Higher Returns at Lower Risk Performance Beyond the Market Cap-Weighted Model © 2014 Frans Waterlander - March, 2014

Summary Investment objectives, investment phase (accumulation, transition, early retirement, etc.) and tolerance for risk are major factors in deciding on the allocation between the various asset classes in a portfolio. Many in the financial world recommend a broad diversification of stocks and bonds to closely represent the overall stock and bond markets, the so-called market cap-weighted model approach. Common wisdom says that such portfolios minimize risk and those that don't follow that model may increase returns, but do so at the expense of increased risk. But is that really true? I decided to analyze the interactions between asset classes and find out what happens to returns and risks, and why. The analysis produced useful insights and identified higher-return, lower-risk alternatives to portfolios that follow the broadly-diversified, market cap-weighted model. In all, 38 model portfolios on the Efficient Frontier, from the very conservative to the very aggressive, have been identified.

Analysis method The Historical time-pathing statistical model is used to create future, long-term risk/return scenarios based on historic returns over the period of 1928 through 2011 of ten U.S. asset classes included in the 2012 Ibbotson Stocks, Bonds, Bills and Inflation (SBBI) Classic Yearbook. Fixed allocation ratios, achieved through yearly re-balancing, are used. The Ibbotson yearbook contains the longest year-by-year history of returns on the capital markets in the U.S. that I could find. It covers the period of 1926 through 2011, but it is missing data on growth and value stocks for the years 1926 and 1927 and for that reason those years are not including in the analysis. It covers a long period and includes many ups and downs like the Great Depression, the 1972-1974 bear market, the 1980s and 1990s bull markets and the 2007-2009 financial crisis. This long-term history gives us higher confidence in the future risk/return scenarios derived from it than those based on much shorter periods, like the pasts 20, 25 or even 40 years. It should be noted that the returns, standard deviations and 1, 5, 10 and 20 year losses for the various combinations of the ten asset classes are calculated from the year-by-year returns of each combination of asset classes, and not from the use of correlation coefficients between asset classes and standard deviations of the asset classes by themselves. Correlation between asset classes varies dramatically from year to year. Calculating and subsequently using "average" correlation coefficients between asset classes for the whole historic period - in our case 1926 through 2011 - will result in inaccuracies. Because all calculations are based on year-by-year returns of each combination of asset classes, no correlation calculations are needed and no inaccuracies are introduced as a result. Along the same line, the standard deviation and 1, 5, 10 and 20 year losses are calculated from the year-toyear returns of each combination of asset classes, preventing inaccuracies introduced if mathematical approximations were derived from using the standard deviations of the asset classes by themselves.

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For starters, Table 1 details returns and risks for the ten asset classes by themselves. Table 1: The returns and risks of ten asset classes, 1928 through 2011, before expenses Risk Biggest loss Standard Ibbotson asset class Return deviation 1 Year 5 Years 10 Years 20 Years Large Company Stocks (LCS) Large Growth Stocks (LGS) Large Value Stocks (LVS) Small Company Stocks (SCS) Small Growth Stocks (SGS) Small Value Stocks (SVS) Long-Term Corporate Bonds (LTCB) Long-Term Government Bonds (LTGB) Intermediate-Term Government Bonds (ITGB) U.S. Treasury Bills (USTB)

9.46% 8.72% 10.73% 11.91% 9.05% 13.87% 6.03% 5.65% 5.41%

20.34% 20.21% 27.79% 32.85% 33.19% 32.77% 8.45% 9.89% 5.73%

43% 37% 58% 58% 49% 52% 8% 15% 5%

49% 51% 73% 80% 74% 76% 11% 10% no loss

13% 17% 44% 44% 10% 35% no loss 1% no loss

no loss no loss no loss no loss no loss no loss no loss no loss no loss

3.59%

3.13%

no loss

no loss

no loss

no loss

The returns and risks of some recommended stock/bond ratio portfolios What happens when you use some of the stock/bond ratios recommended for the market cap-weighted model, but use various combinations of stocks and bonds asset classes in the Ibbotson data base, thereby violating the broadly-diversified, market cap-weighted model? Graph 1 shows what happens to return and risk, expressed as standard deviation, when you create portfolios with different allocations among the 6 stock and 3 bond asset classes identified above. The different ratios show wide, overlapping ranges for both returns and standard deviations. In fact, those numbers are all over the place and very few of the portfolios are on or come close to what is called the Efficient Frontier. This Efficient Frontier curve represents the best possible combination of return and risk for any and all combinations of the 10 asset classes used in this paper. Any point on this curve indicates the lowest possible risk for any given return. The Efficient Frontier will be discussed in more detail later. Other risk factors (biggest 1, 5, 10 and 20 year losses) are also discussed and brought to bear in decision making later on. Graph 1: Risks/returns for some stock/bond ratios

14% 13%

Return

12% 11% 10% 9% 8% 7% 0%

5%

10%

15% 20% Standard Deviation

80/20

70/30

60/40

50/50

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25%

30%

Efficient Frontier

35%

The returns and risks of many stock, bond and bill portfolios We clearly need a more systematic approach. For starters, let's plot the results for combinations of stocks and bonds, stocks and bills, and bonds and bills. Graph 2 shows more than 1,000 of such combinations included in my analysis. These results are somewhat more useful, but don't tell what really happens. Graph 2: Mixing stocks, bonds and bills

14% 12%

Return

10% 8% 6% 4% 2% 0%

5%

10%

15% 20% Standard Deviation

Combinations in analysis

25%

30%

35%

Efficient Frontier

What exactly happens when we mix stocks and bonds Let's take the next step and look in more detail at what happens when we construct a portfolio consisting of both stocks and bonds. In order to study the interactions between stocks in general and bonds in general, Large Company Stocks data has been used as a proxy for stocks and Long-Term Government Bonds data as a proxy for bonds. Returns and standard deviations are represented by the blue dots in Graph 3. Graph 3: What happens when you mix stocks and bonds? 11% 100% Stocks

10%

Return

9% 8% 7% 6% 100% Bonds

5% 7%

9%

11%

13% 15% Standard Deviation Straight line

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17%

19%

21%

Each dot represents a portfolio of 100% stocks or 100% bonds. What would happen if we start with 100% bonds and then decrease that percentage and use stocks to make up the difference until we reach the point where the portfolio consisted of stocks alone? Would we move along a straight line, as shown? No, we wouldn't. What really happens is shown in Graph 4. Graph 4: What really happens when you mix stocks and bonds 11% 100% Stocks

10%

Return

9% 8% 7% 6% 100% Bonds 5% 7%

9%

11%

13% 15% Standard Deviation Straight line

17%

19%

21%

Actual line

Each dot represents a change of 5% in the allocation between stocks and bonds. What really happens is that we get a curved line, the so-called Risk/Return Curve, causing returns to have lower standard deviations than what would be the case if the response were a straight line. This curvature must be caused by the socalled synergy between stocks and bonds, where stocks do better when bonds don't and vice versa, reducing volatility, right? Well, yes and no. There definitely is a stock/bond synergy component to what happens here, but it is small compared to the main, mathematical reason. When you calculate the resulting returns and standard deviations for different ratios of two randomly-variable, un-correlated groups, you get a curve similar to the one in Graph 4.

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So let's take a look at Graph 5 to see how much synergy between stocks and bonds there really is. Graph 5: The real synergy between stocks and bonds 11% 100% Stocks 10%

Return

9% 8% 7% 6% 100% Bonds 5% 7%

9%

11%

Straight line

13% 15% Standard Deviation Actual line

17%

19%

21%

Data randomized

The line with the blue dots represents the original data and shows the total effects of math and the synergy between stocks and bonds. The line with the yellow dots shows the math component only because the original data has been randomized to remove the stock/bond synergy component or correlation. The conclusion is that stock/bond synergy helps, but it is not the major component of the better-than-linear character of the risk/return curve. I refer to the total curvature as represented by the line with the blue dots as Pseudo-Synergy, the curvature caused by mathematics and represented by the line with yellow dots as Math Synergy, and the difference between the two as Real Stock/Bond Synergy.

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Plotting more Risk/Return Curves Graph 6 shows the Risk/Return Curves between the six stock, three bond and one bill asset classes, as well as part of the Risk/Return Curve between Small Value Stocks, Intermediate-Term Government Bonds and U.S. Treasury Bills. The latter curve meanders all over the place because three asset classes are involves and for that reason only that part is shown that falls above all the others. None of the other possible combinations of three or more asset classes falls above the curves shown. It clearly shows that the highest returns for any given risk are parts of only three Risk/Return Curves: Small Value Stocks/Long-Term Government Bonds, Small Value Stocks/Intermediate Term Government Bonds and Small Value Stocks/Intermediate-Term Government Bonds/U.S. Treasury Bills.

Graph 6: All risk/return curves 16% Small Value Stocks

14% Small Company Stocks

12%

Return

Large Value Stocks

10% Large Company Stocks Small Growth Stocks

8%

Large Growth Stocks Long-Term Corporate Bonds

6%

Long-Term Government Bonds Intermediate-Term Government Bonds

4% U.S.Treasury Bills

2% 0%

5% 10 Asset classes

10%

15% 20% Standard Deviation

SVS+ITGB

SVS+LTGB

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25%

30% SVS+ITGB+USTB

35%

The Efficient Frontier Those top-3 Risk/Return Curves form the Efficient Frontier, as shown in Graph 7. Any point on the Efficient Frontier represents a model portfolio with the lowest possible risk for any given return. Graph 7: The top-3 risk/return curves and the Efficient Frontier 16% Small Value Stocks

14%

Return

12%

10% Points with white marker fill are not included in the Efficient Frontier Table

8%

Long-Term Government Bonds

6%

Intermediate-Term Government Bonds

4% U.S.Treasury Bills

2% 0%

5%

10%

Efficient Frontier

15% 20% Standard Deviation SVS+ITGB SVS+LTGB

25%

30%

35%

SVS+ITGB+USTB

This Efficient Frontier is specific to the asset classes covered and the 1928-2011 period. Because the covered period is so long, the resulting model portfolios should be more reliable indicators of future long-term results when compared to model portfolios based on much shorter historic periods, like the past 20, 25 or even 40 years. After all, the longest bull market in history, the 1980s and 1990s, dominates such relatively short time periods, resulting in overly optimistic forecasts.

The Efficient Frontier Table All points on the Efficient Frontier have been entered into Table 2a as Model Portfolio 1 through 38. These portfolios range from very conservative to very aggressive. The table can be used to create new portfolios depending on one's tolerance for risk, or to find alternatives to existing portfolios. Certain less-than-optimal ITGB and LTGB choices are included because their results are close and sticking with one or the other offers the advantage of not having to switch funds when allocations are changed as you move through the various investment phases (accumulation, transition, early retirement, etc.). Fund 1 in the table is not one of my model portfolios, but the most aggressive fund of seven, commonly recommended funds analyzed in Tables 3 and 4. Note that there are 7 model portfolios that have higher returns and the same or lower risk than this Fund 1!

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The use of my model portfolios is a radical departure, not only from the broadly-diversified, market capweighted approach, but also "tilting". "Tilting" occurs when you start out with a broadly-diversified, market cap-weighted portfolio and then "tilt" the allocation modestly, say by 5, 10 or 15%, towards any asset class.

Table 2a: Efficient Frontier Table Allocation Model Stocks Bonds Portfolio # LCS SVS ITGB LTGB 38 100% 37 95% 5% 36 95% 5% 35 90% 10% 34 90% 10% 33 85% 15% 32 85% 15% 31 80% 20% 30 80% 20% 29 75% 25% 28 75% 25% 27 70% 30% 26 70% 30% 25 65% 35% 24 65% 35% 23 60% 40% 22 60% 40% 21 55% 45% 20 55% 45% 19 50% 50% 18 50% 50% 17 45% 55% 16 45% 55% 15 40% 60% 14 40% 60% 13 35% 65% 12 35% 65% 11 30% 70% Fund 1 80% 20% see above 10 30% 70% 9 25% 75% 8 20% 80% 7 15% 85% 6 10% 90% 5 10% 80% 4 10% 40% 3 5% 35% 2 5% 5% 1 LCS = Large Company Stocks SVS = Small Value Stocks ITGB = Intermediate-Term Government Bonds LTGB = Long-Term Government Bonds USTB = U.S. Treasury Bills

Bills USTB

10% 50% 60% 90% 100%

Risks Biggest loss 5 Yrs 10 Yrs 76% 35% 73% 27% 73% 27% 70% 20% 70% 19% 67% 11% 67% 11% 63% 3% 63% 2% 60% no 59% no 56% no 55% no 51% no 51% no 47% no 46% no 42% no 42% no 38% no 37% no 33% no 32% no 28% no 26% no 22% no 21% no 17% no

Return before expenses 13.87% 13.72% 13.70% 13.54% 13.49% 13.33% 13.26% 13.10% 13.00% 12.83% 12.71% 12.54% 12.39% 12.22% 12.05% 11.87% 11.69% 11.49% 11.30% 11.10% 10.89% 10.67% 10.45% 10.22% 9.99% 9.74% 9.50% 9.24%

Standard Deviation 32.77% 31.10% 31.11% 29.44% 29.46% 27.79% 27.80% 26.15% 26.16% 24.53% 24.51% 22.92% 22.88% 21.34% 21.25% 19.78% 19.63% 18.26% 18.02% 16.78% 16.42% 15.36% 14.85% 14.00% 13.30% 12.75% 11.80% 11.62%

1 Yr 52% 50% 49% 47% 47% 45% 44% 43% 42% 40% 39% 38% 37% 36% 35% 33% 32% 31% 30% 29% 27% 26% 25% 24% 22% 22% 20% 19%

9.11%

16.42%

36%

36%

no

no

8.99% 8.46% 7.90% 7.32% 6.71% 6.53% 5.81% 5.00% 4.43% 3.59%

10.34% 8.97% 7.72% 6.66% 5.91% 5.55% 4.44% 3.56% 3.26% 3.13%

17% 15% 12% 10% 7% 7% 6% 3% 2% no

15% 10% 4% no no no no no 1% no

no no no no no no no no no no

no no no no no no no no no no

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20 Yrs no no no no no no no no no no no no no no no no no no no no no no no no no no no no

Translating the Efficient Frontier Table into real-world portfolios Now that we know what combinations of the various asset classes give us the optimum returns and risks, it's time to decide what investments to choose to create the portfolios in the Efficient Frontier Table. I recommend using index mutual funds as much as possible to insure the best possible match between the asset classes and the actual investments, and keep expenses at a minimum. Obviously, Vanguard mutual funds are one way to realize this. My recommendations are spelled out in Table 2b.

Table 2b: recommended Vanguard funds for the various asset classes Asset Class Code

Asset Class

LCS LGS LVS SCS SGS SVS LTCB LTGB ITGB USTB

Large Company Stocks Large Growth Stocks Large Value Stocks Small Company Stocks Small Growth Stocks Small Value Stocks Long-Term Corporate Bonds Long-Term Government Bonds Intermediate-Term Government Bonds US Treasury Bills

Recommended Vanguard fund Vanguard 500 Index Fund Vanguard Growth Index Fund Vanguard Value Index Fund Vanguard Small-Cap Index Fund Vanguard Small-Cap Growth Index Fund Vanguard Small-Cap Value Index Fund Vanguard Long-Term Investment-Grade Fund (1) Vanguard Long-Term Treasury Fund (2) Vanguard Intermediate-Term Treasury Fund (3) Vanguard Short-Term Treasury Fund

1: Vanguard Long-Term Corporate Bond Index Fund is potentially a better match for this asset class; it's relatively new (inception date 2010) and small, and expenses are a tad high at this time; as it grows over time, expenses should come down at which time you may want to make the switch 2: Vanguard Long-Term Government Bond Index Fund is potentially a better match for this asset class; it's relatively new (inception date 2010) and small, and expenses are a tad high at this time; as it grows over time, expenses should come down at which time you may want to make the switch 3: Vanguard Intermediate-Term Government Bond Index Fund is potentially a better match for this asset class; it's relatively new (inception date 2010) and small, and expenses are a tad high at this time; as it grows over time, expenses should come down at which time you may want to make the switch

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Finding better alternatives to some existing balanced index funds Let's now take a look at an existing broadly diversified, market cap-weighted, balanced index fund, in this case the Vanguard Balanced Index Fund and find an alternative portfolio that has a higher return and lower or the same standard deviation and biggest 1, 5, 10 and 20 year losses. The results are shown in Table 3. The historic data in the Ibbotson yearbook were used to calculate the long-term analysis numbers. Large Company Stocks were used as a proxy for the stock content of the Vanguard funds, and Intermediate-Term Government Bonds as a proxy for the bond content. Table 2a was used to find the best alternative model portfolio and Table 2b was used to populate it with existing Vanguard funds. The short-term actual results are calculated from published Vanguard data. Calculating actual biggest losses for 5, 10 and 20 years over a 15 year period is either statistically not significant or impossible. Table 3: Balanced Index Funds and my model portfolio Long-Term Analysis, 1928 through 2011, before expenses LCS SVS ITGB LTGB Return: Advantage: Standard deviation: Advantage: Biggest 1 year loss: Advantage: Biggest 5 year loss: Advantage: Biggest 10 year loss Biggest 20 year loss

Vanguard Balanced Index Fund 60% 40% 8.38% 12.39% 27% 22% no loss no loss

Model Portfolio 12 35% 65% 9.50% 1.12% 11.80% 0.59% 20% 7% 21% 1% no loss no loss

Short-term actual results, 15 years, 1999-2013 15 year return: Advantage: Standard deviation: Advantage: Biggest 1 year loss: Advantage:

5.74%* 11.64%* 22%*

*: Investor share data was used for 1999 and 2000; Admiral share data was used for 2001 and later

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7.64% 1.89% 5.48% 6.16% 2% 20%

Finding better alternatives to some recommended portfolios Let's now take a look at broadly-diversified, market cap-weighted portfolios recommended in John Bogle's Basic Allocation Model and in The Vanguard Retirement Investing Guide and Vanguard's Investing During Retirement. The historic data in the Ibbotson yearbook were used to calculate the long-term analysis numbers. Large Company Stocks were used as a proxy for the stock content of the Vanguard funds, Long-Term Government Bonds as a proxy for bonds, and U.S. Treasury Bills as a proxy for reserves. Table 2a was used to find the best alternative model portfolios and Table 2b was used to populate those with existing Vanguard funds. The short-term actual results are calculated from published Vanguard data. Calculating actual biggest losses for 5, 10 and 20 years over a 15 year period is either statistically not significant or impossible. Table 4: Some recommended portfolios and my model portfolios Long-Term Analysis, 1928 through 2011, before expenses Fund 1

LCS SVS ITGB LTGB USTB Return: Advantage: Standard deviation: Advantage: Biggest 1 year loss: Advantage: Biggest 5 year loss: Advantage: Biggest 10 year loss Biggest 20 year loss

Model Portfolio 17

80%

Fund 2

Model Portfolio 15

70% 45%

Fund 3

Model Portfolio 13

60% 40%

Fund 4

Model Portfolio 10

50% 35%

Fund 5

Model Portfolio 9

40% 30% 70%

Fund 6

20% 25% 75%

40% 20% 7.36%

Model Portfolio 7 15% 85%

20%

55%

30%

60%

40%

65%

50%

9.11%

10.67% 1.56%

8.85%

10.22% 1.37%

8.54%

9.74% 1.20%

8.18%

8.99% 0.81%

16.42%

16.42%

14.58%

14.00% 0.58%

12.87%

12.75% 0.12%

11.36%

10.34% 1.02%

9.15%

8.97% 0.18%

7.35%

6.66% 0.69%

36%

26% 10%

32%

24% 8%

28%

22% 6%

24%

17% 7%

19%

15% 4%

12%

10% 2%

36%

33% 3%

29%

28% 1%

23%

22% 1%

16%

15% 1%

10%

10%

no losss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

no loss

8.46% 1.10%

60% 20% 6.43%

7.32% 0.89%

Short-term actual results, 15 years, 1999-2013 15 year return: Advantage: Standard deviation: Advantage: Biggest 1 year loss: Advantage:

5.65%

8.80% 3.15%

6.00%

8.56% 2.56%

6.25%

8.31% 2.06%

6.42%

7.36% 0.93%

5.97%

7.06% 1.10%

5.98%

6.44% 0.46%

13.91%

7.09% 6.83%

11.22%

6.67% 4.55%

8.72%

6.55% 2.17%

6.62%

4.73% 1.89%

4.92%

4.18% 0.74%

5.26%

4.04% 1.22%

25%

3% 22

19%

4% 15%

13%

4% 9%

7%

1% 6%

4%

2% 3%

1%

2% -1%

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A graphical look at picking better alternatives Graph 8 shows the long-term analysis of the funds in tables 3 and 4. Note that all my model portfolios have higher returns and lower standard deviations and are on the Efficient Frontier. Graph 8: Long-Term Analysis

14% 12%

Return

10% 8% 6% Red dots: Vanguard funds and recommended stock/bond/bill ratios Green dots: my model portfolios

4% 2% 0%

5%

10%

15% 20% Standard Deviation

25%

30%

35%

Efficient Frontier

Graph 9 shows the short-term results. Again, all my portfolios have higher returns and lower risk. Graph 9: Short-term actual results 1999-2013 14% 12%

Return

10% 8% 6% 4%

Red dots: Vanguard funds and recommended stock/bond/bill ratios Green dots: my model portfolios

2% 0%

5%

10%

15% 20% Standard Deviation Efficient Frontier

Page 12 of 15

25%

30%

35%

The Small Value Stocks Premium As we have seen, small value stocks have the highest return of all covered asset classes over the covered period of 1928 through 2011 and were among the highest in terms of risk. However, when combined with various bond and bill asset classes to work the magic of synergy, they occupy most of the points on the Efficient Frontier, while no other stock asset class garnered a spot on this curve. The higher return of small value stocks is commonly referred to as the small value stock premium. Let's take a look at how $1 invested at the beginning of 1928 would have grown over time if invested in large company stocks, small company stocks and small value stocks. Graph 10 tells the story. Graph 10: growth of $1, 1928 through 2011

$100,000

$54,864 $12,687

$10,000

$1,984

$1,000

$100

$10

$1 1920

1930

1940

1950

1960

1970

1980

1990

2000

2010

2020

$0 Large Company Stocks

Small Company Stocks

Small Value Stocks

17

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Over time, small company stocks have significantly outperformed large company stocks and small value stocks have done even better by a wide margin. While small value stocks haven't outperformed the other two stock classes each and every year, over the long haul the difference in results is significant. Another way to look at this is to plot the ratio of cumulative results of small value stocks and large company stocks, as is done in Graph 11. Graph 11: The Small Value Stock Premium

100

10

1 1920

1930

1940

1950

1960

1970

1980

1990

2000

2010

2020

0 Cumulative Small Value Stocks to Large Company Stocks Ratio

Trendline

While nobody can predict with certainty that small value stocks will continue to outperform the other stock classes, chances are that they will.

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Conclusions This paper has analyzed the interactions between stocks and bonds in general, and between the ten asset classes covered here in particular. A fundamental understanding of these interactions has been gained and powerful analytical tools like Risk/Return Curves, Pseudo-Synergy, Math Synergy, Real Stock/Bond Synergy, the Efficient Frontier and the Efficient Frontier Table have been discussed. Better performing alternatives to broadly-diversified, market cap-weighted portfolios have been identified. The alternative portfolios not only show improved performance over the 1928 through 2011 timeframe, but also have shown a much stronger actual performance over the past 15 years. In all, 38 model portfolios on the Efficient Frontier, from the very conservative to the very aggressive, have been identified. Small value stocks have the highest return of all covered asset classes and have among the highest risk. However, when combined with bond and bill asset classes to work the magic of synergy, they are part of almost all model portfolios on the Efficient Frontier.

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