# To rebalance or not to rebalance

I have not been able to understand the concept of rebalancing in investing. To me its a form of the gambler's fallacy.

Let:

• s_n = stock price at time n
• s_{n+1} = stock price at time n+1
• E[x] is expected value of x and E[x|y] = expected value of x, given y
• c, k are some constants

Then,

1. if E[s_{n+1}|s_n=c] = c, in my opinion rebalancing does not make any sense
2. However, if E[s_{n+1}|s_n=c] = E[s_{n+1}] = E[s_n] = k, then rebalancing totally makes sense

Notes:

1. Case 1 is a random walk, case 2 isn't

2. Case 2 is what I understand as reversion/regression to mean

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Great question and I would like to understand it to. Is it possible to provide a little more background about the context? I am guessing this is used in portfolio theory, but I can be wrong – gyurisc Feb 25 '10 at 8:46
Yes I'm intrigued, but also lacking understanding of the question still. Are you referring to rebalancing within a single stock (i.e. dollar cost averaging?) or across a portfolio (multiple stocks or funds)? Could you explain your notation some more? By E[...] are you referring to the expected value of the stock price? c? k? Thank you! And welcome to the site. – Chris W. Rea Feb 25 '10 at 12:36
@morpheus updated your question with the assumptions you clarified in your answer. – Zephyr Feb 26 '10 at 18:27

An asset allocation formula is useful because it provides a way to manage risk. Rebalancing preserves your asset allocation. The investment risk of a well-diversified portfolio (with a few ETFs or mutual funds in there to get a wide range of stocks, bonds, and international exposure) is mostly proportional to the asset class distribution. If you started out with half-stocks and half-bonds, and stocks surged 100% over the past few years while bonds have stayed flat, then you may be left with (say) 66% stocks and 33% bonds. Your portfolio is now more vulnerable to future stock market drops (the risk associated with stocks). (Most asset allocation recommendations are a little more specific than a stock/bond split, but I'm sure you can get the idea.)

Rebalancing can be profitable because it's a formulaic way to enforce you to "buy low, sell high". Massive recessions notwithstanding, usually not everything in your portfolio will rise and fall at the same time, and some are actually negatively correlated (that's one idea behind diversification, anyway). If your stocks have surged, chances are that bonds are cheaper.

This doesn't always work (repeatedly transferring money from bonds into stocks while the market was falling in 2008-2009 could have lost you even more money). Also, if you rebalance frequently, you might incur expenses from the trading (depending on what sort of financial instrument you're holding). It may be more effective to simply channel new money into the sector that you're light on, and limit the major rebalancing of the portfolio so that it's just an occasional thing.

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"'Buy and Hold' Is Still a Winner: An investor who used index funds and stayed the course could have earned satisfactory returns even during the first decade of the 21st century." by By Burton G. Malkiel in The Wall Street Journal on November 18, 2010:

"The other useful technique is "rebalancing," keeping the portfolio asset allocation consistent with the investor's risk tolerance. For example, suppose an investor was most comfortable choosing an initial allocation of 60% equities, 40% bonds. As stock and bond prices change, these proportions will change as well. Rebalancing involves selling some of the asset class whose share is above the desired allocation and putting the money into the other asset class. From 1996 through 1999, annually rebalancing such a portfolio improved its return by 1 and 1/3 percentage points per year versus a strategy of making no changes."

Mr. Malkiel is a professor of economics at Princeton University. This op-ed was adapted from the upcoming 10th edition of his book "A Random Walk Down Wall Street," out in December by W.W. Norton.

http://online.wsj.com/article/SB10001424052748703848204575608623469465624.html

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This stuff is all well and good, as long as your risk assessment is accurate. In 2011, anyone putting serious weight on bonds being a low-risk investment is simply delusional. – duffbeer703 Jan 16 '11 at 17:15

Rebalancing is, simply, a way of making sure your risk/reward level is where you want it to be.

Let's say you've decided that your optimal mix is 50% stocks and 50% bonds (or 50% US stocks, 50% international, or 30/30/30 US large-cap/US small-cap/US midcap...). So you buy \$100 of each, but over time, the prices will of course fluctuate. At the end of the year, the odds that the ratio of the value of your investments is equal to the starting ratio is nil. So you rebalance to get your target mix again.

Rebalance too often and you end up paying a lot in transaction fees. Rebalance not often enough and you end up running outsize risk.

People who tell you that you should rebalance to make money, or use "dollar cost averaging" or think there is any upside to rebalancing outside of risk management are making assumptions about the market (mean regressing or some such thing) that generally you should avoid.

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In theory, investing is not gambling because the expected outcome is not random; people are expecting positive returns, on average, with some relationship to risk undertaken and economic reality. (More risk = more returns.) Historically this is true on average, that assets have positive returns, and riskier assets have higher returns. Also it's true that stock market gains roughly track economic growth.

Valuation (current price level relative to "fundamentals") matters - reversion to the mean does exist over a long enough time. Given a 7-10 year horizon, a lot of the variance in ending price level can be explained by valuation at the start of the period. On average over time, business profits have to vary around a curve that's related to the overall economy, and equity prices should reflect business profits.

The shorter the horizon, the more random noise. Even 1 year is pretty short in this respect.

Bubbles do exist, as do irrational panics, and milder forms of each. Investing is not like a coin flip because the current total number of heads and tails (current valuation) does affect the probability of future outcomes. That said, it's pretty hard to predict the timing, or the specific stocks that will do well, etc.

Rebalancing gives you an objective, automated, unemotional way to take advantage of all the noise around the long-term trend. Rather than trying to use judgment to identify when to get in and out, with rebalancing (and dollar cost averaging) you guarantee getting in a bit more when things are lower, and getting out a bit more when things are higher. You can make money from prices bouncing around even if they end up going nowhere and even if you can't predict the bouncing.

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Yes E[x] is expected value of x. E[x|y] = expected value of x, given y. c, k are some constants

Let E[s_{n+1}|s_n=c] = c, but if E[s_{n+1}|s_n,s_{n-1},...,s_{n-m}] ->some constant k as m->\infty (call this equation 1) then rebalancing makes sense.

Notes:

1. The gambler's fallacy is to believe in equation 1 when in fact it is not true.
2. Equation 1 is really regression towards the mean.
3. So, if rebalancing really helps in case of the stock market, then in this case maybe the gambler's hypothesis is true after all.
4. What we are saying in above is that although in the immediate short term, stock price is likely to stay at its current value c, in the very long term its price will come down/go up to its intrinsic value k, no matter what the stock history is (do you believe that?). Rationale perhaps being that in long term, people will come to their senses.
5. Consider E[s_{n+1}|s_n,s_{n-1},...,s_{n-m}] = weighted average of s_n,s_{n-1},...,s_{n-m}. Should you rebalance in this case?
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The main difference between the stock market and just gambling is that gambling is a pretty random event, while stocks and bonds are priced according to how much money people anticipate they will generate for the holder (and in generally they do so, over time). It's not very random in the long term at all... and asset allocation strategies are most useful over the long term. – fennec Feb 26 '10 at 7:03