analysis of six vanguard funds

CFRM 462 FINAL

PROJECT Analysis of Six Vanguard Funds

ABSTRACT Computational analysis of six vanguard funds

including exploratory analysis, descriptive statistics,

value at risk, rolling analysis, portfolio analysis and targeted efficient portfolios.

Charles Paoletti CFRM 462

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Analysis of Six Vanguard Funds

1 CONTENTS

2 Executive Summary .....................................................................................................................2

2.1 Data Set ..............................................................................................................................2

2.2 Description of Mutual funds .................................................................................................2

2.3 Main Findings ......................................................................................................................3

3 Return calculations and Sample Statistics .....................................................................................5

3.1 Monthly Prices, Continuously Compounded Returns and Equity Curve....................................5

3.2 Four Panel Plots ...................................................................................................................7

3.3 Univariate Descriptive Statistics .......................................................................................... 10

3.4 Sharpe Ratio ...................................................................................................................... 10

3.5 Standard Error and %95 Confidence Intervals of Mean and Standard deviation ..................... 11

3.6 Annualized Estimates ......................................................................................................... 11

3.7 Pairwise plots and Covariance ............................................................................................ 12

3.8 Correlation ........................................................................................................................ 13

4 Value-at-Risk Calculations .......................................................................................................... 14

4.1 Normal VaR ....................................................................................................................... 14

4.2 One Year Normal VaR......................................................................................................... 14

4.3 Empirical VaR .................................................................................................................... 14

5 Rolling Analysis of the CER Model Parameters............................................................................. 15

5.1 Rolling Analysis Graphs ...................................................................................................... 15

5.2 Rolling Correlation ............................................................................................................. 17

6 Portfolio Theory ........................................................................................................................ 17

6.1 Global Minimum Variance Portfolio .................................................................................... 17

6.1.1 Portfolio Weights Graph.............................................................................................. 18

6.2 Annual Sharpe ................................................................................................................... 18

6.3 VaR ................................................................................................................................... 18

6.4 Global Minimum Variance Portfolio with No Shorts ............................................................. 18


6.4.2 Annual Sharpe ............................................................................................................ 19

6.4.3 VaR ............................................................................................................................ 19

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6.5 Efficient Portfolio Frontier .................................................................................................. 20

6.5.1 Efficient Portfolio Frontier Graph................................................................................. 20

6.6 Tangency Portfolio ............................................................................................................. 20


6.6.2 Expected Return, Variance and Standard Deviation of Tangency Portfolio...................... 21

6.6.3 Sharpe Ratio............................................................................................................... 21

6.6.4 Graph of Tangency Portfolio........................................................................................ 21

6.6.5 Annualized Expected Return and Standard Deviation of Tangency Portfolio ................... 22

6.7 Efficient Portfolio Frontier with No Short Sales .................................................................... 23

6.7.1 Graph ........................................................................................................................ 23

6.8 Tangency Portfolio No Short Sales ...................................................................................... 23

6.8.1 Expected Return and Standard Deviation ..................................................................... 24

7 Asset Allocation......................................................................................................................... 24

7.1 6% Target Portfolio ............................................................................................................ 24

7.2 Monthly Standard Deviation and Value at Risk..................................................................... 24

7.3 12% Target Portfolio without Short Sales ............................................................................ 24

2 EXECUTIVE SUMMARY

2.1 DATA SET The data set used for analysis in this project are 5 years of monthly closing price data from

January 2011 through the end of January 2016.

2.2 DESCRIPTION OF MUTUAL FUNDS S&P 500 index: vfinx: The investment seeks to track the performance of a benchmark index that measures the

investment return of large-capitalization stocks. The fund employs an indexing investment approach designed to track the performance of the Standard & Poor's 500 Index, a widely recognized benchmark of U.S. stock market performance that is dominated by the stocks of

large U.S. companies. The advisor attempts to replicate the target index by investing all, or substantially all, of its assets in the stocks that make up the index, holding each stock in approximately the same proportion as its weighting in the index. European stock index: veurx

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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 investment seeks to track the performance of a benchmark index that measures the investment return of stocks issued by companies located in emerging market countries. The

fund employs an indexing investment approach designed to track the performance of the FTSE Emerging Markets All Cap China A Transition Index, an interim index that will gradually increase exposure to small-capitalization stocks and China A-shares while proportionately reducing exposure to other stocks based on their weightings in the index. The index is a market-capitalization-weighted index. Long-term bond fund: vbltx The investment seeks to track the performance of a market-weighted bond index with a long-term dollar-weighted average maturity. The fund employs an indexing investment approach

designed to track the performance of the Barclays U.S. Long Government/Credit Float Adjusted

Index. This index includes all medium and larger issues of U.S. government, investment-grade corporate, and investment-grade international dollar-denominated bonds that have maturities

of greater than 10 years and are publicly issued. All of the fund's investments will be selected through the sampling process, and at least 80% of the fund's assets will be invested in bonds held in the index. Short-term bond fund: vbisx The investment seeks to track the performance of a market-weighted bond index with a short-term dollar-weighted average maturity. The fund employs an indexing investment approach designed to track the performance of the Barclays U.S. 1-5 Year Government/Credit Float Adjusted Index. This index includes all medium and larger issues of U.S. government, investment-grade corporate, and investment-grade international dollar-denominated bonds that have maturities between 1 and 5 years and are publicly issued. All of the fund's

investments will be selected through the sampling process, and at least 80% of the fund's assets will be invested in bonds held in the index.

Pacific stock index: vpacx The investment seeks to track the performance of a benchmark index that measures the

investment return of stocks issued by companies located in the major markets of the Pacific region. The fund employs an indexing investment approach by investing all, or substantially all, of its assets in the common stocks included in the FTSE Developed Asia Pacific All Cap Index. The FTSE Developed Asia Pacific All Cap Index is a market-capitalization-weighted index that is made up of approximately 2,196 common stocks of large-, mid-, and small-cap companies

located in Japan, Australia, South Korea, Hong Kong, Singapore, and New Zealand.

2.3 MAIN FINDINGS All the prices and returns of the six funds had a downturn during 2011, which is explained by the

August 2011 stock markets fall which was precipitated by concerns that the European sovereign

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debt crisis would continue to spread. The prices and returns then increased steadily until the

end of 2016 where the stock funds experienced another downturn.

The bond funds have lower volatility than the stock funds except veiex.

Veiex has the highest volatility amongst the group but does not provide the highest return. Vfinx

has the highest expected return but not the highest volatility.

Vbisx has the lowest volatility with the lowest positive return.

Only vbltx appears normally distributed and vfinx, veiex and vpacx all have outliers

Sharpe’s Ratio measures excess return per unit of risk. Vfinx has the highest Sharpe’s Ratio 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 as the standard deviation.

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

a negative return.

There is a strong positive linear correlation between pairs of the stock funds and a weaker

positive correlation between the two bond funds. There is a weak negative correlation between

vfinx,veurx and vbltx,vbisx.

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. The same observation applies for the one -year

investment horizon. The VaR estimate are useful but not as accurate as one would like. This

should be kept in mind when considering exposure and liquidity.

Rolling estimates of mean and standard deviation shows that none of the assets have stable

means or standard deviations. Vbisx seems to have the most stable.

By creating a diversified portfolio we can significantly reduce risk and maintaining an acceptable

expected return

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 return of tangency portfolio with no short is slightly larger than the one allowing for short

sales however there is significant increase in risk.

To achieve a target return of 6% an efficient portfolio is made up of vfinx, vbltx and vbisx.

It is not possible to target an expected return greater than 10.2% without short sales.

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3 RETURN CALCULATIONS AND SAMPLE STATISTICS

3.1 MONTHLY PRICES, CONTINUOUSLY COMPOUNDED RETURNS AND EQUITY CURVE

The four stock funds have a downward trend from 2011 to 2012 most likely associated with the

European Financial crisis. The Bond funds do not appear to have been affected significantly. All of the

stock funds have another downward trend at the end of 2015 and beginning of 2016. Veiex is the only fund with a negative overall return.

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The CC returns show that the returns possibly could be normally distributed with overall regression to a mean, however the volatility looks like it is probably changing over time for all six asset.

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Form the Equity curve we see vfinx with the highest return followed by vbltx. Veiex has significant

negative return. Vbisx appears the most stable but only has modest growth.

3.2 FOUR PANEL PLOTS

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From the four panel plots it looks like vbltx is the only asset with a normal distribution. The other five

assets have varying degrees of skew and kurtosis which is also reflected in the deviation from the

normal line in the qqplots. Vfinx, veiex and vpacx have outliers in their respective boxplots. None of the assets appear to have significant linear time dependence.

Again we see vfinx has the highest mean and veiex has the lowest. Vbisx has the narrowest distribution followed be vbltx while veiex has the widest. Vfinx, veiex and vpacx all have outliers.

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3.3 UNIVARIATE DESCRIPTIVE STATISTICS

Vfinx has the highest mean while veiex has the lowest and a negative mean indicating a loss. Vbltx has

the next highest mean after vfinx which is surprising as it is a bond fund. Vpacx has the smallest positive

mean. Veiex has the largest standard deviation indicating the most risk followed by veurx then vfinx.

Vbisx has the lowest standard deviation followed by vbltx. This makes sense as they are bond funds and

bonds tend to have lower risk. The descriptive statistics confirm that vbltx is the most normally distributed with the smallest skew but with some negative kurtosis indicating fatter tails.

3.4 SHARPE RATIO

Sharpe ratio is a measure of return per unit of risk. Vfinx has the highest Sharpe ratio followed closely by

vbisx and vbltx. However, the standard error is quite high for all assets indicating it isn’t very accurately

measured.

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3.5 STANDARD ERROR AND %95 CONFIDENCE INTERVALS OF MEAN AND STANDARD DEVIATION

The means are not estimated very precisely. The smallest standard error is about %38 of the mean and

many of the standard errors are larger than the means. The 95% confidence intervals of the means are

wide and most contain positive and negative values. The standard errors are more precisely estimated

but still not as accurately as we would like. They range from about 9% to 8% of the standard deviation.

The confidence intervals of the standard deviation are much narrower and contain only positive values.

3.6 ANNUALIZED ESTIMATES

Annualizing the Sharpe ratio does not affect the ranking of the assets. As indicated in the equity curve,

vfinx has the best return followed by vbltx while veiex has a loss. Veurx, vbisx and vpacx have similar gains after five years.

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3.7 PAIRWISE PLOTS AND COVARIANCE

We see a positive correlation between pairs of stock funds and a slight negative correlation between the

bond funds and vfinx and veurx. There is also a positive correlation between the two bond funds but it is not as strong as between pairs of stock funds.

All of the covariance between pairs of stock funds are positive as well as the covariance between the

bond funds. The covariance is negative between bond funds and stock funds.

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3.8 CORRELATION

There are positive correlation of varying degree between the four stock funds. There is also a positive

correlation between the two bond funds. The positive correlations between the stock funds range from

0.87 to 0.74. In decreasing order they are vfinx and veurx, veurx and veiex, vpacx and veiex then veiex

and vfinx. The bond funds have a positive correlation of 0.65. The strongest negative correlation is between vbltx and vfinx. There will be some diversification benefit.

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4 VALUE-AT-RISK CALCULATIONS

4.1 NORMAL VAR

Vbisx has the least Value at Risk and veiex has the most.

Veiex still has the highest VaR and vbisx still has the lowest. The standard error is about 12%-15% of the

5% VaR and 8%-10% of the 1% VaR indicating the 1% VaR is slightly more accurate. We would like the

standard error to be less but it is still a useful measurement but we should keep it in mind when considering what our actual Value at Risk might be.

4.2 ONE YEAR NORMAL VAR

Obviously the amount each asset has at risk has increased however the order is unaffected.

4.3 EMPIRICAL VAR

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The empirical VaR differ from the Normal VaR. Some of the empirical VaR are less than the normal VaR

and some more. This is another indication that are estimates are inaccurate and we should be careful

when relying on the normal VaR when considering our exposure.

5 ROLLING ANALYSIS OF THE CER MODEL PARAMETERS

5.1 ROLLING ANALYSIS GRAPHS

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None of the assets appear to have stable mean or standard deviation. Vbisx varies the least, while veurx

varies the most.

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5.2 ROLLING CORRELATION

The rolling correlation is not stable overtime. It is lowest near the end of the first quarter of 2013 at

around -0.65 then has a steady upward trend before peaking around the end of 2015. Afterwards it has

a significant drop back to -0.25 then returns to an upward trend into 2016 ending at about 0.

6 PORTFOLIO THEORY

6.1 GLOBAL MINIMUM VARIANCE PORTFOLIO

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Veurx, veiex and vbltx are shorted with the 108.5% in vbisx, 5.6% in vfinx and 0.8% in vpacx.

6.1.1 Portfolio Weights Graph

6.2 ANNUAL SHARPE The annual return is 1.6% with an annualized standard deviation of .0091. This gives a return slightly

higher than veiex, vbisx or vpacx but with a much lower standard deviation than any individual asset.

This is reflected in the fact that its Sharpe ratio is much higher than any of the individual assets.

6.3 VAR The annual 5% and 1% VaR for an investment of $100,000 Global Minimum Variance Portfolio is -

$298.51 and $-477.74 respectively. This is much less than any of the other assets even though it has a higher expected return than vpacx, vbisx or veiex.

6.4 GLOBAL MINIMUM VARIANCE PORTFOLIO WITH NO SHORTS

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The Global Minimum Variance Portfolio with no short sales as a slightly higher expected return but an almost 50% increase in risk. This is also apparent in the increased value at risk.


6.4.2 Annual Sharpe

The annual expected return is 1.6% and annual standard deviation of .001230. The expected return is

still more than vpacx, vbisx and veiex. The annual Sharpe ratio is 0.9205 which is still higher than any of the individual assets.

6.4.3 VaR

The annual 5% and 1% VaR for an investment of $100,000 Global Minimum Variance Portfolio is

-$447.83 and $-689.76 respectively. This is more than the Global Minimum Variance portfolio allowing

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short sales but still is less than any of the other assets even though it has a higher expected return than vpacx, vbisx or veiex.

6.5 EFFICIENT PORTFOLIO FRONTIER

6.5.1 Efficient Portfolio Frontier Graph

6.6 TANGENCY PORTFOLIO

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The Tangency portfolio also includes stocks to short namely veurx and veiex. This is not surprising since

veiex has a negative expected return and veurx does not have a significant higer expected return given

its corresponding risk level.


Veurx and veiex are both shorted.

6.6.2 Expected Return, Variance and Standard Deviation of Tangency Portfolio

The expected return is 0.38% with a standard deviation of 0.0050. This is a better return with a lower level of risk than any individual asset.

6.6.3 Sharpe Ratio

The Tangency portfolio has a Sharpe ratio of 0.672 which is much more than any of the individual assets and also more than the Global Minimum Variance portfolio.

6.6.4 Graph of Tangency Portfolio

Show the tangency portfolio as well as combinations of T-bills and the tangency portfolio on a plot with

the Markowitz bullet. That is, compute the efficient portfolios consisting of T-bills and risky assets.

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Here the blue line is the efficient frontier, the green line is the combination of the Tangency portfolio and t-bills. The yellow dot is the Global Minimum Variance portfolio.

6.6.5 Annualized Expected Return and Standard Deviation of Tangency Portfolio

The annual expected return 4.56% with a standard deviation of 0.01746 and an annual Sharpe ratio of

2.325. This is three times the annual return of the Global Minimum variance portfolio with only about 50% more risk.

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6.7 EFFICIENT PORTFOLIO FRONTIER WITH NO SHORT SALES

6.7.1 Graph

6.8 TANGENCY PORTFOLIO NO SHORT SALES

Compared to the Tangency Portfolio that allows short sales the monthly expected return has increased about 10% however the risk has almost doubled.

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6.8.1 Expected Return and Standard Deviation

The expected return is .43% with a standard deviation of 0.00995 and a Sharpe ratio of 0.388.

6.8.1.1 Annualized Expected Return Standard Deviation and Sharpe Ratio

The annual expected return is 5.1% with an annualized standard deviation of 0.03447 and an annual

Sharpe ratio of 1.342. This gives a higher return than all assets except vfinx and vbltx with less risk than all of the assets except vbisx.

6.8.1.2 Tangency Portfolio Short Sales vs no Short Sales

The expected return is only 10% more than the Tangency portfolio allowing Short sales but with almost

twice the risk. Consequently the Sharpe ratio is almost half of the Tangency portfolio allowing Short

sales.

7 ASSET ALLOCATION

7.1 6% TARGET PORTFOLIO

To achieve an efficient target of 6% a year with no short sales the portfolio would be divided between

vfinx, vbltx and vbisx with the bulk of the portfolio in the bond funds. 28.6% in vfinx, 34.1% in vbltx and

37.3% in vbisx. The 6% portfolio has slightly less of an expected return then vbltx but with less than half the risk of vbltx.

7.2 MONTHLY STANDARD DEVIATION AND VALUE AT RISK The monthly standard deviation is 0.0118 with VaR from a $100,000 investment at -$1439.68 with 5% risk and -$2243.84 at 1% risk. This portfolio has less VaR than any of the individual assets except vbisx.

7.3 12% TARGET PORTFOLIO WITHOUT SHORT SALES We were asked to find an efficient portfolio with a target return of 12% without short sales. This is not

possible as the highest expected return is vfinx at 10.2%. Without the ability to borrow from a lower yielding asset to invest in a higher yielding asset that is the maximum expected return achievable.

analysis of six vanguard funds

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