Risk-Constrained Kelly: Growth Subject to a Drawdown Limit
How does a drawdown constraint alter the capital fraction suggested by Kelly sizing?
Set the probability that wealth falls below a specified fraction of initial capital, then solve for the largest growth-seeking stake satisfying that bound under the assumed return distribution. This differs from a high-water-mark or intraday prop-firm rule.
The convex program uses a conservative bound on the initial-wealth drawdown probability; it is not an exact high-water-mark prop-firm rule.
Evidence map
| Aspect | Finding |
|---|---|
| What it is | The reformulation that makes the growth-optimal criterion usable by adding an explicit constraint on the probability of loss, and shows that the resulting problem is still tractable. |
| Key result / formula | Busseti, Ryu and Boyd add to the Kelly objective a constraint bounding the probability that wealth ever falls to a given fraction of its starting value. |
| Why it matters for backtesting | This converts a rule of thumb into a stated trade-off, which a decision aid could present after its actual implementation is verified. |
What it is
It is the constructive answer to the standard objection that full Kelly is unbearable.
Key result / formula
The key technical step bounds the probability of crossing that initial-wealth floor. Substituting the bound for the original path-probability constraint gives a conservative convex optimization problem: a solution satisfying the surrogate satisfies the stated risk limit under the assumed outcome distribution, although the surrogate need not be the exact optimum of the original constraint. The selected fraction may be smaller than full Kelly as the permitted drawdown risk tightens. The authors compare the constrained solution with fractional Kelly. The result depends on the assumed outcome distribution; neither approach repairs a misspecified edge. They also show the cost in growth of tightening the constraint, which is the quantity a user actually wants to see.
Why it matters for backtesting
An agent need not ask a user to pick a Kelly fraction; it can ask for the drawdown they can tolerate and the probability they accept of exceeding it, then report the growth given up to respect it. That framing is compatible with the rest of this folder: the drawdown constraint concerns wealth falling below a fraction of its initial value. A prop-firm rule may instead use a moving high-water mark, an intraday breach or a finite deadline, so its pass probability needs a separate path model (see [Prop-Firm Challenges: First-Passage View vs Sharpe]). The caution is that the inputs remain estimates: the constraint is no better than the return distribution it is computed from, and that distribution is the backtest's, with all the selection that implies. A sensible practice is to impose the constraint on a deliberately pessimistic version of the distribution.
Source
Busseti, Ryu & Boyd, "Risk-Constrained Kelly Gambling", Journal of Investing 25(3), 2016, 118-134. Primary source