1. It is the same as the expectancy formula in Alexander Elder's trading book which I haven't read in over 10 years (and I lent it out and never got it back). The formula is, if I remember correctly, win rate times average win minus loss rate times average loss. Same math as the Michaud example, at least in year 1.
2. It is a cousin if not half brother to the Omega ratio except that we are skipping over standard deviation (sort of) and skew and kurtosis. And we are not integrating anything. But its the same thing: wins and probability vs losses and probability with respect to a threshold (say zero).
3. The first lesson from my former hedge fund partner was that trading success is a function of risk return and the probability of return. That is the same as #1 and we generally analyzed it the same way.
So binary it is. To do that I take the return series and split it into two: positive and negative. Then I figure out the average positive, the average loss, and the win and loss rate. For example for the SnP for the annual series 1928-2015 I figure the win rate is ~72.7% and the loss rate is 27.3% and the average win is 20.1% and the average lose is -13.7%. The average arithmetic return is 11.4% and the "expectancy" from the binary is 11.4%. So far so good. That's a good single period expectation based on the historical data. Then I did the same thing for the 100% 10-year, the 40/60 50/50 and 30/70 mixes, and for my systematic alt risk strategy, although there I had to use a monthly series and project to an annual horizon which creates some complications.
Then I plug the binary win/loss and P1/P2, each in it's turn, into the geometric game table I used in the last post plus I added a final calc for the expected geometric return at infinity using the point distribution formula Michaud used in his article [see table 2 at end] . And when I do that, remembering that we are using binary outcomes, not the full return series, the results look like this:
Table 1 - The results of the game
The game version of return in N=1 for the "alt" doesn't match the E(r) because of how it's being annualized which is where this analysis goes a little south but we'll hang in there anyway because for the "game" it's just a game. So I can see it better let's chart out N=1 thru 5 with infinity tacked on the end. Looking at just the 30/70, 40/60, 50/50 and alt, it looks like this:
So that's the game version. A little flawed and frankly not all that interesting unless you believe the cross-over of alt and 40/60, which I don't because of the mis-match of the series' intervals, or you didn't know that for N being large, expected geometric returns get lower, which we did. Now let's look at the real historical annual return series with alt being annualized from monthly. I'll calculate a) expected mean returns, and b) the expected geometric return at infinity. For alt I'll do it two ways: 1) using the Meucci method for projecting returns to a horizon and 2) the approximation formula G = r - V/2.
Intermediate Conclusions...
Originally I thought for this post that I'd have a ton of conclusions at this point but I don't. Geo returns are lower at distant horizons. Yawn. My alt risk thing might or might not have a cross-over but the annualization/projection/estimation is so sketchy that nothing can really be concluded I think. Since the whole point of the alt-risk strategy was to reduce vol in the face of consumption -- and the geometric return, while it implicitly considers vol, it explicitly does not consider consumption -- let's look at a head-to-head simulation of 40/60 vs Alt. I picked 40/60 because they look so similar. I'm using the simulated CRRA utility of terminal wealth (in the future I really want to look at U[consumption] but I haven't gotten to that yet) as a metric so that we can factor in consumption (in terms of resulting end-wealth) and see how it works against different volatility profiles for different strategies. When we do that, by adding in Alt's return profile, in 10% increments, to a 40/60 portfolio it looks like this if I got everything right:
But this is more or less the same result as the last post where I put a big fat asterisk on the assumptions. If you can tolerate the assumptions, though, and I'm not sure I can, then heavy doses of the alt risk strategy could perhaps be heaped on a more traditional portfolio even though the geometric return analysis above was either negative or inconclusive. My guess is that sequence of returns risk in simulation favored the lower volatility strategy but I'm not 100% sure.
Conclusions?
Not really other than this post was like riding a bike in first gear: a ton of motion and energy without much forward progress. If pressed I might throw things like this out there:
- The geometric return seems pretty sensitive to how data is projected to a horizon,
- Volatility plays an important but limited role in the geometric analysis because spending is not explicitly part of the calculation. There may be math to do that I just don't have it so I have to simulate my way out of this,
- Cross-overs may exist but are probably an artifact of doing summary and projection stats. I'd have to ask a quant...[see note 1, I changed my mind on this right after my post],
- It looks like the binary form of the game over-estimates long term geometric return, probably due to it's blunt-force use of less than granular return data,
- Using 40 months of data to project anything into the future should, as always, be part of the criminal code.
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[1] Actually on reflection, after I posted this, I realized that some crossovers must exist. For example if Michaud's example in the last post were to be real with a 50% chance of 100% return and a 50% chance of a -50% return the line would go from 25% on the left to 0% on the right and cross through everything. I'd still be wary of how the stats are being done but it's clear that looking at the geometric return is likely a worthwhile endeavor. Pinpointing the N where there is a cross-over, though, is probably a difficult and maybe even pointless effort. I say maybe because retirement is a game of finite duration which means it may actually matter. I'd still test it against spending though. In addition, I remembered that the other approximations for G that are explicit about vol make portfolio combos really interesting in a mean-variance context and when looking at a geo-mean frontier. But my point here was not performing some type of integrated analysis it was just to see what the binary game looked like with some real data over a small number of Ns.
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Table 2 - The geometric return game
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