Some Things I've Learned Recently about the Standard Deviation of Returns [updates]

…or rather "Some things I've learned about calculating and projecting standard deviation of returns from a short frequency measurement (monthly) to a longer frequency horizon (year)." 

[ Note: since I clearly don't know this material nearly as well as I think I do, this post will continue to be a work in progress which I will update as I learn and assimilate more over time.]

If you asked me how many people out of, say, 400 million in the US were interested in standard deviation projections I'd have to guess the sub-population is relatively small, maybe 10s of thousands and that may be generous though it may be more.  If, of that sub-population, one were to ask how many are really, truly interested in whether using the square root of 12 (SQ12) is "good enough" to project a standard deviation using monthly return data to an annual horizon or if there is something better or more correct, well then, we are down, if I am not being overly dramatic, to maybe hundreds (maybe more). I know there is at least one because he is writing this sentence and probably more than one because I am reacting to an article or two on this subject. 

Weekend Links - 3/31/2017

QUOTE OF THE DAY

"The bucket approach differs from the view we have taken in this book, particularly with regard to the design of decision support systems (DSS). Specifically we seek systems that are robust with respect to innumerable possible scenarios -- many more scenarios than anyone would seriously suggest forming physical or mental accounts.  In a given situation there are many possibilities that can affect the supply of, and demand for, an investors wealth.  We can perhaps itemize a representative set of these possibilities, but what lies beyond our enumeration capabilities is the myriad of possible combinations with respect to the timing of these.  Perhaps some unfavorable event happens early in the investor's trajectory, or almost at retirement time, or during retirement. Or two unfavorable events happen back to back. Or a favorable or unfavorable market event happens in close proximity to some favorable or unfavorable event affecting the need for ready cash. The possible combinations of these--over many years until retirement and/or in retirement--can be astronomical."

Markowitz, Risk-Return Analysis Vol. 2, 2016 p 252.


CHART OF THE DAY

RETIREMENT FINANCE AND PLANNING

Tontines, NYT.  

Time Segmentation as the Compromise Solution for RetirementIncome, Pfau.  The Financial Planning Association (FPA) divides retirement income strategies into three categories: systematic withdrawals, time-based segmentation and essential-versus-discretionary income. Time-based segmentation provides a middle ground between the two extremes represented by systematic withdrawals (relying on a total-return investment portfolio for all distributions) and essential-versus-discretionary (using insurance-based products to implement a lifetime-income floor before considering investments). In occupying this middle ground, time segmentation is wildly popular in practice and it goes by many different names. But it is also the least studied retirement income approach. 

Relative Risk re Terminal Wealth and Asset Allocation One Last Time...

I've been playing around with a relative risk utility function and simulation lately and I have been using the distribution of simulated terminal wealth (including negative sim wealth) as the distribution against which I am judging utility.  I'm thinking that that is wrong.  It is easy to see because, while the terminal wealth calc (and "utility") factors in spending in the sense that spending would zero out wealth (and utility) if it were too high,  if I thought I wanted to game utility maximization doing it the way I've been doing it, I'd just roll back spending as much as I possibly could and it would kick up expected utility by quite a bit.  I'd rather not do that.  I'd rather optimize consumption vs. the risk of running out of money which would require a more complex process to evaluate the utility of consumption subject to some risk constraint and/or some magnitude factor.  Basically the thing is that I want to die with my last red penny clutched in my cold dead fingers (no legacy plan for me but maybe some charity if things work out) while having lived with the lowest risk of disaster I could have done over the intervening years. I've seen a ton of research move this direction and I will too but it seems complex. Some have called this one of the most complex economic subjects there is so I probably have my work cut out for me as a amateur retire-ologist.

Playing the Geometric Return "Game" with Some Real Data and My Systematic Alt-risk Strategy

In the last post I was trying to decode Michaud's simple binary example of geometric returns (two possible outcomes with probabilities 1 and 2) to make sure I understood what he was doing.  Now I want to try the same thing on some real data just to see what it looks like.  At first I thought the binary approach was dumb or an oversimplification, which it might be, but: a) it makes it easy to do calculations over a few time periods, and b) it's exactly the same thing as three other things I've run into over the years:

Another Lesson in Geometric Returns Over Multiple Periods

Or another lesson for me anyway.  I went back today to the article by Richard Michaud "A Practical Framework For Portfolio Choice" Journal of Investment Management 2003.  There was another example he used (vs another post I did) to illustrate geometric returns that I wanted to figure out because I wasn't quite sure where the numbers came from.  He can roll over this stuff...he has a PhD. in math.  Me, I gotta go slow.  Here is his example on page 5 (if you are a quant this is probably like doing coloring books in pre-school but me, I need to know): 

Suppose an asset with two equally probable outcomes in each investment period: 100% or −50%. What is the expected geometric mean return for investing in this asset over the investment horizon? In general it is not 0%. A correct answer requires more information. Suppose we plan to invest in this asset for only one period. The expected return of the investment is 25% not 0%. Suppose you are considering investing in the asset for two or three investment periods. The expected geometric mean return is 12.5% over two periods and 8.26% over three periods. For any finite horizon, the investment has a variance as well as an expected return. It is only at the limit, when the number of investment periods is very large, that the expected growth rate of investing in this asset is 0%.

I got the 25% thing he's talking about because it's easy. I got the 0% thing because I had used the formula for the limit of the geometric mean before. It was the 12.5% and 8.26% numbers that I didn't quite get...or rather that I could not quite duplicate.  But let's back up a bit. There are two ways to go about this kind of thing: 1) the dumb way, and 2) the I-won't-even-call-it-smart-because-it's-obvious-in-retrospect way.  #1 is to go back and try to use a decision tree used in a prior project that was probably flawed in conception and execution and then screw around with it for an hour trying to understand why you couldn't replicate the results. # 2 is to take the paragraph literally at face value and use simple math to calculate the result.  Guess which path I went down?  uh, yeah, that's right.

Weekend Links - 3/24/2017

QUOTE OF THE DAY

Past a certain level of income, what you need is just what sits below your ego. Morgan Housel 

CHART OF THE DAY

RETIREMENT FINANCE AND PLANNING

Here’s an ‘income menu’ that could help retirees make theirsavings last, Marketwatch.com.  Here’s the idea: Just as retirement savers need to be diversified and consider how their savings are “allocated” so their money doesn’t disappear in a market free fall, people in retirement need diversified sources of income. The study’s authors call these “retirement income generators,” or RIGs. Those RIGs — or, at least, information about them — should be part of employers’ retirement plans, according to Steve Vernon, a research scholar at the Stanford Center on Longevity and a co-author of the study. 

Foundations in Research for Regulatory Guidelines on the Design& Operation of Retirement Income Solutions in DC Plans, Steve Vernon Stanford. Joint research sponsored by the Stanford Center on Longevity (SCL) and the Society of Actuaries’

Committee on Post-Retirement Needs and Risks (SOA-CPRNR) concluded that it would be very desirable for defined contribution (DC) retirement plans to include an organized approach to providing retirement income with the potential to last the life of a plan participant. 

Adding a Third Asset Class (Systematic Alt Risk) to the Utility of Simulated Terminal Wealth

 In several prior posts (here here here) I looked at the utility of terminal wealth in a basic simulation context. In those posts I created a simple world with a 2-asset-class model, 25 periods and several spend rates ranging from 3 to 5%.  The assumptions were generic [1]. The conclusion, such that it was, based on a CRRA utility function, for spending in the 3-4% range with moderate risk aversion, was to perhaps allocate conservatively to risk somewhere in the 40-70% range to based only on the results in that fake-sim-world. [1a]

Option Skew: Tail risk indicator or Income Opportunity?

David Varadi at Blue Sly Asset Management put out a good post on skew as an indicator of tail risk.  He put some good analytics behind it.  For a long only investor I can see that rising skew could be a bad sign or as he puts it "Skew is a measure of the upside versus downside potential for a given market. Mathematically, negative skew is associated with higher tail risk while positive (or less negative skew) is associated with lower tail risk...Our analysis shows that the CBOE Skew Index is a useful indicator for assessing market potential for up to two and a half months following new highs or new lows. New highs tend to predict below-market returns while new lows tend to predict above-average returns for up to 50-days later. However, more data is needed to determine whether the CBOE Skew Index is an effective predictor of tail risk events."

Harry, Paul, and max E log(1+R)

Here is Harry Markowitz's riff on maximum E Log(1+Rt) (MEL, max geometric return criterion proposed by Kelly in 1956 and embraced or articulated by Latane, Brieman, Markowitz, etc.) contra Samuelson's utility maximizing:

"Theorem: If Harry repeatedly invests in a portfolio whose E log (1+R) is greater than that of Paul, then -- with probability 1.0 -- there will come a time (t0) when Harry's wealth exceeds Paul's and remains so forever thereafter." [emphasis in original] p.150 Risk-Return Analysis, vol. 2.

Of course, as happens in the buying or selling options, what happens within the intermediate time-frames gets interesting, too.

Utility of Terminal Sim Wealth, Now With Risk Aversion

In the last post on using utility in a simulator as implemented by an amateur I used Log(Wt) where Wt is the terminal wealth in the output of the various simulation runs, one for each of 11 allocations between risk and non-risk assets and then those 11 repeated three times for different spend rates for a given set of assumptions[1]. Since Log(Wt) is supposedly the same as a constant relative risk aversion utility function when the risk aversion "gamma" value is 1 [2] (that means someone willing to take risk), I thought I'd see if I could get a visual of utility for a more risk averse retiree (let's say gamma=4, see note 2) for at least a couple of different spend rates.

Utility of Sim Wealth in an Amateur Experiment

I am obviously not an economist or a working finance professional and it would be embarrassing for me to claim that I knew what I am doing with utility functions because I decidedly don't but I thought I'd take a shot to see what something might look like for some sim data I ran recently with whatever little I do know (I'm running with scissors, mom).  The sims were for a $1M portfolio for a 65 year old with at least two spend rates, 3 and 4% over 25 years tested against 11 allocation choices.   In a recent post on "retirement Omega" I looked at the same data and asset allocations and looked at what might be optimal with respect to the allocations by using an omega-ratio approach to analyzing the distribution of final wealth. The answer there was to spend less and if spending less, allocate conservatively...somewhere in the middle between stocks and bonds...more or less...kind of....  Fuzzy, but that's the way it is.

What About a "Retirement Omega?"

In evaluating investment strategies, some analysts will use the Omega Ratio because it captures more moments of the return distribution than just mean and variance. It is calculated as one area of the cumulative distribution function (above a threshold) divided by another (below).  The math looks like this:

Since simulated terminal wealth distributions are, well, distributed, we should be able to play the same game to come to some conclusions about retirement strategies...I think...this may be the exact point where I start to go wrong...

Here is the density function and empirical cumulative distribution of terminal wealth for different allocations (100 to 0% risk asset in 10% increments vs. safe asset; 50/50 is red; 100% stocks is the dispersed distribution of course) for a given spend rate (4%) and a given set of simulator assumptions [1].

A "Bench Strategy" by the Numbers

This post is roughly the same as the "40 months" post I just did but I realized that to some people a 7.4 percent return might sound meager so I thought I'd give some context by explaining what I originally set out to do and how I think about it and why I try to stay in this particular game.

10 years ago, before an early retirement, I wondered if I could get equity-like returns for bond like volatility.  My goal was not day trading it was portfolio "efficiency." In a retirement context this is especially important since spending against a volatile portfolio can have negative consequences.  If I could get, on the margin, the same return with less volatility or something more mean-variance efficient than what I do already then that is a game I want to play.  The purpose of the 40 month post was to tell myself that I had gotten to where I wanted to go after all those years but I had provided no context.  Here is the basic idea by the numbers.

Weekend Links - 3/17/17

QUOTE OF THE DAY

Luck is the second-most important factor for retirement planning; second only to withdrawal rate. Retirementoptimizer.com

CHART OF THE DAY

RETIREMENT FINANCE AND PLANNING

Psychology Of Retirement Income Satisfaction, American College of Fin Services.  Studies comparing the levels of satisfaction among retirees with different levels of guaranteed retirement income found those with higher levels of guarantees were more satisfied than those with less. What is rather surprising, however, was the revelation that a retiree’s income satisfaction does not have a linear relationship with the amount of wealth he or she has saved for retirement… “Once we start viewing money as wealth, as a stock of money and not necessarily as a flow, then we seem to get less happiness out of spending it than we do from money that’s automatically turned into a flow,” Dr. Finke said.   

Retirement Income Library, Portfolio Charts.   

Calculating Perpetual Withdrawal Rates, Jim Otar.  

Let Me Convince You To Save Money. Morgan Housel. ...since you can build wealth without a high income but have no chance without a high savings rate, it’s clear which one matters more. 

40 months

I have been trading since around 2005.  It takes a long time to learn trading especially when one is not in an institutional setting. On one's own it can take a few years to lose money a few more to break even and a few more to limp along.  That was absolutely me.  By my calculation the process took about seven years.  While my current systematic-alt-risk strategy is now 62 months old with a continuous coherent time series to match, 40 months ago would have been about my "seven year mark." So, while a look-back of 12 years is impossible, 5 years still includes the last couple years of the "training" phase, and one or two years don't really say much...40 months should be about right to evaluate what I can consider a mature phase of trading.  I may not have world class hedge fund results but on the other hand they don't look too bad over those 40 months. It takes some time and attention to do this kind of thing so we'll see if it is sustainable.

> Return.annualized(z, scale = 12, geometric = T)[1]
[1] 0.0735794
> StdDev.annualized(z, scale = 12)[1]
[1] 0.03936469
> SharpeRatio.annualized(z,Rf=Rf_m)[1]
[1] 1.336638
> SortinoRatio(z,MAR=0)[1]
[1] 1.235226
> Omega(z,L=0)[1]
[1] 3.940805
>

Some Thoughts on a Geometric Mean Frontier

I know very, very little about geometric means and returns but I'm starting to think that I should know more.  I have tried to read a little bit about this but it can be a tough slog through some of this stuff.  That means that this post is not a demonstration of any kind of comprehensive knowledge. It is, rather, more of an attempt to convince myself that I know anything at all from my recent journey into geo land and an attempt to try to consolidate some of it "on paper" if I do.  As always, let me know if there are any major errors or omissions. 

Asset Allocation, Ruin and Geometric Return

In "A Practical Framework For Portfolio Choice" Michaud 2003, with a footnote reference to Block 1969, he gives this example of asset allocation and geometric return.

6.3 Asset allocation strategies that lead to ruin.
Suppose an investor invests 50% of assets in risky securities in each time period. Either the return matches the investment or it is lost. Both events are equally likely. This is a fair investment game similar to an asset mix investment policy of equal allocation to risky stocks and riskless bonds with rebalancing. In this case, investment policy leads to ruin with probability one. This is because the likely outcome of every two periods results in 75% of original assets. However, the investment is always fair in the sense that the expected value of your wealth at the end of each period is always what you began with. For two periods the expected geometric mean return is negative and declines to the almost sure long-term limit of −13.4%, which is found using (2). This example vividly demonstrates the difference between the expected and median terminal wealth of an investment strategy. It shows that the expected geometric mean return implications of an investment decision are often of significant interest.

I wasn't quite following this so I thought I'd try to figure it out.  I had two main questions:

Market tuition: picked off by a dark pool by being stupid

I've been trading for more than 15 years and have paid a few dues over that time.  Every once in a while, though, I get another tuition bill for my continuing-ed.  Over the years I have moved away from using orders...at all (except for limit to get in and out). Generally non-limit-orders do not favor retail investors and can be gamed by those insidious forces on the other side.  Retail investors have advantages that institutions don't such as small scale for taking relatively illiquid and high spread instruments, no institutional pressure to be "in" all the time, no strict long side bias, no perverting compensation incentives, etc etc. Another big one is that retail investors can be patient and can usually wait and see. 9 times out of 10, except for issues related the efficiency of labor and attention, orders (except for taking oneself in with a limit) are not necessary. Since I have a systematic approach where attention and labor are, on the margin, important, I sometimes execute automated trades structured with layered orders.  For those times that I do use orders they are usually simple stop limits which are also sometimes set up in conjunction with time parameters.  I have gotten lazy though. For certain trend-following trades I sometimes structure a negative scalp where I get out on a trend signal and then get back in systematically if the counter-trend move fails but with a small real-dollar loss and a tax-inefficient hit more often than not.  It's right there that I got weak and lazy. I was stop-limiting out on the sell (when long) and stopping in on the buy.  I knew that was wicked because stop orders become market orders and market orders can be murderous. And so it happened to pass...

Look at a chart of PCEF, a composite CEF income ETF (ignore the question of why that etf) for March 10.  See that giant spike up in price? Ever wonder sometimes who buys at the top of those chart bars? Well....that would be me.  Here's how it goes down.  I stop-limit out yesterday. Then on the market open today I had a standing stop-buy that was open (bad me). What happened? The stop on the signal price became a market order when the market opened and I got taken for a ride by a dark pool where the order got filled at 23.499 where my buy-stop was at 22.77.  It was 300 shares so it was only a "take" of about $220 which is not the end of the world but I consider that $220 part of my kids real school tuition, not my pseudo market tuition, and a type of theft though it was really only my stupidity.  That dark pool is now in my crosshairs though I doubt there is much impact I could ever have on them.  But maybe the day will come and my memory is long in these few years before the dementia starts to kick in...

Weekend Links - 3/10/17

QUOTE OF THE DAY

But in investing, it doesn’t work that way. More resources, more time spent, more effort, etc to do not necessarily lead to better outcomes... If it were simply a case of effort, then the people who trade all day, every day, and never turn off the screens would be the wealthiest.  Josh Brown 

CHART OF THE DAY

RETIREMENT FINANCE AND PLANNING

ObamaCare vs Early Retirement, nyt.  

Annuities: All or Nothing, Dirk Cotton. Economic studies are much better at teaching us how things work than at predicting outcomes for an individual household…Milevsky [2007] shows that the optimal age to buy an All-or-Nothing annuity is when the retiree values the return on an annuity the same as she values the return that she expects from her investments. 

Longevity Risk: To Bear or to Insure? Boon Briere Werker. Tilburg Univ. The collective agreement yields marginally higher individual welfare than an annuity contract priced at its best estimate, and the annuity provider is incapable of adequately compensating its equity holders for bearing longevity risk. Therefore, market-provided annuity contracts would not co-exist with collective schemes. 

The Role That "Return Threshold Asymmetries" Might Play In Retirement Success

In a previous post I pointed out that attempting to reduce return volatility is probably a constructive activity for a retiree due to the pernicious effects of sequence of returns risk. What I failed to mention is that it's not just about volatility it is also about either the asymmetry [1] of volatile returns with respect to a threshold return level and/or about mean-variance efficiency in terms of seeking the same level of returns but with lower volatility. Sometimes those two are the same thing.  Either way, in retirement the game is usually to mitigate the bad (bad returns, fail rates, bankruptcy risk, etc) and stay open to the good (returns, median terminal wealth, increased spending, max geometric return…whatever).  That's a pretty neat trick, especially if you can pull it off without options where asymmetry is the main game.  And I'll say "especially without options" because long options are generally pretty expensive. In fact sometimes the demand for options, both for upside capture and downside hedging, can be so intense that I want to sell options to try to capture some of that hope and fear.  That's been a good business for me for at least the last five years. No point in giving that up. But I think there are ways to both reduce standard return volatility and also to wind it up a bit asymmetrically so that it leans a little bit in the direction we want.  I'm not saying 100% of investors are going to do that because I don't think that necessarily sounds like a healthy "market" in the aggregate.  But I do think it can be done within certain boundaries.  

Impact on Fail Rates of Volatility Reduction Strategies

I'm sure this has been done before but it was easier to code it out than look for the related articles. I wanted to see what happens to simulated fail rates when nothing changes but return volatility.  This might be kind of obvious because, in the absence of random simulated inflation, if return vol were to be zero for example (and spend rate < return rate), simulated retirement would almost certainly be successful. But the idea here, suggested by a reader, was: using a simulator "what is the nature of the effect on fail rates of adding to a portfolio either asset classes or tactical allocation or systematic rule-based strategies that might have a significant, material effect on portfolio volatility (when returns are held constant)?"  The answer can probably be intuited before the last sentence is completely read. Since sequence-of-returns risk, where drawing from a portfolio when returns are bad, is a bad deal for retirees, reducing the scale and frequency of the down-returns through volatility reduction strategies is likely to be a good deal. Increased vol would have an opposite effect.  To cut to the chase, here is what it looks like in fake sim-world:

To do this I replaced the part of the simulator that used historical data to generate returns with a programmatic function that samples fake returns using a similar mean return, standard deviation, and skew. The historical data, when allocated 50/50 to risk/safe and run through a simulator 10k times using some generic assumptions^[1], has a mean return of .076, a standard deviation of .097, a skew of -.53, and a fail rate of .165.  That was the base case.  Replacing the historical returns with a "return function" tuned to the same distribution shape produced similar fail-rate results and similar moments of the return distribution.  Then it was just a matter of varying the standard deviation to see what happens which is what is shown above. It was interesting (to me anyway) that toggling the skew parameter to zero skew had almost no effect at all but I'll play around with that later.

Weekend Links - Fri 3/3/17

QUOTE OF THE DAY 

There is, however, another form of bear market, and, in my opinion, it is the most pernicious one of all.  That is the bear market that you inflicted it on yourself.  That is because, riddled with fear of all the terrible things that could happen to your portfolio, you never invested at all, suffering a massive drawdown in opportunity cost.  Those foregone gains can never be made good, as time, your main ally as an investor, is a non-renewable resource, and you have squandered what time was allotted you.  Lawrence Hamtil  

CHART OF THE DAY

RETIREMENT FINANCE AND PLANNING

Life Cycle Goal Achievement or Portfolio VolatilityReduction? Dempster et al. Making use of the same data and market calibrated Monte Carlo stochastic simulation for all the alternative portfolio strategies, we find that flexibility turns out to be of key importance to individuals for both portfolio and spending decisions. The performance of the adaptive dynamic goal-based portfolio strategy is found to be far superior to all the industry’s Markowitz-based approaches.  [ Probably useful and interesting but disappointing since it is more or less unreadable by all but academics and advanced retirement income researchers/practitioners. That's fine but the distance between a paper like this and real life retirees and what they need and know is vast and it doesn't have to be -- something that I keep hoping will change.  Implementation would likely to require 999/1000 retirees to place their trust and outcomes in the hands of a machine, incomprehensible jargon, and a very sophisticated planner. I'm not sure I'm on board with that except for maybe the affluent. Adapting the plan to life as it unfolds, which is part of the paper but not as clear as it could be, is what most people do anyway but without the advanced jargon-filled academic papers. ]  

Using Reverse Mortgages In A Responsible Retirement IncomePlan, W Pfau.  For those thinking about it, please note that the reverse mortgage should be used as part of a responsible plan. It allows homeowners to borrow against the value of their home, creating liquidity from an otherwise illiquid asset, and grants the flexibility to defer repayment until they have permanently left the home. But if this liquidity creates the temptation to use the proceeds in an unwise manner, then you may be better off avoiding it.