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I'm using Markowitz Mean-Variance Optimization to rebalance my portfolio of 30 holdings. I'm trying to select the best balance of funds that will return 10% with the minimum risk. I understand that rebalancing once a year is better than doing it once every two years. Likewise, rebalancing every month allows me to minimize risk better than once a year.

Is there a limit to this? Obviously at some point the cost of trading and the bid/ask gap will overwhelm the alpha I generate by rebalancing, and there are tax consequences, but assuming these are reasonably easy to calculate, is there a formula that will tell me what the optimal frequency is?

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    Side note to your question - self-selecting 30 investments and managing them is a significant obligation you are putting on yourself, considering that managed funds (or even non-managed low-fee index funds) exist to provide diversification at low-effort. A question like this suggests some newness to investing which suggests you might be getting in a bit over your head with all of this. Oct 7, 2021 at 13:40
  • Thanks for the concern.
    – Quark Soup
    Oct 7, 2021 at 22:21
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    Don't you think this question is more appropriate for e.g. economics.SE?
    – swineone
    Oct 8, 2021 at 1:13
  • Or more like Quantitative Finance given what has been subsequently revealed in comments
    – AakashM
    Oct 8, 2021 at 7:51
  • Given the excellent answers given here I closed the cross-post on Quant.SE.
    – Bob Jansen
    Oct 9, 2021 at 16:34

3 Answers 3

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You are applying theoretical financial concepts to the practical world. There are two (potentially significant) factors you haven't mentioned at all which can create significant 'slippage':

  1. Transaction costs. Any transaction costs at all in a small portfolio will tend to benefit from less frequent balancing.

  2. Tax consequences. Assuming your investments are in regular taxable accounts, doing solely what you are suggesting would get very nasty indeed. Consider that by mathematical law, every rebalancing includes a sale of something that gained, and a purchase of something that lost. So you would continually pay taxes on your gains, and never have losses to offset those gains. In the US and some other jurisdictions, these gains might even be short term instead of long-term [ie: higher tax rate], because of the shorter duration of holding.

In addition to these two direct impacts, consider also the level of personal obsession you would need to perform daily rebalancing. The time investment alone would be significant (even 'just 10 minutes a day' would be about an hour a week, which at something like $10 / hour in minimum-wage-equivalent could be considered about $500 of additional time cost per year - let alone the additional 'always thinking about it stress' that comes along with 'actively managing' your account). Given that typical fund movement would be < 2% on an average day, if you did this daily, that significant time investment would be to move 2% of your portfolio one way, and 2% another way.

Whether to rebalance (and how often) is not something with a concrete objective answer; however a simple rule of thumb would be to not do this more frequently than you are already setting aside time to revisit your financial goals. Consider lining this with an annual review of your long-term investment planning, or perhaps at most a quarterly review of your financial statements received from your broker.

  • Update #2 *

Now you're saying you have a real-time automated system. In that case, "Is there a formula for how often I should rebalance?" is the wrong question. Given that most of the inputs to this formula are dynamic, looking for a static solution is sub-optimal. It would make more sense to calculate on the fly, "How much more money would I make by rebalancing now?", and also, "How much would it cost to rebalance now?" Presumably "how much would I make?" is an estimate, a prediction, based on your theoretical model. "How much will it cost?" can be calculated by seeing how many trades would have to be made multiplied by the cost of each trade, how many trades would result in a short-term capital gain that could at least potentially be deferred to make a long-term capital gain, whatever other costs may exist. Then you could say that, for this particular set of data, rebalancing now either is or is not cost effective.

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  • I mostly agree, but I'd quibble about the tax point. Yes, if you turn a long-term gain into a short-term gain, that's going to cost you something. But I don't see how having more taxable events of itself would hurt you. If you buy X for $1000 and sell a year later for $1100, then buy Y for $1100 and sell a year later for $1200, versus you buy X for $1000 and sell 2 years later for $1200, either way you have a total of $200 in capital gains. Assuming your marginal tax rate hasn't changed -- which may be a big assumption -- your taxes would be the same either way.
    – Jay
    Oct 7, 2021 at 17:33
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    @Jay But rebalancing expressly accelerates recognition of income, and defers recognition of loss. Higher frequency = faster acceleration. If asset A drops 20% to June, then rises back up to original value by December, you will have ultimately rebought at lower and lower prices [recognizing taxable gains on your other assets which hopefully are diversified and in some cases showing coincidentally offsetting gains], then sold some or none up to December [depending on whether your portfolio as a whole had compensating changes]. Oct 7, 2021 at 18:59
  • First, let me level set the conversation. I'm building this for a mass audience, not my personal trading. I have an AI network capable of not only optimizing and rebalancing, but generating the trades and checking sophisticated compliance rules nearly instantaneously. This includes IMA level checks like cash balances, issuer concentration and the like. So we've got the technical side of this covered.
    – Quark Soup
    Oct 7, 2021 at 22:37
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    @Quarkly: When you ask questions on a site about "Personal Finance," you are going to get consumer-grade answers, i.e. answers that are intended to help with personal finance, not commercial finance. What did you expect to happen?
    – Kevin
    Oct 8, 2021 at 4:08
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    @Quarkly I'm going to go ahead and gently suggest that a belief that you have an "AI Trading Algorithm" doesn't exactly line up with "what's the formula to tell me when to start and stop rebalancing?" Naive questions receive simple answers. Oct 8, 2021 at 12:49
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I'm not familiar with the particular algorithm you're using to balance your portfolio. It's probably irrelevant to my answer so I haven't bothered to look it up. Apologies if that assumption is wrong.

Every time you re-balance you face certain costs:

  1. There may be trading fees, depending on what broker you're using. These fees could swallow any gain or loss if you trade too frequently. Like there's a brokerage I use for mutual funds that charges some modest sales fees. They'll also trade individual stocks for me, but their minimum fee per trade is $60. I'm a little guy so my typical stock trade is $1000. If I had to pay $60 to buy and another $60 to sell, that would subtract 12% from any gain right off the bat. (That's why I don't use them for individual stocks, I use another broker that charges zero for most trades.)

  2. Tax rates are considerably different for long term capital gains for sort term. If you balance too frequently you're going to be converting long term gains into short term, and thus increasing your taxes.

  3. Your time. How much time are you spending calculating this and executing the trades? And how much is your time worth? I'm suddenly reminded of a friend of mine who decided that to save money she would grow her own vegetables in her back yard. She said that when harvest time came, she spent an entire day picking, cleaning, and canning the vegetables. And when she was done, she had $11 worth of canned vegetables. For like 12 hours work. Not even counting the time she spent planting and cultivating, or the cost of seeds and canning supplies and whatever.

So even if it was true that by re-balancing every day you would increase your profits ... how much time would that take to increase your profits by how much?

You could probably construct a formula that would take trading fees into account, but the tax difference is more complicated and the value of your time is subjective. So I think it would be tough to come up with a formula that you could really solve for x and get a definitive answer.

ADDENDUM

Okay, IN PRINCIPLE the calculation is simple. Calculate how much it costs to do the rebalancing, between transaction costs, value of your time, tax implications, maybe you can come up with other costs. Calculate how much more money you make by doing a rebalancing. Presumably the more frequently you do it the more profitable. That is, perhaps if you rebalance once a year this makes you $1000 more than if you never rebalanced. (Just making up numbers here, of course.) If you rebalance twice a year you make $1500 more. Rebalance 4 times a year you make $2500 more. Etc. I'd assume that with more frequent rebalances each one would be worth less but the total would be larger. A graph would rise steely at first but then taper off, asymptotically approaching some theoretical maximum if you rebalanced an infinite number of times.

The optimum frequency of rebalancing then is the point where the profit from each rebalance is equal to the cost of rebalancing.

The catch is that neither function is easy to calculate. How do you know how much more money you will make by rebalancing one more time? How do you calculate exactly how many trades a rebalance will require, other than empirically? Etc.

So I think in practice it would not be possible to write a formula and get a single answer. You could only solve the problem by experimentation with real-world data, and then get an approximate solution assuming that your customer's data resembles your own reasonably closely.

And just by the way, all of this assumes that your "system" really works and consistently produces higher profits than simply leaving your money alone.

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    @Quarkly maybe add this information to the question? Though I somehow doubt it then still fits to "personal finance". Oct 7, 2021 at 23:12
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    @Quarkly Actually whether you do it with software or do it with pen and paper makes a great deal of difference to the final answer. At least, if you agree that the time required to do the calculation and the value of your time is relevant. Presumably -- hopefully -- doing it with your software product is significantly faster than doing it with pen and paper.
    – Jay
    Oct 8, 2021 at 3:59
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    @Quarkly Theoretically we should be able to calculate the expected return based on historical data Ah yes, validating hypotheses with the same data that created them. I strongly suggest running simulations against real time data - especially if the safety of other people's savings and investments are going to ride on the same train.
    – Stian
    Oct 8, 2021 at 6:50
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    @Jay Just FYI,you can assume most climate scientists actually aren't idiots. What you're proposing is completely standard for weather models (say, tune parameters on data from 1921-1970 and then predict and validate on 1971-2020). Just blindly starting from current data and predicting the future would get you laughed at.
    – TooTea
    Oct 13, 2021 at 7:58
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Unless you are an institutional investor, transaction cost and time will cost more than you'll earn from optimization.

As you've pointed out yourself bid/ask spreads and taxes will eat up some of your gains. You'll also have transaction fees and the time it takes to do the rebalance. You'll pay taxes on capital gains the moment your "paper" stock is converted to money

Even if you're on a "free-trading" platform like Robinhood, the trades aren't really free. Robinhood and other free brokers make money off of the bid-ask spread by going through "market makers", who buy stocks, jack up the price, and resell them. The fee is priced into each stock you buy.

Finally, even if there was a truly free platform, your time is not worthless. Unless you're managing millions of dollars, optimizations may not net you much benefit. Even with automated help you'll still have to make sure there's a profit margin bigger than the bid-ask spread, and that you'll see a profit after paying taxes. That takes time to setup and any automated tool will need a lot of hand-holding.

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