Munger said: "When the world gives you an opportunity, smart investors make big bets. When they have a great chance of winning, they make big bets. The rest of the time, all they do is wait. That's it." Simple." Buffett also said: "When gold falls from the sky, remember to use a bucket to catch it, not a thimble." In fact, Buffett not only said this, but also did it. Look at Buffett's purchase of Coca-Cola shares in 1988. He invested 1/3 of his capital in it; when investing in American Express, he invested 40% of the net value of the partnership. These investments are classic applications of the Kelly formula in investment.
In fact, the Kelly formula has been cited many times by Buffett, Charlie Munger and Bill Gross. It is a formula for calculating the optimal betting ratio based on odds to obtain the highest investment returns.
Introduction
**Kelly formula (**Kelly criterion, Kelly strategy or Kelly bet), also known as Kelly equation, is originally a formula that calculates the optimal proportion of funds for each bet to all gambling capital based on the probability of winning or losing in gambling. , proposed by AT&T Bell Laboratories physicist John Larry Kelly in reference to Shannon's work on long-distance telephone noise, and published in the Bell System Technical Journal in 1956. In addition to maximizing long-term growth, this equation does not allow for the possibility of losing all available capital on any gamble. Kelly's formula was later applied to blackjack and the stock market by another of Shannon's colleagues, Edward Thorpe.
There are two forms of Kelly's formula:
- Gambling version: When you lose your bet, your bet will return to 0
f ∗ = b p − q b f^* = \frac{bp - q}{b} f∗=bbp−q
in:
f* is the capital ratio that should be bet
b is the odds available for betting, that is, the profit-loss ratio, average profit/average loss
p is the winning rate, that is, the probability of winning
q is the defeat rate, that is, 1 - p
If the odds have no advantage, that is, b < q / p, and the result of the formula is negative, then the formula recommends not to place a bet. If the odds are negative, that is, b < 0, it implies that the bet should be on the other side.
In addition, in the case of losing all or doubling the bet, b=1, the formula is simplified as follows: f* = 2p-1, this formula is also mentioned in the book "Warren Buffett's Investment Portfolio".
- Investment version: When an investment error occurs, the investment can be stopped in advance without returning to zero.
f ∗ = p r l − q r w f^* = \frac{p}{rl} - \frac{q}{rw} f∗=rlp−rwq
in:
f* is the capital ratio that should be bet
p is the winning rate. The probability that the selected target will rise, that is, the probability that the target will hit the take-profit price first
q is the defeat rate, that is, 1 - p. The probability of the selected target falling, that is, the probability of the target hitting the stop loss price first
rw is the winning rate rwin, which is the rate of profit when you win. For example, if the asset increases from 1 to 1+b, and the take-profit increase is 5%, then rw is 5%.
rl is the loss rate rloss, which is the rate of loss when you lose. For example, if the asset is reduced from 1 to 1-c, and the stop loss is 2%, then rl is 2%.
It can be seen that b=rw/rl in the simplified version of Kelly's formula. When rl=100%, the investment version of Kelly's formula becomes the gambling version of Kelly's formula. In some places, it is written in the following format, which is the same as above, but with different letters. Among them, Pwin corresponds to p, Ploss corresponds to q, b corresponds to rw, and c corresponds to rl.
f ∗ = P w i n c − P l o s s b f^* = \frac{Pwin}{c} - \frac{Ploss}{b} f∗=cP w i n−bPloss
Application of Kelly formula in investment
- The Kelly formula is more beneficial for fund management, can fully consider opportunity costs, and is suitable for asset allocation.
- As Buffett advises, investors should wait patiently until the best opportunity presents itself, and then dare to make big bets
- Investments with negative odds are not worth participating in
- The Kelly formula targets the proportion of bets you can afford to lose, not the total assets you own.
- The Kelly formula can be timed. Even for companies with investment value, there are times when they are overvalued and undervalued, and the corresponding winning rates and odds will be very different. You can use the Kelly formula to compare timings.
- The Kelly formula can be used not only in the stock and commodity markets, but also in venture capital. For example, if, as a venture capitalist, you find a start-up company in the angel round, there is a 70% probability that it will double its valuation and earn 200% income; there is only a 30% probability that you will encounter If you lose money, your investment will be in vain and you won’t get back a penny. How much money should you invest when you have 100 million to participate in investing in this company?
Disadvantages/Limitations of the Kelly Formula
- Both the winning rate p and the odds b are relatively subjective. Each investor may recognize p and b differently for the same target. Optimistic and aggressive investors will estimate the winning rate and odds higher.
- The winning rate p and odds b cannot be measured accurately. Using historical data for backtesting is one way, but I don’t believe it. Because history does not represent the future, the future is unknowable
- Both the winning rate p and the odds b change dynamically in investment. An unexpected event may greatly change the winning rate and odds.
- The investment version of the Kelly formula cannot resist black swan events. This is very different from that in a casino. The casino owner limits the maximum odds at the gaming table, but the market is different, because when the odds become more favorable, the proportion of bets obtained by the investment version of the Kelly formula will become larger and larger. In the event of a black swan event, it is easy to liquidate your position, especially if you increase leverage. Even with careful money management like the Turtles, they lost 65% in 1987
- It does not consider the correlation between multiple investments in the portfolio, but multiple targets in the market are correlated, which means that investors' risks will be further increased.
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