The Equation Everyone in Finance Show Know, but Many Probably Don’t!
Here it is:
… With thanks to codecogs.com which makes it really easy to write equations for the web.
This simple matrix equation is extremely powerful. This is really two equations. The first is all you really need. The second is just merely there for illustrative purposes.
This formula says how the variance of a portfolio can be computed from the position weights wT = [w1 w2 … wn] and the covariance matrix V.
σii ≡ σi2 = Var(Ri)
σij ≡ Cov(Ri, Rj) for i ≠ j
The second equation is actually rather limiting. It represents the smallest possible example to clarify the first equation — a two-asset portfolio. Once you understand it for 2 assets, it is relatively easy to extrapolate to 3-asset portfolios, 4-asset portfolios, and before you know it, n-asset portfolios.
Now I show the truly powerful “naked” general form equation:
This is really all you need to know! It works for 50-asset portfolios. For 100 assets. For 1000. You get the point. It works in general. And it is exact. It is the E = mc2 of Modern Portfolio Theory (MPT). It at least about 55 years old (2014 – 1959), while E = mc2 is about 99 years old (2014 – 1915). Harry Markowitz, the Father of (M)PT simply called it “Portfolio Theory” because:
There’s nothing modern about it.
Yes, I’m calling Markowitz the Einstein of Portfolio Theory AND of finance! (Now there are several other “post”-Einstein geniuses… Bohr, Heisenberg, Feynman… just as there are Sharpe, Scholes, Black, Merton, Fama, French, Shiller, [Graham?, Buffet?]…) I’m saying that a physicist who doesn’t know E = mc2 is not much of a physicist. You can read between the lines for what I’m saying about those that dabble in portfolio theory… with other people’s money… without really knowing (or using) the financial analog.
Why Markowitz is Still “The Einstein” of Finance (Even if He was “Wrong”)
Markowitz said that “downside semi-variance” would be better. Sharpe said “In light of the formidable
computational problems…[he] bases his analysis on the variance and standard deviation.”
Today we have no such excuse. We have more than sufficient computational power on our laptops to optimize for downside semi-variance, σd. There is no such tidy, efficient equation for downside semi-variance. (At least not that anyone can agree on… and none that that is exact in any sense of any reasonable mathematical definition of the word ‘exact’.)
Fama and French improve upon Markowitz (M)PT [I say that if M is used in MPT, it should mean “Markowitz,” not “modern”, but I digress.] Shiller, however, decimates it. As does Buffet, in his own applied way. I use the word decimate in its strict sense… killing one in ten. (M)PT is not dead; it is still useful. Diversification still works; rational investors are still risk-averse; and certain low-beta investments (bonds, gold, commodities…) are still poor very-long-term (20+ year) investments in isolation and relative to stocks, though they still can serve a role as Markowitz Portfolio Theory suggests.
Wanna Build your Own Optimizer (for Mean-Return Variance)?
This blog post tells you most of the important bits. I don’t really need to write part 2, do I? Not if you can answer these relatively easy questions…
What is the matrix expression for computing E(Rp) based on w?
What simple constraint is w subject to?
How does the general σp2 equation relate to the efficient frontier?
How might you adapt the general equation to efficiently compute the effects of a Δw event where wi increases and wj decreases? (Hint “cache” the wx terms that don’t change,)
What other constraints may be imposed on w or subsets (asset categories within w)? How will you efficientlydeal with these constraints?
Most of the important financial industry tests (Series 6, 7, 24, 26, CFA I, CFA II, CFA III, etc) only have two possible binary outcomes: PASS or FAIL. Failure is a waste of time and money. Over-studying, however, can also waste time. (Studying for a PASS/FAIL test is investing in a binary “real option.”)
All of the material is worth knowing for someone, but some information is simply not relevant to everyone. For example, investment advisor reps don’t necessarily need to know all of the rules for broker dealer agents (and vise versa). Knowing the stuff that is relevant to you is more valuable than simply allowing you to pass a test.
That said, the goal is to PASS. And you’ve got a million other things to do. So what’s a quant to do? Get quantitative of course!
Quantitative Test Prep
Step 1: Find representative sample tests. All else hinges on this. Obtaining sample tests from multiple independent sources may help.
Step 2: Determine your average score on practice tests.
Step 3: Determine the standard deviation of your scores.
Step 4: Calculate the probability of achieving a passing score given your mean score and standard deviation.
Step 5: Decide the risk/reward and whether more study provides sufficient ROI.
Assuming normal distributions, I use the 68/95/99.7 rule. Regardless of the standard deviation, if your practice average is the same as the minimum score your chance of success is only 50%. Naturally, if your mean practice score is 1-sigma above the threshold for passing, your chance on the real test is 84% [1-(1-0.68)/2]. If your mean score is plus 2 sigma, your chance of passing is almost 98% [1-(1-0.95)/2].
This little exercise shows two possible ways to improve your expected pass rate. The obvious way is getting better with the material. The less obvious way is reducing your standard deviation. Can this second way be achieved? If so how?
Keeping in mind the four-answer multiple-choice format, the mean deviation is:
MD = 2*p*(1-p)
Where p is the probability of answering a particular question correctly. Per-question deviation (PQD) is highest for p=0.5 at 0.5. PQD is lowest when p=1 at 0. For random guessing, PQD is 0.375.
Increasing your pq to from random guess 0.25 to 0.5 for a given category q will increase your expected score, but will also increase sigma. Taking the first derivative of MD with respect to p gives: 2-4p. Because the range of p is [0,1] (arguably [0.25,1)) the best incremental decrease in MD is greatest near p=1.
Now, the test candidate must decide what the the d/dt(pqc(t)) is for each question category (where t is time spent studying that category). Studying the categories (qc) with the highest d/dt(pqc(t)) will most efficiently improve the expected score. Further studying the categories with the maximum d/dt(pqc(t))*(4p-2) will reduce PQD and hence reduce test standard deviation.
Deeper Analysis of the Meta Problem
Naturally, this analysis only scratches the tactical surface of the “binary-test optimization meta problem.” [The test itself is the problem, the tactics are part of the meta-problem of optimizing generalized multiple-choice test prep]. Improving from p=0.8 to p=0.9 is clearly better than improving from p=0.4 to p=0.5 in terms of PQD reduction, and equal in terms of increase of expected score.
Also of relevance is PQD (modified) downside semi-deviation, which I will call PQDd. I’ll spare you the derivation; it turns out that:
PQDd = p*sqrt(2*(1-p))
This value peaks at p=2/3 with a value of 0.5443. PQDd slowly ascends as p goes from 0.25 up to 0.667, then falls pretty rapidly for values of p>0.8.
We care about the random variable S which represents the actual test score. S is a function of the mean expected score μ and standard deviation σ… in a normal distribution. What we really care about is Pr(S>=Threshold), the probability that our score meets or exceeds the minimum passing score.
PQD = PQDd only when p = 0, 0.5, or 1. For p in (0,0.5) PQDd<PQD and for p in (0.5,1) PDQd>PQD. Even though it seems a bit strange for discrete binary distribution, p in (0,0.5) has positive skewness and p in (0.5,1) negative skewness.
In the “final” analysis the chance of passing, pr(S=>Threshold), depends on score mean, μ, and downside deviation, σd. In turn σd depends on PQD and PQDd.
Summary and Conclusions
Theoretically, one’s best course of action is to 1) increase the average expected score and 2) reduce σd. If practical, the best and most efficient way to achieve both objectives simultaneously is to improve areas that are in the 60-75% range (p=0.6 to 0.75) to the mid to high 90% range (p>=0.95). This may seem counter-intuitive, but the math is solid.
Caveats: This analysis is mostly an exercise in showing the value of statics, variance, and downside variance in an area outside of finance. It shows that there is more than one way to approach to a goal; in this case passing a standardized test.
The red and green “clover” pattern illustrates how traditional risk can be modeled. The red “leaves” are triggered when both the portfolio and the “other asset” move together in concert. The green leaves are triggered when the portfolio and asset move in opposite directions.
Each event represents a moment in time, say the closing price for each asset (the portfolio or the new asset). A common time period is 3-years of total-return data [37 months of price and dividend data reduced to 36 monthly returns.]
When a portfolio manager considers adding a new asset to an existing portfolio, she may wish to see how that asset’s returns would have interacted with the rest of the portfolio. Would this new asset have made the portfolio more or less volatile? Risk can be measured by looking at the time-series return data. Each time the asset and the portfolio are in the red, risk is added. Each time they are in the green, risk is subtracted. When all the reds and greens are summed up there is a “mathy” term for this sum: covariance. “Variance” as in change, and “co” as in together. Covariance means the degree to which two items move together.
If there are mostly red events, the two assets move together most of the time. Another way of saying this is that the assets are highly correlated. Again, that is “co” as in together and “related” as in relationship between their movements. If, however, the portfolio and asset move in opposite directions most of the time, the green areas, then the covariance is lower, and can even be negative.
It is not only the whether the two assets move together or apart; it is also the degree to which they move. Larger movements in the red region result in larger covariance than smaller movements. Similarly, larger movements in the green region reduce covariance. In fact it is the product of movements that affects how much the sum of covariance is moved up and down. Notice how the clover-leaf leaves move to the center, (0,0) if either the asset or the portfolio doesn’t move at all. This is because the product of zero times anything must be zero.
Getting Technical: The clover-leaf pattern relates to the angle between each pair of asset movements. It does not show the affect of the magnitude of their positions.
If the incremental covariance of the asset to the portfolio is less than the variance of the portfolio, a portfolio that adds the asset would have had lower overall variance (historically). Since there is a tenancy (but no guarantee!) for asset’s correlations to remain somewhat similar over time, the portfolio manager might use the covariance analysis to decide whether or not to add the new asset to the portfolio.
Semi-Variance: Another Way to Measure Risk
After staring at the covariance visualization, something may strike you as odd — The fact that when the portfolio and the asset move UP together this increases the variance. Since variance is used as a measure of risk, that’s like saying the risk of positive returns.
Most ordinary investors would not consider the two assets going up together to be a bad thing. In general they would consider this to be a good thing.
So why do many (most?) risk measures use a risk model that resembles the red and green cloverleaf? Two reasons: 1) It makes the math easier, 2) history and inertia. Many (most?) textbooks today still define risk in terms of variance, or its related cousin standard deviation.
There is an alternative risk measure: semi-variance. The multi-colored cloverleaf, which I will call the yellow-grey cloverleaf, is a visualization of how semi-variance is computed. The grey leaf indicates that events that occur in that quadrant are ignored (multiplied by zero). So far this is where most academics agree on how to measure semi-variance.
Variants on the Semi-Variance Theme
However differences exist on how to weight the other three clover leaves. It is well-known that for measuring covariance each leaf is weighted equally, with a weight of 1. When it comes to quantifying semi-covariance, methods and opinions differ. Some favor a (0, 0.5, 0.5, 1) weighting scheme where the order is weights for quadrants 1, 2, 3, and 4 respectively. [As a decoder ring Q1 = grey leaf, Q2 = green leaf, Q3 = red leaf, Q4 = yellow leaf].
Personally, I favor weights (0, 3, 2, -1) for the asset versus portfolio semi-covariance calculation. For asset vs asset semi-covariance matrices, I favor a (0, 1, 2, 1) weighting. Notice that in both cases my weighting scheme results in an average weight per quadrant of 1.0, just like for regular covariance calculations.
Financial Industry Moving toward Semi-Variance (Gradually)
Semi-variance more closely resembles how ordinary investors view risk. Moreover it also mirrors a concept economists call “utility.” In general, losing $10,000 is more painful than gaining $10,000 is pleasurable. Additionally, losing $10,000 is more likely to adversely affect a person’s lifestyle than gaining $10,000 is to help improve it. This is the concept of utility in a nutshell: losses and gains have an asymmetrical impact on investors. Losses have a bigger impact than gains of the same size.
Semi-variance optimization software is generally much more expensive than variance-based (MVO mean-variance optimization) software. This creates an environment where larger investment companies are better equipped to afford and use semi-variance optimization for their investment portfolios. This too is gradually changing as more competition enters the semi-variance optimization space. My guestimate is that currently about 20% of professionally-managed U.S. portfolios (as measured by total assets under management, AUM) are using some form of semi-variance in their risk management process. I predict that that percentage will exceed 50% by 2018.
In this post I explain how less is more when it comes to using “big data.” The best data is concise, meaningful, and actionable. It is both an art and a science to turn large, complex data sets into meaningful, useful information. Just like the later paintings of Monet capture the impression of beauty more effectively than a mere photograph, “small data” can help make sense of “big data.”
There is beauty in simplicity, but capturing simplicity is not simple. A young child’s drawings are simple too, but they very unlikely to capture light and mood like Monet did.
Worry not. There will be finance and math, but I will save the math for last, in an attempt to retain the interest of non “mathy” readers.
The point of discussing impressionist painting is show that reduction — taking things away — can be a powerful tool. In fact, filtering out “noise” is both useful and difficult. A great artist can filter out the noise without losing the fidelity of the signal. In this case, the “signal” is emotion and color and light as as perceived by a master painter’s mind.
Applying Impressionism to Finance
Massive amounts of data are available to the financial professional. Two questions I have been asking at Sigma1 since the beginning are 1) How to use “Big Compute” to crunch that data into better portfolios? 2) How to represent that data to humans — both investment pros and lay folk whose money is being invested? After considerable thought, brainstorming, listening, and learning, I think we are beginning to construct a preliminary picture of how to do that — literally.
While not a beautiful as a Monet painting, the picture above is worth a thousand words (and likely many thousands of dollars over time) to me. The assets above constitute all of the current non-CASH building blocks of my personal retirement portfolio. While simple, the above image took considerable software development effort and literally millions of computations to generate [millions is very do-able with computers].
This simple-looking image conveys complex information in an easy-to-understand form. The four colors — red, green, blue, and purple — convey four asset types: fixed income, US stocks, international stocks, and convertible securities. The angle between any two asset lines conveys the relative correlation between the pair. In portfolio construction larger angles are better. Finally the length of the line represents the “effectiveness” with which each asset represents its “angular position” within the portfolio (in addition to other information).
With Powerful Data, First Comes Humility, Next Comes Insight
I have applied the same visualizations to other portfolios, and I see that, according to my software, many of the assets in professionally-managed portfolios exhibit superior “robustness” to my own. As someone who prides myself in having a kick-ass portfolio, this information is humbling, and took some time to absorb from an ego standpoint. But, having gotten over it, I now see potential.
I have seen portfolios that have a significantly wider angle than my current portfolio. What does this mean to me? It means I will begin looking for assets to augment my personal portfolio. Before I do that let me share some other insights. The plot combines covariance matrix data for the 16 assets in the portfolio, as well as semi-variance data for each asset. Without getting to “mathy” yet, the data visualization software reduces 136 pieces of data down to 32 (excluding color). The covariance matrix and semi-variance calculation itself are also a reducers in that they combines 5 years monthly total-return data — 976 data points down to 120 unique covariance numbers and 16 semi-deviation numbers. Taking 976 down to 32 results in a compression ratio of 30.5:1.
Finally, as it currently stands, the visualization software and resulting plot say nothing about expected return. The plot focuses solely on risk mitigation at the moment. Naturally, I intend to change that.
Time for the Math and Finance — Consider Yourself Warned
I mentioned a 30.5:2 (71:2) compression ratio. Just as music and other data, other information, including financial information can be compressed. However, only so much compression can be achieved in lossless manner. In audio compression researchers have learned which portions of music and other audio can be “lost” without the listener telling the difference. There is a field of psychoacoustics around doing just that — modeling what the human ear (and brain) can hear, and what gets “masked” by various physiological factors.
Even more important that preserving fidelity is extracting meaning. One way of achieving that is by removing “noise.” The visualization software performs significant computation to maintain as much angular fidelity as possible. As it optimizes angles, it keeps track of total error vis-a-vis the covariance matrix. It also keeps track of individual assets error (the reciprocal of fitness — fit versus lack of fit).
The real alchemy comes from the line-length computation. It combines semi-variance data with various fitness factors to determine each asset line length.
Just like Mercator projections for maps incur unavoidable error when converting from a 3-D globe to a 2-D map, the portfolio asset visualizations introduce error as well. If one thinks of just the correlation matrix and semi-variance data, each asset has a dimensionality of 8.5 (in the case of 16 assets). Reducing from 8.5-D to 2-D is a complex process, and there are an infinite number of ways to perform such an operation! The art and [data] science is to enhance the “signal” while stripping away the “noise.”
The ultimate goals of portfolio data visualization technology are:
1) Transform raw data into actionable insight
2) Preserve sufficient fidelity of relevant data such that the “map” can be used to reliably get to the desired “destination”
I believe that the first goal has been achieved. I know what actions to take… trying various other securities to find those that can build a “higher-angle”, and arguably more robust, more resilient investment portfolio.
However, the jury is still out on the degree [no pun intended] to which goal #2 has or has not been achieved. Does this simple 2-D map help portfolio builders reliably and consistently navigate the 8+ dimensional portfolio space?
What about 3-D Modelling and Visualization?
I started working with 2-D for one key reason — I can easily share 2-D images with readers and clients alike. I want feedback on what people like and dislike about the visuals. What is easy to understand, what is not? What is useful to them, and what isn’t? Ironing out those details in 2-D is step 1.
Of course I am excited by 3-D. Most of the building blocks are in my head, and I can heavily leverage the 2-D algorithms. I am, however, holding off for now. I am waiting for feedback from readers and clients alike. I spend a lot of time immersed in the language of math, statistics, and finance. This can create a communication gap that is best mitigated through discussion with other people with other perspectives. I wish to focus on 2-D for a while to learn more about market needs.
That being said, it is hard to resist creating a 3-D portfolio asset visualizer. The geek in me is extremely curious about how much the error terms will reduce when given a third degree of freedom to work with.
The bottom line is: Please give me any feedback: positive, negative, technical, aesthetic, etc. This is just the start. I am extremely enthusiastic about where this journey will take me and my company.
Disclosure and Disclaimer
Securities mentioned in this post are holdings in my personal retirement accounts (e.g. 401K, IRA, Roth IRA) as of the day of initial publication of this post. The purpose of this post is to illustrate features of Sigma1 Financial software. This is NOT investment advice, and NOT a recommendation to buy, sell, or hold any securities. Please refer to the “Disclaimer” Tab of the main page of this site for further information.
I start with a hypothetical. You are considering between three portfolios A, B, and C. If you could know with certainty one of the following annual risk measures, which would you choose:
For me the choice is obvious: max drawdown. Variance and semi-variance are deliberately decoupled from return. In fact, we often say variance as short-hand for mean-return variance. Similarly, semi-variance is short-hand for mean-return semi-variance. For each variance flavor, mean-returns — average returns — are subtracted from the risk formula. The mathematical bifurcation of risk and return is deliberate.
Max drawdown blends return and risk. This is mathematically untidy — max drawdown and return are non-orthogonal. However, the crystal ball of max drawdown allows choosing the “best” portfolio because it puts a floor on loss. Tautologically the annual loss cannot exceed the annual max drawdown.
My revised answer stretches the rules. If all three portfolios have future max drawdowns of less than 5 percent, then I’d like to know the semi-variances.
Of course there are no infallible crystal balls. Such choices are only hypothetical.
Past variance tends to be reasonably predictive of future variance; past semi-variance tends to predict future semi-variance to a similar degree. However, I have not seen data about the relationship between past and future drawdowns.
Research Opportunities Regarding Max Drawdown
It turns out that there are complications unique to max drawdown minimization that are not present with MVO or semi-variance optimization. However, at Sigma1, we have found some intriguing ways around those early obstacles.
That said, there are other interesting observations about max drawdown optimization:
1) Max drawdown only considers the worst drawdown period; all other risk data is ignored.
2) Unlike V or SV optimization, longer historical periods increase the max drawdown percentage.
3) There is a scarcity of evidence of the degree (or lack) of relationship between past max drawdowns and future.
(#1) can possibly be addressed by using hybrid risk measures such as combined semi-variance and max drawdown measures. (#2) can be addressed by standardizing max drawdowns… a simple standardization would be DDnorm = DD/num_years. Another possibility is DDnorm = DD/sqrt(num_years). (#3) Requires research. Research across different time periods, different countries, different market caps, etc.
Also note that drawdown has many alternative flavors — cumulative drawdown, weighted cumulative drawdown (WCDD), weighted cumulative drawdown over threshold — just to name three.
The bottom line is that early adopters have embraced semi-variance based optimization and the trend appears to be snowballing. For instance, Morningstar now calculates risk “with an emphasis on downward variation.” I believe that drawdown measures, either stand-alone or hybridized with semi-variance, are the future of post post modern portfolio theory.
Bye PMPT. Time for a Better Name! Contemporary Portfolio Theory?
I recommend starting with the the acronym first. I propose CPT or CAPT. Either could be pronounced as “Capped”. However, CAPT could also be pronounced “Cap T” as distinct from CAPM (“Cap M”). “C” could stand for either Contemporary or Current. And the “A” — Advanced, Alternative — with the first being a bit pretentious, and the latter being more diplomatic. I put my two cents behind CAPT, pronounced “Cap T”; You can figure out what you want the letters to represent. What is your 2 cents? Please leave a comment!
Back to (Contemporary) Risk Measures
I see semi-variance beginning to transition from the early-adopter phase to the early-majority phase. However, my observations may be skewed by the types of interactions Sigma1 Financial invites. I believe that semi-variance optimization will be mainstream in 5 years or less. That is plenty of time for semi-variance optimization companies to flourish. However, we’re also looking for the nextnext big thing in finance.
Suppose you have the tools to compute the mean-return efficient frontier to arbitrary (and sufficient) precision — given a set of total-return time-series data of asset/securities. What would you do with such potential?
I propose that the optimal solution is to “breach the frontier.” Current portfolios provide a historic reference. Provided reference/starting point portfolios have all (so far) provided sufficient room for meaningful and sufficient further optimization, as gauged by, say, improved Sortino ratios.
Often, when the client proposes portfolio additions, some of these additions allow the optimizer to push beyond the original efficient frontier (EF), and provide improved Sortino ratios. Successful companies contact ∑1 in order to see how each of their portfolios:
1) Land on a risk-versus-reward (expected-return) plot
2) Compare to one or more benchmarks, e.g. the S&P500 over the same time period
3) Compare to an EF comprised of assets in the baseline portfolio
Our company is not satisfied to provide marginal or incremental improvement. Our current goal is provide our client with more resilient portfolio solutions. Clients provide the raw materials: a list of vetted assets and expected returns. ∑1 software then provides near-optimal mix of asset allocations that serve a variety of goals:
1) Improved projectedrisk-adjusted returns (based on semi-variance optimization)
2) Identification of under-performing assets (in the context of the “optimal” portfolio)
3) Identification of potential portfolio-enhancing assets and their asset weightings
We are obsessed with meaningful optimization. We wish to find the semi-variance (semi-deviation) efficient frontier and then breach it by including client-selected auxiliary assets. Our “mission” is as simple as that — Better, more resilient portfolios
Disclosure: The purpose of this post is to show how I, personally, use the HALO Portfolio Optimizer software to manage my personal portfolio. It is not investment advice! I use my personal opinions about which assets to select and expected one-year returns into the optimizer configuration. The optimizer then provides an efficient frontier (EF) based on historic total-return data and my personal expected-return estimates.
I use other software (User Tuner) to approach the EF, while limiting the number and size of trades (minimizing churn and trading costs). Getting exactly to the EF would require trading (buying or selling) every asset in my portfolio — which would cost approximately $159 in trading costs for 18 trades. Factoring in bid/ask spreads the cost would be even higher. However, by being frugal about trades, I was able to limit the number of trades to 6 while getting much closer to the EF.
Past performance is no guarantee of future performance, nor is past volatility necessarily indicative of future volatility. Nonetheless, I am making the personal decision to use past volatility information to possibly increase the empirical diversification of my retirement portfolio with the goal of increasing risk-adjusted return. Time will tell whether this approach was successful or not.
In my last post I blogged about reallocating my entire retirement portfolio closer to the MVO efficient frontier computed by the HALO Portfolio Optimizer. The zoomed in plot tells the story to date:
The “objective space” plot is zoomed in and only shows a small portion of the efficient frontier. As you can see the black X is closer to the efficient frontier than the blue diamond, but naturally the dimensions are not the same. Using a risk-free rate of 0.5% the predicted Sharpe ratio has improved from 0.68 to 0.75 – a marked increase of about 10.3%. [If you crunch the numbers yourself, don’t forget to annualize σ.]
While a 10.3% Sharpe ratio expected improvement is very significant, there is obviously room for compelling additional improvement. An expected Sharpe ratio of just north of 0.8 is attainable.
The primary reason the portfolio has not yet moved even closer to the efficient frontier is due to 18.6% of the retirement portfolio being tied up in red tape as a result of my recent voluntary severance or “buy-out” from Intel Corporation. [ Kudos to Intel for offering voluntary severance to all of my local coworkers and me. It is a much more compassionate method of workforce reduction than layoffs! I consider the package offered to me reasonably generous, and I gladly took the opportunity to depart and begin working full time building my start up.]
Time to Get Technical
I won’t finish without mentioning a few important technical details. The points in the objective space (of monthlyσ on the horizontal and expected annual return on the vertical) can be viewed as dependent variables of the (largely) independent variables of asset weights. Such points include the blue diamond, the black X, and all the red triangles on the efficient frontier. I often call the (largely) independent domain of asset allocation weights the “search space”, and the weightings in the search space that result in points on the efficient frontier the “solution space.”
One way to measure the progress from the blue diamond to the X is via improvement in the Sharpe ratio, which implicitly factors in the CAL, or the CML for the tangent CAL. As “X” approaches the red line visually it also approaches the efficient frontier quantitatively and empirically. However, X canmake significant progress towards the efficient frontier, say point EF#9 specifically, with little or no “progress” in the portfolio weights from the blue diamond to the black X.
“Progress” in the objective space is reasonably straight forward — just use Sharpe ratios, for instance. However measuring “progress” in the asset allocation (weight) space is perhaps less clear. Generally, I prefer the use of the L1-norms of differences of the asset-weight vectors Wo (corresponding to original portfolio weight; e.i. the blue diamond), Wx, and Wef_n. The distance of from the blue diamond in search space to the red triangle #9 is denoted as |Wef_9 – Wo|1 while the distance from X in the search space is |Wef_9 – Wx|1. Interestingly, the respective values are 0.572 and 0.664. Wx is, by this measure, actually further from Wef_9 in search space, but closer in objective space!
I sometimes refer to these as the “Hamming distances” (even though “Hamming distance” is typically applied to differences in binary codes or character inequality counts of two strings of characters.) It is simply easier to say the “Hamming distance from Wx to Wef_9” than the “ell-one norm of the difference of Wx and Wef_9.”
I have been working on an utility temporarily called “user tuner” that makes navigating in both the search space and the objective space quicker, easier and more productive. More details to follow in a future post.
Why Not Semi-Variance Optimization?
Frequent readers will know that I believe that mean semi-variance optimization (MSVO or SVO) is superior to vanilla MVO. So why am I starting with MVO? Three reasons:
To many, MVO is less scary because it is somewhat familiar. So I’m starting with the familiar “basics.”
I wanted to talk about Sharpe ratios first, because again they are more familiar than, say, Sortino ratios.
I wanted to use “User Tuner”, and I originally coded it for MVO (though that is easily remedied).
However, asymptotically refining allocation of my entire portfolio to get extremely close to the MVO efficient frontier is only phase 1. It is highly likely I will compute the SVO efficient frontier next and use a slightly modified “User Tuner” to approach the mean semi-variance efficient frontier… Likely in the next month or two, once my 18.6% of assets are freed up.
I am happy to announce that the latest version of the HALO Portfolio-Optimization Suite is now available. Key features include:
Nativeasset constraint support
Nativeasset-category constraint support
Dramatic run-time improvements of 2X to over 100X
Still supported are user-specified risk models, including semi-variance and max-drawdown. What has been temporarilyremoved (based on minimal client interest) is 3-D 2-risk modelling and optimization. This capability may be re-introduced as a premium feature, pending client demand.
Here is a quick screenshot of a 20-asset, fixed-income portfolio optimization. The “risk-free” rate used for the tangent capital allocation line (CAL) is 1.2% (y-intercept not shown), reflecting a mix of T-Bills and stable value funds. Previously this optimization took 18 minutes on an $800 laptop computer. Now, with the new HALO software release, it runs in only 11 seconds on the same laptop.
To date I’ve invested approximately 800 hours developing and testing the heuristics and algorithms behind HALO. Finding exact solutions (with respect to expected-return assumptions) to certain real-world portfolio-optimization problems can be solved. Finding approximate solutions to other real-world portfolio-optimization problems is relatively easy, but finding provably optimal solutions is currently “impossible”. The current advanced science and art of portfolio optimization involves developing methods to efficiently find nearly optimal solutions.
I believe that HALO represents a significant step forward in finding nearly-optimal solutions to generalized risk models for investment portfolios. The primary strengths of HALO are in flexibility and dimensionality of financial risk modeling. While HALO currently finds solutions that are almost identical to exact solutions for convex optimization problems; the true advantage of HALO is in the quality of solutions for non-convex portfolio-optimization problems
Do you know if your particular optimization metric can be articulated in canonical convex notation? I argue that HALO does not care. If it can be, HALO will find a near-optimal solution virtually identical to the ideal convex optimization solution. If it cannot be, and is indeed non-convex, HALO will find solutions competitive with other non-convex optimization methods.
It could be argued that “over-fitting” is a potential danger of optimal and near-optimal solutions. However, I argue that given a sufficiently diverse and under-constrained optimization task, over-fitting is less worrisome. In other words, the quality of the inputs greatly influences the quality of the outputs. One secret is to supply high-quality (e.g. asset expected return) estimates to the optimization problem.
Unlike variance, there a several different formulas for semivariance (SV). If you are a college student looking to get the “right” answer on test or quiz, the formula you are looking for is most likely:
The question-mark-colon syntax simply means if the expression before the “?” is true then the term before the “:” is used, otherwise the term after the “:” is used. So a?b:c simply means chose b if a is true, else chose c. This syntax is widely used in computer science, but less often in the math department. However, I find it more concise than other formulations.
Another common semivariance formula involves comparing returns to a required minimum threshold rt. This is simply:
Classic mean-return semivariance should not be directly compared to mean-return variance. However a slight modification makes direct comparison more meaningful. In general approximately half of mean-adjusted returns are positive and half are negative (exactly zero is a relatively rare event and has no impact to either formula). While mean-variance always has n terms, semi-variance only uses a subset which is typically of size n/2. Thus including a factor of 2 in the formula makes intuitive sense:
Finally, another useful formulation is one I call “Modified Drawdown Only” (MDO) semivariance. The name is self-explanatory… only drawdown events are counted. SVmdo does not require ravg (r bar) nor rt. It produces nearly identical values to SVmod for rapid sampling (say for anything more frequent than daily data). For high-speed trading it also has the advantage of not requiring all of the return data a priori, meaning it can be computed as each return data point becomes available, rather than retrospectively.
Why might SVmdo be useful in high-speed trading? One use may be in put/call option pricing arbitrage strategies. Black–Scholes, to my knowledge, makes no distinction between “up-side” and “down-side” variance, and simply uses plain variance. [Please shout a comment at me if I am mistaken!] However if put and call options are “correctly” priced according to Black–Scholes, but the data shows a pattern of, say, greater downside variance than normal variance on the underlying security, put options may be undervalued. This is just an off-the-cuff example, but it illustrates a potential situation for which SVmdo is best suited.
Pick Your Favorite Risk Measure
Personally, I slightly favor SVmdo over SVmod for computational reasons. They are often quite similar in practice, especially when used to rank risk profiles of a set of candidate portfolios. (The fact that both are anagrams of each other is deliberate.)
I realize that the inclusion of the factor 2 is really just a semantic choice. Since V and (classic) SV, amortized over many data sets, are expected to differ by a factor of 2, standard deviation, σ, and semideviation, σd, can be expected to differ by the square root of 2. I consider this mathematically untidy. Conversely, I consider SVmod to be the most elegant formulation.