Kelly Bet explained

There are two reasons for being risk aversion.

The first reason is about “Utility Theory”. If you only have $1000, losing $100 is a big deal, but if you have $1,000,000, losing $100 is nothing. So the same amount of money is less valuable when you have more money. So gaining 10% always gives less value than losing 10%, because the bigger the asset, the less the value of the same amount of incremental asset.

The second reason is the asymmetry of gain and loss. If you lose 50%, you have to gain 100% to get back to even. This is because the returns are multiplicative, not additive. In another word, we want to maximize the expected log-asset, not expected asset.

The basic Kelly bet formula is just  capital_allocation_percentage = mean/variance. (“Mean” is expected excess return)

However, we can’t simply use it as is for the following reasons:

1. Kelly bet only gives you an “upper bound”, and that upper bound is very big. 

Example: If you are playing a binary game (double or lose everything), and you have 60% chance win and 40% chance lose, your expected return is 0.6 – 0.4 = 0.2. Then your variance is 0.6*1*1 + 0.4 *(-1) * (-1) – 0.2*0.2 = 0.96, and the percentage of capital allocation is 0.2 / 0.96 = 19%. In another word, you should bet 20% of all your money in one single bet. That is really aggressive.

This number is still valuable though, since it tells you no matter how aggressive you are, more than this number will only bring you worse result, not better. It also tells you if you have a very large number of bets at the same odds, this is the optimal bet size. Less than this bet size will NOT make it safer for you, only give you less returns.

Now what it means for stocks? Assume BRKB is much undervalued, and there is 50% upside. The mean return is 50%. Now assume BRKB is so safe that the standard deviation is only 35% of stock price. The kelly bet is 0.5 / (0.35 * 0.35) = 4  or 400% of your capital!!

This means no only you should put all your money into it. You should also borrow 3 times of your capital to bet on it. That is assuming there is no liquidation calls when it goes down.

Apparently this number is too big. We can’t use this number in practical case, and almost nobody uses it except in high frequency trading.

That said, it does give a general sense about how the capital should be allocated. For example, if a stock is twice more risky, then we should only put 25% of the position size.

For example, if BRKB is $130 right now, and we think its eventual share price in a year is likely to be in $100 to $180 (assume equal chance in this range), kelly bet gives about 25% of capital size. Some people choose to use half kelly bet, or 12.5%. For an value investor, this size is still too much, since the mean is $140, so only 7.6% upside. Nobody should put 12% of capital into a stock with only 7.6% upside right? Especially when the likely downside is 12% (half of the maximum drop).

Even though Kelly bet is too big for practical cases, the math here remains true. Meaning we should use a bet that is inversely proportional to the square of risk, and proportional to the mean return.

2. Kelly bet assumes you get large number of opportunities within a reasonable time frame, plus with known fixed odds.

3. We have to be careful about using leverage and the maximum downside.

If it can ever get to complete loss (due to leverage usually), then since anything times zero is zero, the eventual result is zero. This is certainly not the optimal result. So whatever the distribution we use, we must make sure it can never gets to zero.  Even the distribution we use doesn’t lead to a zero, in real life, the distribution is never fixed, so it is always possible to lead to a zero when distribution changes temporarily.

In math, when it gets to zero or negative number, the log function applied on it will be undefined.

The beauty of Kelly Formula is that it gives an optimal bet no matter how much risk appetite a person has. So it is irrelevant to risk aversion factor or utility function.

However, if it could ever gets to zero (no matter how small the chance is), the final result is zero. Apparently, in this case, it does matter to be more risk averse.

In another word, in the classic Kelly Setup, you take “no risk” if you play the game long enough, since final result is almost a given. But if it could end up with a zero at any point, it will have a chance to be zero, the final result is not a given. In fact, if you play infinite number of times, the final result is always zero!

Another way to think this, Kelly is try to maximize the expected log-asset, once that number is zero, log(0) is undefined.

So the question is why do we have such a big difference between Kelly bet and realistic bet we should use? Didn’t I just say less than Kelly bet will not make it safer, only reduce overall returns?

Here are the reasons:

a. Kelly bet assumes you can bet many times (> 100 or at least > 40) within a reasonable time frame, and each bet has same known odds. This is OK for casino case, but not for long term investing. For trading, you can do many times, but still each time it may have different odds since market is always changing. If you can’t bet many times, lets say you bet only 3 times within 10 years, you will not have “the law of large numbers” to help you, and then the fluctuations caused by bad luck will be really damaging to you. So in that case, you would need to be much more cautious. In real life, we don’t know the odds and we could be overly optimistic especially when it is fundamental investing, not technical trading, as there is no historical data to backup our estimate. Even if we know the odds, the odds will be different for different stocks at different times. However, among all the three factors (unknown odds/variance, small number of available bets, different odds over time), the unknown odds and its variance is the most significant factor usually, that means we better use a pretty conservative odds/variance estimation before we can apply Kelly Bet!

 

b. The mean/variance formula is not a precise formula, it is approximate, but usually it is a good enough approximate number. However, during high leverage case, it may get more complicated. Since the kelly bet is optimizing the expected log asset, if asset gets to zero or negative, it is undefined!! So anytime when we apply leverage, there is a chance for asset to go to zero, and therefore kelly bet may not apply. That is why using kelly formula can be risky or conceptually wrong when used in high leverage bet.

c. The psychological challenge is too big if the fluctuation is too big.

d. As mentioned above, we have to care about the possibility of changed distribution and that may cause a wipe-out event. Any possibility of wipe-out event will break the promise of optimal result given by Kelly Bet.

What about for multiple stocks in a portfolio? The good news is that kelly bet is additive. So if for BRKB, kelly bet says you should put in 20%, and for UBNT, kelly bet says you need to put in 10%, you should just do so. This is assuming the sum of both proportion ratio is less than 100%. In this case, the sum is 20% + 10% = 30%.

What if Kelly bet says to put 80% to BRKB, and 70% to UBNT? If the sum goes beyond 100% and you don’t want to use leverage, you can scale it down accordingly. So use 80% / 1.5 = 53% for BRKB and 47% for UBNT. However, this is not really optimal in long term growth. For optimal long term growth, usually we need to put more into the one that gives higher expected excess return, and in this case, we may need to put more into UBNT since it gives higher return. But this would certainly increase risk. Linearly scale-down usually gives better/optimal risk profile within single period, but sacrifices the long term growth.

The above is assuming there is no correlations between stocks, if you have strongly correlated stocks such as two banks in the same industry, the correct formula is to multiply the expected return vector by an inverse of covariance matrix. That math is getting a bit more complicated.

Another word of truth unrelated to this: since the average correlation between stocks is about 15%, math shows that over-diversification will not help. Having a portfolio of 7 stocks is not much different from having a portfolio of 700 stocks, in terms of risk diversification, assuming these 7 stocks have average correlation (15%), and equal sized positions.

 

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