Allocating Capital Between Independent Strategies with Kelly Growth
Summary
The document considers how to divide a bankroll between two independent strategies to maximize expected logarithmic growth. Each strategy has two possible outcomes: an 8% gain or a 5% loss, with different probabilities of a gain. The allocation weights must sum to one, so the payoff in each joint outcome depends on both strategy weights.
The proposed method is to enumerate the four combinations of wins and losses, calculate the resulting bankroll for each, weight each log payoff by its joint probability, and maximize the expected value. The stated allocation to the first strategy is 0.813. This is an example-specific answer, not a general allocation rule; the result depends on the assumed probabilities, returns, independence, and requirement to invest the full bankroll across these two strategies.
Key ideas
- With two independent binary-outcome strategies, there are four joint outcome scenarios.
- Compute the bankroll in each scenario from both allocation weights and the corresponding gains or losses.
- Maximize the probability-weighted log of terminal bankroll to apply the Kelly criterion.
- The document reports a first-strategy weight of 0.813 for its stated assumptions.
- The allocation depends on the assumed outcomes and independence of the strategies.
Tags
Full text
# finding optimal weight using Kelly criterion # finding optimal weight using Kelly criterion Question: Suppose you have two strategies. Strategy 1 gains 8% with probability p, and loses 5% with probability 1-p, where p = 0.53. Strategy 2 gains 8% with probability q, and loses 5% with probability 1-q, where q = 0.51. The outcomes of the two strategies are independent. You assign weights (which sum to one) for the two strategies. What will the optimal weight be for Strategy 1 (to nearest 0.001), if you goal is to maximize your expected growth rate (i.e. maximize log utility, i.e. the Kelly criterion)? Answer: 0.813 What I have tried: using log utility function I tried to set up an equation for both strategies and maximize the alpha. But clearly, my equation is wrong. Please help! ## Answer by user68819 (score 1) https://quant.stackexchange.com/a/82302 years since you asked this, but was doing something similar myself recently. The solution is really to consider how much bankroll to dedicate to each game. Let f1 be the amount invested in strategy 1, by the constraint you know, 1-f1 will be the amount invested in strategy 2. Lets assume your bank roll is $1. you have 4 scenarios: - up in both cases, your PnL is $1 + f1 * 8$% + (1-f1) * 8% (with probability 0.53 * 0.51) - up in case 1, down in case 2, your PnL is $1 + f1 * 8$% - (1-f1) * 5% (with probability 0.53 * 0.49) so and so forth for all four scenarios. Apply logs, to the payoff and solve for the maximum...and you should get to f1=0.813.
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