Principles of Portfolio Optimization Software

Explaining technical investment concepts in a non-technical way is critical to having a meaningful dialog with individual investors.  Most individual investors (also called “retail investors”, or “small investors”) do not have the time nor the desire to learn the jargon and concepts behind building a solid investment portfolio.  This is generally true for most individual investors regardless of the size of their investment portfolios.  Individual investors expect investment professionals (also called “institutional investors”) to help manage their portfolios and explain the major investment decisions behind the management of their individual portfolios.

In the same way that a good doctor helps her patient make informed medical decisions, a good investment adviser helps her clients make informed investment decisions.

I get routinely asked how the HALO Portfolio Optimizer works.  Every time I answer that question, I face two risks: 1) that I don’t provide enough information to convince the investment profession or their clients that HALO optimization provides significant value and risk-mitigation capability and 2) I risk sharing key intellectual property (IP) unique to the Sigma1 Financial HALO optimizer.

This post is my best effort to provide both investment advisers and their clients with enough information to evaluate and understand HALO optimization, while avoiding sharing key Sigma1 trade secrets and intelectual property.  I would very much appreciate feedback, both positive and negative, as to whether I have achieved these goals.

First Principle of Portfolio Optimization Software

Once when J.P. Morgan was asked what the market would do, he answered “It will fluctuate.”  While some might find this answer rather flippant, I find it extremely insightful.  It turns out that so-called modern portfolio theory (MPT) is based understanding (or quantifying) market fluctuations. MPT labels these fluctuations as “risk” and identifies “return” as the reward that a rational investor is willing to accept for a given amount of risk.  MPT assumes that a rational investor, or his/her investment adviser will diversify away most or all “diversifiable risk” by creating a suitable investment portfolio tailored to the investor’s current “risk tolerance.”

In other words, the primary job of the investment adviser (in a “fiduciary” role), is to maximize investment portfolio return for a client’s acceptable risk.  Said yet another way, the job is to maximize the risk/reward ratio for the client, without incurring excess risk.

Now for the first principle: past asset “risk” tends to indicate future asset “risk”.  In general an asset that has been previously more volatile will tend to remain more volatile, and and asset that has been less volatile will tend to remain less volatile.  Commonly, both academia and professional investors have equated volatility with risk.

Second Principle of Portfolio Optimization Software

The Second Principle is closely related to the first.  The idea is that the past portfolio volatility tends to indicate future portfolio volatility. This thesis is so prevalent that it is almost inherently assumed.  This is evidenced by search results that reaches beyond volatility and looks at the hysteresis of return-versus-volatility ratios, papers such at this.

Past Performance is Not Necessarily Indicative of Future Results.

Third Principle of Portfolio Optimization Software

The benefits of diversification are manifest in risk mitigation.  If two assets are imperfectly correlated, then their combined volatility (risk) will be less than the weighted averages of their individual volatilities.  An in-depth mathematical description two-asset portfolio volatilities can be found on William Sharpe’s web page.  Two-asset mean-variance optimization is relatively simple, and can be performed with relatively few floating-point operations on a computer.  This process creates the two-asset efficient frontier*.  As more assets are added to the mix, the computational demand to find the optimal efficient frontier grows geometrically, if you don’t immediately see why look at page 8 of this paper.

A much simpler explanation of the the third principle is as follows.  If asset A has annual standard deviation of 10%, and asset B an annual standard deviation of 20%, and A and B are not perfectly correlated, then the portfolio of one half invested in A and the other half invested in B will have a annual standard deviation of less than 15%.  (Non-perfectly correlated means a correlation of less than 1.0).  Some example correlations of assets can be found here.

In so-called plain English, the Third Principle of Portfolio Optimization can be stated: “For a given level of expected return, portfolio optimization software can reduce portfolio risk by utilizing the fact that different assets move somewhat independently from each other.”

Forth Principle of Portfolio Optimization Software

The Forth Principle of Portfolio Optimization establishes a relationship between risk and return.  The classic assumption of modern portfolio theory (MPT) is that so-called systematic risk is rewarded (over a long-enough time horizon) with increased returns.  Portfolio-optimization software seeks to reduce or eliminate unsystematic risk when creating an optimized set of portfolios.  The portfolio manager can thus select one of these optimized portfolios from the “best-in-breed” list created by the optimization software that is best suited to his/her client’s needs.

Fifth Principle of Portfolio Optimization Software

The 5th Principle is that the portfolio manager and his team adds value to the portfolio composition process by 1) selecting a robust mix of assets, 2) applying constraints to the weights of said assets and asset-groups, and 3) assigning expected returns to each asset.  The 5th Principle focuses on the assignment of expected returns.  This  process can be grouped under the category of investment analysis or investment research.  Investment firms pay good money for either in-house or contracted investment analysis of selected securities.

Applying the Portfolio Optimization Principles Together

Sigma1 Financial HALO Software applies these five principles together to help portfolio managers improve or fine-tune their proprietary-trading and/or client investment portfolios.  HALO Portfolio Optimization software utilizes the assets, constraints, and expected returns from the 5th Principal as a starting point.  It then uses the 4th Principal by optimizing away systematic risk from a set of portfolios by taking maximum advantage of varying degrees of non-correlation of the portfolio assets.  The 3rd Principle alludes to the computational difficulty of solving the multi-asset optimization problem.  Principles 1 and 2 form the bedrock of the concepts behind the use of historical correlation data to predict and estimate future correlations.

The Fine Print

Past asset volatility of most assets and most portfolios is historically well correlated with future volatility. However, not only are assets increasingly correlated, there is some evidence that asset correlations tend to increase during times of financial crisis. Even if assets are more correlated, there remains significant value in exploiting partial-discorrelation.
(*) The two-asset model can be represented as two parametric functions of a single variable, “t”, ER(t), and var(t).  t simply represents the investment proportion invested in asset 0 (aka asset A).  For three variables, expected return becomes ER(t0,t1) as does var(t0,t1).  And so on for increasing numbers of assets.  The computational effort required to compute ER(t0…tn) scales linearly with number of assets, but var(t0…tn) scales geometrically.
Optimizing efficiently within this complex space benefits from creative algorithms and heuristics.

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Inverted Risk/Return Curves

Over 50 years of academic financial thinking is based on a kind of financial gravity:  the notion that for a relatively diverse investment portfolio, higher risk translates into higher return given a sufficiently long time horizon.  Stated simply: “Risk equals reward.”  Stated less tersely, “Return for an optimized portfolio is proportional to portfolio risk.”

As I assimilated the CAPM doctrine in grad school, part of my brain rejected some CAPM concepts even as it embraced others.  I remember seeing a graph of asset diversification that showed that randomly selected portfolios exhibited better risk/reward profiles up to 30 assets, at which point further improvement was minuscule and only asymptotically approached an “optimal” risk/reward asymptote.  That resonated.

Conversely, strict CAPM thinking implied that a well-diversified portfolio of high-beta stocks will outperform a marketed-weighted portfolio of stocks over the long-term, albeit in a zero-alpha fashion.  That concept met with cognitive dissonance.

Now, dear reader, as a reward for staying with this post this far, I will reward you with some hard-won insights.  After much risk/reward curve fitting on compute-intensive analyses, I found that the best-fit expected-return metric for assets was proportional to the square root of beta.  In my analyses I defined an asset’s beta as 36-month, monthly returns relative to the benchmark index.  Mostly, for US assets, my benchmark “index” was VTI total-return data.

Little did I know, at the time, that a brilliant financial maverick had been doing the heavy academic lifting around similar financial ideas.  His name is Bob Haugen. I only learned of the work of this kindred spirit upon his passing.

My academic number crunching on data since 1980 suggested a positive, but decreasing incremental total return vs. increasing volatility (or for increasing beta).  Bob Haugen suggested a negative incremental total return for high-volatility assets above an inflection-point of volatility.

Mr. Haugen’s lifetime of  published research dwarfs my to-date analyses. There is some consolation in the fact that I followed the data to conclusions that had more in common with Mr. Haugen’s than with the Academic Consensus.

An objective analysis of the investment approach of three investing greats will show that they have more in common with Mr. Haugen than Mr. E.M. Hypothesis (aka Mr. Efficient Markets, [Hypothesis] , not to be confused with “Mr. Market”).  Those great investors are 1) Benjamin Graham, 2) Warren Buffet, 3) Peter Lynch.

CAPM suggests that, with either optimal “risk-free”or leveraged investments a capital asset line exists — tantamount to a linear risk-reward relationship. This line is set according to an unique tangent point to the efficient frontier curve of expected volatility to expected return.

My research at Sigma1 suggests a modified curve with a tangent point portfolio comprised, generally, of a greater proportion of low volatility assets than CAPM would indicate.  In other words, my back-testing at Sigma1 Financial suggests that a different mix, favoring lower-volatility assets is optimal.  The Sigma1 CAL (capital allocation line) is different and based on a different asset mix.  Nonetheless, the slope (first derivative) of the Sigma1 efficient frontier is always upward sloping.

Mr. Haugen’s research indicates that, in theory, the efficient frontier curve past a critical point begins sloping downward with as portfolio volatility increases. (Arguably the curve past the critical point ceases to be “efficient”, but from a parametric point it can be calculated for academic or theoretical purposes.)  An inverted risk/return curve can exist, just as an inverted Treasury yield curve can exist.

Academia routinely deletes the dominated bottom of the the parabola-like portion of the the complete “efficient frontier” curve (resembling a parabola of the form x = A + B*y^2) for allocation of two assets (commonly stocks (e.g. SPY) and bonds (e.g. AGG)).

Maybe a more thorough explanation is called for.   In the two-asset model the complete “parabola” is a parametric equation where x = Vol(t*A, (1-t)*B) and y = ER( t*A, (1-t)*B.  [Vol == Volatility or standard-deviation, ER = Expected Return)].   The bottom part of the “parabola” is excluded because it has no potential utility to any rational investor.  In the multi-weight model, x=minVol (W), y=maxER(W), and W is subject to the condition that the sum of weights in vector W = 1.  In the multi-weight, multi-asset model the underside is automatically excluded.  However there is no guarantee that there is no point where dy/dx is negative.  In fact, Bob Haugen’s research suggests that negative slopes (dy/dx) are possible, even likely, for many collections of assets.

Time prevents me from following this financial rabbit hole to its end.  However I will point out the increasing popularity and short-run success of low-volatility ETFs such as SPLV, USMV, and EEMV.  I am invested in them, and so far am pleased with their high returns AND lower volatilities.

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NOTE: The part about W is oversimplified for flow of reading.  The bulkier explanation is y is stepped from y = ER(W) for minVol(W) to max expected-return of all the assets (Wmax_ER_asset = 1, y = max_ER_asset_return), and each x = minVol(W) s.t. y = ER(W) and sum_of_weights(W) = 1.   Clear as mud, right?  That’s why I wrote it the other way first.

 

Variance, Semivariance Convergence

In running various assets through portfolio-optimization software, I noticed that for an undiversified set of assets there can be wide differences between portfolios with the highest Sharpe ratios versus portfolios with the Sortino ratios.  Further, if the efficient frontier of ten portfolios is constructed (based on mean-variance optimization) and sorted according to both Sharpe and Sortino ratios the ordering is very different.

If, however, the same analysis is performed on a globally-diversified set of assets the portfolios tend to converge.  The broad ribbon of of the 3-D efficient surface seen with undiversfied assets narrows until it begins to resemble a string arching smoothly through space.  The Sharpe/Sortino ordering becomes very similar with ranks seldom differing by more than 1 or 2 positions.  Portfolios E and F may rank 2 and 3 in the Sharpe ranking but rank      2 and 1 in the Sortino ranking, for example.

Variance/Semivariance divergence is wider for optimized portfolios of individual stocks.  When sector-based stock ETFs are used instead of individual stocks, the divergence narrows.  When bond- and broad-based index ETFs are optimized, the divergence narrows to the point that it could be considered by many to be insignificant.

This simple convergence observation has interesting ramifications.  First, a first-pass of faster variance optimization can be applied, followed by a slower semivariance-based refinement to more efficiently achieve a semivariance-optimized portfolio.  Second, semivariance distinctions can be very significant for non-ETF (stock-picking) and less-diversified portfolios.  Third, for globally-diversified, stock/bond, index-EFT-based portfolios, the differences between variance-optimized and semivariance-optimized portfolios are extremely subtle and minute.

 

 

Semivariance-Based and Hybrid Financial Ratios

When evaluating investment portfolios, a higher Sharpe ratio is typically better than a lower Sharpe ratio.  Similarly, a higher Sortino ratio is better than a lower one.  Finally, for portfolios with identical expected returns, the one with a lower standard deviation (sigma, σ) is better.

My best estimate is that less than 2% of portfolio assets are currently optimized for semivariance of any kind (mean-return semivariance, return semivariance, semivariance with respect to minimum acceptable return, etc.).   Meanwhile at least 60% of portfolio assets take variance into account.  According to the laws of supply and demand, the variance contribution (often related to beta) of an asset is strongly priced into its market value.  Conversely, the asset’s semivariance is essentially not priced in.

If, as I believe,  semivariance-based models continue to play an increasing roll in portfolio optimization strategies then asset-semivariance contributions will increasingly price in.   If this prediction holds true, then early adopters of semivariance optimization will front-run asset pricing changes by buying assets with superior price-to-semivariance properties and shunning assets that detract from, say, Sortino ratios.

Sharpe and Sortino ratios differ only in the denominator (σ_p versus σ_d) .  For portfolio-optimization, I prefer to modify σ_p and σ_d such that they would be identical for a normal distribution.  In such a scenario, each sigma is the square root of its corresponding (return) variance.  Because downside variance (σ_d ), in a normal distribution, is half of total variance I multiply downside variance by 2.   For the same reason, I tend to avoid anything mathematically arbitrary such as minimum acceptable return (MAR), and use zero or mean-return instead.  Empirically, I have found that zero works quite well.

My preliminary data shows is that for virtually any portfolio σ_p ≠ σ_d.  While σ_p and σ_d are positively correlated, their ex-post correlation is as low as 0.5.   In some cases, for optimized portfolios of the same return,  increasing  σ_p  by one unit (say 0.01) reduces σ_d by a similar amount  and vice versa.   This is an astounding trade off — it means variance and semivariance compete head to head.

I don’t believe that the investment industry will switch from variance to semivariance models overnight.  Similarly individual investment houses desire to shift gradually (each on their own timescale).  Thus I propose the use of hybrid variance/semivariance ratios.  With hybrid ratios, financial professionals can choose just how much they want semivariance to factor into their optimization… from zero to 100%, and everything in between.

The shifting portfolio-optimization market is large enough large enough for several players to make healthy profits.  For those players that can and do make a shift to semivariance, I recommend a contiguous hybrid model.

 

 

 

Beta Software, First Month

This marks the first month (30 days) of engagement with beta financial partners.  The goal is to test Sigma1 HAL0 portfolio-optimization software on real investment portfolios and get feedback from financial professionals.  The beta period is free.  Beta users provide tickers and expected-returns estimates via email, and Sigma1 provides portfolio results back with the best Sharpe, Sortino, or Sharpe/Sortino hybrid ratio results.

HAL0 portfolio-optimization software provides a set of optimized portfolios, often 40 to 100 “optimal” portfolios, optimized for expected return, return-variance and return-semivariance.   “Generic” portfolios containing a sufficiently-diverse set of ETFs produce similar-looking graphs.  A portfolio set containing SPY, VTI, BND, EFA, and BWX is sufficient to produce a prototypical graph.  The contour lines on the graph clearly show a tradeoff between semi-variance and variance.

 

Portfolio Optimization Graph

 

Once the set of optimized portfolios has been generated the user can select the “best” portfolio based on their selection criteria.

So far I have learned that many financial advisers and fund managers are aware of post-modern portfolio theory (PMPT) measures such as semivariance, but also a bit wary of them.  At the same time, some I have spoken with acknowledge that semivariance and parts of PMPT are the likely future of investing.  Portfolio managers want to be equipped for the day when one of their big investors asks, “What is the Sortino ratio of my portfolio? Can you reduce the semi-variance of my portfolio?”

I was surprised to hear that all of Sigma1 beta partners are interested exclusively in a web-based interface. This preliminary finding is encouraging because it aligns with a business model that protects Sigma1 IP from unsanctioned copying and reverse-engineering.

Another surprise has been the sizes of the asset sets supplied, ranging from 30 to 50 assets. Prior to software beta, I put significant effort into ensuring that HAL0 optimization could handle 500+ asset portfolios. My goal, which I achieved, was high-quality optimization of 500 assets in one hour and overnight deep-dive optimization (adding 8-10 basis points of additional expected-return for a given variance/semi-variance). On the portfolio assets provided to-date, deep-dive runtimes have all been under 5 minutes.

The best-testing phase has provided me with a prioritized list of software improvements. #1 is per-asset weighting limits. #2 is an easy-to-use web interface. #3 is focused optimization, such as the ability to set max variance.  There have also been company-specific requests that I will strive to implement as time permits.

Financial professionals (financial advisers, wealth managers, fund managers, proprietary trade managers, risk managers, etc.) seem inclined to want to optimize and analyze risk in both old ways (mean-return variance) and new (historic worst-year loss, VAR measures, tail risk, portfolio stress tests, semivariance, etc.).

Some Sigma1 beta partners have been hesitant to provide proprietary risk measure algorithms.  These partners prefer to use built-in Sigma1 optimizations, receive the resulting portfolios, and perform their own in-house analysis of risk.  The downside of this is that I cannot optimize directly to proprietary risk measures.  The upside is that I can further refine the HAL0 algos to solve more universal portfolio-optimization problems.  Even indirect feedback is helpful.

Portfolio and fund managers are generally happy with mean-return variance optimization, but are concerned that semivariance-return measures are reasonably likely to change the financial industry in the coming years.   Luckily the Sharpe ratio and Sortino ratio differ by only the denominator (σ_p versus σ_d) .  By normalizing the definitions of volatility (currently called modified-return variance and modified-return semivariance) HAL0 software optimizes simultaneously for both (modified) Sharpe and Sortino ratios, or any Sharpe/Sortino hybrid ratios in-between.  A variance-focused investor can use a 100% variance-optimized portfolio.  An investor wanting to dabble with semi-variance can explore portfolios with, say, a 70%/30% Sharpe/Sortino ratio.   And an investor, fairly bullish on semivariance minimization, could use a 20%/80% Sharpe/Sortino hybrid ratio.

I am very thankful to investment managers and other financial pros who are taking the time to explore the capabilities of HAL0 portfolio-optimization software.  I am hopeful that, over time, I can persuade some beta partners to become clients as HAL0 software evolves and improves.  In other cases I hope to provide Sigma1 partners with new ideas and perspectives on portfolio optimization and risk analysis.  Even in one short month, every partner has helped HAL0 software become better in a variety of ways.

Sigma1 is interested in taking on 1 or 2 additional investment professionals as beta partners.  If interested please submit a brief request for info on our contact page.

 

Portfolio Risks: Risk Analysis, Optimization and Management

With news like JPMorgan losing $9 billion dollars in a quarter due to trading losses, it’s no wonder that risk management software is seen as increasingly important.  I appears that the highest level executives have no clue how to assess the risks that their traders are taking on.  No clue, that is, until they are side-swiped by massive losses.

To begin to fathom the risk exposure from proprietary-trading (and hedging) it is necessary to have near-real-time data for the complete portfolio of securities, derivatives, and other financial positions and obligations.  This is a herculean, but achievable task for more vanilla securities positions such as long and short positions in stocks, bonds, ETNs, options and futures.  All of these assets have standardized tickers, trading rules, and essentially zero counterparty risk.  Further these financial assets have thorough, easily-accessible, real-time data for price, volume, bid and ask.  Even thinly traded assets like many option contracts have sufficient data to at least estimate their current liquidation value with tolerable uncertainty (say +/- 10%).

OTC trades, contracts, and obligations pose a much greater challenge for risk managers.  Lets think about credit-default swaps on Greek bonds.  Believe it or not there is uncertainty over the definition of “default”.  If European banks agreed to take a 50% haircut on Greek debt, does that constitute a default.  Most accounts I have read say no.  So even if a savvy European bank hedged its Greek bond exposure with CDS contracts, they lose.  Their hedge really wasn’t.

Sigma1 doesn’t (currently) attempt to assess risk for exotic OTC contracts and obligations.  What Sigma1 HAL0 software does do is better model standardized financial asset portfolios.  A tag line for HAL0 software could be “Risk: Better Modelling, Sounder Sleep”.

My goal is to continuously improve risk management and risk optimization in the following ways:

1. Risk models that are more robust and intuitive.
2. Enhanced risk visualization.  Taking the abstract and making it visible
3. Optimizing downside risk (minimizing downside risk) with sophisticated heuristic algorithms.

I prefer the term “optimize” (in most contexts) to “minimize” or “maximize” because it is clear what optimize means.  Naturally portfolio optimization means finding the efficient frontier of minimized risk returns (or return-maximized risks).  Either way optimization usually involves concurrent minimization and maximization of various objective functions.

HAL0 portfolio optimization is best suited for optimizing the following types of funds and portfolios  1) individual investment portfolios, 2) endowment portfolios, 3) pension funds, 4) insurance company portfolios, 5) traditional (non-investment bank) bank portfolios, 6) company investment portfolios (including bond obligations).

While the core HAL0 optimization algorithm is designed to optimize more than 3 objective functions, I have been increasingly focused on optimizing for 3 concurrent objectives.   In the most common usage model, I envision one expected return function, one risk function, a third objective function.   The third objective function can be another risk model, diversification metric, investment-style metric or any other quantitative measure.

For example, HAL0 can optimize from a pool of 500 investments to create a 3D efficient frontier surface.  The z axis is, by convention, always the expect return.  The x axis is generally the primary risk measure, such as 3-year monthly semivariance.  The y axis, depth, can be another risk measure such as worst 5-year quarterly return.

Looking at this surface gives perspective on the tradeoffs between the various return and risk metrics.  It is particularly elucidating to plot a point representing one’s current investment pool or portfolio.  If it is on the surface, it is optimal (or near optimal).  However, if it is under the surface it is sub-optimal.  Either way, looking north, south, east, or west show the nearby alternatives — trading of various risks and rewards.

So the nascent marketer in me asks:  Can your financial optimization software optimize and display in 3 dimensions?  Can it optimize non-standard functions (such as worst-case quarterly return over 5 years)?  Is your current portfolio optimization software written from the ground up to be specifically optimized for financial optimization challenges?

HAL0 is.   It is the financial software that I would buy (and will personally use) to optimize my financial portfolio.  It is so compelling that it is the first project that is causing me to seriously consider quitting my day job with excellent benefits, vacation, and a six-figure salary for.   To borrow a baseball analogy software development and finance are in my wheelhouse.  I am considering giving up the comfort and security of a solid job in electrical engineering to pursue my dream and my truest talents.  Many in my industry would “kill” for my current position.  To me it feels largely intellectually unchallenging. In contrast, developing and enhancing HAL0 has taken every spare ounce of my creativity, knowledge, and passion.  In essence, HAL0 is a labor of love.

I passionately want to redefine financial risk.  I also want to modestly redefine financial return.  I see the current financial model and flawed in major and minor (yet significant) ways and hope to reinvent it.   It’s about leveraging the best of the past (Markowitz’s core ideas including semivariance) and the best of the now (fast, networked, parallel compute technology).  To accomplish this requires great software, the beta version of which, called HAL0, is residing on my Linux server.

Benchmarking Financial Algorithms

In my last post I showed that there are far more that a googol permutations of portfolio of 100 assets with (positive, non-zero) weights in increments of 10 basis points, or 0.1%.    That number can be expressed as C(999,99), or C(999,900) or 999!/(99!*900!), or ~6.385*10¹³⁸.  Out of sheer audacity, I will call this number Balhiser’s first constant (K_β1).  [Wouldn’t it be ironic and embarrassing if my math was incorrect?]

In the spirit of Alan Turing’s 100th birthday today and David Hilbert’s 23 unsolved problems of 1900, I propose the creation of an initial set of financial problems to rate the general effectiveness of various portfolio-optimization algorithms.  These problems would be of a similar form:  each having a search space of K_β1. There would be 23 initial problems P₁…P₂₃.  Each would have a series of 37 monthly absolute returns.  Each security will have an expected annualized 3-year return (some based on the historic 37-month returns, others independent).  The challenge for any algorithm A to score the best average score on these problems.

I propose the following scoring measures:  1) S”(A) (S double prime) which simply computes the least average semi-variance portfolio independent of expected return.  2) S'(A) which computes the best average semi-variance and expected return efficient frontier versus a baseline frontier.  3) S(A) which computes the best average semi-variance, variance, and expected return efficient frontier surface versus a baseline surface.  Any algorithm would be disqualified if any single test took longer than 10 minutes.  Similarly any algorithm would be disqualified if it failed to produce a “sufficient solution density and breadth” for S’ and S” on any test.  Obviously, a standard benchmark computer would be required.  Any OS, supporting software, etc could be used for purposes of benchmarking.

The benchmark computer would likely be a well-equipped multi-core system such as a 32 GB Intel  i7-3770 system.  There could be separate benchmarks for parallel computing, where the algorithm + hardware was tested as holistic system.

I propose these initial portfolio benchmarks for a variety of reasons.  1)  Similar standardized benchmarks have been very helpful in evaluating and improving algorithms in other fields such as electrical engineering.  2)  Providing a standard that helps separate statistically significant from anecdotal inference. 3)  Illustrate both the challenge and the opportunity for financial algorithms to solve important investing problems. 4)  Lowering barriers to entry for financial algorithm developers (and thus lowering the cost of high-quality algorithms to financial businesses).  5)  I believe HAL0 can provide superior results.

The Equation that Will Change Finance

Two mathematical equations have transformed the world of modern finance.  The first was CAPM, the second Black-Scholes.  CAPM gave a new perspective on portfolio construction.  Black-Scholes gave insight into pricing options and other derivatives.  There have been many other advancements in the field of financial optimization, such as Fama-French — but CAPM and Black-Scholes-Merton stand out as perhaps the two most influential.

Enter Semi-Variance

Modified Semi-Variance Equation, A Financial Game Changer

When CAPM (and MPT) were invented, computers existed, but were very limited.  Though the father of CAPM, Harry Markowitz, wanted to use semi-variance, the computers of 1959 were simply inadequate.  So Markowitz used variance in his ground breaking book “Portfolio Selection — Efficient Diversification of Investments”.

Choosing variance over semi-variance made the computations orders of magnitude easier, but the were still very taxing to the computers of 1959.  Classic covariance-based optimizations are still reasonably compute-intensive when a large number of assets are considered.  Classic optimization of a 2000 asset portfolio starts by creating a 2,002,000-entry (technically 2,002,000 unique entries which, when mirrored about the shared diagonal, number 4,000,000) covariance matrix; that is the easy part.  The hard part involves optimizing (minimizing) portfolio variance for a range of expected returns.  This is often referred to as computing the efficient frontier.

The concept of semi-variance (SV) is very similar to variance used in CAPM.  The difference is in the computation.  A quick internet search reveals very little data about computing a “semi-covariance matrix”.  Such a matrix, if it existed in the right form, could possibly allow quick and precise computation of portfolio semi-variance in the same way that a covariance matrix does for computing portfolio variance.  Semi-covariance matrices (SMVs) exist, but none “in the right form.” Each form of SVM has strengths and weaknesses. Thus, one of the many problems with semi-covariance matrices is that there is no unique canonical form for a given data set.  SVMs of different types only capture an incomplete portion of the information needed for semi-variance optimization.

The beauty of SV is that it measures “downside risk”, exclusively.  Variance includes the odd concept of “upside risk” and penalizes investments for it.  While not  going to the extreme of rewarding upside “risk”, the modified semi-variance formula presented in this blog post simply disregards it.

I’m sure most of the readers of this blog understand this modified semi-variance formula.  Please indulge me while I touch on some of the finer points.   First, the 2 may look a bit out of place.  The 2 simply normalizes the value of SV relative to variance (V).  Second, the “question mark, colon” notation simply means if the first statement is true use the squared value in summation, else use zero.  Third, notice I use r_i rather than r_i – r_avg.

The last point above is intentional and another difference from “mean variance”, or rather “mean semi-variance”.  If R is monotonically increasing during for all samples (n intervals, n+1 data points), then SV is zero.  I have many reasons for this choice.  The primary reason is that with  r_avg the SV for a straight descending R would be zero.  I don’t want a formula that rewards such a performance with 0, the best possible SV score.  [Others would substitute T, a usually positive number, as target return, sometimes called minimal acceptable return.]

Finally, a word about r_i — r_i is the total return over the interval i.  Intervals should be as uniform as possible.  I tend to avoid daily intervals due to the non-uniformity introduced by weekends and holidays.  Weekly (last closing price of the trading week), monthly (last closing price of the month), and quarterly are significantly more uniform in duration.

Big Data and Heuristic Algorithms

Innovations in computing and algorithms are how semi-variance equations will change the world of finance.  Common sense is why. I’ll explain why heuristic algorithms like Sigma1’s HALO can quickly find near-optimal SV solutions on a common desktop workstation, and even better solutions when leveraging a data center’s resources.  And I’ll explain why SV is vastly superior to variance.

Computing SV for a single portfolio of 100 securities is easy on a modern desktop computer.  For example 3-year monthly semi-variance requires 3700 multiply-accumulate operations to compute portfolio return, R_p, followed by a mere 37 subtractions, 36 multiplies (for squaring), and 36 additions (plus multiplying by 2/n).  Any modern computer can perform this computation in the blink of an eye.

Now consider building a 100-security portfolio from scratch.  Assume the portfolio is long-only and that any of these securities can have a weight between 0.1% and 90% in steps of 0.1%.  Each security has 900 possible weightings.  I’ll spare you the math — there are 6.385*10¹³⁸ permutations. Needless to say, this problem cannot be solved by brute force.  Further note that if the portfolio is turned into a long-short portfolio, where negative values down to -50% are allowed, the search space explodes to close to 10²⁰⁰⁰.

I don’t care how big your data center is, a brute force solution is never going to work.  This is where heuristic algorithms come into play.  Heuristic algorithms are a subset of metaheuristics.  In essence heuristic algorithms are algorithms that guide heuristics (or vise versa) to find approximate solution(s) to a complex problem.   I prefer the term heuristic algorithm to describe HALO, because in some cases it is hard to say whether a particular line of code is “algorithmic” or “heuristic”, because sometimes the answer is both.  For example, semi-variance is computed by an algorithm but is fundamentally a heuristic.

Heuristic Algorithms, HAs, find practical solutions for problems that are too difficult to brute force.  They can be configured to look deeper or run faster as desired by the user.  Smarter HAs can take advantage of modern computer infrastructure by utilizing multiple threads, multiple cores, and multiple compute servers in parallel.  Many, such as HAL0, can provide intermediate solutions as they run far and deep into the solution space.

Let me be blunt — If you’re using Microsoft Excel Solver for portfolio optimization, you’re missing out.  Fly me out and let me bring my laptop loaded with HAL0 to crunch your data set — You’ll be glad you did.

Now For the Fun Part:  Why switch to Semi-Variance?

Thanks for reading this far!  Would you buy insurance that paid you if your house didn’t burn down?   Say you pay $500/year and after 10 years, if your house is still standing, you get $6000. Otherwise you get $0. Ludicrous, right?  Or insurance that only “protects” your house from appreciation?  Say it pays 50 cents for every dollar make when you resell your house, but if you lose money on the resale you get nothing?

In essence that is what you are doing when you buy (or create) a portfolio optimized for variance.   Sure, variance analysis seeks to reduce the downs, but it also penalizes the ups (if they are too rapid).  Run the numbers on any portfolio and you’ll see that SV ≠ V.  All things equal, the portfolios with SV < V are the better bet. (Note that classic_SV ≤ V, because it has a subset of positive numbers added together compared to V).

Let me close with a real-world example.  SPLV is an ETF I own.  It is based on owning the 100 stocks out of the S&P 500 with the lowest 12-month volatility.  It has performed well, and been received well by the ETF marketplace, accumulating over $1.5 billion in AUM.  A simple variant of SPLV (which could be called PLSV for PowerShares Low Semi-Variance) would contain the 100 stocks with the least SV.  An even better variant would contain the 100 stocks that in aggregate produced the lowest SV portfolio over the proceeding 12 months.

HALO has the power to construct such a portfolio. It could solve preserving the relative market-cap ratios of the 100 stocks, picking which 100 stocks are collectively optimal.  Or it could produce a re-weighted portfolio that further reduced overall semi-variance.

[Even more information on semi-variance (in its many related forms) can be found here.]

 

Flaws in Stock and ETF Charts

Traditional Stock Charts can Mislead

Almost every stock chart presents incomplete data for a security’s total return.  Simply put, stock charts don’t reflect dividends and distributions.  Stock charts simply show price data.  A handful of charts superimpose dividends over the price data.  Such charts are an improvement, but require mental gymnastics to correctly interpret total return.

At the end of the year, I suspect the vast majority of investors are much more interested in how much money they made than whether their profits come from asset appreciation, dividends, interest or other distributions.  In the case of tax-differed or tax-exempt accounts (such as IRA, Roth IRAs, 401k, etc. accounts) the source of returns is unimportant.  Naturally, for other portfolios, some types of return are more tax-advantaged than others.  In one case I tried to persuade a relative that MUB (iShares S&P National AMT-Free Muni Bd) was a good investment for them in spite of it’s chart, because the chart did not show the positive tax impact of tax-exempt income.

Our minds see what they want to see.  When we compare two stocks (or ETFs) we often have a slight bias towards one.  If we see what we want in a stock’s chart, we may look past the dividend annotations and make a incorrect decision.

This 1-year chart comparing two ETFs illustrates this point.  These two ETFs track each other reasonably well until Dec 16th, where there is a sharp drop in PBP.  This large dip reflects the effect of a large distribution of roughly 10%.  Judging strictly by the price data, it at first appears that SPY beats PBP by 7%.  When factoring the yield of PBP, about 10.1%, and SPY, roughly 1.9%, shows a 1.2% 1-year out-performance by PBP.  First appearances show SPY outperforming;  a little math shows PBP outperforming.

Yahoo! Finance provides raw data adjusted for dividends and distributions.  Using the 1-year start and end data shows SPY returning a net 3.77%, and PBP returning a net 4.96%.  The delta shows a 1.19% out performance by PBP.  Yahoo! Finance’s table have all the right data;  I would love to see Yahoo! add an option to display this adjusted-price data graphically.

Total return is not a new concept.  Bill Gross was very insightful in naming PIMCO’s “Total Return” lineup of funds over 25 years ago.  Many mutual funds provide total return charts.  For instance, Vanguard provides total return charts for investments such as Vanguard Total Stock Market Index Fund Admiral Shares.  I am pleased to see Fidelity offering similar charts for ETFs in research “performance” reports for its customers.  Unfortunately, I have not found a convenient way to superimpose two total-return charts.

While traditional stock and ETF charts do not play a large roll in my investment decisions, I do look at them when evaluating potential additions to my investment portfolio.  When I do look at charts, I’d prefer to have the option of looking at total return charts rather than “old fashioned” price charts.

That said, I prefer to use quantitative portfolio analysis as my primary asset allocation technology.  For such analysis I compute total return data for each asset from price data and distribution data, assuming reinvestment.  Reformatting asset data in this way allows HAL0 portfolio-optimization software to directly compare different asset classes (gold, commodities, stock ETFs, bond ETFs, leveraged ETFs, etc).  Moreover, such pre-formatting allows faster computation of risk for various asset allocations within a portfolio.

A large part of my vision for Sigma1 is revolutionizing how investors and money managers visualize and conceptualize portfolio construction.  The key pieces of that conceptual revolution are:

1. Rethinking return to always mean total return.
2. Rethinking risk to mean something other than variance or standard deviation.

Many already think of total return as the key measure of raw portfolio performance.  It is odd, then, that so many charts display something other than total return.  And some would like to measure, manage, and model risk in more robust ways.  A major obstacle to alternate risk measures is a dearth of financial portfolio optimization tools that work with PMPT models such as semi-variance.

HAL0 is designed from the ground up to address the goals of optimizing portfolios based on total return and a wide variety of advanced, more-robust risk models.  (And, yes, total return can be defined in terms of after-tax total return, if desired.)

Disclosure:  I have long positions in SPY, the Vanguard Total Stock Market Index, and PBP.

 

 

 

 

Seeking a Well-Matched Angel Investor (Part I)

Most of the reading I have done regarding angel investing suggests that finding the right “match” is a critical part of the process.  This process is not just about a business plan and a product, it is also about people and personalities.

Let me attempt to give some insight into my entrepreneurial personality.  I have been working (and continue to work) in a corporate environment for 15 years. Over that time I have received a lot of feedback.  Two common themes emerge from that feedback.  1)  I tend to be a bit too “technical”.  2)  I tend to invest more effort on work that I like.

Long Story about my Tech Career

Since I work in the tech industry, being too technical at first didn’t sound like something I should work on.  I eventually came to understand that this wasn’t feedback from my peers, but from managers.   Tech moves so fast that many managers simply do not keep up with these changes except in the most superficial ways.  (Please note I say many, not most).  While being technical is my natural tendency, I have learned to adjust the technical content to suite the  composition of the meeting room.

The second theme has been a harder personal challenge.  Two general areas I love are technical challenges and collaboration.  I love when there is no “smartest person in the room” because everybody is the best at at least one thing, if not many.  When a team like that faces a new critical issue — never before seen — magic often occurs.  To me this is not work; it is much closer to play.

I have seen my industry, VLSI and microprocessor design, evolve and mature.  While everyone is still the “smartest person in the room”, the arrival of novel challenges is increasingly rare.   We are increasingly challenged to become masters of execution rather than masters of innovation.

Backing up a bit, when I started at Hewlett-Packard, straight out of college, I had the best job in the world, or darn near.  For 3-4 months I “drank from a fire hose” of knowledge from my mentor.  After just 6 months I was given what, even in retrospect, was tremendous responsibilities (and a nice raise).  I was put in charge of integrating “logic synthesis” software into the lab’s compute infrastructure.  When I started, about 10% of the lab’s silicon area was created via synthesis; when I left 8 years later about 90% of the lab’s silicon was created via logic synthesis.  I was part of that transformation, but I wasn’t the cause — logic synthesis was simply the next disruptive technology in the industry.

So why did change companies?  I was developing software to build advanced “ASICs”.  First the company moved ASIC manufacturing overseas, then increasingly ASIC hardware design.  The writing was on the wall… ASIC software development would eventually move.  So I made a very difficult choice and moved into microprocessor software development.  Looking back now, this was the likely the best career choice I have ever made.

Practically overnight I was again “drinking from a fire hose.”   Rather than working with software, my former teammates and I had built from scratch, I was knee-deep in poorly-commented code that been abandoned by all but one of the original developers.  In about 9 months my co-developer and I had transformed this code into something that resembled properly-architected software.

Again, I saw the winds of change transforming my career environment: this time, microprocessor design.  Software development was moving from locally-integrated hardware/software design labs to a centralized software-design organization.  Seeing this shift, I moved within the company, to microprocessor hardware design.  Three and a half years later I see the pros and cons of this choice.  The largest pro is having about 5 times more opportunities in the industry — both within the company, and without.  The largest con, for me, is dramatically less software development work.  Hardware design still requires some software work, perhaps, 20-25%.  Much of this software design, however, is very task-specific.  When the task is complete — perhaps after a week or a month — it is obsolete.

A Passion for Software and Finance

While I was working, I spent some time in grad school. I took all the EE classes that related to VLSI and microprocessor design. The most interesting class was an open-ended research project. The project I chose, while related directly to microprocessor design, had a 50/50 mix of software design and circuit/device-physics research. I took over the software design work, and my partner took on most of the other work. The resulting paper was shortened and revised (with the help of our professor and third grad student) and accepted for presentation at the 2005 Society of Industrial and Applied Mathematics (SIAM) Conference in Stockholm, Sweden.  Unfortunately, none of us where able to attend due to conflicting professional commitments.

Having exhausted all “interesting” EE/ECE courses, I started taking grad school courses in finance.  CSU did not yet have a full-fledged MSBA in Financial Risk Management program, but it did offer a Graduate Certificate in Finance, which I earned.  Some research papers of note include “Above Board Methods of Hedging Company Stock Option Grants” and “Building an ‘Optimal’ Bond Portfolio including TIPS.”

Software development has been an interest of mine since I took a LOGO summer class in 5th grade.  It has been a passion of mine since I taught myself “C” in high school.  During my undergrad in EE, I took enough CS electives to earn a Minor in Computer Science along with my BSEE.   Almost all of my elective CS courses centered around algorithms and AI.   Unlike EE, which at times I found very challenging, I found CS courses easy and fun.  That said, I earned straight A’s in college, grad and undergrad, with one exception: I got a B- in International Marketing.  Go figure.

My interest in finance started early as well.  I had a paper route at the age of 12, and a bank account.  I learned about compound interest and was hooked.  With help from my Dad, and still 12 years old, I soon had a money market account and long-maturity zero-coupon bond.  My full-fledged passion for finance developed when I was issued my first big grant of company stock options.  I realized I knew quite a bit about stocks, bonds, CD’s and money market funds, but I knew practically nothing about options.  Learning about options was the primary reason I started studying Finance in grad school.  I was, however, soon to learn about CAPM and MPT, and portfolio construction and optimization.  Since then, trying to build the “perfect” portfolio has been a lingering fascination.

Gradually, I began to see flaws in MPT and the efficient-markets hypothesis (EMH).  Flaws that Markowitz acknowledged from the beginning!  [Amazing what you can learn from going beyond textbooks, and back to original sources.]   I read in some depth about the rise and demise of Long-Term Capital Management.  I read about high-frequency trading methods and algorithms.  I looked into how options can be integrated into long-term portfolio-building strategies.  And finally, I started researching the ever-evolving field of Post-Modern Portfolio Theory (PMPT.)

When I finally realized how I could integrate my software development skills, my computer science (AI) background, my graduate EE/ECE work and my financial background into a revolutionary software product, I was thunderstruck. I can and did build the alpha version of this product, HAL0, and it works even better than I expected.  If I can turn this product into a robust business, I can work on what I like, even what I love.  And that passion will be a strength rather than a “flaw”.   Send me an angel!