To fix ideas, we begin with the simplest quantitative model of stock returns: the one factor model which states that

- Excess return of a stock over the risk free rate =
*alpha*+*beta*times the excess return of the market over the risk free rate, which is equivalent to - Excess return of a stock over the market =
*alpha*+*(beta -1 )*times the excess return of the market over the risk free rate,

where alpha and beta are numerical parameters determined from the historical data. Clearly, the parameter beta magnifies the losses with respect to the market during the market downturns and, so, sometimes it is associated with risk (though within the framework of the classical theory actual measure of risk is the standard deviation of the stock returns).

As a test of the efficacy of this model for empirical data, I analyzed the performance of fifty five stocks that appear to have been presented for consideration for further research for going long in articles published during the six month period of March thru August 2015 by the author of the "Great Beta Hoax" article (seekingalpha.com/author/chuck-carnevale/... ). The following stocks were excluded from the analysis:

- the stocks mentioned in the articles whose titles suggested that they were used for illustrating general concepts,
- the stocks that the title suggested to have been overvalued, and
- the stocks whose dates of inception were later than Dec. 31, 2009 (as the stock returns from 2010 to-date were needed for the calculations).

For each stock the following computations were performed:

- the total return of the stock if purchased at closing on the date of publication of the article (or the next day if the market was closed on that date) and held to-date (2016/1/26) ,
- the total return of SPY if purchased at closing on the date of publication of the article (or the next day if the market was closed on that date) and held to-date,
- one factor beta of the stock on the basis of the monthly return data for the five year period 2010-2014, both inclusive. The ten year treasury rate was used as the risk free rate for this calculation.

The following plot depicts the excess return of the stock (difference between the returns of the stock and that of SPY) against its excess beta (i.e beta -1).

From the plot it is clear that higher the beta, lower is the excess return, which is as it should be, since the effect of positive excess beta is to magnify the gains as well as the losses. This of course holds, by definition, for the period of 2010-2014 as the historical return data from this period was used to calculate beta. However, the plot suggests that the result also holds for the out of sample data (in this case for 2015-2016), which is the main conclusion of this analysis.

A more quantitative and statistically satisfactory analysis to ensure that the observed data is not due to chance is based on the following contingency table, which summarizes the data in the form of a 2X2 matrix, with, for example, there being ten stocks with beta<1 which had negative excess return.

Beta <1 | Beta >=1 | |

Negative Excess Return | 10 | 20 |

Positive Excess Return | 20 | 5 |

The classical statistical methods for the analysis of a contingency table such as this can be used to determine the probability that the observed difference between the excess returns of low beta and high beta stocks is due to random effects or chance. For the table above, this probability - the so called p-value - turns out to be 0.05%. So obviously the differences observed here are not due to chance, and the result is statistically significant. Thus at least this data supports the conclusion that beta is definitely a good indicator of how the stock will perform in relation to the market.

Our results are consistent with the well known (but obviously not universally accepted) concepts related to beta:

- High beta stocks are likely to suffer higher draw-downs than the market during market downturns,
- As a consequence, during early phases of long term market downturns (if indeed such periods can be identified), it is not advisable to add high beta stocks to a portfolio, and
- Low beta stocks are generally preferable for a long term buy and hold portfolio if the investor is concerned about potentially high draw-downs during market corrections.

To fix ideas, we begin with the simplest quantitative model of stock returns: the one factor model which states that

- Excess return of a stock over the risk free rate =
*alpha*+*beta*times the excess return of the market over the risk free rate, which is equivalent to - Excess return of a stock over the market =
*alpha*+*(beta -1 )*times the excess return of the market over the risk free rate,

where alpha and beta are numerical parameters determined from the historical data. Clearly, the parameter beta magnifies the losses with respect to the market during the market downturns and, so, sometimes it is associated with risk (though within the framework of the classical theory actual measure of risk is the standard deviation of the stock returns).

As a test of the efficacy of this model for empirical data, I analyzed the performance of fifty five stocks that appear to have been presented for consideration for further research for going long in articles published during the six month period of March thru August 2015 by the author of the "Great Beta Hoax" article (seekingalpha.com/author/chuck-carnevale/... ). The following stocks were excluded from the analysis:

- the stocks mentioned in the articles whose titles suggested that they were used for illustrating general concepts,
- the stocks that the title suggested to have been overvalued, and
- the stocks whose dates of inception were later than Dec. 31, 2009 (as the stock returns from 2010 to-date were needed for the calculations).

For each stock the following computations were performed:

- the total return of the stock if purchased at closing on the date of publication of the article (or the next day if the market was closed on that date) and held to-date (2016/1/26) ,
- the total return of SPY if purchased at closing on the date of publication of the article (or the next day if the market was closed on that date) and held to-date,
- one factor beta of the stock on the basis of the monthly return data for the five year period 2010-2014, both inclusive. The ten year treasury rate was used as the risk free rate for this calculation.

The following plot depicts the excess return of the stock (difference between the returns of the stock and that of SPY) against its excess beta (i.e beta -1).

From the plot it is clear that higher the beta, lower is the excess return, which is as it should be, since the effect of positive excess beta is to magnify the gains as well as the losses. This of course holds, by definition, for the period of 2010-2014 as the historical return data from this period was used to calculate beta. However, the plot suggests that the result also holds for the out of sample data (in this case for 2015-2016), which is the main conclusion of this analysis.

A more quantitative and statistically satisfactory analysis to ensure that the observed data is not due to chance is based on the following contingency table, which summarizes the data in the form of a 2X2 matrix, with, for example, there being ten stocks with beta<1 which had negative excess return.

Beta <1 | Beta >=1 | |

Negative Excess Return | 10 | 20 |

Positive Excess Return | 20 | 5 |

The classical statistical methods for the analysis of a contingency table such as this can be used to determine the probability that the observed difference between the excess returns of low beta and high beta stocks is due to random effects or chance. For the table above, this probability - the so called p-value - turns out to be 0.05%. So obviously the differences observed here are not due to chance, and the result is statistically significant. Thus at least this data supports the conclusion that beta is definitely a good indicator of how the stock will perform in relation to the market.

Our results are consistent with the well known (but obviously not universally accepted) concepts related to beta:

- High beta stocks are likely to suffer higher draw-downs than the market during market downturns,
- As a consequence, during early phases of long term market downturns (if indeed such periods can be identified), it is not advisable to add high beta stocks to a portfolio, and
- Low beta stocks are generally preferable for a long term buy and hold portfolio if the investor is concerned about potentially high draw-downs during market corrections.

**1. Fidelity Select Funds**

This is a slight modification of the strategy that I introduced at the beginning of 2015 ( http://seekingalpha.com/instablog/709762-varan/3652756-a-five-fund-basket-for-the-long-haul ), which returned 3% for the year - not quite satisfactory, but slightly better than SPY.

I have replaced the Select Consumer Staples Fund (FDFAX) by the Select Retailing Fund (FSRPX), and the Real State Fund (FRESX) by the Select Construction Fund (FSHOX), due to the better overall historical performance of the in-coming funds. The strategy entails rebalancing the basket at the beginning of every year by means of a modified version of the risk parity method that utilizes the variance matrix of the five funds based upon the daily returns of the prior year, as well as the total returns of the five funds during the prior year.

The back test performance of the strategy is compared in the figure above with the market indices. The figure depicts the compound annual growth rates of portfolios initially funded at the beginning of any year since 1988 and held to date. (The results prior to 2004 were computed by using VUSTX instead of TLT.)

The performance metrics for the period 1988-2015 are as follows:

CAGR 14.6%, Maximum Drawdown 21.4%, Sharpe Ratio .98, Sortino Ratio 1.98

For 2016, the portfolio consists of the following:

FSRPX 33% FSCSX 23% FSHOX 20% FBIOX 17% TLT 7%.

**2. Nasdaq 12**

This is a momentum based strategy applied to the basket of 100 stocks in the NASDAQ 100 index ( http://dark-liquidity.com/Varan.php ). In 2015, the NASDAQ 12 returned 8.9%, and its hedged version returned 3.9%, both handily beating SPY. It is noteworthy that of the over forty lists of stocks selected by various financial entities and tracked here (

http://dark-liquidity.com/StockPicksSummary.php), these strategies were two of the only six which did not return a loss for 2015.

For 2016. the following stocks make up the NASDAQ 12:

NFLX, AMZN, ATVI, NVDA,

SBUX, EXPE, ALTR, AVGO,

GOOG, EQIX, MNST, FB

For the hedged version of the NASDAQ 12, 5.4% is allocated to each of the twelve stocks listed above, and the rest to TLT.

**3. Utilities**

This (http://seekingalpha.com/instablog/709762-varan/2550271-an-annually-updated-portfolio-of-utility-stocks ) is a momentum based strategy coupled with allocation based on the maximum diversified portfolio (M.D.P.) algorithm ( http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1895459 ) that seeks to maximize a measure of the diversification of the portfolio.

For 2015, this portfolio returned the negligible 0.2%, but that was enough to be better than the returns of many utility funds ( XLU [ -4.9%], IDU [ -4.8%], RYU [ -4.1%], FXU [ -6.5%], FSUTX [-10.9%],VPU [ -4.8%] ).

For 2016, the strategy yields the following portfolio:

NGG 26% ARTNA 20% YORW 16% ED 10% CTWS 10% MSEX 9% WR 7% AWK 2%

**4. Leveraged funds**

This strategy (http://seekingalpha.com/instablog/709762-varan/4587756-winning-buffetts-bet ) is designed to minimize the volatility generally associated with leveraged funds while simultaneously retaining at least some of the performance enhancement due to leverage. The strategy entails annual risk-parity based rebalancing of QLD ( the leveraged version of the NASDAQ 100 fund) together with UBT, the leveraged version of long term treasury funds. I will be using this strategy for the first time in 2016.

For 2016, the strategy requires 53% allocation to UBT and 47% to QLD.

**5. Contra-momentum**

Generally, for a basket of stocks which yields consistently good performance in terms of total return, periodically investing in the laggards may actually improve the returns or drawdowns or both. I started with the basket of following stocks (which was constructed while exploring the performance of individual stocks based counterparts of various portfolios containing Fidelity Select Sector Funds)

MKL, BRK-A, BLK, BAM, O, SKT, PSA, HCP, NNN, SRE, WEC, NGG, AWR, AWK, WTR, ARTNA, CTWS, MSEX, SJW, YORW, ROST, TJX, ANCTF, AZO,CVS, WBA, ESRX, GILD, AMGN, CELG, DVA, NVO, MSFT, INTC, AAPL, GOOGL, FB

containing equities in conglomerates, REITs, utilities, retail, health, and technology. This basket would have yielded very good performance during the last ten years or more. However, just investing, at the beginning of every year, in the equally weighted portfolios of a few of these equities whose performance was the worst in the prior year would have led to generally better returns than investment in the whole basket. The following figure depicts the compound annual growth rate of the whole basket together with the growth rates of various *contra-momentum* portfolios initially funded at the beginning of any year since 2003 and held to date (with, for example, the **Contra 5** portfolio being the portfolio of five yearly updated laggards ).

The performance metrics for various portfolios are given in the following table.

Contra 5 | Contra 10 | Whole Basket | |

CAGR | 22.50% | 19.80% | 19.20% |

Max. Drawdown | 31% | 26% | 28% |

Max. Annual Loss | None | 0.40% | 17% |

The ten laggards for 2016 are: SRE, BRK-A, SKT, HCP, DVA, SJW, BAM, AAPL, BLK, INTC. I will be using this strategy for the first time in 2016.

All the strategies that I have described here suffer from selection, survivor, and data-snooping biases to varying degrees, and so these biases also have to be adequately considered by anyone contemplating their implementation for personal portfolios.

]]>**1. Fidelity Select Funds**

This is a slight modification of the strategy that I introduced at the beginning of 2015 ( http://seekingalpha.com/instablog/709762-varan/3652756-a-five-fund-basket-for-the-long-haul ), which returned 3% for the year - not quite satisfactory, but slightly better than SPY.

I have replaced the Select Consumer Staples Fund (FDFAX) by the Select Retailing Fund (FSRPX), and the Real State Fund (FRESX) by the Select Construction Fund (FSHOX), due to the better overall historical performance of the in-coming funds. The strategy entails rebalancing the basket at the beginning of every year by means of a modified version of the risk parity method that utilizes the variance matrix of the five funds based upon the daily returns of the prior year, as well as the total returns of the five funds during the prior year.

The back test performance of the strategy is compared in the figure above with the market indices. The figure depicts the compound annual growth rates of portfolios initially funded at the beginning of any year since 1988 and held to date. (The results prior to 2004 were computed by using VUSTX instead of TLT.)

The performance metrics for the period 1988-2015 are as follows:

CAGR 14.6%, Maximum Drawdown 21.4%, Sharpe Ratio .98, Sortino Ratio 1.98

For 2016, the portfolio consists of the following:

FSRPX 33% FSCSX 23% FSHOX 20% FBIOX 17% TLT 7%.

**2. Nasdaq 12**

This is a momentum based strategy applied to the basket of 100 stocks in the NASDAQ 100 index ( http://dark-liquidity.com/Varan.php ). In 2015, the NASDAQ 12 returned 8.9%, and its hedged version returned 3.9%, both handily beating SPY. It is noteworthy that of the over forty lists of stocks selected by various financial entities and tracked here (

http://dark-liquidity.com/StockPicksSummary.php), these strategies were two of the only six which did not return a loss for 2015.

For 2016. the following stocks make up the NASDAQ 12:

NFLX, AMZN, ATVI, NVDA,

SBUX, EXPE, ALTR, AVGO,

GOOG, EQIX, MNST, FB

For the hedged version of the NASDAQ 12, 5.4% is allocated to each of the twelve stocks listed above, and the rest to TLT.

**3. Utilities**

This (http://seekingalpha.com/instablog/709762-varan/2550271-an-annually-updated-portfolio-of-utility-stocks ) is a momentum based strategy coupled with allocation based on the maximum diversified portfolio (M.D.P.) algorithm ( http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1895459 ) that seeks to maximize a measure of the diversification of the portfolio.

For 2015, this portfolio returned the negligible 0.2%, but that was enough to be better than the returns of many utility funds ( XLU [ -4.9%], IDU [ -4.8%], RYU [ -4.1%], FXU [ -6.5%], FSUTX [-10.9%],VPU [ -4.8%] ).

For 2016, the strategy yields the following portfolio:

NGG 26% ARTNA 20% YORW 16% ED 10% CTWS 10% MSEX 9% WR 7% AWK 2%

**4. Leveraged funds**

This strategy (http://seekingalpha.com/instablog/709762-varan/4587756-winning-buffetts-bet ) is designed to minimize the volatility generally associated with leveraged funds while simultaneously retaining at least some of the performance enhancement due to leverage. The strategy entails annual risk-parity based rebalancing of QLD ( the leveraged version of the NASDAQ 100 fund) together with UBT, the leveraged version of long term treasury funds. I will be using this strategy for the first time in 2016.

For 2016, the strategy requires 53% allocation to UBT and 47% to QLD.

**5. Contra-momentum**

Generally, for a basket of stocks which yields consistently good performance in terms of total return, periodically investing in the laggards may actually improve the returns or drawdowns or both. I started with the basket of following stocks (which was constructed while exploring the performance of individual stocks based counterparts of various portfolios containing Fidelity Select Sector Funds)

MKL, BRK-A, BLK, BAM, O, SKT, PSA, HCP, NNN, SRE, WEC, NGG, AWR, AWK, WTR, ARTNA, CTWS, MSEX, SJW, YORW, ROST, TJX, ANCTF, AZO,CVS, WBA, ESRX, GILD, AMGN, CELG, DVA, NVO, MSFT, INTC, AAPL, GOOGL, FB

containing equities in conglomerates, REITs, utilities, retail, health, and technology. This basket would have yielded very good performance during the last ten years or more. However, just investing, at the beginning of every year, in the equally weighted portfolios of a few of these equities whose performance was the worst in the prior year would have led to generally better returns than investment in the whole basket. The following figure depicts the compound annual growth rate of the whole basket together with the growth rates of various *contra-momentum* portfolios initially funded at the beginning of any year since 2003 and held to date (with, for example, the **Contra 5** portfolio being the portfolio of five yearly updated laggards ).

The performance metrics for various portfolios are given in the following table.

Contra 5 | Contra 10 | Whole Basket | |

CAGR | 22.50% | 19.80% | 19.20% |

Max. Drawdown | 31% | 26% | 28% |

Max. Annual Loss | None | 0.40% | 17% |

The ten laggards for 2016 are: SRE, BRK-A, SKT, HCP, DVA, SJW, BAM, AAPL, BLK, INTC. I will be using this strategy for the first time in 2016.

All the strategies that I have described here suffer from selection, survivor, and data-snooping biases to varying degrees, and so these biases also have to be adequately considered by anyone contemplating their implementation for personal portfolios.

]]>For every month during the period of interest, with the assumption that the portfolio was initiated at the last trading day of the prior month, and held till the end of the period, the CAGR of the portfolio was computed.

The fund portfolio was rebalanced to be equal weight at the end of every year. For years prior to 2002 (the date of inception of FPHAX) only the other two funds were used.

From the figure it is clear that no matter which month the portfolios were initiated in, the CAGR of the funds portfolio was higher than that of JNJ, very often by significant amounts.

]]>For every month during the period of interest, with the assumption that the portfolio was initiated at the last trading day of the prior month, and held till the end of the period, the CAGR of the portfolio was computed.

The fund portfolio was rebalanced to be equal weight at the end of every year. For years prior to 2002 (the date of inception of FPHAX) only the other two funds were used.

From the figure it is clear that no matter which month the portfolios were initiated in, the CAGR of the funds portfolio was higher than that of JNJ, very often by significant amounts.

]]>1. Withdraw 4% of the portfolio value on January 1 of the year of retirement.

2. In subsequent years, at the beginning of the year, prior to rebalancing the portfolio if necessary, withdraw the previous year's withdrawal amount incremented by the previous year's inflation rate.

The portfolios considered are:

1. SP500 with dividends (VFINX).

2. Mid cap value (FDVLX)

3. 30% VFINX, 30% FDVLX, 40% Aggregate Bond Fund (VBMFX)

4. 30% VFINX, 30% FDVLX, 20% VBMFX, 20% Long Term Treasury Fund (VUSTX)

5. 60% FDVLX 40% VUSTX

The first figure depicts the compound annual growth rate (Net CAGR) of the portfolio net of withdrawals computed for the period starting in the year of retirement thru 2014. Note that except for the VFINX, Net CAGR is always positive. Thus, although the VFINX portfolio would have sustained the withdrawal schedule, for retirement beginning in some of the years around 2000, the portfolio value at the end of 2014 would have been lower than its starting value. For every other portfolio, if the retirement commenced in any year starting in 1987-2014, at the end of 2014 the portfolio value would be higher than the starting value.

In the following figure are shown the maximum excess (beginning of the year) drawdowns for various portfolios. This quantity is defined to be the difference between the initial starting value of the portfolio and its lowest value at the time of withdrawal in any of the subsequent years, less 4%, as at best 4% will be the drawdown of the portfolio at the beginning of retirement. Clearly the excess drawdowns of the all equity portfolios will be unacceptable for most retirees. However, for the 60/40 portfolios, the excess drawdowns appear to be a good measure of what can be expected from well constructed portfolios. (For example, the portfolio of JNJ, MCD, CL, KO and PG with the same withdrawal schedule would have had the maximum excess drawdown of about 24%, similar to the 20% for the 60/40 portfolios.). *(click to enlarge)*.

I

]]>1. Withdraw 4% of the portfolio value on January 1 of the year of retirement.

2. In subsequent years, at the beginning of the year, prior to rebalancing the portfolio if necessary, withdraw the previous year's withdrawal amount incremented by the previous year's inflation rate.

The portfolios considered are:

1. SP500 with dividends (VFINX).

2. Mid cap value (FDVLX)

3. 30% VFINX, 30% FDVLX, 40% Aggregate Bond Fund (VBMFX)

4. 30% VFINX, 30% FDVLX, 20% VBMFX, 20% Long Term Treasury Fund (VUSTX)

5. 60% FDVLX 40% VUSTX

The first figure depicts the compound annual growth rate (Net CAGR) of the portfolio net of withdrawals computed for the period starting in the year of retirement thru 2014. Note that except for the VFINX, Net CAGR is always positive. Thus, although the VFINX portfolio would have sustained the withdrawal schedule, for retirement beginning in some of the years around 2000, the portfolio value at the end of 2014 would have been lower than its starting value. For every other portfolio, if the retirement commenced in any year starting in 1987-2014, at the end of 2014 the portfolio value would be higher than the starting value.

In the following figure are shown the maximum excess (beginning of the year) drawdowns for various portfolios. This quantity is defined to be the difference between the initial starting value of the portfolio and its lowest value at the time of withdrawal in any of the subsequent years, less 4%, as at best 4% will be the drawdown of the portfolio at the beginning of retirement. Clearly the excess drawdowns of the all equity portfolios will be unacceptable for most retirees. However, for the 60/40 portfolios, the excess drawdowns appear to be a good measure of what can be expected from well constructed portfolios. (For example, the portfolio of JNJ, MCD, CL, KO and PG with the same withdrawal schedule would have had the maximum excess drawdown of about 24%, similar to the 20% for the 60/40 portfolios.). *(click to enlarge)*.

I

]]>Obviously the future performance of the portfolio relative to SPY may be entirely different, but the results suggest that this basket may prove to be a good long term investment, a conclusion that is also supported by the various performance metrics shown below.

CAGR (%) | MaxDD (%) | Sharpe | Sortino | Volatility (%) | |
---|---|---|---|---|---|

Eq. Wt. | 14.3 | 33.2 | 0.91 | 1.69 | 12.6 |

Risk Parity | 16.2 | 34.8 | 1.04 | 1.85 | 12.7 |

VFINX | 9.9 | 50.9 | 0.54 | 0.88 | 14.5 |

BRK-A | 15.7 | 44.5 | 0.68 | 1.36 | 20.6 |

This is the allocation for the annually rebalanced portfolio according to a modified risk parity method for the current year 2015 (YTD return : 4.4%, 2014 return : 27.7%)

FBIOX 13.3%

FRESX 24.4%

TLT 43.8%

FDFAX 16.9%

FSCSX 1.6%

]]>Obviously the future performance of the portfolio relative to SPY may be entirely different, but the results suggest that this basket may prove to be a good long term investment, a conclusion that is also supported by the various performance metrics shown below.

CAGR (%) | MaxDD (%) | Sharpe | Sortino | Volatility (%) | |
---|---|---|---|---|---|

Eq. Wt. | 14.3 | 33.2 | 0.91 | 1.69 | 12.6 |

Risk Parity | 16.2 | 34.8 | 1.04 | 1.85 | 12.7 |

VFINX | 9.9 | 50.9 | 0.54 | 0.88 | 14.5 |

BRK-A | 15.7 | 44.5 | 0.68 | 1.36 | 20.6 |

This is the allocation for the annually rebalanced portfolio according to a modified risk parity method for the current year 2015 (YTD return : 4.4%, 2014 return : 27.7%)

FBIOX 13.3%

FRESX 24.4%

TLT 43.8%

FDFAX 16.9%

FSCSX 1.6%

]]>