*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.

Sector | Symbol | Market Weight |
Number Stocks |
Top Holding |
---|---|---|---|---|

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.

**Asset Return Statistics**

`[1] “Anualized Mean Returns & Volatilities”`

VCR VDC VDE VFH VHT VIS VGT VAW

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

VNQ VOX VPU

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”`

VCR VDC VDE VFH VHT VIS VGT VAW

-0.6007 -1.0632 -0.6709 -1.1483 -0.7636 -0.8216 -0.6175 -0.8935

VNQ VOX VPU

-1.3777 -0.9316 -1.4867

All sectors showed negative skewness.

`[1] “Excess Kurtosis”`

VCR VDC VDE VFH VHT VIS VGT VAW VNQ VOX

2.3000 2.2469 0.6616 3.1354 1.5546 2.1091 0.6555 2.6129 5.6900 1.1349

VPU

2.9224

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”`

VCR VDC VDE VFH VHT VIS VGT

0.05270 0.15431 0.08879 -0.05473 0.06312 0.04663 0.02891

VAW VNQ VOX VPU

0.06770 0.05106 0.07175 0.12409

Highest Sharpe ration was Consumer Staples VDC.

`[1] “Correlation Matrix”`

VCR VDC VDE VFH VHT VIS VGT VAW VNQ

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

VOX VPU

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.

**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:

VCR VDC VDE VFH VHT VIS VGT VAW VNQ VOX

0.0000 0.7355 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

VPU

0.2645

`[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.

**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:

VCR VDC VDE VFH VHT VIS VGT VAW VNQ VOX

0.0000 0.7762 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

VPU

0.2238

`[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.

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.

**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.