Analyzing Assets of Stock Index Funds
Dainese Chandra
An index fund is a portfolio of stocks or bonds designed to mimic the composition and performance of a financial market index.
Table of Contents
- Executive Summary
- Return Calculation and Sample Statistics
- Value-at-Risk Calculations
- Portfolio Theory
- Asset Allocation
- Bibliograpby
0. Executive Summary
In this project, I will analyze 5 years of monthly closing price data from the end of October 2014 through the end of October 2019.
Description of funds
S&P 500 index: vfinx - market-capitalization-weighted index of the 500 largest U.S. publicly traded companies. Benchmark of US’s stock performance
European stock index: veurx - mutual fund which invests in all stocks in the MSCI (Morgan Stanley Capital International) Europe index, which has stocks of companies located in Europe.
Emerging markets fund: veiex - mutual fund which invests in all stocks in the MSCI Emerging Markets index, which has stocks of companies located in emerging markets/ nations with rapid growth and industrialization.
Long-term bond fund: vbltx - mutual fund which tracks performance of a market-weighted bond index called the Barclays Capital U.S. Long Government/Credit Float Adjusted Index, which includes government and high-quality corporate bonds that have an average maturity of 15-30 years.
Short-term bond fund: vbisx - mutual fund which tracks performance of a market-weighted bond index called the Barclays Capital U.S. 1-5 Year Government/Credit Float Adjusted Index, which includes government and high-quality corporate bonds that have an average maturity of 1-5 years.
Pacific stock index: vpacx - mutual funds that attemps to replicate performance of MSCI Pacific Index, which has stocks of comapnies located in Japan, Australia, Hong Kong, Singapore, and New Zealand.
- Main findings of Analysis:
- The mutual funds that index groups of countries all had prices which collectively rose and peaked at 2018, before dipping through the end of 2018 and 2019. The two bond funds had prices which experienced a steady increase from 2015 to 2019.
- The mutual funds have returns which look relatively normally distributed.
- Country stock indexes SP500 (vfinx), European (veurx), and Pacific (vpacx) funds perform poorly as they have lower return and higher risk than the two bond funds (vbltx) and (vbisx).
- Correlations between the vanguard assrts are all positives, making diversification difficult as mutual funds cannot be shorted.
- Short-term bond fund (vbisx) has lowest value-at-risk, while emerging markets fund (veiex) has highest value-at-risk
- Global minimum portfolio without short sale has higher stadard deviation and expected return than the global minimum portfolio with short sale.
- Tangency portfolio with shorting has higher expected return, lower standard deviation and higher Sharpe Ratio than the portfolio without shorting.
1. Return Calculation and Sample Statistics
1.1 Monthly prices, cc (continuously compounded) returns, and future values 
Mutual funds like the S&P 500 (vfinx),European stock index (veurx), emerging markets fund (veiex) and Pacific stock index (vpax) have prices and continuously compounded returns that move together and seem to display similar trends. For example, prices collectively rose and peaked at 2018, before dipping through the end of 2018 and 2019. The continuously compounded return for these four funds also fluctuated around their respective means with little drastic changes.
The two bond funds, long-term bond fund (vbltx) and short-term bond fund (vbisx) also display similar patterns. Their prices experienced a steady increase from 2015 to 2019. The continuously compounded returns for both funds also moved together, remaining steady except for a drop in returns in 2017.
The highest future value is for the S&P 500 (vfinx) funds, which is not surprising as it contains a high proportion of stocks of many large US companies.
1.2 Normality of Monthly Continuosuly Compounded Returns In this section, there are four panel diagnostic plots for each asset containing histograms, smoothed density plots, boxplots and qq-plots, which are used to determine if these returns are normally distributed.
The returns look relatively normally distributed. There does not seem to be any evidence of linear time dependence.
1.3 Constant Expected Return (CER) Model Parameters
| vfinx | veurx | veiex | vbltx | vbisx | vpacx | |
|---|---|---|---|---|---|---|
| Mean | 0.008 | 0.003 | 0.002 | 0.005 | 0.001 | 0.004 |
| Std Dev | 0.035 | 0.040 | 0.044 | 0.025 | 0.004 | 0.038 |
| Skewness | -0.521 | -0.227 | 0.265 | 0.208 | 0.365 | -0.235 |
| Excess Kurtosis | 0.877 | -0.970 | -0.247 | 0.802 | 0.115 | 0.236 |
| 1% Quantile | -0.080 | -0.075 | -0.080 | -0.048 | -0.007 | -0.086 |
| 5% Quantile | -0.062 | -0.063 | -0.062 | -0.032 | -0.005 | -0.054 |
S&P 500 index (vfinx) has the highest average returns, while short-term bond fund (vbisx) has the lowest average returns. Emerging markets funds (veiex) has the highest standard deviation (meaning it is the riskiest), while short-term bond funds (vbisx) has the lowest standard deviation (least risky). We can see that country stock indexes SP500 (vfinx), European (veurx), and Pacific (vpacx) funds perform poorly as they have lower return and higher risk than the two bond funds (vbltx) and (viscx)
Short-term bond fund (viscx) looks the most normally distributed as it has an excess kurtosis closest to zero. European (veurx) fund looks the least normally distributed.
1.4 95% Confidence Intervals of Mean and Standard Deviation Estimates
| mu.lower.95 | mu.upper.95 | sigma.lower.95 | sigma.upper.95 | |
|---|---|---|---|---|
| vfinx | -0.001 | 0.018 | 0.029 | 0.042 |
| veurx | -0.007 | 0.013 | 0.032 | 0.047 |
| veiex | -0.010 | 0.013 | 0.036 | 0.053 |
| vbltx | -0.002 | 0.011 | 0.021 | 0.030 |
| vbisx | 0.000 | 0.003 | 0.003 | 0.005 |
| vpacx | -0.006 | 0.014 | 0.031 | 0.044 |
The mean and standard deviation estimates are quite precise, with the mean estimates being more precise than the standard deviation estimates. The mean 95% CIs contain both positive and negative values, signifying more uncertainty as th economic interpretation of a negative mean and positive mean are very different. From the table, we can see that short-term bond funds (vbisx) have the lowest volatility.
1.5 Annualized Returns & Annualized Sharpe Ratios
| x | |
|---|---|
| vfinx | 0.101 |
| veurx | 0.036 |
| veiex | 0.021 |
| vbltx | 0.054 |
| vbisx | 0.017 |
| vpacx | 0.048 |
| x | |
|---|---|
| vfinx | 0.122 |
| veurx | 0.137 |
| veiex | 0.154 |
| vbltx | 0.088 |
| vbisx | 0.015 |
| vpacx | 0.130 |
SP 500 (vfinx) has the highest annual mean, which makes sense as it takes up a big portion of stocks of the largest US companies. European index (veiex) has the highest volatility, showing that it is the riskiest asset.
| x | |
|---|---|
| vfinx | 0.786 |
| veurx | 0.229 |
| veiex | 0.105 |
| vbltx | 0.564 |
| vbisx | 0.850 |
| vpacx | 0.329 |
## annualized_sharpe SharpeRatios
## vfinx 0.786 0.2269
## veurx 0.229 0.0662
## veiex 0.105 0.0303
## vbltx 0.564 0.1627
## vbisx 0.850 0.2453
## vpacx 0.329 0.0948Table compares annualized sharpe ratios with monthly sharpe ratios. The rankings for annualized sharpe ratios and the rankings of monthly sharpe ratio are the same.
Future Value of $1 if same average annual return for 5 years
## vfinx veurx veiex vbltx vbisx vpacx
## 1.50 1.18 1.11 1.27 1.09 1.24## vfinx veurx veiex vbltx vbisx vpacx
## vfinx 1.24e-03 1.10e-03 1.08e-03 -8.36e-05 -3.99e-05 1.08e-03
## veurx 1.10e-03 1.57e-03 1.30e-03 -6.77e-07 -2.23e-05 1.23e-03
## veiex 1.08e-03 1.30e-03 1.98e-03 1.08e-04 7.53e-06 1.42e-03
## vbltx -8.36e-05 -6.77e-07 1.08e-04 6.39e-04 8.56e-05 -5.72e-06
## vbisx -3.99e-05 -2.23e-05 7.53e-06 8.56e-05 1.80e-05 -2.02e-05
## vpacx 1.08e-03 1.23e-03 1.42e-03 -5.72e-06 -2.02e-05 1.41e-03
All of the scatterplots seem to have am upward tilt, indicating positive covaraicne and small positive correlations.
Sample correlation matrix of the return 
The corrplot shows the correlation matrix visually. The darker the numbers and ellipses, the more positive is the correlation. Here, we see that European stock index (veiex) and Pacific stock index (vpacx) have the largest correlation at 0.85 and short-term bond fund (vbisx) and S&P 500 stock index (vfinx) have the weakest at -0.27. Based on the estimated correlation values, diversification will reduce risk as there are more positively correlated assets than negatively correlated assets.
2. Value-at-Risk Calculations
2.1 VaR of $100,000 Over Different Investment Horizons
| asset | VaR.1. | VaR.5. |
|---|---|---|
| vfinx | -7094 | -4835 |
| veurx | -8520 | -6019 |
| veiex | -9668 | -6889 |
| vbltx | -5285 | -3638 |
| vbisx | -837 | -550 |
| vpacx | -7991 | -5609 |
| asset | VaR.1. | VaR.5. |
|---|---|---|
| vfinx | -16728 | -9503 |
| veurx | -24616 | -17231 |
| veiex | -28621 | -20720 |
| vbltx | -13877 | -8579 |
| vbisx | -1654 | -665 |
| vpacx | -22477 | -15299 |
Short-term bond fund (vbisx) has lowest value-at-risk, while emerging markets fund (veiex) has highest value-at-risk.
2.2 Bootstrapped Standard Errors and 95% Confidence Intervals for 1% and 5% Value-at-Risk
## asset VaR.1. bootstrap.std.error.1. VaR.5. bootstrap.std.error.5.
## 1 vfinx -7094 1063.0 -4835 810.0
## 2 veurx -8520 831.0 -6019 710.0
## 3 veiex -9668 827.0 -6889 706.0
## 4 vbltx -5285 638.0 -3638 502.0
## 5 vbisx -837 92.5 -550 73.6
## 6 vpacx -7991 929.0 -5609 797.0
## X95..Confidence.Interval
## 1 (-6568, -3267)
## 2 (-7459, -4726)
## 3 (-8410, -5591)
## 4 (-4624, -2756)
## 5 (-708, -413)
## 6 (-7208, -4141)The short-term bond (vbisx) has the lowest value at risk, while the emerging markets fund( veiex) has the highest value at risk. Since the estimated 5% VaR is within the confidence interval for all six assets, the 5% VaR for all assets are statistically significant and are reliable measures.
2.3 Bootstrapped Standard Errors for 1% and 5% Empirical Value-at-Risk
| Emp..VaR.1. | Emp..VaR.5. | |
|---|---|---|
| vfinx | -7714 | -6058 |
| veurx | -7256 | -6120 |
| veiex | -7692 | -6011 |
| vbltx | -4691 | -3149 |
| vbisx | -691 | -473 |
| vpacx | -8283 | -5275 |
The values for Empirical VaR and normal VaR are similar, except that some empirical values (such as vbltx and vpacx) are out of the 95% confidence interval for the normal values.
3. Portfolio Theory
3.1 Global Minimum Variance Portfolio
Emerging markets fund (veiex) and long-term bond fund (vbltx) have negative weights. Since there are negative weights in the portfolio, it means this global minimum variance portfolio was constructed by shorting mutual funds, which cannot be shorted. THerefore, this global min portfolio that allows short sales is not replicable.
3.2 Annualized Sharpe Ratio
| x |
|---|
| 1.31 |
| x | |
|---|---|
| vfinx | 0.786 |
| veurx | 0.229 |
| veiex | 0.105 |
| vbltx | 0.564 |
| vbisx | 0.850 |
| vpacx | 0.329 |
Portfolio sharpe ratio is lower (-3.72) than the sharpe ratio of the assets. This means that the risk-free rate is greater than the portfolio’s return,o r the portfolio’s return is expected to be negative.
3.3 Value-at-Risk of Global Minimum Variance Portfolio
## VaR.of.Portfolio
## 1 204## VaR.of.Assets
## vfinx -4956
## veurx -6208
## veiex -7138
## vbltx -3706
## vbisx -552
## vpacx -5772Value-at-risk for portfolio is way lower than value-at-risk for the assets separately.
3.4 Global Minimum Variance Portfolio without Short Sales This is relevant as you cannot short mutual funds in your 401k account.
3.5 Comparing Short Sales and No Short Sales portfolio
## Short Sales Sharpe No Short Sales Sharpe
## [1,] 1.31 1.18## Short Sales SD No Short Sales SD
## [1,] 0.00895 0.0136Annual sharpe ratio for portfolio with no short sales is higher than the sharpe ratio for portfolio with short sales and it is closer to zero.
3.6 Efficient Portfolio Frontier: Markowitz Bullet Efficient Frontier Allowing Short Sales 
3.7 Efficient Portfolio Frontier allowing short sales vs not allowing short sales 
3.8 Tangency Portfolio
## Sharpe.Ratio
## vfinx 0.2269
## veurx 0.0662
## veiex 0.0303
## vbltx 0.1627
## vbisx 0.2453
## vpacx 0.0948Sharpe ratio of portfolio (0.517) is greater than the sharpe portfolio of the individual assets. There is higher expected reutnr to risk for the portfolio as opposed to individual assets.
3.9 Efficient Frontier with no short sale vs with short sale
3.10 Portfolio with target volatility of 2% per month
## [1] -0.0107The expected return for investing in a no short sale portfolio is 0.0045. The expected return for investing in a short sale portfolio is 0.0063.
3.11 Tangency portfolio with no short sales
## Call:
## tangency.portfolio(er = muhat.vals, cov.mat = cov.mat, risk.free = rf,
## shorts = FALSE)
##
## Portfolio expected return: 0.00217
## Portfolio standard deviation: 0.0045
## Portfolio Sharpe Ratio: 0.39
## Portfolio weights:
## vfinx veurx veiex vbltx vbisx vpacx
## 0.103 0.000 0.000 0.000 0.897 0.000
The sharpe ratio for the no-short sale tangency portfolio is 0.39. The annual sharpe ratio for the no-short sale tangency portfolio is 1.65. The sharpe ratio for the tangency portfolio including short sale is higher (0.517) than the portfolio with no short-sale (0.39). The expected return and standard deviation between the two portfolios are similar, although the portfolio with no short-sales only have positive weights.
4. Asset Allocation
4.1 Targeted Mean Efficient Portfolio
Efficient Portfolio with target expected return of 6% per year (which corresponds to an expected return of 0.5% per month) using only the risky assets and no short sales. Recall, you cannot short a mutual fund.
| x | |
|---|---|
| vfinx | 12.326 |
| veurx | -0.391 |
| veiex | -0.342 |
| vbltx | 0.615 |
| vbisx | 0.063 |
| vpacx | -11.272 |
The efficient portfolio achieves 6% expected return with 6 risky assets and short sale. Not possible to have no short sale in this case.
## vfinx veurx veiex vbltx vbisx vpacx
## vfinx 2.50e-03 2.18e-03 2.11e-03 1.64e-04 2.17e-03 2.50e-03
## veurx 2.18e-03 3.20e-03 2.60e-03 -1.35e-06 2.52e-03 2.18e-03
## veiex 2.11e-03 2.60e-03 3.87e-03 1.92e-05 2.84e-03 2.11e-03
## vbltx -1.64e-04 -1.35e-06 1.92e-05 1.25e-03 -1.15e-06 -1.64e-04
## vbisx -7.48e-05 -4.24e-05 1.41e-05 1.60e-04 -3.87e-05 -7.48e-05
## vpacx 2.17e-03 2.52e-03 2.84e-03 -3.87e-05 2.89e-03 2.17e-03
## mu.vec
## vfinx 0.00841 1
## veurx 0.00304 1
## veiex 0.00176 1
## vbltx 0.00453 1
## vbisx 0.00146 1
## vpacx 0.00397 1| x | |
|---|---|
| vfinx | 25.839 |
| veurx | -0.391 |
| veiex | -0.342 |
| vbltx | 0.615 |
| vbisx | 0.063 |
| vpacx | -24.785 |
The portfolio with a 12% target expected return has similar weights with the 6% portfolio, except that it places more weight on S&P 500 (vfinx) (25.839%) as compared to the portfolio with a 6% targeted return (12.326%). It also shorts more on the Pacific index (vpacx) (-24.785%) as opposed to the 6% portfolio (-11.272%).
4.2 Monthly Standard Deviation and Value-At-Risk of a $100,000 investment
Below is the monthly 1% VaR and 5% VaR of the efficient portfolio.
## X1..VaR.of.Assets X5..VaR.of.Assets
## vfinx -7358 -4956
## veurx -8905 -6208
## veiex -10168 -7138
## vbltx -5429 -3706
## vbisx -840 -552
## vpacx -8329 -5772The 1% VaR is larger than the 5% VaR.
5. Bibliography
- Zivot, E. (2017). Computational finance and financial econometrics. CRC Press.
- Kenton, W. (2021, May 24). S&P 500 Index. Investopedia. https://www.investopedia.com/terms/s/sp500.asp#:~:text=Key%20Takeaways-,The%20S%26P%20500%20Index%2C%20or%20the%20Standard%20%26%20Poor’s%20500%20Index,shares%20available%20for%20public%20trading.