Maximise expected log wealth in a fictional repeated game; compare fractional Kelly, overbetting, uncertain probabilities and joint portfolio allocation.
Kelly criterion
The Kelly criterion chooses a stake or portfolio allocation to maximise the expected logarithm of wealth. For repeated opportunities with a known, stable probability law, it targets long-run compound growth. It is a particular objective for position sizing: a positive expected return alone does not tell you how much wealth to expose.
The distinction matters because maximising expected wealth and maximising expected log wealth can produce very different decisions. Kelly's original paper illustrates that tension with repeated bets; a portfolio application must additionally describe joint returns, funding and feasible exposures. Kelly (1956), pp. 919–920.
A favourable game that punishes oversized bets
Consider a fictional even-money game, with independent plays and known, unchanged win probability p=0.6. Each unit staked earns one net unit on a win and loses one unit on a loss. Stake a constant fraction f of current wealth before each play; leave the rest idle at zero interest. Wealth is divisible, and there are no costs, borrowing, withdrawals or minimum stakes. For this one-sided game take 0≤f<1.
If Kn of the first n plays are wins, the exact wealth is
Wn=W0(1+f)Kn(1−f)n−Kn.
The realised count Kn is random; it need not equal 0.6n. The law of large numbers gives Kn/n→0.6 almost surely, so
n1logW0Wn⟶g(f)=0.6log(1+f)+0.4log(1−f).
Here and throughout, log is the natural logarithm. Differentiate:
g′(f)=1+f0.6−1−f0.4=1−f20.2−f.
The derivative vanishes at f∗=0.2. Since g′′(f)=−0.6/(1+f)2−0.4/(1−f)2<0, this is the unique maximum. The fraction is reset after every outcome: starting with 100 game units means staking 20; after a loss, wealth is 80 and the next stake is 16.
Fraction of current wealth
Expected log growth per play g(f)
0.10: half Kelly
0.015042
0.20: full Kelly
0.020136
0.40: double Kelly
−0.002447
These are log-growth coefficients, not annual returns or guaranteed returns on an individual play. The limiting geometric wealth multiplier per play is eg(f).
The fictional 60–40 even-money game has maximum expected log growth 0.02014 at f equals 0.2; half Kelly at 0.1 gives 0.01504, double Kelly at 0.4 gives minus 0.00245, and the positive zero-growth boundary is about 0.38939.
Original calculation for the fictional iid game. Expected natural-log growth peaks at 20% staked; the nonzero zero-growth boundary is about 38.939%, below double Kelly.
Oversizing can therefore produce negative long-run growth in a positive-expectation game. At f=0.4, wealth tends to zero almost surely under these assumptions. It stays strictly positive after every finite number of plays, however: approaching zero is different from hitting zero. The exact zero-growth threshold is approximately 0.38939; the familiar “twice Kelly” boundary comes from a quadratic approximation and is not exact here.
Why expected wealth gives another answer
In this game, independence implies
E[Wn]=W0(1+0.2f)n.
That quantity increases with f. If an all-in stake f=1 is permitted, it maximises expected terminal wealth, giving W0(1.2)n. Yet the first loss eliminates the entire bankroll: survival through n plays requires winning every time, with probability 0.6n. The small chance of a huge outcome can raise the average while most players have nothing.
Log wealth assigns an infinite penalty to zero wealth. It produces a different objective, not a promise to win over a finite horizon or an answer to every investor's preferences. This expected-wealth versus compound-growth distinction is central to Kelly's original example. Kelly (1956).
Even full Kelly can experience deep drawdowns. Five consecutive losses leave 0.85=32.768% of the wealth held before that run. Half Kelly leaves 0.95=59.049%. These are conditional losing-run calculations, not forecasts of when such a run occurs. Minimum stakes, margin liquidation or a capital floor can end participation before any mathematical asymptotic result becomes useful.
Fractional Kelly and uncertain inputs
Fractional Kelly uses f=λf∗ with 0<λ<1. In the known game above, half Kelly retains about 74.7% of full Kelly's expected log growth, while exposing less wealth on each play. The fraction is an explicit trade-off; half Kelly is not universally optimal for a different objective.
When the win probability is estimated, the distinction is sharper. Suppose the chosen estimate is p^=0.60, but the true probability is 0.55, with all other game assumptions unchanged. The estimated full-Kelly stake remains 0.20, but the true optimum is 2(0.55)−1=0.10:
Chosen stake
Log growth evaluated at the true p=0.55
f=0.20
−0.000138 per play
f=0.10
0.005008 per play
This is a sensitivity scenario, not an estimated market edge. Reducing a stake can cushion an optimistic estimate; it cannot repair a nonexistent edge, omitted loss state or changing payoff law. Thorp discusses fractional Kelly specifically in relation to uncertain inputs and the consequences of overbetting. Thorp, §7.3.
For unequal odds, if a win earns b>0net units per unit staked and a loss costs the whole stake, maximising plog(1+bf)+(1−p)log(1−f) gives
f∗=bbp−(1−p).
With betting only on that side and cash allowed, a nonpositive result means no bet under this model. The formula is about the fraction exposed to the specified payoff. A trade returning only a small percentage on its notional has a different payoff mapping; its win probability cannot simply be inserted while treating notional as an even-money stake.
Allocating across several strategies
Several simultaneous bets require their joint return law. Computing standalone fractions and adding them ignores correlation and shared loss states.
To state the exact objective, let X be the vector of one-period risky returns measured relative to a deterministic cash account. With risky weights w and the residual in cash, the cash-normalised wealth multiplier is 1+w⊤X. Choose
w∈CmaxE[log(1+w⊤X)],1+w⊤X>0almost surely.
The set C records feasible funding, shorting and exposure constraints; the expectation must be well defined. Repeating a fixed feasible allocation in iid periods with integrable log returns connects this objective to long-run growth. General changing opportunities require a conditional, dynamic formulation.
There is a useful approximation. With μ=EX] and [covariance matrix Σ, a second-order expansion gives
E[log(1+w⊤X)]≈w⊤μ−21w⊤(Σ+μμ⊤)w.
If portfolio returns are sufficiently small, higher-order terms are controlled, and the mean-square term is negligible relative to variance, this becomes w⊤μ−21w⊤Σw. Its unconstrained stationary solution is w≈Σ−1μ, requiring positive-definite Σ. Binding constraints change that solution. A normal model for discrete simple returns does not make this exact: its unbounded downside can violate positive wealth. Boyd and coauthors give the expansion and its assumptions in §4.2. Multi-Period Trading via Convex Optimization, p. 28.
The owned Mean–Variance Optimisation lesson develops the covariance, cash and budget-constraint calculations behind that comparison; its preview is public and its full teaching requires membership. It supplies quadratic portfolio mathematics, while the exact Kelly objective uses the full joint payoff distribution.
Costs, discrete rebalancing, uncertain correlations, funding terms and tail losses all affect the feasible growth calculation. A fraction derived for the fictional game does not identify suitable real-world leverage.
Practice: a low hit rate with favourable odds
A second fictional iid game wins with probability 0.40. A win earns two net units per unit staked; a loss loses one. There are no costs or interest, wealth is divisible, and only this side or cash is allowed. Is the log-growth-maximising stake zero because most plays lose?
Answer. No. Here b=2, so f∗=[2(0.40)−0.60]/2=0.10 of current wealth. The winning multiplier is 1.20 and the losing multiplier is 0.90. The hit rate alone misses the larger winning payoff; using 2p−1 would incorrectly apply even-money odds.
Sources
J. L. Kelly Jr., A New Interpretation of Information Rate (1956), pp. 919–920: repeated fractional stakes, exponential growth and the contrasting expected-wealth objective. The paper uses base-two logarithms; this article uses natural logarithms, which preserve the maximising fraction.
