Retireyoung wrote:Can anybody back-test buying low selling high with long term moving average (maybe even using Bolinger-bands) ? It will give more accurate answer about the Bonus .

boggler wrote:Basically, the "bonus" relies upon the assumption that the markets are mean-reverting. If they are, then buying low and selling high works, otherwise, it doesn't.

The rebalancing bonus has nothing to do with "mean reversion" (which no one in this thread is using correctly) or with momentum effects.

The reason you rebalance is because rebalancing causes your geometric returns (i.e. what you get) to be closer to the arithmetic returns of the portfolio components. Depending on volatility, there is some "drag" that pulls the returns you see down from the arithmetic average. So you get a bonus from rebalancing by eliminating this drag.

As was explained:

rmelvey wrote:Its kind of ironic that people keep saying that rebalancing reduces risk, but does not increase return. Holding arithmetic returns constant, decreasing risk increases the geometric return.

In summary, rebalancing increases returns because it decreases risk. Buying low and selling high has absolutely nothing to do with the bonus I am talking about.

Put another way, your allocation has some arithmetic return. Rebalancing will bring your geometric return closer to this arithmetic average because it reduces your risk and therefore the drag on your returns.

Re: this, from earlier in the thread:

nisiprius wrote:Look, someone had better state clearly and unambiguously what the proposition is, because what tends to happen these discussions is that different advocates of rebalancing put forward different claims for what it is supposed to be doing, never quite clearly stated.

(I'm not quoting the whole thing to save space)

You are overlooking the fact that for any risky asset there is a maximum safe allocation. Going beyond this allocation will reduce your geometric returns and increase your risk. Going too far beyond will make your risk of ruin approach certainty. (This is a really subtle point that very few people seem to understand fully.) The reason for this is because gains and losses are asymetric. To get back what you lost, you need gain = ((1/(1-loss%))-1. I.e. A 50% loss requires and offsetting 100% gain. So if you over-allocate to a particular risk, then you hurt your geometric returns because the higher arithmetic gains don't offset the volatility the additional allocation added.

In the case of a single coin-flipping game,

p = the probability of winning on a given bet.

B = the ratio of the amount won to the amount lost

f = the percent of your assets risked on each flip

So the growth rate will be:

G(f) = p * ln(1+B*f) + (1-p)*ln(1-f)

The maximium safe allocation is:

f = mathematical expectation / B = ((B+1)*p-1)/B

So for example, a 50/50 flip with a B of 1.25, has a maximum f = .1

For f = .1, G(f) = .6% per flip.

G(.05) = .46% per flip.

G(.15) = .46% per flip

G(.2) = 0% per flip

So if you invested more than 20% in this scheme, you'd lose money even though it has positive expectancy.

Did this help?

nisiprius wrote:The thing about arithmetic and geometric means is really simple: for any unit time period, the right thing to do is to take the arithmetic means of the returns of the assets in the portfolio; for periods spanning more than one unit, you take the geometric mean of those arithmetic means.

Now, all this stuff, like the Monty Hall problem and so many things in probability, is tricky enough to think about that it is genuinely hard to be sure you've got it right, particularly if it's offered up in some new guise, hence the endless debate.

There are a fair number of people, though, who really believe that rebalancing magically ratchets up returns because it is an automatic mechanical way of buying low and selling high, and that the effect depends simply on the fact that it is rebalancing, and doesn't depend on mean reversion. As opposed to saying that if you rebalance at periodic intervals, you may actually catch a small benefit if the rebalancing interval happens to resonate with the mean reversion period.

No mean reversion, no rebalancing bonus. You always have the opportunity to shape the distribution of outcomes and remold it nearer to the heart's desire, but which shape is best depends on one's personal heart's desire--that is to say, one's appetite for risk and which direction one prefers to have the outcomes skewed.

There are two related things going on here:

First off, there's the point that you can actually take too much risk. Up to a point, more risk increases your returns. Beyond that point, you may have higher arithmetic returns, but the growth rate of your capital will decline or go negative.

Second off, there's the mathematically true point that rebalancing makes a given portfolio's growth rate better track the arithmetic returns of the underlying assets. This has nothing to do with mean reversion (either actual mean reversion or the sort of thing people in this thread have in mind when they use that expression).

swaption wrote:You know, this hypothetical example seemingly ends up in a dead end when applied ot the real world. Or does it? On the surface, this geometric 100% gain and 50% loss does not really exist, at least in the world of coin flips. Even if one flip, it's a bad bet for the house and no such thing exists in Vegas or anywhere. Vegas is a world with no expected return. But of course Kelly and Shannon were dealing with a different kind of world. They were dealing in a world where there was some sort of advantage, something that gave an expectation of a return. The Kelly Criteria was essentially an approach to sizing bets so that the advantage could be realized without going bust.

Well, the calculations tell you how much to risk at a given time to optimize your growth rate for a given draw-down risk. The coin flipping example is just a really simple, easy-to-understand model of what is going on. You can do the calculations with actual investments, but it involves solving some nasty calculus problems, and the math would obscure the point the example is intended to prove.

For example:

camontgo wrote:Over longer horizons, the result remains the same. The likelihood of the average returns being close increases with a longer horizon..but the "realized return difference" window where you get a bonus shrinks....so the average bonus over many trials is still zero.

Where have I gone wrong with this analysis?

If you increase the volatility of your assets to make the bonus more extreme, it will show up more obviously. Similarly, you could add a cash component and rebalance between all three at the optimal f to make it as apparent as possible. (It may even be that your simulation is having problems because it is over-leveraged without a "cash" component. In which case the rebalancing can't help you since you are taking too much risk to begin with.)