Saha, Final Project
- 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
- 2. Saha | 2 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
- 3. Saha | 3 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.
- 4. Saha | 4 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
- 5. Saha | 5 • 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.
- 6. Saha | 6 • 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.
- 7. Saha | 7 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.
- 8. Saha | 8 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.
- 9. Saha | 9 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.
- 10. Saha | 10 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.
- 11. Saha | 11 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.
- 12. Saha | 12 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.
- 13. Saha | 13 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.
- 14. Saha | 14 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.
- 15. Saha | 15 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
- 16. Saha | 16 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%.
- 17. Saha | 17 Rolling Analysis of the CER Model Parameters I. 24 Month Rolling estimates VFINX VEURX VEIEX VBLTX VBISX VPACX
- 18. Saha | 18 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.
- 19. Saha | 19 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%.
- 20. Saha | 20 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.
- 21. Saha | 21 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
- 22. Saha | 22 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.
- 23. Saha | 23 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.
- 24. Saha | 24 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
- 25. Saha | 25 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
- 26. Saha | 26 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.