- There is, of course, an efficient frontier of portfolio risk and return where for a given level of return there is a minimum variance or for a given level of variance there is a maximum return,
- The supposedly optimal portfolio, as is known, is the one that is where a line coming from the risk free rate is tangent to the efficient frontier,
- According to Markowitz (2016) the optimal MV portfolio and max utility are likely the same under some bounded assumptions
- If an investor seeks a higher level of return than the tangency portfolio for whatever reason, then levering the tangency portfolio is more efficient than selecting a portfolio allocation that is riskier than the optimal portfolio
- The effect of volatility and time can seriously take the shine off of the single-period expected returns of even optimal portfolios and that can usually be seen in multi-period (or limit of) geometric return estimates if not the final realized returns
- Spending, in collusion with time and volatility, can lay waste to a portfolio especially when returns are bad at the wrong time or bad all the time or when spending is at or above unsustainable levels for too long, or........
And all of that got me thinking[1] about trying to simulate different spending rates over long periods of time against a levered theoretically-correct low-volatility portfolio. This seemed not just vaguely interesting, it seemed, according to what we are supposed to know, mandatory if not essential. I don't think this is new or that is has not been done before, but I have not seen it addressed by anyone myself and I wanted to see what it looked like. First, let's ignore pesky things[2] like estimation error (if not actual facts) in coming up with returns and variance or things like constraints on leverage, though that is supposedly a reason for the low-volatility anomaly: people can't lever low vol assets to get higher return if they need it so they take more risk and in the process bid up high risk assets driving down expected returns below where they might otherwise be. In fact, even though I'll try to use some realistic data after ignoring the previous stuff, let's keep in mind that all of this is hypothetical and is ready and willing to fall apart on close examination.
First, let's build an efficient frontier and select an optimal portfolio. I'm going to use 1928-2015 market data from the Stern school to estimate returns and variance. Stocks will be S&P500 (no international!? gasp) and bonds will just be the 10year (total return). These are just thrown in for some plausible reality.
I am not going to re-sample ala Michaud. I'm also not going to use excess returns, though I believe that is recommended, but just straight up returns. I'll ignore that the mean return estimate for stocks coming from my data seems really high relative to both the current expected future as well as history of capital returns over the last 1000 years. I'll roll with it for now. I will also assume that the risk free rate really is 2% for no real reason other than it is an easy-to-work-with number and creates a tidy tangency point for this game.
Now I will somehow come up, by hook or crook, with estimates for mean (arithmetic) returns, standard deviations, and covariance. In this case the data say to me 11.4 mean return (see, it's high...maybe I did it wrong...or need a Bayesian prior...but let's keep moving) for stocks and 5.2 for bonds with standard deviations of 19.8 and 7.8 respectively. The correlation coefficient if I trust excel is -.0258. Using that data and Markowitz math and some other math I'll do three things: 1) generate a two-asset efficient frontier, 2) create another two-asset portfolio made up with combinations of the risk-free asset and the spoiler-alert 30/70 tangency portfolio to create and draw a capital market line, and then 3) I will use geometric return estimation to look at alternative multi-period frontiers and capital market lines that might result from the effects of time and volatility on multi-period returns. For #3 I'm going to use the formula G = E - V/2 because, even though it is supposedly not the best estimator, it's really easy and the stakes are low. Then, like a good blogger, I will render it thus:
Figure 1. Mean Variance space for the low-vol game
Now what I want to do is take that 30/70 portfolio and add leverage to push it up (or down) the CML and then, while I am doing that, I want to layer on a simulation of spending over time to see what happens with the levered portfolios. I've never tried that before. Usually I am just simulating different asset allocations constrained by allocation weights summing to 1. But why let that stop us when we're having fun. The problem though is that I don't trust my simulator with the job (and I wrote it, how's that for bad juju?). I'm not totally sure it can handle negative weights and do the proper volatility curve ball when it is using those weights (I'll try it some other day). Instead I'll come into this problem by a side door. Rather than simulate with asset weights in the sim what I'll try to do is use the Markowitz math to calculate the return and var for the CML I have drawn and then create a function in R to sample from each MV distribution along that line as we simulate. It's almost as if, in the sim, I were to be investing in an different ETF along the CML where each ETF is levered to different degrees against a 30/70 portfolio (someone should make one of those I was thinking after I did most of this post) and has the same statistical mean and variance characteristics. The trick with the function is coming up with the proper structure. I realize that one can't assume distributions are normal in finance. I also now realize, after way too much reading this year, that one can't even assume they are lognormal. But this is a game so: normal. Just to do a visual check I overlaid a density function from the stern data for a 30/70 against the R function that I decided to go with. That overlay looks like this:
Figure 2. return distribution vs R sampling distribution
Not perfect but I'm an amateur hack so perfect enough for now. Now for the sim. What I'll do here is do some sequential trial and error. I'll run 10k sims at various intervals specified for different levels of leverage when using the risk free asset (and I am also assuming I can borrow at the risk free rate. Let's blow off tax issues while we are at it). I'll run this sequence from 100% risk free asset to -160%, always keeping the other part of the allocation at a constant 30/07 mix, using my pseudo-ETF side-door approach while hoping that I haven't lost my mind doing it this way. Each sim at each interval in the sequence will calculate fail rates, median fail duration, and some utility math on the present value of terminal wealth as it stands at the end of each sim-life. When I am done with all that I'll also run a sim for each asset allocation for a traditional two-asset portfolio that happens to line up with the risk of each levered portfolio in the sequence. This particular effort won't go below a 30% allocation to stocks because, for example, at 20% the two-asset portfolio std dev is below the minimum variance portfolio and it won't make sense. That means that in the chart we'll see a second (dotted) line that goes from, in two-asset world, ~30% risk to almost 100% risk or just part of the way up the CML, basically from tangency to full risk. Oh, and we'll do it for two different spend rates: 3% and 4% constant inflation adjusted[3]. When we do all of this, it looks like:
Figure 3. Levered Low-Vol fail rates.
Figure 4. Median Fail duration.
Figure 5a. Utility of the PV of terminal wealth using levered low-vol...
So there you have it. Different degrees of leverage of a hypothetical low-vol portfolio done in a simulator that can't hack it the right way using a method that might not stand scrutiny. Can we actually make any sense of this or come to any conclusions? Probably not in the real world. In low-vol-levered-within-a-sim-game world we can say whatever we want. If we want we might say things like:
- As always: spending less has an out-sized impact before we even get to optimization considerations
- It appears, on paper at least, that levering low vol portfolios actually works to reduce risk and improve outcomes in multi-period retirement scenarios. If we consider the theory this should not be surprising. If we think in real-world practical terms we are most certainly missing something. Like maybe how to "lever" if we are not an institution. Sure, for us retail investors we can perhaps say use margin debt or maybe asset assigned lines of credit (depending on the rules which are not friendly, I think the SEC forbids this kind of thing) but in day-to-day practice it's really pretty hard. Like I said, there should be an ETF. Then there are always transaction costs, fees and taxes to ruin a day.
- Ignoring the tax issues of re-configuring a portfolio to pursue a new strategy it looks, last point notwithstanding, as if one could actually increase spending while keeping fail risk constant or potentially lower (especially if one starts out with a high risk portfolio) by switching to a levered low-vol approach. Those cross-over points are interesting.
- There is no real material gain, as far as I can see, when looking at the fail magnitudes. Taking on risk in any form means that the blowups are always going to get bigger.
- Net of the new thoughts on levering low vol, this analysis looks like it confirms my bias that there is a broad but finite range of allocations that make sense in terms of retirement multi-period risk. The only thing here is that it finally dawned on me that any allocation below the MV min-var portfolio is going to be a problem (that should not have been news to anyone including me) and anything up the MV frontier past where the n-period geometric math starts to bend expectations away from the MV arithmetic frontier are going to be something to be wary of as well (again not really news).
- I'll have other comments. Like previous posts, I always think of something because these posts are not wikis they are just me writing about what I am learning and reading and thinking about. I'll update as I think about this stuff.
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[1] In what has not become my near normal standard disclaimer: I'll bet in at least half my posts I write somewhere "I hope I got this right." In the other half I know I'm thinking it. This post is no different. Maybe especially this one because I am stretching a little bit with a topic of which I do not have a solid command. But that doesn't matter because this is just an exploration. Anyone who thinks this is an answer is taking their own risks.
[2] Let's also ignore:
- knowledge that my simulator is pretty rudimentary and cuts some corners
- market data used is not well matched to the current era
- No doubt there will be errors and omissions
- that I can actually borrow at the risk free rate
- that borrowing at the risk-free rate is hard to do for most of the world that I know
[3] Other sim assumptions:
- $1M endowment
- age 65-90, fixed not stochastic
- Bonds are 100% 10 year with no mix with bills
- Stocks are SnP with no cap for weird years in 20th C and no suppression either
- SS and annuity income are off
- Risk aversion coefficient is 4 when used
- Spend shocks, variability, and trending are turned off
- other...
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