In Part 2 of this series, we show that classical metrics like the Sharpe ratio, win rate and Brier score failed to properly evaluate prediction market strategies. Then in Part 3, we show that the KL Divergence $D_{\mathrm{KL}}(\theta || p)$ between the truth and the market was the optimal growth rate for a repeated binary bet. Now, we'll use the concepts from last time to provide proper evaluation metrics.

The Geometric Sharpe Ratio

The Sharpe ratio measures excess return per unit of risk, using arithmetic returns and their standard deviation. The concept of a risk-adjusted return is reasonable, but the particular concepts in the traditional definition don't work in prediction markets for multiple reasons: the underlying Bernoulli process and multiplicative compounding both violate its assumptions. So we'll replace it with an appropriate form of risk-adjusted return.

Wealth compounds multiplicatively, so instead of considering returns and wealth $R_t,W_t$ directly, it is more appropriate to consider log-returns and log-wealth $\ln R_t, \ln W_t$ . Define the geometric Sharpe ratio as

$S_g = \frac{g}{\sigma_g}$

where $g = \mathbb E[\ln R_t]$ is the growth rate and $\sigma_g = \sqrt{\mathrm{Var}[\ln R_t]}$ is the standard deviation of log-returns.

For one binary contract at price $p$ , true probability $\theta$ , and bet fraction $f$ , recall that each contract has log-return $\ln R_t$ that takes one of two values:

$\ln R_t = \begin{cases} \ln\!\left(1 + \frac{f(1-p)}{p}\right) & \text{with probability } \theta \\ \ln(1-f) & \text{with probability } 1-\theta \end{cases}$

where $\Delta = \ln\!\left(1 + \frac{f(1-p)}{p}\right) - \ln(1-f)$ is the log-odds swing: the difference between a win and a loss on that trade in log space. Because the outcome is Bernoulli, the variance has a clean form:

$\sigma_g^2 = \theta(1-\theta)\,\Delta^2$

The geometric Sharpe then becomes

$S_g = \frac{\theta\, \ln\!\left(1 + \frac{f(1-p)}{p}\right) + (1-\theta)\,\ln(1-f)}{\sqrt{\theta(1-\theta)}\;\Delta}$

Classical Sharpe has many properties that this model fixes

Instead of using arithmetic means of normal returns, it reflects the compounding by geometric means of log-returns.
Instead of using the variance of Gaussian returns, the variance $\theta(1-\theta)\,\Delta^2$ reflects the Bernoulli nature of a binary market.
Since prediction markets are zero-sum, the appropriate metric is to "subtract" the actual market rate $R_f = 0.$

At the Kelly fraction $f^* = (\theta - p)/(1-p)$ , the log-odds swing is reduced to a difference in logits ( $\mathrm{logit}(x)=\ln(x/(1-x))$ ):

$\Delta^* = \ln\frac{\theta(1-p)}{p(1-\theta)} = \mathrm{logit}(\theta) - \mathrm{logit}(p)$

and the numerator of the geometric Sharpe is $g^* = D_{\mathrm{KL}}(\theta || p)$ . Therefore, the Kelly-optimal geometric Sharpe is

$S_g^* = \frac{D_{\mathrm{KL}}(\theta || p)}{\sqrt{\theta(1-\theta)}\;[\mathrm{logit}(\theta) - \mathrm{logit}(p)]}$

One important fact that this derivation highlights is that the edge ultimately and exclusively comes down to novel information. If the market is perfect and $p=\theta,$ no strategy will have a positive geometric Sharpe ratio.

Drawdown and Ruin

Growth rate represents the average rate that wealth compounds. But averages are agnostic to the path, so even a strategy with expected positive growth could take large intermediate losses and get wiped out. To survive this, we must also consider the drawdown.

Peak-to-Trough in Log-Space

Define log-wealth as $L_t = \ln W_t$ . The drawdown at time $t$ is the gap between the running maximum and the current value:

$D_t = \max_{s \leq t} L_s - L_t$

and the maximum drawdown over $T$ trades is $\mathrm{MDD}_T = \max_{t \leq T} D_t$ .

In classical markets, prices move continuously, and so does the drawdown. But prediction markets do not because of the contractual nature. Each resolved contract either adds the winnings (denoted $w$ ) or subtracts the losses ( $\ell$ ) to log-wealth. So drawdowns are discrete with jumps of size $|\ell| = |\ln(1-f)|$ for each loss. Consequently, a losing streak of length $k$ results in a drawdown of $k|\ell|$ , which in dollars comes out to a wealth loss of $(1-f)^k$ of the peak. The minimum drawdown is $|\ln(1-f)|$ ; in continuous markets, drawdowns are only bounded by tick size.

Ruin Bounds and Survival

We model the log-wealth $L_t$ using a random walk with drift $g > 0$ and step variance $\sigma_g^2 = \theta(1-\theta)\Delta^2$ . In this case, the probability of a maximum drawdown of depth $d$ is bounded by

$P(\mathrm{MDD} \geq d) \leq \exp\!\left(-\frac{2gd}{\sigma_g^2}\right)$

The ratio $2g/\sigma_g^2$ controls how rapidly the tail probability decays with drawdown depth. Even slightly larger growth exponentially reduces the chance of a bad drawdown.

Still, at full Kelly this bound allows for a surprisingly high chance of ruin. As an example, losing half the bankroll ( $d = \ln 2$ ) under full Kelly betting is close to coinflip odds. This is a huge conflict between growth-rate optimization and survival: full Kelly maximizes $g$ , but also increases $\sigma_g^2$ enough that ruinous drawdowns are common.

Recall from Part 3 that estimation error $\epsilon = p^* - \theta$ introduced a growth penalty $\Delta g = \frac{1}{2}\mathbb E\!\left[\frac{\epsilon^2}{\theta(1-\theta)}\right],$ proportional to the Fisher information. This penalty, which is unavoidable in normal circumstances, reduces the expected return from the random walk without reducing variance, making the exponent $2g/\sigma_g^2$ worse and ruin more likely. So errors in modelling $\theta$ have a double whammy: they hurt growth, and increase the chance of ruin.

Given this, it is natural to ask: what is the largest bet fraction that keeps the drawdown $d$ under probability $\delta$ ? $P(\mathrm{MDD} \geq d) \leq \delta$ is equivalent to $2gd/\sigma_g^2 \geq \ln(1/\delta)$ . Since $g$ grows linearly in $f$ while $\sigma_g^2$ grows quadratically (for small $f$ ), the ratio $g/\sigma_g^2$ is expected to have a peak above zero, and using a small- $f$ approximation,

$f^D = \frac{2dp(\theta - p)}{\theta(1-\theta)\!\left[\ln(1/\delta) + d\right] + d(\theta - p)^2.}$

This is smaller than the Kelly fraction $f^*$ , which maximizes growth $g$ alone while $f^D$ caps growth to ensure survival. We can call $f^D$ a risk-adjusted Kelly fraction, and the corresponding $g(f^D)$ a risk-adjusted growth rate.

The interaction between a risk-adjusted Kelly fraction and market structure (Part 0) is worth noting:

Binary markets have a single source of variance, so this analysis works out of the box.
Categorical / winner-take-all markets are broken up across $n$ mutually exclusive contracts. The growth rate now involves the joint distribution of sizing across a portfolio, and we must account for vig reducing the available edge.
PDF / bucket markets bet on the shape of the distribution. The analysis now depends on how many buckets are simultaneously in play, producing a portfolio within a single market.
CDF markets create correlated positions, and the monotonicity constraint (Part 0) caps the variance of a combined position but also limits diversification.

In all cases, more complicated markets require more complicated analysis, but the underlying idea remains the same.

The Benchmarking Ladder Revisited

In Part 1, we introduced a ladder of benchmarks that any prediction market strategy can clear:

Beating equity markets > Beating risk-free rate > Breaking even > Beating the median

At the time, we lacked the tools to make these comparisons precise. Now we do.

Beating the median means $g > g_{\mathrm{median}}$ . In a zero-sum market, the median participant has $g = 0,$ which turns negative after fees and vig. Skew in trader skill amplifies this effect. Simply beating a median user isn't enough to ensure returns.

Breaking even means $g \geq 0$ : the strategy makes money. From the KL decomposition in Part 3, this means $D_{\mathrm{KL}}(\theta || p) > D_{\mathrm{KL}}(\theta || p^*)$ , i.e. the trader must be less wrong than the market.

Beating the risk-free rate $r_f$ requires $g \geq r_f / N$ on a per-trade basis, where $N$ is the number of trades per year. Unlike traditional Sharpe, this comparison is made in log-space, comparing the growth rates of the strategy and of the risk-free bond, making a fair comparison.

Beating equity markets requires $g \geq g_{\mathrm{equity}}$ , where $g_{\mathrm{equity}}$ is the growth of the equity benchmark. This is the real hurdle for justifying capital allocation to prediction markets in itself, rather than using it as part of an inter-market strategy or passive index investment.

The geometric Sharpe can then rank two strategies that clear the same hurdle. The one with the larger $S_g$ is preferred because it achieves that growth with less variance and risk of ruin.

Further Topics

Over the past few blogs, we have developed metrics that respects the structure of prediction markets: the geometric Sharpe ratio $S_g$ for risk-adjusted performance, drawdown analysis in log-space for path risk, and the risk-adjusted Kelly fraction for reconciling growth with survival, alongside the growth rate decomposition from Part 3.

However, we've focused on a single market of contracts. In future posts, we look at portfolios: allocating capital across simultaneous contracts, the effects of correlation, and how the different market structures can affect growth. We shall also examine how prediction markets can be a way to hedge within strategies in other markets.