Sunday, April 3, 2011

Optimizing the DOW 30, for April 1, 2011

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Optimizing the Dow

Earlier this week I distributed some internal notes to GGT subscribers which outlined concepts on picking stocks out of major indexes with the intent of beating the index but not taking on a huge amount of risk in the process.  The following describes this process a bit more.

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Here are some terms that you will need to understand, at least in basic concept:

Risk-Free Asset:  think Government bonds, typically 3 months in duration.  The principal in guaranteed, as is the rate of return.  For all intensive purposes the variability of returns is 0.  As of this writing I'm using 0.25% as the annualized rate of return for risk-free assets.

Volatility:  many people equate risk and volatility as the same thing.  They are not, but one does approximate the other over various time frames.  Typically, they are directly proportional -- as volatility goes up, risk goes up.  As you will see, this does not mean that gains go up in direct proportion; it's possible to have incredibly high risk but overall poor return in a security, which obviously is a bad place to have your money.

Return or Excess Return:  "Return", by itself, is how much gain you have on a security in some particular time frame.  "Excess Return" is the same thing, but you subtract the return of the Risk-Free Asset described above.  In most of my plots I will plot "Return", as we are most familiar with it.

Correlation:  this is a mathematical value between -1 and +1 that describes how well two securities, or a security and an index, move with respect to each other.  If something has a correlation of +1, they move exactly the same, e.g., if the broad index S&P500 goes up 1%, the ETF SPY will also go up 1% by design, and the correlation is +1.  Conversely, if two securities are negatively correlated with a value of -1, then if the S&P500 goes up +1%, the ETF SH will drop by -1%, again by design, because they have a correlation of -1.  Stocks and bonds tend to be negatively correlated between 0 and -1, stocks (in general) are positively correlated with other stocks and their correlation is generally between 0 and +1.  When two securities have a correlation of 0 they do not move at all with respect to each other.

Capital Allocation Line:  if all your money is in a risk-free asset, then your return is the risk free asset return and the volatility is 0.  If all your money is 100% in one security, then your return is that security's return, and the volatility (risk) is the volatility of that security.  Hence, the line connecting a security's return/risk value to that of a risk free asset represents all the combinations possible for owning the security (100%), owning a portion of the security (e.g. 40% security, 60% risk-free asset), or owning 100% of the risk-free asset.  Typically, the line connects the risk-free-asset return (volatility = 0) to the return/risk of an index that you are comparing to.

Efficient Frontier:  In the Capital-Allocation-Line (CAL) description above, where we have the choice of 100% in a risk-free asset, some percentage X in the risk free asset and the balance (1-X) in a security, or 100% in the security, the CAL describes a curve called the Efficient Frontier.  In this two-security case (one risk-free with 0 variance), the Efficient Frontier is a straight line.  In the case where no risk-free asset exists, two securities combine to have variable rates of return as well as variable levels of risk.  This rates of return and the risks are related by the term correlation as described above, AND ARE NOT LINEAR (a straight line).  The more securities that are introduced that have positive and negative correlations, the more "bullet shaped" or "hyperbolic"-looking the Efficient Frontier becomes.

Portfolio:  a combination of securities, and as I use it in my work, all have variance (risk) > 0.  This means that while I'll compare to a risk-free-asset, I'll not actually invest in a risk-free asset in my portfolio until I understand how my portfolio behaves (more on this later).  The basket of 30 stocks comprising the Dow Jones Industrials is considered a portfolio.

Tangency Portfolio:  You'll see this below in a figure, but it represents where the Efficient Frontier of all securities being considered intersects the Capital Allocation Line.  The point of intersection represents the best possible combination of return and risk, given the alternative of simply investing in an index AND a risk-free asset.  At the point of intersection, which is where the line is tangent to the curve, you are 100% invested in stocks in some ratio that represents an optimum reward/risk level.  More on this below.



The diagram above was taken from the Wiki link on Modern Portfolio Theory.  Here, they use "Standard Deviation" on the x-axis, where I normally use "Volatility".  The two are related -- all things equal, standard deviation is the square root of volatility.  Many people use either term interchangeably, which is mathematically incorrect, but the concepts are equivalent.

Starting on the left of the figure above, where Standard Deviation = 0, we have the Risk Free Rate (RFR) as described above.  As we introduce a security to the equation -- just 1 -- we have a choice of some proportion of our money in the RFR and some remaining proportion in the security.  This is described by the "Best Possible Capital Allocation Line".  It is assumed that the single security being considered has a rate of return greater than the RFR -- why would we invest in the security if we could get a higher rate of return at no risk by investing 100% in the RFR?  This guarantees that the line moves from the lower left of the figure towards the upper right.

The golden dots of Individual Assets shows their performance, over some period in the past, of Standard Deviation and Expected Return.  Let's make this more concrete in the figure below.




Here, I'm using the RFR (risk free return) = 0.25%, and you see two stocks:  AA and IBM.

  • The unmarked yellow diamond in the middle represents some arbitrary portfolio ratio between some percentage in AA and the remaining percentage in IBM.  In this case it was 50/50.  Note that the yellow diamond is ON THE BLUE CURVE.
  • The blue curve is constructed by running through all the combinations of owning IBM and AA.  This is why you see that a 50/50 ratio between the two stocks falls on the blue curve.   It is a coincidence that AA lies on the Efficient Frontier of these two portfolios; that is not normally the case. 
  • The red line is the CAL that connects the RFR and the unknown portfolio.  At the point of intersection (purple diamond) the percentage in IBM and AA is unknown, but common sense would tell you that if the yellow diamond is 50/50, and AA is to the right of the yellow diamond and IBM to the left, that the intersection of the two lines is where we have "more" in IBM and "less" in AA.
It's important to understand that above where the CAL intersects the EF curve (purple diamond), RISK IS GETTING WORSE FASTER THAN GAIN IS IMPROVING.  YOU ARE TAKING ON MORE RISK FOR A SMALLER AND SMALLER INCREASE IN GAIN.   If you hold a 50/50 ratio between AA and IBM, you are accepting more risk than you should!  If we want to be 100% invested, e.g., no cash, then the best reward/risk point is where the purple diamond is located, e.g., where the CAL and EF intersect.

The obvious question now is how to find the exact amounts of the proportion where we are 100% invested in AA and IBM but not taking on more risk for some desired gain.  I'm not going to go into the details here, but download this presentation if you are interested in more details on how to solve the two-security problem as well as the n-security problem.  This presentation forms the basis of how I'm doing what I'm doing now.

Of course, you could say, "I want 30% gain out of these two stocks", then put a constraint in the system and it would come up with the ratios to get there.  For just AA and IBM, you'd be looking at a 1-year volatility of greater than 24% ... what's YOUR ulcer index?  Alternatively, you could say "include the RFR amount so that I have THREE positions -- RFR, AA, and IBM", and it would generate those proportions.  The choice is completely up to you.

Note here too that past performance is no guarantee of future performance.  If the behavior observed is the result of the past 252 trading days, then we have a reasonable expectation that day 253 will perform within some boundary of the performance of the previous 252 days.  The same goes for 260 days, as well as 290 days.  All this means is that we have to re-evaluate periodically, and other literature that I have suggests that we should rebalance at least quarterly, and some of my own work shows that rebalancing monthly should suffice to keep us on track.

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So, with the background established, let's take a look at the DOW 30, Yahoo! symbol ^DJI.




The figure above shows the Efficient Frontier for all the stocks of the Dow 30, complete with their historical volatility and return over the past two years.  
  • Again, I'm using the risk-free-return value as 0.25% per year, based on 3-month Treasury notes.
  • The unmarked yellow diamond represents the portfolio performance with 100 shares held in each stock (arbitrary).
  • The close proximity of the yellow diamond and the purple diamond is pure co-incidence; if I weighted the stocks by equal dollar value the yellow diamond would have moved to a different location on the blue curve.
The important concept in the picture above of the Dow 30 is that there are numerous stocks which have poor return and low volatility (not so dangerous but would underperform over the long haul, e.g., KFT and TRV), poor gain and higher volatility (dangerous, still underperform and would give you ulcers, e.g. INTC and PFE), and higher gain/higher volatility (generally desired if you are going after risk).  The key here is to understand which stocks we would want to consider if we were to "beat the Dow Jones 30 Index"...

Take a look at this next graph:



Here, I've drawn artificial lines over the optimum location where the CAL intersects the DJ30 EF boundary. These lines divide the figure into quadrants -- let's take a look at what each quadrant tells us.
  1. Quadrant I.  These stocks have above average gain than our optimum point, which is what we want, but above average volatility, which is what we do NOT want.  Despite this, we expect that higher volatility (risk) means that we should have higher gain, and these stocks meet that criteria.
  2. Quadrant II:  There are no stocks here, by definition, because the origin (crossing of the two axis lines) lies directly on the Efficient Frontier boundary, and there is no combination of stocks that gives us higher gain than the optimum point but lower volatility than the optimum point.  Of course, there are stocks that fall into Quadrant II -- but they are NOT part of the Dow Jones 30.  These would be GREAT stocks to find, relative to the stocks in the DJ30, because they would boost gain but lower risk.  Note that for future reference, we want stocks that have historically returned more than 20.9% over the last year and have volatility below 13.1%; this is where the "optimum" point lies.
  3. Quadrant III:  These stocks have lower volatility, but poor gain.  There is no way that adding these stocks to a portfolio can increase gain -- all they can do is reduce portfolio volatility AND reduce gain at the same time.  They may be candidates for balancing reward/risk, but in general, they are to be avoided.
  4. Quadrant IV:  These stocks have above average volatility and below average gain than our ideal point. Again, adding these to a portfolio cannot result in any form of increase in gain, all they can do is lower it.  Furthermore, adding these to a portfolio cannot reduce volatility, all they can do is increase it.  Hence, stocks in Quadrant IV MUST be avoided.
What you should conclude here is that stocks in Quadrant I are the only stocks we should consider once we know their relative volatility and returns, compared to a target weighting.  As we search for our basket of stocks we would love to have stocks that fall into Quadrant II, but for now, that subset is empty.

What we need now is a method to select only those stocks that perform well.  Enter the Sharpe Ratio.

The Sharpe Ratio gives us a metric to evaluate individual stocks in terms of return, risk-free return, and volatility.  The optimum CAL/EF point has a 12-month return of 20.88%, a volatility of 13.10%, so using the risk-free rate of 0.25%, we have the SR = (20.88 - 0.25) / 13.10 = 1.58 [just as a point of reference, this is an amazing SR value.  If we were to consider multi-year time frames the value would be much less, because we would be including the drawdown period of 2008/early 2009 combined with much higher volatilities and RFRs.  This shorter time frame is justified on many fronts, but namely, it is based on work conducted by RiskMetrics].

With a basket like the DJ30, and when restricting ourselves to just those stocks in Quadrant 1, we're left with 15 total stocks out of the 30.  Let's simply get a benchmark of what those stocks, when properly combined and weighted, will reveal in terms of our previous results.



The figure above shows all 15 stocks of Quadrant I, combined as a portfolio.  Note the location of the purple diamond:  25.08% return, 15.97% volatility, and a Sharpe Ratio of 1.55.  Given that the original portfolio had a return of 20.08% and a volatility of 13.1%, yielding a Sharpe Ratio of 1.57, we have achieved nearly the same reward/risk ratio (1.55 vs. 1.57) while improving the gain 5% per annum.

We can say that for data through April 1, 2011, that we if restrict our investment decisions to this basket of 15 stocks of the DJ30, we can significantly improve our returns over investing in the ETF DIA without introducing significant risk to the portfolio.

I understand that 15 stocks still is a high number of stocks; the obvious question is whether we can further increase our gains while not taking on any additional risk, or at a minimum, keep the Sharpe Ratio constant near a value of 1.55 to 1.57?

We know that uncorrelated assets, when added together, reduce volatility.  It's not unreasonable to conclude that if we start with the highest return asset (AA, to maximize gain with n = 1) and then select the most uncorrelated asset shown above (HPQ), and repeat this process, slowly adding the next uncorrelated asset to the portfolio while watching how far off we are from our target Sharpe Ratio with n =15, we can make a reasonable estimate at the marginal contribution of each asset towards our target Sharpe Ratio.

To do this, I generate a correlation matrix of how each stock behaves compared to the other stocks.  Here's the matrix for the 15 stocks:



In the chart above, it's obvious that AA when correlated with itself will return a 1.0.  Since AA has the highest return, we can see that the stock with the lowest correlation to AA is HPQ @ 0.35.  We can create a portfolio containing just AA and HPQ and take the measurements.  Once we have this, and knowing what our target at n=15 stocks is, we can then determine the error.  This is called a marginal contribution analysis.  Here's the table:



The table is constructed by taking AA+HPQ and determining the composite return, volatility, Sharpe Ratio, and known error from our goal of n=15.  In this case we see that the n = 2 combination, comprised of AA and HPQ, which are the two poorest correlated stocks to be considered, result in over 18% error from the target return/risk ratio.  This obviously is unacceptable and I never would trade such a portfolio.

Next we add HD, so that the portfolio looks like AA+HPQ+HD.  You can see how "Error from Goal" improved from 18% to just under 10%, and this is due to diversification with n = 3 stocks.  You also see an entry in the column "Marginal Contribution", which simply shows that the Sharpe Ratio improved 10.2%.

Continuing, you see the trend.

What is important here is that at n = 11 stocks we are less than 1% from our target reward/risk ratio when n = 15 stocks.  At n = 11, returns with the basket shown above are about 26.0%, with 16.6% volatility.  The Sharpe Ratio only improves 0.54% by adding 4 more stocks to complete all the stocks that were in Quadrant I, so we have effectively reduced the basket from n = 30 to n = 11 and have taken on incremental risk while significantly adding return to the basket.

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But wait!  There's more!

Simply jumping in to this basket of stocks is detrimental to your portfolio's health.  If we take the top 11 stocks as listed above and put them on the GGT dashboard, we get the following recommendations:



Here, we see that 3 of the 11 stocks are not being recommended for entry at the present moment.  Correspondingly, it is prudent to wait until these flash a New Long signal, at which time I would evaluate entry.

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The final aspect of this is to determine the relative weightings of positions.  This is NOT a equal-dollar-into-11 -stocks; each stock is position-sized based upon relative gain and volatility within the portfolio.  For April 1, here are the allocations of the 11 stocks:



Note that JPM indicating 0% allocation is not a typo.  The Efficient Frontier calculations, with this set of stocks, have established that JPM does not add anything in terms of return or risk reduction when bundled with these other stocks AND given their recent behavior.  This is a bonus ... and then there were TEN stocks!

These allocations will be valid for the month of April unless I discover an error (always possible, as this is new work and I don't have anybody looking over my shoulder that I can bounce ideas/calculations off of).

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Summary

I've presented a considerable amount of material here which represents several weeks of work.  Results are preliminary, but I feel that we can use this methodology to meet/beat indexes.  The question is whether this is able to be put into practice.  What have we learned?
  1. The concept that there exists an optimum reward/risk ratio given a set basket of stocks.  We can find this optimum reward/risk ratio using standard tools.
  2. The concept that we can eliminate under-performing stocks in a given basket, based upon historical return and volatility, and improve the basket performance (in hind sight, of course)
  3. The concept that we can selectively pick stocks from a subset of stocks that comprise an index and improve our chances of outperforming the index without taking significant additional risk.
  4. That we can take these new candidates, apply GGT timing methods, and the result will improve overall performance because we will not be participating in the stock when it is underperforming.
Of course, it must be stated again that we cannot predict the future, and because of this, we must rebalance.  Presently, it appears that monthly rebalancing is all that will be necessary in order to implement this strategy.  This is subject to change, further testing is required.  For now, it appears that it will work "good enough".

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As always, you are responsible for your own investment decisions, and I am not.  Please do your diligence, and please take ownership for your actions.

Regards,

pgd