1.
ECONOMICS
424
Computational
Finance
and
Financial
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 àFV= 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
Expected Return 0.0018
Annualized Return 0.0216
Standard Deviation 0.0039
Annualized SD 0.0135
1% VaR -$1,268.52
5% VaR -$260.42
Sharpe’s Ratio 0.354
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
Expected Return 0.012
Annualized Return 0.144
Standard Deviation 0.0147
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%)
Expected Return 0.006
Annualized Return 0.072
Standard Deviation 0.0068
Annualized SD 0.0236
Variance 0.0000462
Sharpe’s Ratio 0.822
Weights
Vfinx 0.3669
Veurx -0.1041
Veiex -0.0998
Vbltx 0.1413
Vbisx 0.6908
Vpacx 0.0049
Sharpe’s
Ratio
Vfinx 0.3114
Veurx 0.0733
Veiex 0.0184
Vbltx 0.2914
Vbisx 0.3013
Vpacx 0.0910
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
Vfinx 0.278
Veurx 0.000
Veiex 0.000
Vbltx 0.384
Vbisx 0.338
Vpacx 0.000
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.
Expected Return 0.00681
Annualized Return 0.08172
Standard Deviation 0.0112
Annualized SD 0.0388
Variance 0.0001254
Sharpe’s Ratio 0.569

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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.
Expected Return 0.005
Annualized Return 0.06
Standard Deviation 0.0091
Annualized SD 0.0317
1% VaR -$1,627.00
5% VaR -$1,004.00
Expected Return 0.01
Annualized Return 0.12
Standard Deviation 0.0224
Annualized SD 0.0776
1% VaR -$4,117.37
5% VaR -$2,645.15
Weights
Vfinx 0.228
Veurx 0.000
Veiex 0.000
Vbltx 0.315
Vbisx 0.277
Vpacx 0.000
Weights
Vfinx 0.646
Veurx 0.000
Veiex 0.000
Vbltx 0.354
Vbisx 0.000
Vpacx 0.000

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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.