BEO2431 Risk Management Model Semester 2

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Presented by:

ZHIYI ZHOU 3923070 KAIYUAN ZHANG 3920263

Presented to:

Chinthana Hatangala

Tutorial No.3

Victoria University College of Business

BEO2431 Risk Management Model

Semester 2 2014

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5

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$

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15

Returns-US$

Returns-US

Time

R

et

u

rn

s-U

S

$

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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5 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

No

v-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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5

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

No

v-Compared with United States and Australian financial markets, fluctuation scope of

Market

Volatility _Low Volatility

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5 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

No

v-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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5

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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22

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

Skewness -0.562719367 -0.612193408 -0.368007992 -0.1023303 -0.274207174 0.060597263

CV 22.49215927 5.501743278 7.499730498 137.109706 7.724172784 10.12102738

Z-Score -0.044459938 -0.181760571 -0.133338125 -0.00729343 -0.129463702 -0.098804199

Prob of Z-Score 48.40% 42.86% 44.83% 49.60% 44.83% 46.02%

1. Arithmetic Mean

The formula of arithmetic mean is shown below:

AM=

∑

i=1

n Ri n

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2. Geometric mean

The formula of geometric mean is:

GM=

_√

n(RR1)(RR2)(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:

CWIn=WI0(RR1) (RR2)…(RRn)

CWIn: The cumulative wealth index as of the end of the period n WI0: 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:

σ=

√

_NN −1

∑

_t₌₁

T

(X_t− ´X)2

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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:

S= n

(n−1)(n−2)

∑

t=1

T

(Xt− ´X)3/σ3

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:



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.

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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 z-score. The equation of calculating negative z-score is:

z=0−´_σx(Pr=(R− ´_σR))

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

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(c) Compare risk and return from these investments using the appropriate plot.

Table Ac1 Risk

(STD)

Returns (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

0.03 0.04 0.04 0.05 0.05 0.06 0.06 0.07 0.07

0 0 0 0 0 0.01 0.01 0.01 0.01 0.01

Risk and Return

Risk

R

e

tu

rn

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.

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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

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.

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-0.1

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

-0.1

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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

-0.2

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5. FTSE Return

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

-0.2

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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 non-normal. (Use Excel)

Table E1

　 Mean Median Kurtosis Skewness

US$ 0.001431609 0.002485244 2.274434

5

-0.562719367

SP500 0.007625755 0.011111111 1.227791 8

-0.612193408

ASX 0.005147363 0.0089982 0.344470

7

-0.368007992

TOPIX 0.000411864 0.000883392 0.874432 1

-0.102330303

FTSE 0.005446177 0.007740687 0.513573

3

-0.274207174

ST 0.006474094 0.010557855 3.416980

8

0.060597263

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

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(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

Table Ba1

US$ 1

SP500 0.377035 1

ASX 0.386429 0.62947 1

TOPIX 0.28858 0.445345 0.464591 1

FTSE 0.326469 0.788593 0.647592 0.430079 1

ST 0.133918 0.102732 0.047323 0.097533 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:

The value of the correlation is always located between -1 and 1. (-1AB 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

ρ

AB=

Cov

(

R

A, RB)

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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.

Table Bb1

US$ 0.001034

SP500 0.000508 0.001755

ASX 0.000479 0.001016 0.001486

TOPIX 0.000523 0.001052 0.00101 0.003179

FTSE 0.000441 0.001387 0.001048 0.001018 0.001764

ST 0.000282 0.000282 0.000119 0.00036 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

(

R

i

,R

j

)

_:

The formula of the variance:

It can be seen from the table that all of covariance data is positive which means the

COV

(

R

_i

, R

_j

)=

∑

t=1

n

(

R

_i,t

−

R

_i

)

−

(

R

_j,t

−

R

−_j

)

n

−

1 σ

2 =

∑

t=1

n

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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 ut

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 weak-form 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

rj=

∑

t=j+1 T

(

Xt−X

)(

Xt−j−X

)

∑

t=1

T

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Table Bc1 US

$

SP5 00

AS X

TO PI

X

FT SE

ST

US$ (t-1)

0.0 49 76

SP50 0(t-1)

0.0 646 45

ASX( t-1)

0.1 198 41

TOP IX(t-1)

0.0 859 23

FTS E(t-1)

-0.0 181 7

ST(t-1)

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

Deviation

Correlation

Coefficient

ASX 0.00515 0.5 0.03860

0.647592

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Expected Return:

E

(

R

_p

)=

∑

W

_i

×

E

(

R

_i

)

E

(

R

)=

0.5 ×

0.00515

+

0.5 ×

0.00545

=

0.0053

=

0.53 %

Standard Deviation:

δ

_p

=

√

W

2 _A

δ

2 _A

+

W

_B

2 δ

_B

2 +

2 ρ

_AB

W

_A

W

_B

δ

_A

δ

_B

δ

_P

=

√

0.5

2

×

0.0386

2

+

0.5

2

×

0.04207

2

+

2 ×

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

Table C2

Share Return Weight Standard

Deviation

Correlation

Coefficient

ST 0.00647 0.45 0.06552

ρ

_{ST , ASX}

₌

0.04732

ρ

_{ST ,SP}

=

0.102732

ρ

_{ASX ,SP}

=

0.62947

ASX 0.00515 0.25 0.03860

SP 0.00763 0.30 0.04195

E

(

R

_p

)=

∑

W

_i

×

E

(

R

_i

)

E

((

R

_P

)=

0.00647

×

0.45 +

0.00515

×

0.25 +

0.00763

×

0.3 =

0.6488%

(22)

δ

_p

=

√

W

2_A

δ

2_A

+

W

_B2

δ

_B2

+

W

_C2

δ

_C2

+

2 ρ

_AB

W

_A

W

_B

+

2 ρ

_AC

W

_A

W

_C

δ

_A

δ

_C

+

2 ρ

_BC

W

_B

W

_C

δ

_B

δ

_C

δ

_P

=

¿

√

0.45

2 ×

0.06552

2 +

0.25

2 ×

0.0386

2 +

0.3

2 ×

0.04195

2 +

2 ×

0.04732

×

0.45 ×

0.25 ¿¿

0.06552

×

0.0386

+

2 ×

0.102732

×

0.45 ×

0.3 ×

0.06552

×

0.04195

+

2 ×

0.62947

×

¿

0.03860

×

0.04195

¿¿

δ

_p

=

0.037107

=

3.71%

¿¿

(23)

Table C3

Share Return Weight Standard

Deviation

Correlation

Coefficient

ASX 0.005147 0.2 0.038604

ρ

_ρ

ASX ,SP

=

0.62947

ASX ,ST

=

0.04732

ρ

_{ASX ,FT}

=

0.64759

ρ

_{ASX ,TO}

=

0.46459

ρ

_{SP ,ST}

=

0.102732

ρ

_{SP ,FT}

=

0.78859

ρ

_{SP ,TO}

=

0.44534

ρ

_{ST ,FT}

=

0.05848

ρ

_{ST ,TO}

=

0.09753

ρ

_FT,TO

=

0.43008

SP500 0.007626 0.2 0.041955

ST 0.006474 0.2 0.065524

FTSE 0.005446 0.2 0.042067

TOPIX

0.000412 0.2 0.056471

E

(

R

_p

)=

∑

W

_i

×

E

(

R

_i

)

=WASX× RASX+WSP500× RSP500+WST× RST+WFTSE× RFTSE+WTOPIX× 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

σP=

√

WASX

2

σ_ASX2+W_SP₅₀₀2σ_SP₅₀₀2+W_ST2σ_ST2+W_FTSE2σ_FTSE2 +WTOPIX2σTOPIX2

+2WASXWSP500ρASX SP500σASXσSP500+2WASXWSTρASX STσASXσST

+2WASXWFTSEρASX FTSEσASXσFTSE

+2W_ASXW_TOPIXρ_{ASX TOPIX}σ_ASXσ_TOPIX+2W_SP₅₀₀W_STρ_SP₅₀₀_STσ_SP₅₀₀σ_ST +2W_SP₅₀₀W_FTSEρ_SP₅₀₀_FTSEσ_SP₅₀₀σ_FTSE

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=

√

0.20

2_×_0.0386042

+0.202×0.0419552+0.202×0.0655242 +0.202×0.0420672+0.202×0.0564712

+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%