Portfolio Splitting and the Kelly Criterion

Suppose the Knicks play the Spurs tomorrow.

A bookmaker believes $P(\text{Knicks win}) = p$. He sells two fractional contracts, each fairly priced under that belief:

  • A $\$1$ Knicks contract that pays $\frac{1}{p}$ if the Knicks win.
  • A $\$1$ Spurs contract that pays $\frac{1}{1-p}$ if the Spurs win.

(Note: Zero vig $\implies$ fair value = $1)

A gambler believes $P(\text{Knicks win}) = q > p$. He has just $1 but wants to grow it aggressively, so he spends:

  • $\$q$ on the Knicks.
  • $\$(1-q)$ on the Spurs.

(This is equivalent to wagering $a = \frac{q-p}{1-p}$ on the Knicks and leaving the rest in cash.)

As the game is about to start, the gambler is feeling good. His expected log-wealth is:

$$E_q[\log(W)] = q \log\left(\frac{q}{p}\right) + (1-q) \log\left(\frac{1-q}{1-p}\right)$$

Does this formula look familiar?

If you’ve taken a course in statistics or machine learning, you will recognize this as the Kullback-Leibler (KL) Divergence, denoted as $D_{\text{KL}}(q \parallel p)$.

Interpretation

  • The Kelly Criterion: The $q, 1-q$ split is algebraically the same as staking $f^* = \frac{q-p}{1-p}$ on the Knicks per the Kelly criterion.
  • Expected Log-Wealth: The highest possible expected log‑return is the Kullback-Leibler divergence between the gambler’s beliefs and the market’s.
  • Actual Payoff: After the game, his actual wealth-multiplier equals the likelihood ratio ($\frac{q}{p}$ or $\frac{1-q}{1-p}$).

Finger-Exercises

Portfolio Replication: Convince yourself that wagering $a = \frac{q-p}{1-p}$ on the Knicks and keeping the rest in cash perfectly replicates the gambler’s $q, 1-q$ split portfolio.

The Bookie’s Perspective: Under the bookie’s beliefs, show that the gambler’s portfolio has fair expected value:

\[ E_p[W] = 1. \]

But also show that the bookie’s expected log-wealth (under the bookie’s beliefs) is:

\[ \begin{aligned} E_p[\log W] &= p \log\left(\frac{q}{p}\right) + (1-p)\log\left(\frac{1-q}{1-p}\right) \\ &= -D_{\text{KL}}(p \parallel q) \\ &\leq 0. \end{aligned} \]

How can both statements be true? What, exactly, is the bookie happy about?