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Today's 10.0% 5assetportfolio weights: AGG 60% SLV 5% JKG 17% SDS 10% SMH 8%: STDDEV 3.4% for last 252 days. May 17, 2011

Today's 10.0% 5assetportfolio weights: SHY 60% SLV 8% OEF 12% JKG 10% SDS 10%: STDDEV 2.6% for last 252 days. OEF replaced IVV. Apr 26, 2011

Today's 10.0% 5assetportfolio weights: SHY 60% SLV 8% IVV 12% JKG 10% SDS 10%: STDDEV 2.7% for last 252 days. Apr 20, 2011
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QLD out, AGG in, Volatility slightly up for the 10% 200 day L1B Portfolio
A (very) short description of the L1B Portfolios
First, what's in a name? I'm calling these portfolios the L1X portfolios, where X is either A, B, or C to represent different “species” of portfolio that I'm tracking. The L1 name prefix comes from the paper by Konno and Yamazaki (1991) who developed a much faster linear programming formulation to solve the mean absolute deviation (MAD) portfolio as an approximation of the modern portfolio theory, and much more computationally expensive, meanvariancerisk portfolio presented by Markowitz (1959).
Here, I use the MAD method is used to quickly scan through hundreds of financial assets to objectively determine the best subset for further analysis. The suffix letter A/B/C represent three different set of constraints: L1A can contain all assets in the universe, L1B can have at most five (5), and the L1C has two (2) assets. The goal here, is to have timely, useful, and actionable information for investors wanting sustainable, smooth returns with minimum volatility. I'll talk more about the different models in future posts.
The L1B portfolio is a subset of the assets from the solution of a noncardinality constrained portfolio (L1A), which can have as many as 41 assets. It typically contains 1225 ETFs. The L1B contains a single cardinality constraint to restrict the number of assets (not asset classes) in the portfolio to no more than five (5). Why five? Five assets seems to be a number that people on the street can recall, digest, track, and count on one hand. Think of it this way, if you're stopped on the street to have a “portfolio conversation” the fingers of one hand can hold reminders of the information you need to make quick, informed decisions about your asset allocation. Each finger can be used to help us kinesthetic learners quickly recall a basic knowledgebase of different asset classes (bonds, stocks, commodities, etc.) and a few important concepts for the entire hand (i.e. maximum weights, transaction costs, etc.) These portfolios are not intended to be endallbeall portfolios with several dozen assets. They're intended to be used as a portfolio navigational aid to help people make better investment (not trading) decisions.
The Two Most Important Portfolio Metrics
The portfolio metrics I'm really interested in tracking are: 1) the annualized target return, and 2) the portfolio volatility, as measured by the standard deviation of the portfolio (not the individual assets in the portfolio). There are plenty of explanations for these two metrics, but if you're interested in computing them for yourself , let me give a very brief example from today's L1B0200010 (five asset, 200 day look back, 10 percent return portfolio). For starters, today's solution for this portfolio is:
Minimum Absolute Deviation Portfolio for Universe from 20100315 to 20101227
Computed on = 20101228 (GMT)
Trading Days Back = 200
Trading Days/Year = 250
Maximum Number of Assets in Portfolio = 5
Actual Number of Assets in Portfolio = 5
Maximum Allocation for Single Asset = 100.00%
Target Portfolio Return (daily) = 0.038131%
Target Portfolio Variance (daily) = 0.000004
Target Portfolio Standard Deviation (daily) = 0.002003
Target Portfolio Return (backtest period) = 7.923035%
Target Portfolio Standard Deviation (backtest period) = 2.831967%
Target Portfolio Return (annualized) = 10.000000%
Target Portfolio Standard Deviation (annualized) = 3.166235%
Name Symbol Return Annualized Std Dev Annualized Weight
iShares Barclays 13 Year Treasury Bond Fund SHY 0.000069 1.731886 0.000737 1.165653 0.529324
iShares Barclays 710 Year Treasury Bond Fund IEF 0.000326 8.483794 0.005028 7.950185 0.000000
iShares Barclays Aggregate AGG 0.000193 4.953866 0.002383 3.767520 0.210559
iShares Silver Trust SLV 0.002841 103.234112 0.018298 28.932085 0.090500
iShares Morningstar Mid Core Index Fund JKG 0.000935 26.310503 0.014138 22.353470 0.000000
ProShares UltraShort S&P500 SDS 0.001218 26.272366 0.023667 37.421021 0.050889
ProShares Ultra Dow30 DDM 0.001103 31.719883 0.021078 33.327128 0.000000
ProShares Ultra QQQ QLD 0.001631 50.277108 0.024815 39.235847 0.000000
SPDR Energy Select Sector Fund XLE 0.000874 24.394134 0.015168 23.982059 0.000000
SPDR Consumer Discretionary Select Sector Fund XLY 0.000919 25.824555 0.013823 21.855914 0.118728
HOLDRS Merrill Lynch Semiconductor SMH 0.001147 33.175128 0.016623 26.283009 0.000000
Mathematically, the portfolio return is defined by the sum of the products between the mean daily return and the weights. For example, today's L1B0200010 portfolio, computed for the minimum absolute deviation weights, based on the previous 200 trading days, where our annualized target return is 10 percent.
To verify these results, multiple the mean daily returns by the weights included in the portfolio (i.e. AGG, EEM, SDS, DDM, and QLD):
0.000069 * 0.529324 + 0.000193 * 0.210559 + 0.002841 * 0.090500 + (0.001218 * 0.050889) + 0.000919 * 0.118728 = 0.0003814
which is the daily return target. To compute the annualized return from the daily return, apply the following formula to the mean daily target return,
100 * ( ( ( 1 + 0.0003814) ^ TDPY )  1 ) = 10.00%
where TDPY is the number of trading days per year (TDPY=250). Actually, the result we show here is 10.00238%, but I truncated the results for brevity.
I should tell you here, do not use the annualized return and standard deviation values. The annualized values simply printed out and are for reporting purposes only and are not used in the calculations. Using the annualized values will yield incorrect results.
The second important metric for the portfolio is the standard deviation. The standard deviation of the portfolio, which represents the volatility of the portfolio, is defined mathematically as,
100 * sqrt( \sum_{i=1}^{N} \sum_{j=1}^{N} ( w[i]*w[j]*cov[i,j] ) ) / sqrt( 1 / TDPY )
where the C[i,j] is the entry from the covariance matrix of the daily returns between asset I and j, w[i] is the portfolio weight for the included asset, and TDPY is the number of trading days per year. I use 250 trading days per year.
To compute the portfolio standard deviation, you need the vector of weights, the covariance matrix for the mean daily returns for all ETFs in the universe (not shown), and the mean daily returns. For this example, the covariance matrix is a 41 x 41 matrix, which is too large to show here, so I'll assuming you're okay with me not presenting it.
Finally, to compute the annualized standard deviation from the standard deviation from the mean daily returns:
100 * SD_{daily} * \sqrt( TDPY ) = SD_{annualized}
where TDPY is the number of trading days per year.
For the example given, the resulting annualized standard deviation was 3.17%, which was considerably lower than the S&P 500 standard deviation for the previous 200 days of nearly 18%.
Changes from Last Week
For last week, the L1B0200010 portfolio solution was:
Minimum Absolute Deviation Portfolio for Universe from 20100309 to 20101220
Computed on = 20101221 (GMT)
Trading Days Back = 200
Trading Days/Year = 250
Maximum Number of Assets in Portfolio = 5
Actual Number of Assets in Portfolio = 5
Maximum Allocation for Single Asset = 100.00%
Target Portfolio Return (daily) = 0.038131%
Target Portfolio Variance (daily) = 0.000004
Target Portfolio Standard Deviation (daily) = 0.001898
Target Portfolio Return (backtest period) = 7.923035%
Target Portfolio Standard Deviation (backtest period) = 2.683680%
Target Portfolio Return (annualized) = 10.000000%
Target Portfolio Standard Deviation (annualized) = 3.000446%
Name Symbol Return Annualized Std Dev Annualized Weight
iShares Barclays 13 Year Treasury Bond Fund SHY 0.000072 1.806581 0.000736 1.164503 0.656503
iShares Barclays 710 Year Treasury Bond Fund IEF 0.000335 8.741430 0.005014 7.928329 0.000000
iShares Barclays Aggregate AGG 0.000188 4.806683 0.002367 3.743038 0.000000
iShares Silver Trust SLV 0.002824 102.379736 0.018344 29.004502 0.092307
iShares Morningstar Mid Core Index Fund JKG 0.000922 25.914516 0.014131 22.342966 0.000000
ProShares UltraShort S&P500 SDS 0.001233 26.546007 0.023662 37.413134 0.102377
ProShares Ultra Dow30 DDM 0.001106 31.828495 0.021059 33.297117 0.000000
ProShares Ultra QQQ QLD 0.001778 55.920375 0.024833 39.265211 0.067604
SPDR Energy Select Sector Fund XLE 0.000847 23.575973 0.015160 23.970018 0.000000
SPDR Industrial Select Sector Fund XLI 0.000925 26.010099 0.014555 23.013661 0.000000
SPDR Consumer Discretionary Select Sector Fund XLY 0.000981 27.780420 0.013823 21.855631 0.081209
HOLDRS Merrill Lynch Semiconductor SMH 0.001126 32.475052 0.016693 26.394550 0.000000
You can see small changes in the two weight vectors, but the target return stays constant, and there's a slight increase in the portfolio volatility from last week, which was 3.16%. So between last week and today, the assets in the portfolio have changed a little (added AGG, removed QLD), and the weights of the remaining assets (SHY, SLY, SDS, and SLV) have changed very little.
To graphically compare the portfolio from the last post, I've also plotted the pie chart containing the weights and the historical performance against the major indexes in google finance.
Compared the last post's portfolio, the only asset that has dropped out of the portfolio is the 2x leveraged ProShares Ultra QQQ (QLD), which had an allocation of about 7% last week. Entering the portfolio is the iShares Barclays Aggregate Bond Fund (AGG), which considering the recent trends in bonds, could present an opportunity for savvy investors, but not right now.
Also, looks like the allocation of ProShares UltraShort S&P500 (SDS) has declined a little as has the Consumer Discretionary Select Sector SPDR ETF (XLY). The total proportion of the portfolio in bonds is still considerable and I suspect that as we continue on our current QE2/recovery path, interest rates will rise, as will bond yields, but will not appreciate as they have over the previous few weeks.
Like last time, I've plotted the performance for today's L1B0200010 portfolio:
Which looks much like last week's portfolio, and has only recently been taken over by the NASDAQ in terms of performance.
Lot's of stuff here today. The bottom line/take home messages are these:
The target return is constant, not “make as much as possible and forget about volatility” portfolios.
If you're like me, I like excitement, just not in my portfolio. Given a choice between a portfolio making a target return, I want the lowest volatility I can get my hands on every time.
The portfolio weights can change each day, but for the most part, they remain relatively stable (for lower yielding portfolios <10%) for long look back periods, but these results can be used to assist people thinking about rebalancing.
Disclosure: I am long SLV.
Introduction
and the two objectives for the portfolios are to:
Disclosure: I am long EEM, SH, IAU, SLV.