# Formula like Kelly criterion for choosing how much of an investment to liquidate?

I made a few investments, and surprise, surprise, a few of them shot up in value. Now I'm faced with the dilemma of the greedy: do I sell now and capitalize on existing profits, or do I wait, risking my existing gains in the hopes of gaining more?

The optimal answer, intuitively at least, is somewhere in the middle. Sell some portion of your newly appreciated investment, capitalizing a little, but leaving some in the pot to hopefully grow. This sounds an awful lot like something there'd be a handy dandy formula for, maybe plug in a few values and you get a mathematically or game-theoretic optimal proportion of the position to sell off.

Does such a formula exist? Perhaps a few that fill this function?

• The efficient frontier roughly approximates this: investopedia.com/terms/e/efficientfrontier.asp Mar 22 at 10:12
• Doesn't the Kelly criterion already apply here? I thought it tells you how much to invest at each point in time.
– usul
Mar 22 at 18:44
• There is no correct answer. But if you're concerned about having a few large positions dominate your returns, you could always sell some of those and add to your other bets. (This general thing would be called 'rebalancing').
– Kaz
Mar 22 at 19:03
• quant.stackexchange.com Mar 23 at 15:20

There is a simple answer: Look at your stocks and decide whether you would buy them now if you did not own them yet.

If your answer is "why should I pay this crazy price?", it is probably time to sell. If it still looks like a good investment, keep it.

• Yup, if the demand isn't there, it won't go up any further. Mar 22 at 16:36
• I do not see how this answers the question in any way. The same issue is present when choosing to buy a stock, namely "What percentage of my portfolio should I invest in a stock I have determined is a good buy?" Mar 22 at 17:57
• @BradyGilg the answer is precisely that this is the same issue as investing in the first place. We know that how you invest exactly is an unsolved problem, but the OP had an investment strategy that has paid off. The answer says to rethink the investment strategy with the information known today and rebalance accordingly.
– csiz
Mar 23 at 2:53
• Weighting of single assets within an asset class is pretty much arbitrary. Market cap is a good indicator as it will bring your return closer to the market average. Equal weight is just as good. In the end, taxation and brokerage fees can influence this weighting as well (e.g. preferable tax treatment for local stocks suggests overweighting them) Mar 23 at 8:50

I don't think the Kelly criterion is something most investors should consider for portfolio allocation. The counterpart is the efficient market hypothesis. I don't know how efficient markets actually are, but they are probably efficient enough that most investors can't apply the Kelly criterion in a meaningful way!

Instead, I would take a look at guidelines for portfolio rebalancing. Rebalancing less frequently lets your winners keep winning for awhile. You should find good advice here and via Google searches about how often you should rebalance.

In addition to rebalancing, you could also decide to change your target portfolio allocation, and this is akin to Manziel's answer.

• "I don't know how efficient markets actually are, but they are probably efficient enough that most investors can't apply the Kelly criterion in a meaningful way!" – What makes you say that? The Kelly criterion and the efficient market hypothesis not in conflict with each other. It's totally possible to have a perfectly efficient market where the Kelly criterion applies. In fact, I can't think of how you could have any kind of market where the Kelly criterion does not apply. Mar 22 at 15:42
• To my understanding the Kelly criterion is based on the assumption that there is a certain predictable risk rate such as a 60% coin toss. In a perfectly efficient market there is no predictablity. Any price can go up or down randomly, or to be precise, relative to the market average performance will be better or worse randomly. If you can't predict anything (perfect efficiency) there is no basis for risk-based allocation because you have no numbers for the risk Mar 22 at 17:06
• @Manziel "In a perfectly efficient market there is no predictablity." – No, that can't be right. Imagine that there's some tree which has exactly a 50% chance of bearing fruit each year, and somebody is selling one of these trees. If, as you say, a perfectly efficient market can't have predictable risks, then the very existence of this tree (and the ability to sell it) constitutes a market inefficiency. That doesn't sound right to me at all. Mar 22 at 17:17
• Maybe I am wrong but to my understanding the practical application of the Kelly criterion requires a mismatch between risk and reward. If I have a perfect coin toss (50% risk, double reward) there is no clever strategy that will on average improve my expected return. Only if I have a mismatch (e.g. 60% head, 100% reward) strategy makes a difference. In a perfectly efficient market there can be no mismatch between risk and reward or there would be an arbitrage opportunity. No matter how risky a single investment may be, this idiosyncratic risk can always be diversified away Mar 22 at 19:32
• Thank you @Manziel for explaining my answer so well! Mar 22 at 21:19

If you divide the market into asset classes and study the long-term returns and covariances of those classes, then a Kelly-like formula will tell you the optimal fractions of your portfolio that belong in each of those classes.

For example, you could decide that you want to invest in a mix of S&P500 stocks and intermediate term government bonds. (Hopefully through low cost ETFs.) Your Kelly analysis will tell you something like: "The optimal investment is 70% stocks and 30% bonds."

Great, now you have a strategy. You might also notice that the optimal peak is broad. It doesn't matter significantly if your ratios slip out of tune by 5% either way.

So now invest your capital accordingly, and revisit it every six months, or even annually, and "retune" your portfolio. Sell and buy enough of each class to move back to your optimal split.

You will be selling off a part of the winners when necessary, just as you felt you should. You will always be nearly optimally invested.

To expand on @Menziel's answer, you have to consider opportunity cost, taxes, and your personal wants and needs.

1. Opportunity cost is what Menziel was leading towards. If you have a fixed amount of money today, would you rather invest in the company you already own or a different company/asset that you think will do better. That said the world is uncertain, so your judgement on which company does better will be fuzzy. Particularly there's a chance the asset you have will do better than any other asset you know, so you have to incorporate that thesis in whether you decide to sell and how much.
2. Taxes are the thing that Menziel disregarded in his answer. However they have a disproportionally large effect the more the asset that you own has increased since you bought it. For a concrete example imagine you held bitcoin since it was worth pennies and you decide to sell now and pay nearly 20% of the value in capital gains taxes. If you have \$100 now then \$20 goes to taxes and \$80 goes to a second investment. If you invest in a different asset and it doubles, you'll have in the end \$144 (\$160 - \$16 new taxes), however if you held on to the bitcoins and bitcoin doubled, you'd have \$160 (\$200 - \$40 taxes). The tax "penalty" means the competing investment opportunity needs to be 20% better.
3. Finally there are personal wants and needs. If you want a fancy car now and you have the money for it from investments, then obviously you could sell stocks and buy the car. Or worse case you or your spouse gets cancer so you have to liquidate to cover treatment... This point is obviously personal preference or circumstances.
• This is good answer (upvoted). Although such an answer cannot reasonably handle every case, there are 2 points that I think may be important to add: (1) whether or not there are capital gains tax likely depends on the taxing laws of the region, (2) since taxes are often not due right away, one could reinvest the entire amount instead of the post-tax amount - until the tax is actually due; doing this would involved increased risk as one could be faced with a tax bill without sufficient assets to pay the bill. Mar 23 at 8:53

To reiterate and expand on gaefan's answer. You set a target portfolio allocation. Periodically you examine the values of the allocations. Cull the high values (take profit!) and distribute the gains to lowest value allocations. This allows you to sell when high and buy low, without trying to time the market. You simply, on a pre-planned take take profits and buy-in to undervalued stocks.

For example you might invest in 3 stocks. And you allocate them evenly 33% 33% 33%. When you compare their values 3 months later you see that they are now 50% 30% 20%. You would then sell off enough of the 1st stock to reduce its total value to 33% of your total investment and then use the cash proceeds to buy into the 2nd and 3rd stocks to get them up to 33% of your total investment.

• That implicitly increases leverage on the worst-performing stocks while decreasing leverage on the best-performing stocks, in effect investing most of your money in the losers. Why is this a good strategy? Mar 23 at 18:33
• You don't own stocks that you expect to go down. You own stocks that you have evaluated and determined to be undervalued. Therefore you are buying into stocks that YOU have designated as having growth potential. Mar 23 at 21:11
• @TheEnvironmentalist Because past performance is an extremely bad predictor of future performance. If a given stock went down over the past quarter, that tells you essentially nothing about what it will do in the coming quarter. As a result, past performance is irrelevant to your strategy. Mar 23 at 23:11

The Kelly Criterion implies you have some sort of estimate on these probabilities when in reality nobody knows this. There would be a ton of variables involved in estimating the probabilities of equities being at certain values and at the end of the day they would never be definitive numbers.

You should ideally be searching for some sort of probability agnostic allocation formula which I'm not sure even exists (or at least to the extent that it can be mathematically proven as being the best possible action like Kelly's derivation makes clear). The efficacy frontier is probably your best bet as someone mentioned.

If you're using options at all and want to maintain certain a predetermined risk profile, you can look into what's called "beta weighting".