George Rebane
Here’s a little something from my financial engineering toy box that one or two RR readers may find interesting. The discussion is a tiny bit technical, but the pictures should be accessible to all. Most investors know that a stock’s beta (Greek β) is a number that indicates how well the security tracks its market, say, the SP500. And rho (Greek ρ) measures the correlation between, say, the price series of two stock’s – the way they both zig and zag together, or how much one zigs while the other zags. I’ll try to keep this short.
From august sources like Modern Portfolio Theory (MPT) by nobelists Markowitz and Sharpe, we learn that if a stock’s β equals one, then it more or less goes up and down in similar percent changes with the market. If β > 1, then the stock’s swings have a greater percentage than the market’s; and if β < 1, then the stock’s swings behave in a more reserved manner than do the market’s. Riskier stocks have the bigger swings, and more conservative stocks have the smaller swings. The referenced formula (here) for beta reflects this with its inclusion of the standard deviation of the price swings.
As stated above, ρ measures in how tight of a formation two stocks will fly. The range of ρ is between +1 and -1, with the positive values indicating tighter formations, and ρ = 1 denoting a perfectly tight formation matching each other’s zigs and zags. A small or zero ρ means that the stocks pretty much act independently of each other. In designing your investment portfolio from a selected shortlist of stocks, MPT teaches that you should try to avoid stocks that pair with high ρ values. Doing probabilistic math recommends that you pick stocks that pair with the lowest ρ values possible to give you a low volatility (i.e. low risk) portfolio.
Now after you understand all that, we’re going to throw a monkey wrench into the whole thing, a little bit of information you will not hear from your well-paid wealth advisor who manages your portfolio. (Drum roll please.) Betas and rhos, calculated from past performance, are pretty much worthless measures of securities performance for portfolio design. Why? Well, that answer depends on what is known as the investor’s ‘investment horizon’. That’s the period into the future over which the investment plan is supposed to hold with numbers such as expected portfolio appreciation, its most likely upper and lower limits, and so forth.
But here’s the big fallacy in using the rhos and betas over any extended investment horizon – they don’t remain constant and vary all over the place. No matter what values you pick for them, they are guaranteed to change significantly over the next several months to years. And portfolio theories such as MPT all assume that beta and rho stay put with their assigned values to the end of your investment horizon. And here’s another little wiggle in using such critical parameters for your investment decisions – both rho and beta are almost always derived from past price histories over chosen time intervals, usually the last couple of quarters or year.
But there’s no guarantee the financial environment (markets, business prospects, monetary and fiscal policies) will remain unchanged over the next such periods. In fact, I can guarantee that they won’t. So, the calculated values of beta and rho are pretty much fiction (‘brown numbers’) in their use as constants for portfolio design. (An interesting anecdote is that it is almost impossible to find an investment professional who uses MPT to design their own portfolios. But they have no problem using it to fashion yours for a handsome fee. Why? Because it’s the best CYA for the investment advisor in the event you decide to sue because your portfolio tanked - ‘Hey, I used MPT with two Nobel prizes under its belt, doesn’t everyone?’
If you’re still with me, you get to look at some pretty pictures of how rho and beta misbehave. I downloaded about ten years of closing price data for the SP500, AAPL (Apple), EOS (high yield income ETF), and XLE (ETF of energy stocks). Then I wrote some code that allowed me to compute the rhos and betas from selectable past time windows, and also compare these over future time horizons. A sample of the outputs are presented for your edification and enjoyment.
The first plot below shows the correlation coefficient rho between AAPL and XLE over a trailing 125 trading-day window that is computed as a rolling value over a period of about ten years. It is clear that these two seemingly independent securities vary as much as possible with the passing of the years. Take any trailing window to compute a prediction of rho over a future investment horizon, and you will see a markedly different rho value that obtains over this future time interval. So, what pairwise rho values are you going to use to select your shortlist of stocks? (For those with investment advisors, show them this plot and give them a link to this post. At a minimum, their response should be entertaining.)
Finally, to drive home the point of this expose, we take a look at the predictability of rho over a short 200 trading day window (here from trading 600 to 800). The figure below shows in blue the behavior of rho, calculated from the last 125 trading days, in that interval. Similarly, in green we see rhoF, the actual future value of rho calculated over the next 125 trading days. The difference between the two is shown in red, and ideally we would like to see this line hover around zero, indicating that our calculated rho would serve well to predict its future value. However, the red trace shows that blatantly this is not the case, substantiating our thesis that using past time series to predict the correlation behavior of two stocks ranges from pretty much useless to possibly harmful to your portfolio’s performance.
It’s important to note here that such results can be seen for any two securities over literally any sliding window of trading days. This effect underlines the usual cautionary ‘past performance is not a guarantee of future results’. So how should we derive rho values for our shortlist of stocks, if not from past data? Well, unfortunately the answer is that such an estimate of rhoF should come from the investor’s own apprehension of economic prospects for the pair over his investment horizon. This process is of necessity highly personalized, and will involve considering many factors depending on the investor’s acumen. And what about beta in view of the above? Since the formula for beta involves rho as a multiplicative factor, then to the extent that rhoF is bogus, so is betaF. A more detailed discussion of developing rho and beta values is out of scope for what has been presented. (Technically, this involves different methodologies for calculating the covariance matrix for the shortlist of stocks as required by most portfolio design models.)
[Addendum] Asset (e.g. stocks) price performance prediction models and portfolio (composed of such assets) design models are distinct instruments yielding different information sets to the investor. The above discussion addressed two common asset performance parameters, calculated from past price histories and often used to predict future assets’ performance. I emphasize ‘asset’ here because rho and beta can be computed for any asset whose market price history is available – e.g. numismatic coins, real estate, commodity contracts, the oeuvre of a rock star, … .
Almost all portfolio design models permit the use of a variety of performance prediction models to obtain the necessary asset price predictions over the investment horizon. The minimal requirements from the prediction models are the expected prices and covariance of the assets in the portfolio designers shortlist of assets. For example, MPT the expected prices and the covariance matrix of the shortlist assets at the investment horizon. From this can be computed its required expected returns and normalized variances and covariances.
Like most design models, MPT then searches the asset-weights space to find the optimum weight or allocation vector for the shortlist assets. MPT actually finds an ‘infinite set’ of optimum portfolio weights which it characterizes in the form of a curve known as the efficient frontier drawn so that the x-axis represents the standard deviation of price variability (MPT’s risk/volatility parameter), and the y-axis represents the expected portfolio return at a given weight vector or allocation fractions. The efficient frontier is made of points that vary the weight vector to yield the minimum sigma (i.e. risk) at a given value of expected return. (Here is a good MPT reference with appropriate graphics.) MPT then defines the single best portfolio that balances return and risk at the point where the so-called ‘capital allocation line’, starting on the y-axis at the current ‘risk-free rate’ (e.g. US Treasuries), just grazes the efficient frontier.
One of the many problems with MPT is that, starting with the same shortlist of assets, the best allocation vector for both you and Jeff Bezos or Elon Musk or … is exactly the same. And everyone knows that is not true, since the tolerance for monetary risk varies widely, depending on your current net assets among other factors. A useful alternative for such portfolio design tools are those that take investors’ monetary utility into account. These then can provide individualized portfolio designs instead of one-size-fits-all portfolios that MPT computes.
At this point it is important to recall that any given portfolio design method will yield different results that depend not only on what asset prediction models are used, but also on the specific parameter values (representing the investor’s beliefs) used in any such model. Complex stuff with a lot of dependers, all of which emphasizes that there is no free lunch.
George, what's missing in truly modern portfolio management is a term accounting for the possibility of the government confiscating your returns, or your principal. Liberals want a higher tax rate (called confiscation), and Cherokee-of-the-month Ms Warren wants some of your capital itself. Sure, but they promise only if your wealth is above $1B. That's just in the U.S. And we ignore China and its ability to do whatever the CCP wants to do. China's stocks are held by quite a few Americans through mutual funds.
Few portfolio managers include an inflation risk due to the federal govt. printing money and adding to the country's debt. Neither the left nor the right have voted to contain spending ever.
I find it disturbing that the Federal Reserve accepts 2% inflation as a given. That means they intentionally reduce your perceived wealth by 50% over your working lifetime. Prof. Mark Perry has a graph showing the 'expertise' of the Fed since its inception in 1913. https://www.aei.org/carpe-diem/monday-afternoon-links-all-graphic-edition/
Chart of the Day V shows that for more than a century from 1800 to 1913 prices in the US were relatively stable based on Federal Reserve data. But since the creation of the Federal Reserve in 1913, its monetary policies increased the CPI by 27.5X, decreasing the value of the US dollar by more than 96% in the process. How is that helping?
Capitalism requires many small skills to deliver the goods or services. Currently, our education system has trouble providing the 3 R's. Who will provide the remaining letters in the capitalist's alphabet - like transportation; machining; ditch-digging; drafting? So far, it's been the skills of smaller entrepreneurs doing that on their own. But we saw what happens to small companies when a pandemic arrives.
Posted by: The Estonian Fox | 06 January 2022 at 04:29 AM
..as the world's worst stock investor, I've always been nervous about any portfolio theory. A lot of the market appears to be in lockstep due to govt-encouraged tax-deferred trading (continuous inflows into index funds via IRA/401(k)). A lot of it is managed by over-powerful firms that can drive money to themselves or receive bailouts (BlackRock and the like). A lot of it is magic (machine trading). Any particularly successful theory is smothered by other people piling on.
Of course, anyone who became a stock true believer in the mid 1980's looks like a genius now. Things stay the same until they don't. It's like Taleb's story of the coin-flipping expert, one of many, who ends up flipping heads many times in a row. The financial press came to his door looking for hints on coin flipping.
Rather than hedging stocks against each other or building some sort of lower risk world by mixing stock risk v. reward, I'd prefer to see asset management that used stocks as a monolith (overall market fund?) with other assets as equals in the mix. Real estate for example. It's probably old fashioned, but physical ownership seems more comfortable rather than numbers in an account.
Given the death of small business and small farming in this country, I admit that the casino is one of the few areas for sopping up capital with (hopefully) some return, but I can't say that I like it.
Posted by: scenes | 06 January 2022 at 06:56 AM
The thoughtful comments of Messrs Efox and scenes are much appreciated. While they may well understand the different functions of asset performance prediction and portfolio design models, I fear that the average will continue to confuse the two. This motivates me to write a short pithy addendum to the above piece - soon.
Posted by: George Rebane | 06 January 2022 at 09:33 AM
"This motivates me to write a short pithy addendum to the above piece - soon."
I was mostly making the point that I think that the financial markets, as a whole, can be thought of as a single point of risk. I rather hate seeing the entire country's retirement hopes pinned on that one thing (with the exception of gubmint employee pension plans backed by guns). In fact, typically only the US financial market. Obviously, anyone who had to get out of the way of the Red Army appreciates risk more than most.
Posted by: scenes | 06 January 2022 at 10:03 AM
“..as the world's worst stock investor, I've always been nervous about….”— Scenes.
I have much to say on the topic, but in a nutshell, my wealth accumulation days are over, albeit I am still creating wealth as a by-product. I am not a stock picker and when I was, it was like going to the casinos and coming back talking about winning 900 bucks… but failing to mention all the times I came back a big Loser, with the gas tank on fumes a more credit card debt.
I feel compelled to clear up any misconception that I am a tax cheat or gaming the system to say that the reason I am considered ‘po folk’ is because all my income is tax free, State and Federal. Basically all income for the near future…until 2030. Then I will raid the dividends on the stock side, then if need be, cash out the REITs, but do not welcome the idea of the capital gains tax due that will be one-two hundred thousand dollars. I just don’t want to cut that check to Uncle Sam and Gavin Newsom…so let it ride.
I am in the period of my life where building up the nest egg is less important than tax avoidance. So, with my little Social Security check and all income being tax free, on paper I qualified for that $600 check from the State. They must go by income tax filings cause it surprised the heck out of me…and I felt it immoral and still do. No, I refuse to accept food stamps or discounted PGE or any aid….at the present. Nor am I interested in growing the wacky tobaccy plant. I have three portfolios. Stocks, Minis, and Real Estate investment private equity trusts.
It’s the minis that have shocked the living daylightsout of me. It’s worthless advice now with the interest rates in the crapper…but timing is everything. The munis are so boring and unexciting, but they have turned out unintentionally as the bees’ knees for me. And the munis are the smallest of the three portfolios..by far.
My munis are ALL invested in school bonds. Yep, staggered school bonds and as they are reaching their maturity this decade. Still paying 5-6-7 percent, free of state and federal taxes. When Stockton went bankrupt, they never once defaulted on school bonds. I don’t care if it is liberal Marin County or ultra blue Contra Costa County or Berkeley School District….those places may have their financial ups and downs like all cities and towns, but never will miss an school bond payment. Sad to see one bond expiring in 2023. It has been paying 7.1 percent, lol. Another paying 6.9% matures in 2026. I will miss them.
It’s not how much you make, it’s how much is the net. Tax free is like a 20-30% increase in revenue. Each year more munis will reach maturity and I have no idea what I will do next. Ain’t buying any new ones, for sure. I
**When I said All my income is tax free, there are always exceptions. Figured I would get hit with a $3,500 tax when I ordered some rebalancing of the stock side last year. But, nooo. That Standard Deduction of 10k or whatever it is absorbed all that. I don’t need keep tabs or receipts for deductions cause the standard deduction is fine.
**It was important to me (and me only) that I clarify why I am pro folk….and legally so in the eyes of the IRS. With inflation, I will be losing wealth…on the bond/cash side for sure. It would be one hell of a lot worse if I had to pay taxes on it. That would be adding insult to injury.
I will discuss the portfolios theories later. Again, since I am not building up the nest egg intentionally right now, I feel unqualified to say much more. I am a John Vogel fan and still am…when I was managing my own portillios for years to save that 1-2 % fees of having managed funds. An good Vanguard index fund cuts the expenses way down, except you really start saving on expenses with a bigger buy-in. Think I was eventually paying 9 cents per thousand in an annual fees, maybe 9 cents per hundred. A penny saved…
A unmanaged no load index fund is designed to pert near match the market, not beat the market. And with all those expensive high flying big name managed accounts out there, the index beats them 6, 7 times out of ten.
After years of research and knowing what I know now, my stock portfolio, including many mutual funds along with many individual stocks, are now being actively managed by one of those expensive big boys names with fees! I simply lost interest in it. Took too much time and rent in my head.
I do not even bother to look at the market or do any research. Sometimes I don’t even read the monthly statements. It’s all on automatic pilot. As Will Rogers quipped, ‘the return OF my investment is more important that the return ON my investment. Will Rogers is probably wrong considering today’s inflation, taxes, management fees.
Good article, Dr. Rebane. If I had an investment club like you, I might be really digging down into it. Right now I enjoy not even bothering to follow it…just as long as that tax free income comes in every month. Later, when that is gone, I will have to participate in my own financial welfare again. The SS checks pays the mortgage, taxes, fire insurance and car insurance because, like many, I went out a bought a house in my later years. My heart bleeds for those folks who bought a home with a few thousand down (like me) in their later working years and now have $150-300 bucks to live on each month. A $600 propane bill would set them back for months of scrimping and eating more beans and rice.
I know many and after SS takes out for Medicare, they are screwed. But, there is no place like home
Posted by: Bill Tozer | 08 January 2022 at 12:14 PM
"that $600 check from the State."
So it's real? I was going to throw it in the trash along with the offers for appliance warranties.
Sweet. Count me in the minimize income camp. Lotsa benefits for the poor out there, from cheap PG&E to cheap health insurance (thanks Obama!) for the sub 65's.
Non-asset based benefits are a weird beast, but guys like Mr. Paul have convinced me that everyone is in it for themselves from here on. I have no interest in paying for Team Blue hobby projects. The American Dream and patriotism are dead ducks until we can form something new and fresh.
It seems to me that the most practical act of morality, concerning money, you could perform is in the form of a will. Steer the bux to something of lasting value.
Posted by: scenes | 08 January 2022 at 12:33 PM
@ 12:33 pm
‘The legacy’ is what I call it. Once one’s revenue source reaches a point where you can’t spend it all after covering expenses, then either start giving it away or stick it in a will.
Note to self again for the umpteenth time: Get off your lazy butt and find a good estate lawyer before I unexpectedly kick the bucket and the State (or County ?) hires a shister to divide it up and grab 30% off the top for expenses…while dragging it out for a couple of years.
That would really chap my hide…almost as bad than finding out I am a active registered Democrat voter long after the worms crawled in, the worms crawled out. A will is part of money management and speaks my words when, like ole John Brown, am rotting in the grave…or out in the back forty. Anybody hear from Biker Bill lately? :). What’s that smell down in the canyon? A will is words from beyond the grave. No marble orchard for me. Too expensive and a frivolous waste of money. We must be frugal (but not stark) and prudent with what will be other people’s money.
Besides, I will just probably catch Covid or Legionnaires or E Bola if I went on cruise to exotic places.
Posted by: Bill Tozer | 08 January 2022 at 01:21 PM
BillT 121pm - After getting the same recommendation from multiple respected sources, we changed from our SoCal attorney to local estate attorney Richard Keene (530-477-2281). We have been very happy with his knowledge, advice, work product, and professionalism.
Posted by: George Rebane | 08 January 2022 at 01:40 PM
Thanks Dr. Rebane. I am now ready to cut the check to some Estate Attorney that is worth his weight in gold in knowledge and experience. The time has come.
Oh, had a question from the Jo Anne part of Scattershots. Does she also keep track of all the birthdays of the grandkids and friends? I bet she does.
Posted by: Bill Tozer | 08 January 2022 at 02:09 PM
BillT 209pm - Indeed she does Mr Tozer, as one of the many hats, besides Finance Manager, that she wears (I'm in charge of income, she handles all the out-gos.) I am a kept man, flaneur-in-training, and still blessed in our 60th year.
BTW, there is no more 'investment club'; the Rebanes have been Lone Rangers with our portfolio for many a year now. As successful entrepreneurs, we have some high tax bills.
Posted by: George Rebane | 08 January 2022 at 03:01 PM