Investing successfully requires an investor to make many decisions. The first and foremost decision is the amount of capital available for investing, the type of asset (e.g. mutual funds, index funds, stocks, gold etc.) to invest in and construct an investment portfolio. Regardless whether capital allocators select investment managers or individual securities, optimal position sizing is paramount to portfolio success. Small allocations to prescient investments minimize their impact while large allocations to poorly performing investments leads to underperformance
Some investors use a simplified model to construct their portfolio where an equal weight allocation is given to all investments not taking into account uncertainty regarding which investments will perform best. The portfolio is then rebalanced on a regular basis to maintain the equal-weight allocation. This equal weighted strategy benefits from simplicity and it prevents the portfolio from being over-allocated to one particular stock. However, the drawback of this strategy includes underweighting exceptional investments and overweighting marginal ideas.
An alternative strategy is to allocate large amount of capitals to the ideas with the most potential upside. This strategy suggests investors should invest proportionally according to their return expectations. Portfolio constructed using this strategy will tend to be concentrated with most of the portfolio being invested in a few high conviction stocks. The advantage of this methodology is matching prospective return to investment size. Many famed value investors such as Bruce Berkowitz, Donald Yacktman, Mohnish Pabrai and Warren Buffett runs concentrated portfolio where they take up huge position in their high conviction stock picks. One method that they could be using to determine the position sizing is the Kelly Criterion. Mohnish Pabrai in his book, The Dhandho Investor, explain exclusively on how Kelly Formula can be used for portfolio allocation and how investors can benefit from using it.
What is Kelly Criterion?
The Kelly Growth Criterion is a simple formula that determines mathematically optimal allocations to maximize long-term portfolio performance given each investment’s probability of success (“edge”) compared to the amount gained or lost (“odds”). This mathematical formula was developed by John Larry Kelly Jr 50 years ago while working at the AT&T Bell Laboratories. The formula assumes a bimodal outcome of success (“base case”) or failure (“stress case”) over a single time period:
How to compute the amount to invest/bet on a stock?
Let’s assume somebody offers you the following odds on a $1 bet and your bankroll is $10,000.
80% chance of winning $21
10% chance of winning $7.5
10% chance of losing it all
According to Kelly Criterion, the edge is equal to 80% x 21 + 10% x 7.5 + 10% x -1( because we have lost our money)=16.8+0.75-0.1=17.45
Odds= Maximum money that we can win= 21
So edge/odds=17.45/21 x10,000= 8309. So you can bet $8309 on the bet.
Applying Kelly Criterion to Investing
Accordingly to GrizzlyRock Capital, the Kelly Criterion has six inputs when used in investing. The first 2 factors are portfolio size and amount of capital the portfolio will risk in the pursuit of gain. The next 4 factors are regarding the investment itself: the probability of gain in a base case, probability of loss in a stress case, percent of projected gain in the base case, and percent of projected loss in the stress case
For example, you have a $100,000 portfolio of US equities and that your maximum portfolio drawdown tolerable is 15% of all assets. You believe that Cisco is undervalued and attractive. Click here on how to identify undervalued stocks. However you are unsure of how to size the position within the portfolio. Your research of Cisco tells you that the projected gain in the base case is 12% and the loss in the stress case to be 8%. The probability of gain is projected to be 55% (base case) and 45% probability of losing (stress case). Based on Kelly Criterion, 3.75% of the portfolio should be allocated to Cisco.
Evidence of Kelly Criterion used by Warren Buffett and Charlie Munger
Mohnish Pabrai, in chapter 10 (page 78 – 81) of The Dhandho Investor discussed that Buffett’s concentrated bets gives considerable evidence that Buffet thinks like a Kelly investor, citing Buffett bets of 25% to 40% of his net worth on single situations. A stock situation where Warren Buffett invested a huge portion of his portfolio was the Salad Oil Scandal where he invested in American Express. He effectively bet 40 percent of his net worth on a scandal-ridden business making negative headlines daily. What were the odds that this bet offered? Mr Buffett shed some light in his letters to partners from 1964 to 1967:
“We might invest up to 40% of our net worth in a single security under conditions coupling an extremely high probability that our facts and reasoning are correct with a very low probability that anything could change the underlying value of the investment. We are obviously only going to go to 40% in very rare situations— this rarity, of course, is what makes it necessary that we concentrate so heavily, when we see such an opportunity.”
The same thoughts and approach was shared by his partner, Charlie Munger. In a speech at the University of Southern California’s Marshall School of Business, Charlie Munger said:
The wise ones bet heavily when the world offers them that opportunity. They bet big when they have the odds. And the rest of the time, they don’t. It’s just that simple. —Charlie Munger
Pros and Cons of Using Kelly Criterion in Investing
1. Kelly Criterion allows investors to bet bigger on their high conviction stocks
2. Using the Kelly Criterion in position sizing allow the investor to consider behavioral bias. Basic questions like, "how much can we make, what is the downside risk, and what are the probabilities of each", must be answered before any asset is placed in the portfolio
1. Kelly Criterion require investors to estimate six inputs. As with any model, the formula is only as good as its input. If investment allocators systematically overestimate the probability of success, long run return will be hampered
The Kelly Criterion is valuable to investors given the systematic, repeatable process and mathematically optimal portfolio structure. When used conservatively, the formula will maximize portfolio growth by allocating capital to the most advantageous investments given both prospective return and risk.
For more detail on the Kelly Criterion, I recommend reading: Fortune’s Formula by William Poundstone