Sunday October 21st 2018

How to Build a Minimum Variance Stock Portfolio with ETFs

BNSF Railway locomotive pulling a freight train. Flagstaff, Dec, 2010. All Photos courtesy Standard Travel Photos

In the previous article, Stock Index Funds vs. Monkeys, we looked at a paper that compared various index construction methods. The Minimum Variance Portfolio MVP had both the lowest volatility and best Sharpe Ratio. The weights of the MVP were selected to minimize the volatility of a portfolio. The MVP is on the efficient frontier, on the tip of the Markowitz bullet. We looked at finding minimum variance portfolios in previous articles.

In the paper, the authors solved for the MVP of 1000 individual stocks. But for small investors, it’s not practical to own 1000 different securities. After all, that’s why index funds were created in the first place. But most low-cost index funds are market cap weighted, which was the worst index-construction method.

We would prefer a higher Sharpe ratio, so how can we approximate the MVP for all U.S. stocks?

Vanguard Sector ETFs
Vanguard divides their Total Stock Market Index ETF (VTI) into ten individual sectors which they offer as separate ETFs. See Vanguard Sector ETFs. The sector ETFs are market-cap weighted within each sector.

In this article, we’ll look at finding the weights for the MVP of the U.S. Stock Market using Vanguard sector ETFs. Finding the weights of sector funds won’t be exactly the same as finding the weights of each individual stock, but maybe it will be close.

Table 1 shows all the Vanguard Sector ETFs and their recent market weights in the Vanguard Total Stock Market ETF (VTI). Note that there are actually eleven sector funds, but one of them, REITs, is not represented in the Total Stock Market and has zero weight.

Table 1. U.S. Total Stock Market Sectors
(as of 03/31/2013)
Sector Symbol Market
1. Consumer Discretionary VCR 12.40 369 Comcast
2. Consumer Staples VDC 9.60 109 Proctor & Gamble
3. Energy VDE 10.10 169 Exxon Mobil
4. Financials VFH 16.80 520 Wells Fargo & Co.
5. Health Care VHT 12.30 290 Johnson & Johnson
6. Industrials VIS 11.20 358 General Electric
7 Information Technology VGT 17.70 415 Apple Inc.
8. Materials VAW 3.80 135 Monsanto Co,
9. REIT VNQ 0.00 121 Simon Property Group
10. Telecommunications Services VOX 2.60 34 Verizon Communications Inc.
9. Utiliities VPU 3.50 78 Duke Energy Corp

I listed the number of stocks in each sector and the top holding to give an idea about the kind of companies in each sector. The sector market weights vary with the ups and downs of the market prices. We’ll be solving for the sectors weights for the MVP portfolio, which will differ from the market weights.

Analysis of Sector-Fund Portfolios
I created a program in R that pulls data for these funds from Yahoo for the period Jan-2004 to Dec-2012, then calculates the global minimum variance portfolio (GMVP), Tangency Portfolio (TP), Equal-Weight Portfolio (EW) and Efficient Frontier (EF).

Here is a plot of the monthly returns for each of the four assets over the 8-year period.

Vanguard Sector Fund Monthly Returns, Jan-2004 to Dec-2012.

Asset Return Statistics

[1] “Anualized Mean Returns & Volatilities”
0.06141 0.08285 0.1005 -0.02024 0.05464 0.05916 0.04512 0.07966
0.19945 0.10823 0.2456 0.23862 0.13556 0.21146 0.20086 0.23304
0.07549 0.06738 0.08027
0.28544 0.17053 0.12858

The highest returning sector was Energy VDE. Financials VFH had negative return. The least volatile was the Consumer Staples VDC, which was less than half as volatile as some of the other sectors.

[1] “Skewness”
-0.6007 -1.0632 -0.6709 -1.1483 -0.7636 -0.8216 -0.6175 -0.8935
-1.3777 -0.9316 -1.4867

All sectors showed negative skewness.

[1] “Excess Kurtosis”
2.3000 2.2469 0.6616 3.1354 1.5546 2.1091 0.6555 2.6129 5.6900 1.1349

All sectors showed excess kurtosis, which means that extreme events occur more frequently than a Gaussian distribution would predict. REITs had the fattest tails.

[1] “Sharpe Ratios”
0.05270 0.15431 0.08879 -0.05473 0.06312 0.04663 0.02891
0.06770 0.05106 0.07175 0.12409

Highest Sharpe ration was Consumer Staples VDC.

[1] “Correlation Matrix”
VCR 1.0000 0.7741 0.5497 0.8619 0.7174 0.9084 0.8660 0.8365 0.8217
VDC 0.7741 1.0000 0.5134 0.7727 0.7835 0.8004 0.7101 0.7011 0.7260
VDE 0.5497 0.5134 1.0000 0.4914 0.4941 0.6621 0.6532 0.7775 0.4841
VFH 0.8619 0.7727 0.4914 1.0000 0.7259 0.8671 0.7213 0.7458 0.8520
VHT 0.7174 0.7835 0.4941 0.7259 1.0000 0.7409 0.7107 0.6943 0.7038
VIS 0.9084 0.8004 0.6621 0.8671 0.7409 1.0000 0.8423 0.8958 0.7970
VGT 0.8660 0.7101 0.6532 0.7213 0.7107 0.8423 1.0000 0.8460 0.6929
VAW 0.8365 0.7011 0.7775 0.7458 0.6943 0.8958 0.8460 1.0000 0.7205
VNQ 0.8217 0.7260 0.4841 0.8520 0.7038 0.7970 0.6929 0.7205 1.0000
VOX 0.7466 0.7053 0.5652 0.6479 0.6708 0.7624 0.7872 0.7494 0.6158
VPU 0.4563 0.5971 0.5934 0.4300 0.5975 0.5390 0.5235 0.5436 0.4902
VCR 0.7466 0.4563
VDC 0.7053 0.5971
VDE 0.5652 0.5934
VFH 0.6479 0.4300
VHT 0.6708 0.5975
VIS 0.7624 0.5390
VGT 0.7872 0.5235
VAW 0.7494 0.5436
VNQ 0.6158 0.4902
VOX 1.0000 0.6446
VPU 0.6446 1.0000

Utilities VPU seems to be least correlated with other sectors, making it a potentially good diversifier.

The scatterplot matrix graphically shows correlation of returns among sectors. Plots that are along a line are more correlated, than plots that look more like a shotgun blast. For example, Consumer Discretionary VCR and Industrials VIS look highly correlated. All the Utilities VPU plots look more like the shotgun blast.

Vanguard Sector Fund Scatterplot showing corelation, Jan-2004 to Dec-2012.

Minimum Variance Portfolio
As a reminder, to calculate the Minimum Variance Portfolio MVP, only variances and covariances are needed. Means are not needed. This is an advantage because estimation errors of the means are huge, while estimation errors of variances are much smaller. The MVP is agnostic with respect to expected returns.

Mean Monthly return, volatility and weights for Global Minimum Variance Portfolio GMVP:

Portfolio expected return: 0.006848
Portfolio standard deviation: 0.03021
Portfolio weights:
0.0000 0.7355 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

[1] “GMVP Sharpe Ratio”
[1] 0.1577

The portfolio weights for the MVP over the period Jan-2004 to Dec-2012 was 74% VDC and 26% VPU. The weight in all the other funds was zero. Optimized results commonly have have zero weight on most of the components. Optimal solutions often puts a lot of weight on assets with the lowest volatilities, only adding higher volatile assets if they are good diversifiers. In this case, only Utilites VPU provided a diversification benefit.

The MVP portfolio weights are plotted in the next chart.

Minimum Variance Portfolio weights for Vanguard Sector Funds, Jan-2004 to Dec-2012.

Efficient Frontier and Tangency Portfolio
Recall that TP is the portfolio with maximum Sharpe ratio. The mean Monthly return, volatility and weights for the Tangency Portfolio TP was:

Portfolio expected return: 0.006856
Portfolio standard deviation: 0.03023
Portfolio weights:
0.0000 0.7762 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

[1] “TP Sharpe Ratio”
[1] 0.1579

The highest Sharpe slope was with 78% VDC and 22% VPU. There was zero allocation to all of the other sector funds. TP is nearly identical to the MVP.

Again, anyone that has run a few portfolio optimizations knows that the results are often highly concentrated portfolios. Optimization tends to heavily weight assets that have high returns, low volatility or low correlation. Results often have zero allocation to most of the assets. That’s the way she goes.

Plotted here is the Efficient Frontier with the Capital Allocation Line CAL, the line from the risk-free rate to the Tangency Portfolio TP. The MVP is the last blue dot on the left. Optimum portfolios would be a combination of TP plus the risk-free asset, along the green line.

Efficient Frontier for Vanguard Sector Funds, Jan-2004 to Dec-2012.

One of the red dots, P1, is a slightly more diversified portfolio. P1 arbitrarily allocates 50% to VDC and 5% to each of the other ten ETFs. The other red dot is the Equal-Weight portfolio EW, which is the most diversified portfolio. It was far off the efficient frontier, in the middle of all the sector funds.

The next set of plots shows the asset weights along the efficient frontier EF. All EF portfolios consist of only three of the eleven funds. The Going from lowest to highest return, it starts with the GMV portfolio which has VDC/VPU. With increasing return, they are phased out in favor of Energy VDE. The highest return was 100% VDE, which is the Energy sector.

Minimum Variance Portfolio weights for Vanguard Sector Funds, Jan-2004 to Dec-2012.

Concentration vs. Diversification
That is our attempt at finding a minimum variance portfolio MVP for U.S. Stocks using Vanguard sector ETFs: 74% Consumer Staples, 26% Utilities and 0% in other sectors. That is a total of 187 different stocks. Is that enough for diversification?

If risk were measured, only by volatility, then the MVP would have the least risk. But for some people, concentrating everything in only two sectors might seem to be a risky bet.

One way to have a less concentrated portfolio would be to include some minimum allocation, like 3% or 5%, to each of the other sectors. Portfolio P1 in the Efficient Frontier chart above was just that. As can be seen, volatility increased somewhat.

A more diverse portfolio is the Equal-weight EW. This adds even more volatility than P1, trading volatility for concentration risk. Recall from the Cass paper that the Equal Weight Index was about the same as the average monkey.

A third solution might be to use Vanguard Total Stock Market (VTI) as a core holding, along with VDC and VPU to tilt the portfolio toward the MVP. That would be very simple to implement with just three funds. But how much to tilt? If there is only a slight tilt, that’s getting closer to the standard market-cap index which we were trying to improve upon.

There is a tradeoff between volatility and how concentrated the portfolio is.

Try It Yourself
You can download the R program PortOpt-10Fund.R. This program solves for the Global Minimum Variance Portfolio GMVP, the Tangency Portfolio TP and plots the Efficient Frontier EF and Capital Allocation Line CAL. I suggest running it under RStudio.

You will also need to download portfolio_noshorts.r from the University of Washington, which has the functions to optimize a portfolio with no short sales.

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