Kelly Versus Target-Based Betting Over Three Rounds

Expectation · Hard · Free problem

A trader plays a three-round betting game with a fair coin. At the start of each round $t = 1, 2, 3$, before the coin is tossed, she chooses a fraction $f_t \in [0, 1]$ of her current wealth to wager on Heads at even odds: if Heads, the staked amount doubles (she gains $f_t W$); if Tails, the staked amount is lost (she loses $f_t W$). Her wealth after a round with fraction $f$ is $W(1 + f)$ on Heads or $W(1 - f)$ on Tails.

(a) Suppose she uses the fraction $f = 0.5$ in every round (the problem's stated Kelly fraction for this game). Derive the full distribution of her terminal wealth $W_3$ as a multiple of her initial wealth $W_0$.

(b) Now consider a strategy that stakes a fixed fraction $g \in (0, 1)$ every round, chosen to maximize the probability that $W_3 \geq 2 W_0$. Express this probability as a function of $g$, and find the value(s) of $g$ that maximize it. You may leave the maximizer as the solution to an explicit condition.

(c) Compare the Kelly strategy ($f = 0.5$) and the optimal target-based strategy from (b) along three dimensions: (i) expected terminal wealth, (ii) expected log-wealth, and (iii) probability of reaching

W_0$. Explain the practical implications for finite-horizon trading versus long-run growth.

Hints

  1. The wealth multiplier after 3 rounds is $(1+f)^k(1-f)^{3-k}$ where $k$ is the number of heads. Start by listing all possible outcomes and their probabilities.
  2. To hit the target W_0$, check which values of $k$ (0, 1, 2, or 3 heads) can produce a multiplier of at least 2. For $k = 2$, find the maximum of $(1+g)^2(1-g)$ over $g \in (0,1)$ and compare it to 2.
  3. For a fair coin at even odds, the expected per-round multiplier is always 1 regardless of $g$, so expected wealth cannot distinguish the strategies. Compare them instead via $E[\log(W_3/W_0)] = \frac{3}{2}\log(1 - g^2)$, which penalizes larger bets.

Worked Solution

How to Think About It: This is really a question about what happens when you optimize for different objectives in a multiplicative wealth game. The Kelly criterion maximizes long-run log growth, but over a short horizon like 3 rounds, other objectives (like hitting a target) can lead to very different strategies. The first thing to notice: with a fair coin at even odds, the expected wealth is the same regardless of how much you bet -- you cannot create expected profit. So the only knobs you are turning are the shape of the distribution and the log growth rate.

Quick Estimate: With $f = 0.5$, each round either multiplies wealth by 1.5 (Heads) or 0.5 (Tails). After 3 rounds, the outcomes depend on how many heads you get. The best case is

.5^3 = 3.375$ and the worst is $0.5^3 = 0.125$. Only the all-heads path ($k = 3$) exceeds
W_0$, so $P(W_3 \geq 2 W_0) = 1/8 = 0.125$ under Kelly. Can we do better by betting less aggressively? Well, for $k = 2$ heads the multiplier is $(1+g)^2(1-g)$, and the maximum of this over $g$ is $32/27 \approx 1.185$, which is less than 2. So no strategy can make $k = 2$ heads reach the target -- you always need all three heads. That pins the maximum probability at
/8$ regardless of $g$, as long as $g$ is large enough for three heads to clear the bar.

Approach: We enumerate all

^3 = 8$ equally likely coin sequences, compute the terminal wealth for each, and then optimize over $g$.

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(a) Distribution of $W_3 / W_0$ under $f = 0.5$

Each round multiplies wealth by

.5$ (Heads) or $0.5$ (Tails). If $k$ heads occur in 3 flips:

$\frac{W_3}{W_0} = 1.5^k \cdot 0.5^{3-k}$

| Heads $k$ | $W_3 / W_0$ | Probability | |-----------|-------------|-------------| | 0 | $0.5^3 = 0.125$ |

/8$ | | 1 |
.5 \cdot 0.25 = 0.375$ | $3/8$ | | 2 |
.25 \cdot 0.5 = 1.125$ | $3/8$ | | 3 | $3.375$ |
/8$ |

The distribution has mean:

$E\left[\frac{W_3}{W_0}\right] = \frac{1}{8}(0.125) + \frac{3}{8}(0.375) + \frac{3}{8}(1.125) + \frac{1}{8}(3.375) = 1.0$

This is no coincidence: each round has $E[\text{multiplier}] = 0.5(1.5) + 0.5(0.5) = 1$, so expected wealth is preserved.

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(b) Target-based strategy: maximize $P(W_3 \geq 2 W_0)$

With a fixed fraction $g$ each round, the terminal wealth after $k$ heads is:

$\frac{W_3}{W_0} = (1+g)^k (1-g)^{3-k}$

We need $(1+g)^k (1-g)^{3-k} \geq 2$ for at least one value of $k \in \{0, 1, 2, 3\}$.

  • $k = 0$: $(1-g)^3 < 1 < 2$ for all $g \in (0,1)$. Never reaches the target.
  • $k = 1$: $(1+g)(1-g)^2 = (1-g^2)(1-g) \leq 1$ for $g \in (0,1)$. Never reaches the target.
  • $k = 2$: $(1+g)^2(1-g)$. Taking the derivative and setting to zero gives the critical point at $g = 1/3$, with maximum value $(4/3)^2(2/3) = 32/27 \approx 1.185 < 2$. Never reaches the target.
  • $k = 3$: $(1+g)^3 \geq 2$ if and only if $g \geq 2^{1/3} - 1 \approx 0.2599$.

Therefore:

$P(W_3 \geq 2 W_0) = \begin{cases} 1/8 & \text{if } g \geq 2^{1/3} - 1 \\ 0 & \text{if } g < 2^{1/3} - 1 \end{cases}$

The maximum probability is

/8$, achieved by any $g \in [2^{1/3} - 1, \, 1)$. The minimum such fraction is:

$g^{*} = 2^{1/3} - 1 \approx 0.2599$

Among all maximizers, $g^{*}$ is the most conservative -- it bets just enough for three consecutive heads to double the wealth.

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(c) Comparison of the two strategies

Let the Kelly strategy use $f = 0.5$ and the target-based strategy use $g^{*} = 2^{1/3} - 1$.

(i) Expected terminal wealth:

For any fixed-fraction strategy on a fair coin at even odds:

$E\left[\frac{W_3}{W_0}\right] = \left(\frac{1+g}{2} + \frac{1-g}{2}\right)^3 = 1$

Both strategies yield $E[W_3] = W_0$. No fixed-fraction strategy can create expected profit with a fair coin at even odds.

(ii) Expected log-wealth:

$E\left[\log\frac{W_3}{W_0}\right] = \frac{3}{2}\log(1 - g^2)$

  • Kelly ($f = 0.5$): $\;\frac{3}{2}\log(0.75) \approx -0.432$
  • Target ($g^{*} \approx 0.26$): $\;\frac{3}{2}\log(1 - 0.0675) = \frac{3}{2}\log(0.9325) \approx -0.105$

The target-based strategy has a much less negative expected log-wealth. In fact, for a fair coin at even odds, the true log-optimal fraction is $g = 0$ (do not bet), since every bet has zero edge but introduces variance in log-wealth. The target strategy, by betting the minimum needed, stays closer to this optimum.

(iii) Probability of reaching

W_0$:

Both strategies achieve $P(W_3 \geq 2 W_0) = 1/8 = 12.5\%$. The Kelly strategy overshoots the target when all three heads land ($3.375 W_0$), while the target strategy barely clears it (

.0 W_0$).

Practical implications:

  • Over a long horizon, maximizing expected log-wealth (the Kelly objective) produces the fastest compounding rate -- but only when you have an edge. For a fair game, Kelly says do not bet at all. The stated $f = 0.5$ fraction actually destroys log-wealth at a rate of $-0.144$ per round.
  • Over a finite horizon with a specific target, the optimal strategy often looks very different from Kelly. Here, the target-based approach achieves the same target probability while suffering far less log-wealth erosion.
  • The core tension: Kelly ignores targets entirely (it optimizes the geometric growth rate), while target-based strategies ignore long-run growth (they optimize a threshold probability). In practice, traders face both concerns: they need to hit P\&L targets within a finite window while avoiding strategies that systematically erode capital.

Answer: (a) $W_3/W_0$ takes values $\{0.125, 0.375, 1.125, 3.375\}$ with probabilities $\{1/8, 3/8, 3/8, 1/8\}$. (b) $P(W_3 \geq 2W_0) = 1/8$ for $g \geq 2^{1/3} - 1$ and $0$ otherwise; the minimum maximizer is $g^{*} = 2^{1/3} - 1 \approx 0.26$. (c) Both strategies share the same expected wealth ($W_0$) and target probability (

/8$), but the target strategy has far better expected log-wealth ($-0.105$ vs $-0.432$), illustrating that aggressive Kelly-like betting in a zero-edge game destroys geometric growth while adding no benefit to either expected wealth or target-hitting probability.

Intuition

This problem exposes a fundamental tension in finite-horizon portfolio management: the Kelly criterion is a long-run champion, but over short horizons it can be wildly suboptimal -- or even irrelevant -- depending on your actual objective. Here, the fair coin at even odds is the starkest possible illustration: there is zero edge, so the true Kelly fraction is zero (do not bet). Betting $f = 0.5$ every round gives you a flashy upside ($3.375 W_0$ with probability

/8$) but systematically erodes your geometric growth rate. The expected log-wealth is deeply negative, meaning your median outcome is well below your starting wealth.

The target-based strategy reveals a different way to think about risk. Instead of maximizing growth, you ask: what is the cheapest bet that gives me a shot at doubling? The answer is $g = 2^{1/3} - 1 \approx 0.26$, which bets the bare minimum for the all-heads path to clear the target. This achieves the same

/8$ target probability while preserving far more of your capital on the other seven paths. In real trading, this maps directly to how you size positions when you have a P&L target and a finite deadline: you want enough exposure to hit the target if your thesis is right, but not so much that you bleed out if it takes longer than expected. The lesson is that matching your strategy to your actual objective -- rather than defaulting to a textbook formula -- is what separates good risk management from reckless betting.

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