Saha, Final Project

ECONOMICS 424 Computat ional F inance and Financia l Econometrics

F i n a l P r o j e c t , W i n t e r 2 0 1 5

Written by: Kanokbhorn (KK) Saha

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Table of Contents

Page

Executive Summary 3

Return Calculations and Sample Statistics 7

Value-at-Risk Calculations 16

Rolling Analysis of the CER Model Parameters 17

Portfolio Theory 19

Asset Allocation 25

Conclusion 26

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Executive Summary

I. Data Set

The data set used for analysis in this project are 5 years of monthly closing price data from the end

of December 2009 through the end of December 2014.

II. Description of Mutual funds

S&P 500 index: vfinx:

The Vanguard 500 Index Investment tracks performance of a benchmark index that measures

investment return of large capitalization stocks. The indexing investment approach is designed to

track performance of the Standard & Poor’s. It attempts to replicate the target index by investing

all (or substantially all) of its assets in stocks that make up the index, holding each stock in about

the same proportion as its weighing in the index. Their net asset is 208.78 billion.

European stock index: veurx

The Vanguard European Stock Index Investment tracks performance of a benchmark index that

measure investment return of stocks issued by companies located in major markets of Europe. The

index is made up of approximately 521 common stocks of companies located in 16 European

countries. Their net asset is 18.70 billion.

Emerging markets fund: veiex

The Vanguard Emerging Markets Stock Index Investment tracks the performance of a benchmark

index measures investment return of stocks issued by companies located in emerging market

countries. The approach is investing approximately 95% of its assets in common stocks included in

the FTSE Emerging Index, while employing a form of sampling intended to reduce risk. Their net

asset is 64.43 billion.

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Long-term bond fund: vbltx

The Vanguard Long-Term Bond Index Investment tracks performance of market-weighted bond

index with long-term dollar-weighed average maturity. It includes all medium and larger issues of

United States government, investment-grade corporate, and investment-grade international dollar-

denominated bonds that have maturities of greater than 1- years and are publicly issues. Their net

asset is 8.85 billion.

Short-term bond fund: vbisx

The Vanguard Short-Term Bond Index Investment tracks performance of market-weighed bond

index with a short-term dollar-weighted average maturity. It includes all medium and larger issues

of United States government, investment-grade corporate, and investment-grade international

dollar-denominated bonds that have maturities between 1 and 5 years and are publicly issues. Their

net asset is 39.07 billion.

Pacific stock index: vpacx

The Vanguard Pacific Stock Index Investment tracks performance of a benchmark index that

measures investment return of stocks issued by companies located in the major markets of the

Pacific region. The indexing investment approach is investing all of its assets in the common

stocks included in the FTSE Developed Asia Pacific Index. Their net asset is 5.76 billion.

III. Main Findings

• All the prices and returns of mutual funds had a certain degree of drop during 2011, which

is in the midst of the financial crisis. The prices and returns then picked up gradually

throughout the second half of 2011 and towards 2012.

• There are a certain degree of volatility for the ones that index groups of countries while the

two bond funds have a lower volatility

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• Veiex has one of the highest volatility amongst the group but in contrary, does not provide

the highest return. Vfinx does not have the highest volatility but does provide the highest

return.

• Vbisx has the lowest volatility with the lowest return

• Most of the funds have a fairly normally distributed return with a slight skew that is

inevitable. All the funds have anomalies.

• Sharpe’s Ratio measures excess return per unit of risk. Vfinx has the highest Sharpe’s Slope

value while veiex has the lowest. The standard errors for all funds are very similar.

• The mean values have a higher standard error compared to the standard deviation which

means that the mean values are not estimated as precisely.

• The growth of $1 shows that vfinx provides the highest growth over 5 years while veiex

provides the lowest.

• There is a strong positive linear correlation between the index funds of countries and a

negative relationship between vfinx and vbltx. There are no clear correlations between the

long and short-term bonds.

• The value-at-risk over a one-month investment horizon is largest (in absolute values) for

veiex, for both 1% and 5%, and lowest for vbisx. Usually the emerging country stock index

has higher VaR than that of bonds. The same observation applies for the one-year

investment horizon.

• Rolling estimates of mean and standard deviation shows that the index funds for countries

appear to have a constant rolling mean and standard deviation. The mean has an upward

trend while the standard deviation has a downward trend. This is the opposite for the

bonds. The mean and standard deviation for the bonds move in a more unified way.

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• The expected return and standard deviation of global minimum variance portfolio are

higher when short sales are not allowed. The VaR is overall larger for the portfolio not

allowing short sales.

• The expected return and standard deviation for the efficient portfolio frontier has an

upward trend when allowed for short sales.

• The return of tangency portfolio with no short is smaller than the one allowing for short

sales.

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Return Calculations and Sample Statistics

I. Monthly Prices and Continuously Compounded Returns

Monthly Prices

The 4 funds that have similar trends are vfinx, veurx, vbltx, and vbisx. They all gradually

increase with minor drops and fluctuations in between. For vfinx, the only substantial drop was

during mid 2011. Veurx experienced 2 substantial drops, the first being around the same time as

vfinx and the second is during the mid of 2014. Vbisx did not experience any major drops and had

a smooth increase over the 5 years. Vbltx experienced 2 major drops with other minor drops.

The other 2 funds, veiex and vpacx, have similar trends with each other that are different

from the other 4. It pertained to more fluctuations in the time horizon with veiex having a drastic

drop towards the end of 2011. Besides that huge drop in its price, there are many smaller drops

throughout the time horizon. There was an overall smallest increase in price compared to the other

5 funds. Vpacx had an overall increase with a drop during mid 2011. There was a streak of smooth

increase at the end of 2012 towards beginning of 2013. One main observation is that all 6 graphs

have different vertical scales, which meant that they all traded at a difference price range; vfinx

having the highest priced range and vpacx with the lowest.

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Returns (cc)

The returns for vfinx, veurx, and veiex are very similar to each other with a moderate

amount of volatility. Vbisx and vbltx are the least volatile amongst the 6 funds. The common

pattern between vfinx, veurx, and veiex is that there is a big drop towards the first quarter of 2011

and the trough of the graph is around August 2011. There is less volatility for all 3 funds early 2013

until the end of 2014. For Vbisx and vbltx, they both have lesser major drops and a more stable

fluctuation throughout the 5-year span. Vbltx and vbisx are both bonds, which could be the

reasoning behind the similarity in the movements for those 2.

From the returns and monthly price graphs, it can be deduced that bonds tend to be less

volatile and have a smoother increase in price over the time horizon.

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Equity Curve (Growth of $1)

The equity curve shows the growth of $1 in each of the funds over the period of 5 years.

Vfinx produced the highest while veiex and vbisx tied for producing the lowest at the end of 2014.

Although vfinx produced the highest growth, it would have been assumed that it would have a

high volatility but this is not true in this case.

II. Four Panel Diagnostic plots

Distributions

Veurx has the largest distribution while vbisx has the most concentrated data about its

mean, i.e. smallest standard deviation.

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S&P 500 Index: vfinx

The returns do not look normally distributed. The histogram is left tail skewed which is

then reflected in the Q-Q plot. The box plot shows that the median is a little above the zero.

According to the ACF, there seems to be more negative than positive numbers but the negative

values are much smaller.

European Stock Index: veurx

The return looks quite normally distributed with two peaks, one being only slightly smaller

than the other. There is almost an equal distribution of positive and negative values, which gives a

smooth density. From the Q-Q plot it looks like there is a right skew to the data.

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Emerging Markets fund: veiex

The return is normally distributed. From the boxplot, the returns are mostly concentrated

around the mean with only 1 anomaly. There is somewhat of an equal distribution of positive and

negative values and a slight right skew from looking at the Q-Q plot.

Long-Term Bond Fund: vbltx

The return is normally distributed with one peak. The boxplot also shows that the data is

mostly concentrated around the mean and the ACF shows that there are a little more positive

values than negative. There is not much skewness in this data.

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Short-Term Bond Fund: vbisx

The return looks the most normally distributed out of the 6 funds. The Q-Q plot shows that there

is almost no skewness in the data and the ACF shows that there is almost an equal distribution of positive

and negative values with negative values being slightly smaller than the positives.

Pacific Stock Index: vpacx

The return looks normally distributed with a right skew in the data. There is also one outlier

according to the box plot and the ACF shows that there are slightly more negative values than positive and

the positive values are smaller than the negative.

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III. Univariate Descriptive Statistics

Mean Variance Standard Deviation

Skewness Kurtosis 1% Quantile

5% Quantile

Vfinx 0.01202 0.00139 0.03726 -0.3986 0.2927 -0.0773 -0.0563

Veurx 0.00454 0.00316 0.05622 -0.3107 0.0752 -0.129 -0.107

Veiex 0.00144 0.00309 0.05561 -0.3927 0.8702 -0.1414 -0.0902 Vbltx 0.00762 0.000611 0.02472 0.0206 -0.2685 -0.0497 -0.0259 Vbisx 0.00161 0.0000157 0.00396 0.0711 -0.3895 -0.00634 -0.00478 Vpacx 0.00429 0.00181 0.04256 -0.5516 0.1462 -0.1010 -0.0792

The overall values of the mean are all in the positives. Vfinx has the highest mean and

veiex has the lowest. Veurx has the highest standard deviation while vbisx has the lowest and this

might be because bonds are usually more stable. Risk-adverse investors would end up choosing

vbisx because it is the safest amongst the 6 but it provides almost the lowest returns. All funds

except vbltx and vbisx are negatively skewed and vbltx is closest to a normal distribution.

Expected Return vs. Risk

IV. Sharpe’s Slope

Sharpe’s Slope Estimated Standard Errors Vfinx 0.3115 0.147 Veurx 0.0734 0.133 Veiex 0.0185 0.129 Vbltx 0.2916 0.134 Vbisx 0.3002 0.133 Vpacx 0.0910 0.134 Above is the Sharpe’s slope using a monthly risk free rate of 0.0004157 per month, which

corresponds to a continuously compounded annual rate of 0.5%. Vfinx has the highest Sharpe’s

slope and also the highest standard error. The slopes are not estimated precisely due to the high

estimated standard error.

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V. Estimated Standard Errors and 95% Confidence Intervals

Mean Standard Deviation

Mean SE Lower Upper SD SE Lower Upper Vfinx 0.01202 0.004810 0.002403 0.02164 0.03726 0.003401 0.03045 0.04406 Veurx 0.00454 0.007258 -0.009975 0.01906 0.05622 0.005132 0.04596 0.06649 Veiex 0.00144 0.007179 -0.012915 0.01580 0.05561 0.005076 0.04546 0.06576 Vbltx 0.00762 0.003191 0.001242 0.01401 0.02472 0.002257 0.02021 0.02923 Vbisx 0.00161 0.000511 0.000583 0.00263 0.00396 0.000362 0.00324 0.00469 Vpacx 0.00429 0.005494 -0.006703 0.01527 0.04256 0.003885 0.03479 0.05033

The mean is not estimated very precisely because the standard error values are quite large

and are larger compared to the standard error values of the standard deviation. Referring to the

95% confidence interval, the mean has both positive and negative values while the standard

deviation only has positive values.

VI. Annualized mean, standard deviation, and Sharpe’s ratio

Annualize Mean Annualize SD Annualize Sharpe’s Ratio Growth of $1 àFV= PV(1+r)n Vfinx 0.1443 0.1291 1.0791 $1.96 Veurx 0.0545 0.1948 0.2543 $1.30 Veiex 0.0173 0.1926 0.0641 $1.09 Vbltx 0.0915 0.0856 1.0101 $1.55 Vbisx 0.0193 0.0137 1.0399 $1.10 Vpacx 0.0514 0.1474 0.3152 $1.28 Vfinx has the highest annual mean and veiex has the lowest. Veurx has the highest

standard deviation, followed by veiex while vbisx has the lowest. Vfinx also has the highest

annualized Sharpe’s ratio while veiex has the lowest. The rankings of the annualize Sharpe’s Ratio

is the same for the monthly Sharpe’s Ratios.

VII. Pair-Wise Scatterplots

There is a positive linear relationship

between vfinx and veurx, veiex, and vpacx.

There is a clear negative relationship

between vfinx and vbltx. There are no

clear linear relationship established

between vbltx and vbisx.

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VIII. Sample Covariance Matrix

Sample Covariance Vfinx Veurx Veiex Vbltx Vbisx Vpacx

Vfinx 0.00139 0.00183 0.00166 -0.000399 -0.0000103 0.00123 Veurx 0.00183 0.00316 0.00264 -0.000479 0.0000183 0.00194 Veiex 0.00166 0.00264 0.00309 -0.000334 0.0000312 0.00194 Vbltx -0.000399 -0.000479 -0.000334 0.000611 0.0000548 -0.000291 Vbisx -0.0000103 0.0000183 0.0000312 0.0000548 0.0000157 0.0000122 Vpacx 0.00123 0.00194 0.00194 -0.000291 0.0000122 0.00181 The covariance measures the direction of the two funds. All the positive covariance values

mean that the two funds move in the same direction. For the ones with negative values, they move

in the opposite direction.

IX. Sample Correlation Matrix

Correlation indicates how assets move in relations to each other and they range from -1 to

1 with -1 being the most negatively correlated and 1 being the most positively correlated. There is a

strong positive relation between vfinx, veurx, veiex, and vpacx while there is a negative relation

between vfinx, vbltx, and vbisx. Vfinx and veurx has the strongest positive relationship while vbltx

and vfinx have the strongest negative relationship. Diversifying the portfolio will reduce risk

because the value of risk will be averaged out.

Sample Correlation

Vfinx Veurx Veiex Vbltx Vbisx Vpacx

Vfinx 1.000 0.8740 0.799 -0.433 -0.0700 0.7748

Veurx 0.874 1.000 0.843 -0.344 0.0821 0.8104

Veiex 0.799 0.8434 1.000 -0.243 0.1416 0.8178

Vbltx -0.433 -0.3443 -0.243 1.000 0.5599 -0.2763

Vbisx -0.070 0.0821 0.142 0.560 1.000 0.0721

Vpacx 0.775 0.8104 0.818 -0.276 0.0721 1.000

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Value-at-Risk Calculations

I. 1% and 5% VaR (Monthly and Annual); Estimated SE and 95% Confidence Intervals

Initial wealth= $100,000

Value-at-Risk

Estimated SE

95% Confidence Interval (For One Month VaR)

One Month One Year Normal Percentile VaR 1% VaR 5% VaR 1% VaR 5% Lower Upper Lower Upper

Vfinx -$7,193 -$4,807 -$14,442 -$6,576 823 -6495 -3268 -6237 -3097 Veurx -$11,861 -$8,418 -$32,873 -$23,345 1139 -10730 -6267 -10736 -6102 Veiex -$12,008 -$8,609 -$35,002 -$25,884 1254 -11238 -6323 -11002 -5983 Vbltx -$4,866 -$3,250 -$10,212 -$4,816 452 -4174 -2401 -4203 -2366 Vbisx -$758 -$490 -$1,257 -$330 71.4 -636 -356 -618 -342 Vpacx -$9,037 -$6,360 -$25,288 -$17,392 982 -8308 -4460 -8240 -4403

Since the 95% confidence intervals are pretty narrow for both normal and percentile, the

estimated values are quite precise. From the estimated standard error values for the one-month

VaR, veiex is the least precise estimation while vbisx is the most precise. For both the one-month

and the one-year 1% and 5% VaR, veiex is the highest while vbisx is the lowest.

II. Empirical 1% and 5% Quantiles of Return Distribution

One Month 1% Quantile 5% Quantile

Vfinx -$7,442 -$5,477 Veurx -$12,076 -$10,166 Veiex -$13,190 -$8,629 Vbltx -$4,845 -$2,558 Vbisx -$632 -$476 Vpacx -$9,605 -$7,613

When comparing the one-month 1% and 5% quantile for the empirical values, there is not a big

difference in the results from those based on the normal distribution. The empirical values are

overall larger except for the values of vbltx and vbisx for both 1% and 5%.

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Rolling Analysis of the CER Model Parameters

I. 24 Month Rolling estimates

VFINX VEURX

VEIEX VBLTX

VBISX VPACX

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The rolling estimates of the mean and standard deviation for vfinx, veurx, veiex, and vpacx

looks similar in the way that it starts out with a wide gap between the mean and the standard

deviation but then the gap narrows as it proceeds into the end of 2014. Vbisx and vbltx has its

rolling mean and standard deviation moving in a more unified pattern in which the gap between it

are fairly constant. The 6 funds overall has a rolling standard deviation higher than the mean.

II. 24 Month Rolling Estimates of Sample Correlation between S&P 500 Index (vfinx) and Long-

Term Bond Index (vbltx).

Rolling Correlation between vfinx and vbltx

The correlation between vfinx and vbltx is not stable overtime because from the graph,

there is an upward trend throughout the investment horizon. The correlation starts with

approximate -0.63 at the beginning of 2012 and ends at approximately 0.05 at the end of 2014. The

highest correlation is at 0.05 and the lowest is at -0.69.

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Portfolio Theory

I. Global Minimum Variance Portfolio, Expected Return, and Standard Deviation

Expected Return 0.00171 Annualized Return 0.02052 Standard Deviation 0.00328

Annualized SD 0.01136 1% VaR -$986.17 5% VaR -$267

Sharpe’s Ratio 0.396

Weights

Vfinx 0.0610 Veurx -0.0344 Veiex -0.0228 Vbltx -0.0743 Vbisx 1.0639 Vpacx 0.0067

Veurx, veiex, and vbltx have negative weights in the global minimum variance portfolio.

The annualized return for the portfolio, which is 2.05%, is less than the annualized return of

individual funds. On the other hand, the annualized standard deviation for the portfolio, which is

0.33%, is also less than that of individual funds. The 1% VaR of the global portfolio is smaller than

the 1% VaR for individual mutual funds and this also applies with the 5%.

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II. Global Minimum Variance Portfolio with No Short-Sales


Annualized SD 0.0135 1% VaR -$1,268.52 5% VaR -$260.42


Weights

Vfinx 0.0183 Veurx 0.0000 Veiex 0.0000 Vbltx 0.0000 Vbisx 0.9817 Vpacx 0.0000

The annualized expected return and standard deviation for no short sales are greater than

the portfolio that allows short sales. There are no negative weights and 4 out of the 6 funds are at

0. The biggest weight portfolio is vbisx.

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III. Efficient Portfolio Frontier Allowing for Short Sales (using Markowitz Algorithm) and Efficient

Minimum Variance Portfolio with Target Return Equal to Maximum of Average Returns

Efficient Frontier


Annualized SD 0.0509 Sharpe’s Ratio 0.79

Weights

Vfinx 0.7965 Veurx -0.2019 Veiex -0.2079 Vbltx 0.4440 Vbisx 0.1670 Vpacx 0.0023

Global minimum

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IV. Tangency Portfolio using Monthly Risk Free Rate (0.5%)


Annualized SD 0.0236 Variance 0.0000462


Weights

Vfinx 0.3669 Veurx -0.1041 Veiex -0.0998 Vbltx 0.1413 Vbisx 0.6908 Vpacx 0.0049

Sharpe’s Ratio


Veurx and veiex are the only 2 assets out of the 6 that have negative weights. The reason for this is

that the annualized expected returns are less than the expected return of the tangency portfolio and

that they are being short sold for other funds. The individual fund’s Sharpe’s Ratio are smaller than

that of the tangency portfolio.

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V. Efficient Portfolio Frontier with No Short Sales

Efficient Frontier with No Short Sale

Short Sale Frontier and No Short Sale Frontier

The portfolio that allows short sales (blue dotted curve) provides a larger return than that

of no short sales (red dotted curve). The portfolio that allows short sales also has a steeper upward

trend while the no short sale has more of a plateau trend towards the end.

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Target Volatility of 0.02 Per Month

Expected Return (Short Sale) Expected Return (No Short Sale)

0.0099 0.0142

Investing in no short sale portfolio will lead to a loss of approximately 0.0043, which is 0.43%, on

the returns

VI. Tangency Portfolio with No Short Sales

Weights


When no short sales are allowed compared to when short sales are allowed, the expected return is

a little bit larger but the Sharpe’s ratio for no short sale is lower.


Annualized SD 0.0388 Variance 0.0001254


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Asset Allocation

I. Target Expected Return of 6% Per Year (0.5% per month) Using Risky Assets and No Short

Sales

To reach the target expected return of 6% per year, or 0.5% per month, the investor will

have to invest in vfinx, vbltx, and vbisx for the following amount shown in the table above.

II. Target Expected Return of 12% Per Year (1% Per Month) Using Risky Assets and No Short

Sales

To reach the target expected return of 12% per year, or 1% per month, the investor will have

to invest in vfinx and vbltx for the following amount shown in the table above. The standard

deviation of the target expected return of 12% is larger than that of the 6% and the same trend

applies to the 1% and 5% value at risk, which means investors are more likely to lose more

money. In the case of the 12%, vbisx is no longer part of the investment.


Annualized SD 0.0317 1% VaR -$1,627.00 5% VaR -$1,004.00


Annualized SD 0.0776 1% VaR -$4,117.37 5% VaR -$2,645.15

Weights


Weights


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Conclusion

In conclusion, this project analyzed 5 years of monthly closing price data, ranging from the end of

December 2009 through the end of December 2014. The funds used are

1. S&P 500 Index: vfinx

2. European Stock Index: veurx

3. Emerging Markets Fund: veiex

4. Long-Term Bond Fund: vbltx

5. Short-Term Bond Fund: vbisx

6. Pacific Stock Index: vpacx

The data and information about these funds were taken from the Yahoo! Finance Site. The

calculations and graphs were produced using the R Financial Program. The codes used for R are from

the Economics 424 class website resource owned by Professor Eric Zivot.

Link: http://faculty.washington.edu/ezivot/econ424/424projectWinter2015.R

From the analysis, research, and calculations made throughout the project, there are a variety of

findings and variables that needs to be taken into account when investing in an asset or a portfolio. Risk,

short sales, efficiency, target rates, etc. all play a key role in working with assets in the financial world.

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Saha, Final Project

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