First off I’d like to thank Jay over at marketfolly.com for posting this article last week (http://www.marketfolly.com/2013/01/position-sizing-utilizing-kelly-growth.html)
I’m going to re-post my article here simply for blog continuity.
Position Sizing Utilizing the Kelly Growth Criterion
One of the more vexing tasks for investment allocators is position sizing. Regardless whether 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 allocators elect to equal-weight investments given uncertainty regarding which investments will perform best. This strategy creates a basket of attractive investments that should profit regardless of which investments in the basket succeed. This method benefits from simplicity and recognizes the future is inherently uncertain. Drawbacks of the strategy include underweighting exceptional investments and overweighting marginal ideas.
Another strategy is to allocate large amounts of capital to the investment ideas with the most potential. This methodology suggests investors should invest proportionally according to their ex-ante return expectations. The advantage of this methodology is matching prospective return to investment size. However, this strategy breaks down when allocators are incorrect about future investment return or risk prospects.
Investment allocators determine their methodology through a combination of portfolio mandate, risk tolerance, and confidence level in investment assessments. While the various approaches implemented are directionally helpful, most are mathematically sub-optimal. There is a better way – the Kelly Growth Criterion.
Kelly Growth 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”). The formula assumes a bimodal outcome of success (“base case”) or failure (“stress case”) over a single time period.
The formula’s power lies in considering both the chance of success and payoff offered. For example the formula would urge a large investment if you had a significant edge given reasonable odds or exceptional odds payoff offsetting a moderate chance of success.
Ok, How Can the Formula be Applied to Investing?
When applied to investing, the Kelly Growth Criterion formula has six inputs. First is simply portfolio size. Second is the amount of capital the portfolio will risk in the pursuit of gain. This amount is also called the maximum tolerable drawdown. For a venture capital group this number will be high while a conservative pension plan would be willing to risk much less. Portfolio size and maximum tolerable drawdown remain constant for each portfolio analyzed regardless of specific investment opportunities.
Next comes four factors 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. The formula is below.
For example, let’s say we are running a $100 million portfolio of US equities. Our mandate is fairly conservative so let’s assume the maximum portfolio drawdown tolerable to be 15% of assets.
We believe General Electric is attractive but are not sure how to size the position within our portfolio. Our team has agreed that our projected gain in the base case is 12% while we would lose 8% in the stress case. Further, the team has agreed the probability of gain to be 55% (base case) and a 45% probability of loss (stress case). Ok, so how much capital do we allocate to General Electric stock?
Given both a strong “edge” (55% probability of success) and advantageous “odds” (12% projected gain in the base case is greater than 8% projected loss in the stress case), the formula suggests we allocate 3.75% of our portfolio to GE. If the “edge” was even, the formula would recommend a 2.50% allocation – again due to the disproportionate odds of success (12% vs. 8%).
What strikes many allocators initially is the magnitude of the size. Is 3.75% really optimal under a scenario with only a 55% probability of success? Mathematically speaking, yes. Why does this seem high?
The recommended 3.75% investment in GE seems high due to the commonality of diversification by funds and investment allocators. Let’s again work an example with our $100 million fund with a 15% maximum tolerable drawdown. Let’s further assume this fund has 100 investments therefore averaging 1.0% per investment. If an allocator takes the view that the “edge” is a coin flip (i.e. probability of success is equal to probability of failure). What would an allocation of 1.00% imply about the expected “odds”?
As shown above, a 1.00% allocation to a position implies just an 11.54% gain in the base case versus a 10.00% decline in the stress case assuming equal odds. These odds are hardly the makings of a scintillating investment.
Why is the Kelly Growth Criterion Rarely Used for Investment Allocation?
Given the formula is mathematically optimal and simple to implement, one might think allocators would embrace the tool. However, investment allocators are not aware of this tool primarily because academic finance has not fully embraced the tool. Secondly, there are a few key weaknesses of the tool.
How Can Inherent Limitations of the Kelly Growth Criterion Formula be Overcome by Investment Allocators?
(1) Ex-ante input assumptions are inherently precise: As with any model, the formula is only as good as its inputs. How can allocators know beforehand whether an investment has a 50% or 55% chance of success? This input must be estimated without an ability to determine the efficacy of the estimate ex-post facto.
The simplicity and power of the formula is a double-edged sword. If investment allocators systematically overestimate the probability of success, long run return will be hampered. The offset of this risk is to estimate projected gains and success probability conservatively. If allocators error on the conservative side, the model will allocate smaller amounts to each investment. This is perfectly acceptable given the model’s proclivity to encourage substantial position sizes.
(2) The formula cannot account for correlation: The Kelly Growth Criterion accounts for an investment’s specific edge and odds. As such, the formula cannot address the relationship between portfolio investments and thus does not account for correlation.
Ask anyone who invested during 2008, correlations rise during a stress environment. If the probabilities of investment success (“edge”) in a given portfolio are correlated, a portfolio allocated strictly according to the Kelly Growth Criterion would be susceptible to risk factors which increase correlation.
There are two mitigants for this risk: (1) Invest in securities with divergent risk factors. If your edge in each investment is not correlated, the formula will provide a strong outcome at a portfolio level. (2) Akin to the mitigants for imprecise input assumptions, estimating a conservative edge and odds for each investment will decrease position sizing in any one security. By avoiding the weaknesses of the Kelly Growth Criterion, the robustness of the formula is enhanced.
(3) The formula assumes a single time period while portfolios are managed more frequently: The Kelly formula assumes a bimodal outcome, success or failure. Portfolio managers often confront prices that meander towards their eventual outcome over time. As prices change, positions sizing will be suboptimal at various times. To compensate for the model’s simplicity, allocators should specify time horizons before entering a position. For example, if hiring a private equity fund manager with an investment horizon of 10 years your Kelly Growth formula will utilize a much longer time frame than if you manage a trading book.
My firm, GrizzlyRock Capital, utilizes long-term, fundamental value methodology. As such, we utilize a period of multiple years when applying the Kelly Growth Criterion. We calculate the value of a business using an upside, base, and stress case and then utilize the base and stress case forecast in the Kelly formula. This conservatism allows the investment to trend towards our base case without our needing to reassess position sizing using the formula.
The Kelly Growth Criterion is valuable to investment allocators given the systematic, repeatable process and mathematically optimal portfolio structure. While a practical tool, the formula is not a silver bullet. When used conservatively, the formula will maximize portfolio growth by allocating capital to the most advantageous investments given both prospective return and risk.
Bill Miller of Legg Mason and Ed Thorp of Princeton Newport Partners (now closed) are investors with stellar track records over decades who embrace and advocate the use of the Kelly Growth Criterion in portfolio allocation. In his recent treatise, Antifragile, Nassim Taleb lavishes praise on the Kelly Growth Criterion: “Kelly’s method requires no joint distribution or utility function. In practice one needs the ratio of expected profit to worst-case return – dynamically adjusted to avoid ruin.”
At GrizzlyRock, we have found utilizing the Kelly formula eliminates our emotional biases towards certain aspects of investing and provides a stable, repeatable investment allocation of capital. Please drop me a line at firstname.lastname@example.org if you wish to discuss further or be added to our distribution list.
Best of luck implementing the formula at your firm!