### Building a Better Robo Advisor

The more we learned about the current crop of robo advisory firms, the more we realized we could do better. This brief blog post hits the high points of that thinking.

### Not Just the Same Robo Advisory Technology

It appears that all major robo advisory companies use 50+ year-old MPT (modern portfolio theory). At Sigma1 we use so-called post-modern portfolio theory (PMPT) that is much more current. At the heart of PMPT is optimizing return versus semivariance. The details are not important to most people, but the takeaway is the PMPT, in theory, allows greater downside risk mitigation and does not penalize portfolios that have sharp upward jumps.

Robo advisors, we infer, must use some sort of Monte Carlo analysis to estimate “poor market condition” returns. We believe we have superior technology in this area too.

Finally, while most robo advisory firms offer tax loss harvesting, we believe we can 1) set up portfolios that do it better, 2) go beyond just tax loss harvesting to achieve greater portfolio tax efficiency.

# Semivariance Excel Example

The most in-demand topic on this blog is for an Excel semivariance example. I have posted mathematical semivariance formulas before, but now I am providing a description of exactly how to compute semivariance in “vanilla” Excel… no VBA required.

The starting point is row D. Cell D\$2 contains average returns of over the past 36 months. The range D31:D66 contains those returns.  Thus the contents of D\$2 are simply:

`=AVERAGE(D31:D66)`

This leads us to the semivariance formula:

`{=SQRT(12)*(SQRT(SUM(IF((D31:D66-D\$2)<0,(D31:D66-D\$2)^2,0))/(COUNT(D31:D66-1))))}`

We will now examine each building block of this formula starting with

`IF((D31:D66-D\$2)<0,(D31:D66-D\$2)^2,0)`

We only want to measure “dips” below the mean return. For all the observations that “dip” below the mean we take the square of the dip, otherwise we return zero. Obviously this is a vector operation, the IF function returns a vector of values.

Next we divide the resulting vector by the number of observations (months) minus 1. We can simply COUNT the number of observations with `COUNT(D31:D66-1)`.  [NOTE 1: The minus 1 means we are taking the semivariance of a sample, not a population. NOTE 2: We could just as easily taken the division “outside” the SUM — the result is the same either way.]

Next is the SUM. The following formula is the monthly semivariance of our returns in row D:

`{=SUM(IF((D31:D66-D\$2)<0,(D31:D66-D\$2)^2,0))/(COUNT(D31:D66-1))}`

You’ll notice the added curly braces around this formula. This specifies that this formula should be treated as a vector (matrix) operation.  The curly braces allow this formula to stand alone.  The way the curly braces are applied to a vector (or matrix) formula is to hit <CTRL><SHIFT><ENTER> rather than just <ENTER>. Hitting <CTRL><SHIFT><ENTER> is required after every edit.

We now have monthly semivariance. If we wanted annual semivariance we could simply multiply by 12.

Often, however, we ultimately want annual semi-deviation (also called semi-standard deviation) for computing things like Sortino ratios, etc. Going up one more layer in the call stack brings us to the SQRT operation, specifically:

`{=SQRT(SUM(IF((D31:D66-D\$2)<0,(D31:D66-D\$2)^2,0))/(COUNT(D31:D66-1)))}`

This is monthly (downside) semi-deviation. We are just one step away from computing annual semi-deviation. That step is multiplying by SQRT(12), which brings us back to the big full formula.

There it is in a nutshell. You now have the formulas to compute semivariance and semi-deviation in Excel.

# Quant Cross-Training

A very astute professor of finance told our graduate finance class that the best way to become a bona fide quant is NOT to get a Ph.D. in Finance!  It is better, he said, to get a Ph.D. in statistics, applied mathematics, or even physics. Why? Because a Ph.D in Finance is generally not sufficiently quantitative. A quant needs a strong background in Stochastic Calculus.

### “Quants for Hire?”

Our company has been described as a “quants for hire” firm. That is flattering. While we currently have 4 folks with master’s of science degrees (and one close to finishing a master’s) what we do is probably more accurately described as “quant-like” or “quant-lite” software and services. However “Quants for Hire” definitely has a nice succinct ring to it.

### Quant-like Tangents to Financial Learning

Most of our quant-like work has been fairly vanilla — back testing trading strategies in Excel, Monte Carlo simulations (also in Excel), factor analysis, options strategy analysis. So far our clients like Excel and are not very interested in R. The main application of R has been to double-check our Excel back tests!

We have attracted fairly sophisticated clients.  They seem reasonably comfortable about talking about viewing portfolios as unit vectors that can be linearly combined.  They tend to understand correlation matrices, Sortino ratios, and in some cases even relate to partial derivatives and gradients. But they tend to push back on explanations involving geometric Brownian motion, Ito’s lemma, and the finer points of  Black-Scholes-Merton. They do, however, appear to appreciate that we “know our stuff.”

I’ve got a decent set of R skills, but I’m looking to take them to the next level. I’m taking a page from my professor in tackling non-financial quantitative problems. My current problem du jour image compression. I came up with an R script that achieves very high compression levels for lossy compression.  It is shorter than 200 lines commented and shorter than 100 lines when stripped of comments and blank (formatting) lines.

It can easily achieve 20X or greater compression, albeit with a loss in quality. In my initial tests my R algorithm (IC_DXB1.1) was somewhat comparable to JPEG (GIMP 2.8) at 20X compression, though I the JPEG clearly looks better in general. I also found an elegant R compressor that is extremely compact R code… the kernel is about 5 lines! Let’s call this SVD (singular value decomposition) for reference. So here’s the bake off results (all ~20X compressed to ~1.5KB):

JPEG:                                                             IC_DXB1.1:

What’s interesting to me is that each algorithm uses radically different approaches. JPEG uses DCT (discrete cosine transform) plus a frequency “mask” or filter that reduces more and more high-frequency components to achieve compression. My ic_dxb1.1 algorithm uses a variant of B-splines. The SVD approach uses singular value decomposition from linear algebra.

Obviously tens of thousands of hours have been invested in JPEG encoding. And, unfortunately, 99%+ of JPEG images are not as compact as they could be due to a series of patent disputes around arithmetic coding. Even thought the patents have all (to the best of my knowledge) expired, there is simply too much inertia behind the alternative Huffman coding at the present. It is worth noting that my analysis of all 3 algorithms is based on Huffman coding for consistency.  All three approaches could ultimately use either Huffman or arithmetic coding.

### So this Image Stuff Relates to Finance How?

Another of my professors explained that, fundamentally, finance is about information. One set of financial interview questions start with the premise that you have immediate (light-speed, real-time) access to all public information. Generally how would you make use of this information to make money trading? Alternatively you are to assume (correctly) that information costs money… how would your prioritize your firm’s information access?  How important is frequency and latency?

Having boat loads of real-time data and knowing what to do with it are two different things. I use R to back test strategies, because it easy to write readable R code with a low bug rate. If I had to implement those strategies in a high-frequency trading environment, I would not use R, I would likely use C or C++. R is fast compared to Excel (maybe 5X faster), but is slow compared to good C/C++ implementations (often 100X slower).

My thinking is that while knowledge is important, so is creativity. By dabbling in areas outside of my “realm of expertise”, I improve my knowledge while simultaneously exercising my creativity.

Both signal processing and quant finance can reasonably be viewed as signal processing problems. Signal processing and information theory are closely related. So I would argue that developing skills in one area is cross-training skills in the other… and with greater opportunity for developing creativity. Finance is inextricably linked to information.

### The Future of Finance Requires Disruptive (Software) Technology

Robo advising will have at least three important disruptive impacts:

1. Accelerating downward pressure on advisory fees
2. Taking of market share and AUM
3. Increasing market demand for investment tax management services such as tax-loss harvesting

Are you ready for the rise of the bots? We at Sigma1 are, and we are looking forward to it. That is because we believe we have the software and skills to make robo advisors work better. And we are not resting on our laurels — we are focusing our professional development on software, computer science, advanced mathematics, information theory, and the like.

# The Equation Everyone in Finance Should Know (MV Optimization: How To, Part 2)

As the previous post shows, it all starts with…

$inline&space;dpi{300}&space;large&space;sigma_{p}^{2}=&space;mathbf{w}^topmathbf{V}mathbf{w}$

In order get close to bare-metal access to your compute hardware, use C.  In order to utilize powerful, tested, convex optimization methods use CVXGEN.  You can start with this CVXGEN code, but you’ll have to retool it…

• Discard the (m,m) matrix for an (n,n) matrix. I prefer to still call it V, but Sigma is fine too.  Just note that there is a major difference between Sigma (the covariance-variance matrix) and sigma (individual asset-return variances matrix; the diagonal of Sigma).
• Go meta for the efficient frontier (EF).  We’re going to iteratively generate/call CVXGEN with multiple scripts. The differences will be w.r.t the E(Rp).
• Computing Max: E(Rp)  is easy, given α.  [I’d strongly recommend renaming this to something like expect_ret comprised of (r1, r2, … rn). Alpha has too much overloaded meaning in finance].
• [Rmax] The first computation is simple.  Maximize E(Rp) s.t constraints.  This is trivial and can be done w/o CVXGEN.
• [Rmin] The first CVXGEN call is the simplest.  Minimize σp2 s.t. constraints, but ignoring E(Rp)
• Using Rmin and Rmax, iteratively call CVXGEN q times (i=1 to q) using the additional constraint s.t. Rp_i= Rmin + (i/(q+1)*(Rmax-Rmin). This will produce q+2 portfolios on the EF [including Rmin and Rmax].  [Think of each step (1/(q+1))*(Rmax-Rmin) as a quantization of intermediate returns.]
• Present, as you see fit, the following data…
• (w0, w1, …wq+1)
• [ E(Rp_0), …E(Rp_(q+1)) ]
• [ σ(Rp_0), …σ(Rp_(q+1)) ]

My point is that —  in two short blog posts — I’ve hopefully shown how easily-accessible advanced MVO portfolio optimization has become.  In essence, you can do it for “free”… and stop paying for simple MVO optimization… so long as you “roll your own” in house.

I do this for the following reasons:

• To spread MVO to the “masses”
• To highlight that if “anyone with a master’s in finance and computer can do MVO for free” to consider their quantitative portfolio-optimization differentiation (AKA portfolio risk management differentiation), if any
• To emphasize that this and the previous blog will not greatly help with semi-variance portfolio optimization

I ask you to consider that you, as one of the few that read this blog, have a potential advantage.  You know who to contact for advanced, relatively-inexpensive SVO software. Will you use that advantage?

# How to Write a Mean-Variance Optimizer: Part 1

### The Equation Everyone in Finance Show Know, but Many Probably Don’t!

… 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:
$inline&space;dpi{300}&space;large&space;sigma_{p}^{2}=&space;mathbf{w}^topmathbf{V}mathbf{w}$
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:

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 efficiently deal with these constraints?
• Is short-selling allowed?  What if it is?
• OK… this one’s a bit tricky:  How can convex optimization methods be applied?

If you can answer these questions, a Part 2 really isn’t necessary is it?

# Clover Patterns Show How Portfolios Manage Risk

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.]

### Plain English

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.

### Covariance Details

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.

# Data Science: Shrinking Big Data into Meaningful Data

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.

# Choosing your Crystal Ball for Risk

### Choose Your “Perfect” Risk Model

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:

1. Variance
2. Semi-variance
3. Max Drawdown

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.

### Cheating Risk

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.

### Semi-Variance Risk Measure Reaching Critical Mass?

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 riskwith 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 next next big thing in finance.

# Surpassing the Frontier?

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 projected risk-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

# Pursuing Alpha with Antivariance

A simple and marginally-effective strategy to reduce portfolio variance is by constructing an asset correlation matrix, selecting assets with low (preferably negative) correlations, and building a portfolio of low-correlation assets.  This basic strategy involves creating a set of assets whose cross-correlations (covariances) are minimized.

One reason this basic  strategy is only somewhat effective is that a correlation matrix (or covariance matrix) only provides a partial picture of the chosen investment landscape.  Some fundamental limitations include non-normal distributions, skewness, and kurtosis to name a few.  To most readers these are fancy words with varying degrees of meaning.

Personally, I often find the mathematics of the work I do seductive like a Siren’s song.  I endeavor to strike a balance between exploring tangential mathematical constructs, and keeping most of my math applied. One mental antidote to the Siren’s song of pure mathematics is to think more conceptually than mathematically by asking questions like:

What are the goals of portfolio optimization?  What elements of the investing landscape allow these goals to be achieved?

I then attempt to answer these questions with explanations that a person with a college degree but without a mathematically background beyond algebra could understand.  This approach lets me define the concept first, and develop the math later.  In essence I can temporarily free my mind of the slow, system 2 thinking generally required for math.

Recently, I came up with the concept of antivariance.  I’m sure others have had similar ideas and a cursory web search reveals that as profession poker player’s nickname.  I will layout my concept of antivariance as it relates to porfolio theory in particular and the broader concept in general.

By convention, one of the key objective of modern portfolio theory is the reduction of portfolio return variance.  The mathematical concept is the idea that by combining assets with correlations of less than 1.0, the return variance is less than the weighted sums of each asset’s individual variance.

Antivariance assumes that there are underlying patterns explain why two or more assets should be somewhat less correlated (independent), but at times negatively correlated.  Consider the affects of major hurricanes like Andrew or Katrina.  Their effects were negative for insurance companies with large exposures, but were arguably positive for companies that manufactured and supplied building materials used in the subsequent rebuilds.  I mention Andrew because there was much more and more rapid rebuilding following Andrew than Katrina.  The disparate groups of stocks of (regional) insurance versus construction companies can be considered to exhibit paired antivariance to devastating weather events.

Nicholas Nassim Taleb coined the the term antifragile, because terms such as robust simply don’t convey the exact mental connections.  I am beginning to use the term antivariance because it conveys concepts not well captured by terms like “negatively correlated”, “less correlated”, “semi-independent”, etc.   In many respects antifragile systems should exhibit antivariance characteristics, and vice versa.

The concept of antivariance can be extended to related concepts such as anticovariance and anticorrelation.