US weekly portfolio recommendation (from 28 Feb to 4 Mar, 2011)

Please, see my first post

http://estimationrisk.blogspot.com/2011/02/why-low-risk-strategies-can-achieve.html

for more details regarding the portfolio strategy.

The low volatility portfolio recommended for this week is (ticker notation):

'MO'    'AMGN'    'BCR'    'BDX'    'CVS'    'CPB'    'CLX'    'ED'    'FDO'   

'GIS'    'HRL'    'JNJ'   'K'    'KMB'    'LH'    'MKC'    'MCD'    'PEP'    'PG'   

'RAI'    'SO'    'WEC'

Although I recommend a portfolio composition every week, it is desirable to maintain this composition for four weeks, and then rebalance with the new composition.

This portfolio consists of 22 stocks. Four weeks ago, the portfolio composition was the same except: now we have added ‘CPB’ and ‘LH’, and now we no longer have ‘AON’. In order to have again equal weights for all the stocks, the turnover respect to previous composition is 19% (due to the portfolio growth in the last month and the change of 3 stocks in the portfolio composition).

Regarding the performance, over the last year (52 weeks), the strategy attained a volatility of 10% (versus 17% of the S&P 500).
The weekly 95%-VaR was 2.5% (versus 4.2% of the S&P 500). See this post for the details about the back-testing.

The last year annualized Sharpe ratio of the low vol strategy was 1.68 (after proportional transaction costs of 40 bps was discounted). On the other hand, the SR of the S&P 500 was 1.12 over the same period.

Finally, the correlation of the low vol strategy with the S&P 500 along the last 52 weeks was 85%.

Again, all these performance results are consistent over time.

In this case, I am going to talk a bit more about the performance results in order to understand better how the low volatility portfolios outperform market indexes.

In the next figure, you can see cumulative return over the last 52 weeks of the low vol portfolio (after transaction costs) and the S&P 500 (no costs).

It can be observed that the S&P 500 attains a bit better final return than that of the low vol portfolio. But we need to analyze this graph with caution. It depends on both the initial and final investment periods. Moreover, we need to focus on the associated risk too, not only on the final return.

For these reasons, I prefer to project the previous 1D trajectories into a 2D world: the risk-return space. That is, each of the two trajectories will be summarized by one point containing their corresponding mean returns and volatilities.

The next graph shows the risk-return space for the two considered portfolios.

Each blue point represents the mean return and volatility of the low vol portfolio over a given period of 52 past consecutive weeks. On the other hand, each red point represents the mean return and volatility of the S&P 500 index over the same periods of 52 consecutive weeks.

We say that, for a given period, the low vol portfolio dominates the index market if its volatility is lower and the return is higher than that of the index.

In total, the graph contains 50 of such periods, corresponding to 50 different trajectories of cumulative returns for the two portfolios.

With this 2D vision, we can compare better the performance of the two strategies. We can see that the volatility of the low vol portfolio remains stable around 10%. The volatility of the S&P 500 is less stable and near 20%. But what happens with the associated return?

The S&P 500 does not attain a larger return. In contrast, most of the time, its return is worse than that of the low volatility portfolio.

More concretely, the volatility of the low vol portfolio is always less than that of the S&P 500. And more than 70% of the time, the corresponding return is better.

As a summary, the low vol portfolio dominates the market index 70% of the time in the risk-return space, showing they attain consistently better risk-adjusted returns. At least, this is true for the last two years.

Spanish weekly portfolio recommendation (from 28 Feb to 4 Mar, 2011)

Please, see my first post

http://estimationrisk.blogspot.com/2011/02/why-low-risk-strategies-can-achieve.html

for more details regarding the portfolio strategy.

The low volatility portfolio recommended for this week is (ticker notation):

'ACS.MC'    'ELE.MC'    'ENG.MC'    'IDR.MC'    'ITX.MC'    'REE.MC'    'TEF.MC'

Although I recommend a portfolio composition every week, it is desirable to maintain this composition for four weeks, and then rebalance with the new composition.

This portfolio consists again of 7 stocks. Indeed, it is the same composition than that of 4 weeks ago. In order to have again equal weights for all the stocks, the turnover respect to previous composition is 4.4% (due to the portfolio growth in the last month). Hence, the transaction costs associated with this rebalancing are low.

Regarding the performance, over the last year (52 weeks), the strategy attained a volatility of 19% (versus 28% of the Ibex 35).

The weekly 95%-VaR was 4.4% (versus 5.2% of the Ibex 35). See this post for the details about the back-testing.

The last year annualized Sharpe ratio of the low vol strategy was 0.47 (after proportional transaction costs of 40 bps was discounted). On the other hand, the SR of the Ibex 35 was 0.19 over the same period.

Finally, the correlation of the low vol strategy with the Ibex 35 along the last 52 weeks was 93%.

Again, all these performance results are consistent over time.

In this case, I am going to talk a bit more about the performance results in order to understand better how the low volatility portfolios outperform market indexes.

In the next figure, you can see cumulative return over the last 52 weeks of the low vol portfolio (after transaction costs) and the Ibex 35 (no costs).

It can be observed that the Ibex 35 attains a bit worse final return than that of the low vol portfolio. But we need to analyze this graph with caution. It depends on both the initial and final investment periods. Moreover, we need to focus on the associated risk too, not only on the final return.

For these reasons, I prefer to project the previous 1D trajectories into a 2D world: the risk-return space. That is, each of the two trajectories will be summarized by one point containing their corresponding mean returns and volatilities.

The next graph shows the risk-return space for the two considered portfolios.

Each blue point represents the mean return and volatility of the low vol portfolio over a given period of 52 consecutive weeks. On the other hand, each red point represents the mean return and volatility of the Ibex 35 index over the same periods of 52 consecutive weeks.

We say that, for a given period, the low vol portfolio dominates the index market if its volatility is lower and the return is higher than that of the index.

In total, the graph contains 50 of such periods, corresponding to 50 different trajectories of cumulative returns for the two portfolios.

With this 2D vision, we can compare better the performance of the two strategies. We can see that the volatility of the low vol portfolio remains stable around 20%. The volatility of the Ibex 35 is less stable and near 30%. But what happens with the associated return?

The Ibex 35 does not attain a larger return. In contrast, all the time, its return is worse than that of the low volatility portfolio.

More concretely, the volatility of the low vol portfolio is always less than that of the Ibex 35. And always again, the corresponding return is better, although this could change a bit over time.

As a summary, the low vol portfolio dominates the market index 100% of the time in the risk-return space, showing they attain consistently better risk-adjusted returns. At least, this is true for the last two years.

US weekly portfolio recommendation (from 21 to 25, Feb 2011)

Please, see my first post

http://estimationrisk.blogspot.com/2011/02/why-low-risk-strategies-can-achieve.html

for more details regarding the portfolio strategy.

The low volatility portfolio recommended for this week is (ticker notation):

    'MO'    'AMGN'    'BCR'    'BDX'    'CVS'    'CPB'    'ED'    'FDO'
 
    'GIS'    'HRL'    'JNJ'    'K'    'KMB'    'LH'    'MKC'    'MCD'
 
    'PEP'    'PG'    'RAI'    'SO'    'WEC'

Although I recommend a portfolio composition every week, it is desirable to maintain this composition for four weeks, and then rebalance with the new composition.

This portfolio consists of 21 stocks. Four weeks ago, the portfolio composition was the same except: now we have added ‘CPB’ and ‘LH’, and now we no longer have ‘CLX’. In order to have again equal weights for all the stocks, the turnover respect to previous composition is 19% (mainly due to the change of 3 stocks in the portfolio composition and to the portfolio growth in the last month).

Regarding the performance, over the last year (52 weeks), the strategy attained a volatility of 10% (versus 17% of the S&P 500).
The weekly 95%-VaR was 2.4% (versus 4.2% of the S&P 500). See this post for the details about the back-testing.

The last year annualized Sharpe ratio of the low vol strategy was 1.91 (after proportional transaction costs of 40 bps were discounted). On the other hand, the SR of the S&P 500 was 1.40 over the same period.

Finally, the correlation of the low vol strategy with the S&P 500 along the last 52 weeks was 85%.

All these performance results are consistent over time. I have omitted the evolution of the performance measures over a long history because they are roughly the same as those in the previous posts.

 As a summary, the back-testing results confirm again that low volatility portfolios continue to have low volatility in the future, and furthermore, they attain better returns (per unit of risk) than those of market indexes like the S&P 500.

Spanish weekly portfolio recommendation (from 21 to 25, Feb 2011)

Please, see my first post

http://estimationrisk.blogspot.com/2011/02/why-low-risk-strategies-can-achieve.html

for details regarding the portfolio strategy.

The low volatility portfolio recommended for this week is (ticker notation):

 'ACX.MC'   'ELE.MC'    'ENG.MC'    'IDR.MC'    'ITX.MC'    'REE.MC'   'TEF.MC'

Although I recommend a portfolio composition every week, it is desirable to maintain this composition for four weeks, and then rebalance with the new composition.

This portfolio consists again of 7 stocks. Indeed, it is the same composition than that of 4 weeks ago. In order to have again equal weights for all the stocks, the turnover respect to previous composition is 5% (due to the portfolio growth in the last month). Hence, the transaction costs associated with this rebalancing are low.

Regarding the performance, over the last year (52 weeks), the strategy attained a volatility of 18% (versus 29% of the IBEX 35).
The weekly 95%-VaR was 4.7% (versus 5.2% of the IBEX 35). See this post for the details about the back-testing.

The last year annualized Sharpe ratio of the low vol strategy was 0.68 (after proportional transaction costs of 40 bps were discounted). On the other hand, the SR of the IBEX 35 was 0.41 over the same period.

Finally, the correlation of the low vol strategy with the IBEX 35 along the last 52 weeks was 92%.

All these performance results are consistent over time. I have omitted the evolution of the performance measures over a long history because they are roughly the same as those in the previous posts.

As a summary, the back-testing results confirm again that low volatility portfolios continue to have low volatility in the future, and furthermore, they attain better returns (per unit of risk) than those of market indexes like the IBEX 35.

Spanish weekly portfolio recommendation (from 14 to 18, Feb 2011)

Please, see my first post for details regarding the low volatility portfolio strategy.

The low volatility portfolio recommended for this week is (ticker notation):

'ACX.MC'    'ELE.MC'    'ENG.MC'    'IDR.MC'    'ITX.MC'    'REE.MC'    'TEF.MC'

This portfolio consists of 7 stocks. Although the strategy recommends different weights for each stock, to further improve the stability properties of the strategy, I recommend using equal weights for all the stocks.

In order to check the performance of this strategy over the last years, the strategy was run every four weeks in the past. In the next figure, the portfolio composition (number of stocks) from 2008 is shown.

It can be observed that the average number of stocks is around 8. This composition is reasonably stable over time.

Regarding the performance, over the last year (52 weeks), the strategy attained a volatility of 18% (versus 28% of the IBEX 35). The weekly 95%-VaR was 4.7% (versus 5.2% of the IBEX 35). See this post for the details about the back-testing.

In the next figure, we can see the evolution of the weekly 95%-VaR from 2009, over rolling 52-weeks periods.

We can see the portfolio VaR of the low volatility strategy is almost always below that of the IBEX 35. This is not surprising because the portfolio composition is focused on minimizing the portfolio risk.

But more surprising is the return performance. The last year annualized Sharpe ratio of the low vol strategy was 0.64 (after proportional transaction costs of 40 bps was discounted). On the other hand, the SR of the IBEX 35 was 0.37 over the same period.

To see if these results are consistent over time, next figure shows the evolution of the annualized SR from 2009, over rolling 52-weeks periods.

We can see most of the time the SR of the low vol strategy is larger than that of the IBEX 35.

Finally, the correlation of the low vol strategy with the IBEX 35 along the last 52 weeks was 92%.

In the next figure, we can see the evolution of the this correlation from 2006, over rolling 52-weeks periods.

We can see the correlation with the market index is between 80% and 90%.

As a summary, these extensive back-testing results confirm that low volatility portfolios continue to have low volatility in the future, and furthermore, they attain returns similar (or even better) than market indexes like the IBEX 35.

US weekly portfolio recommendation (from 14 to 18, Feb 2011)

Please, see my first post for details regarding the low volatility portfolio strategy.

The low volatility portfolio recommended for this week is (ticker notation):

'MO'    'AMGN'   'BCR'   'BDX'    'CVS'  'CPB'   'CLX'    'ED'    'FDO'    'GIS'  'HRL'   

'JNJ'    'K'    'KMB'   'MKC'   'MCD'    'PEP'    'PG'    'RAI'   'SO'    'WEC'

This portfolio consists of 21 stocks. Although the strategy recommends different weights for each stock, to further improve the stability properties of the strategy, I recommend using equal weights for all the stocks.

In order to check the performance of this strategy over the last years, the strategy was run every four weeks in the past. In the next figure, the portfolio composition (number of stocks) from 2005 is shown.

It can be observed that the average number of stocks is around 20. This composition is reasonably stable over time, except during the crisis period where the portfolio composition is drastically reduced.

Regarding the performance, over the last year (52 weeks), the strategy attained a volatility of 10% (versus 17% of the S&P 500). The weekly 95%-VaR was 2.4% (versus 4.2% of the S&P 500). See this post for the details about the back-testing.

In the next figure, we can see the evolution of the weekly 95%-VaR from 2006, over rolling 52-weeks periods.

We can see the portfolio VaR of the low volatility strategy is almost always below that of the S&P 500. This is not surprising because the portfolio composition is focused on minimizing the portfolio risk.

But more surprising is the return performance. The last year annualized Sharpe ratio of the low vol strategy was 1.7 (after proportional transaction costs of 40 bps was discounted). On the other hand, the SR of the S&P 500 was 1.4 over the same period.

To see if these results are consistent over time, next figure shows the evolution of the annualized SR from 2006, over rolling 52-weeks periods.

We can see most of the time the SR of the low vol strategy is larger than that of the S&P 500.

Finally, the correlation of the low vol strategy with the S&P 500 along the last 52 weeks was 84%.

In the next figure, we can see the evolution of the this correlation from 2006, over rolling 52-weeks periods.

We can see the correlation with the market index is above 75%, except during the crisis period where it lowers up to 60%. Note during the crisis period the portfolio composition was reduced to around 5 stocks (less risky).

As a summary, these extensive back-testing results confirm that low volatility portfolios continue to have low volatility in the future, and furthermore, they attain returns similar (or even better) than market indexes like the S&P 500.

Can low risk portfolios achieve high rewards? A statistical view

Welcome to this blog where I will try to explain some ideas regarding the implementation of portfolio strategies based on quantitative techniques (mainly Statistics and Optimization). These strategies are also based on historical data, hence their performance can be evaluated through extensive back-testing.

The ideas start with the pathbreaking work of Markowitz, designed for investors who care only about the mean and variance of a portfolio’s return. But to implement these portfolios in practice, one needs to estimate the means, volatilities and correlations of stock returns.

It is well known that next-period expected returns cannot be forecasted reasonably well from historical data. And if we try to estimate these means by fundamental or technical analysis, the forecasting error is still substantial.

For this reason, it is better to ignore these forecasts from the original framework. As Jagannathan and Ma stated: “the estimation error in the sample mean is so large that nothing much is lost in ignoring the mean altogether”.

Hence, we are going to focus on low volatility portfolios. One may argue that these portfolios will imply a low return too. But surprise: they usually perform better than portfolios that try to optimize the trade-off between risk and return.

Indeed, there exist minimum volatility indices, like the one by MSCI, to serve as relevant benchmark for financial institutions.

In subsequent posts, we will show how these low volatility portfolios continue to have low volatility in the future, and furthermore, they attain returns similar (or even better) than broad indexes like the S&P 500.

This apparent paradox is explained because the financial markets are not perfectly efficient all the time. Then, risk and return at the stock level do not need to be highly correlated in practice. Indeed, there exists evidence of stock portfolios that not always exhibit a significant positive correlation between risk and return.

This means we can increase our portfolio return per unit of risk (the well-known Sharpe ratio) by focusing only on the volatility, and ignoring the expected return.

A major concern among investors and portfolio managers is the transaction costs. In order to reduce them, it is desirable: (i) to hold a relatively small number of stocks, and (ii) to lessen the wealth traded in each period.

We propose the use of optimization techniques together with statistical analysis to select a few stocks from a broader index like the S&P 500.

And to reduce the wealth traded in each period, we propose a reasonably stable portfolio composition that does not change drastically over time. Finally, and to reduce even more the impact of transaction costs, the portfolio strategies will be rebalanced every 4 weeks.

In the next posts, we will recommend low volatility portfolios with the above properties (low volatility and good return after transaction costs), and show the corresponding back-tested results.


The back-test is as follows: at a given week, we compute the portfolio strategy, and a week later we compute the corresponding portfolio return net of transaction costs respect to the last portfolio composition. We repeat this procedure for every week in the last years obtaining the following performance measures: portfolio return mean and volatility, the Sharpe ratio, the Value-at-Risk, the correlation with the market index, etc.

Please, note that all the results reported here are back-tested results: they do not represent results of actual trading.

Finally, these ideas are inspired by the following works:

"Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?”. Review of Financial Studies 22(5), 1915--1953 (2009).

“Portfolio Selection with Robust Estimation''. Operations Research, 57(3), pp. 560-577, 2009. 

“A Generalized Approach to Portfolio Optimization: Improving Performance By Constraining Portfolio Norms''. Management Science, 55(5), pp. 798-812, 2009.