOPIX
Correlation
Coefficient
ρ ASX ,SP=0.62947
ρ ASX ,ST=0.04732
ρ ASX ,FT =0.64759
ρ ASX ,TO=0.46459
ρSP ,ST =0.102732
ρSP ,FT =0.78859
ρSP ,TO=0.44534
ρST ,FT =0.05848
ρST ,TO=0.09753
ρ FT,TO=0.43008
Expected Return:
E( R p )=∑ W i ×E( Ri )
=W ASX × R ASX +W SP500 × RSP 500 + W ST × R ST +W FTSE × R FTSE +W TOPIX × RTOPIX
=0.2×0.005147+0.2×0.007626+0.2×0.006474+0.2×0.005446+0.2×0.000412
= 0.005021
= 0.5021%
Standard deviations
W ASX2 σ ASX2 +W SP500 2 σ SP 500 2+W ST 2 σ ST 2+ W FTSE 2 σ FTSE2
+W TOPIX2 σ TOPIX 2
+2W ASX W SP 500 ρ ASX SP 500 σ ASX σ SP 500 +2 W ASX W ST ρ ASX ST σ ASX σ ST
+2 W ASX W FTSE ρ ASX FTSE σ ASX σ FTSE
σ P=
+ 2W ASX W TOPIX ρ ASX TOPIX σ ASX σ TOPIX +2 W SP 500 W ST ρ SP500 ST σ SP 500 σ ST
+2 W SP500 W FTSE ρ SP500 FTSE σ SP 500 σ FTSE
+2 W SP 500 W TOPIX ρSP 500 TOPIX σ SP 500 σ TOPIX +2 W ST W FTSE ρST FTSE σ ST σ FTSE
+2 W ST W TOPIX ρ ST TOPIX σ ST σ TOPIX +2 W FTSE W TOPIX ρ FTSE TOPIX σ FTSE σ TOPIX
√
Victoria University College of Business
BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
0.202 ×0.038604 2+ 0.202 × 0.0419552+ 0.202 × 0.0655242
+0.202 ×0.0420672 +0.20 2 ×0.0564712
+ 2× 0.20 ×0.20 ×0.62947 × 0.038604 ×0.041955
+ 2× 0.20× 0.20 ×0.047323 ×0.038604 × 0.065524
+2× 0.20 ×0.20 × 0.647592×0.038604 × 0.042067
= +2 ×0.20 ×0.20 × 0.0 .464591× 0.038604 ×0.056471
+2× 0.20 ×0.20 × 0.102732× 0.041955× 0.065524
+2× 0.20 ×0.20 × 0.788593× 0.041955× 0.042067
+2 ×0.20 ×0.20 × 0.445345× 0.041955 ×0.056471
+ 2× 0.20 ×0.20 ×0.058489 × 0.065524 ×0.042067
+2× 0.20 ×0.20 × 0.097533× 0.065524 ×0.056471
+2 ×0.20 ×0.20 × 0.430079× 0.042067 ×0.056471
√
= 0.0331 =3.31%
4. Comment on the results by using the plot of risk and return
Table C4
Portfolio 1
Portfolio 2
Portfolio 3
Return (Mean)
0.53%
0.6488%
0.5021%
STD (Risk)
3.66%
3.71%
3.31%
0.04
0.04
0.04
0.04
0.03
0.03
0.03
Return
Standard Deviation
0.02
0.02
0.01
0.01
0.01
0.01
0
0.5
1
1.5
2
0.01
2.5
3
3.5
Figure C4
By the analysis of the plot of risk and return, it can be proved that higher return is
Victoria University College of Business
BEO2431 Risk Management Model Semester 2 2014
Case Study 1
ZHIYI ZHOU & KAIYUAN ZHANG
always with higher risk. If the investor intends to get increasingly return, they need to
face higher risk.
If investors want to get the risk averse, they will choose the portfolio 3.
If investors want to get the risk neutral, they will choose the portfolio 1.
If investors want to get the risk seeking, they will choose portfolio 2.
Victoria University College of Business