Expected Value

#Definition

Expected Value is the long-run average result of a bet, found by multiplying each possible payoff by its probability and summing across all outcomes.

Mathematically, if a bet has outcomes $x*1, x_2, \dots, x_n$ with probabilities $p_1, p_2, \dots, p_n$ (where all $p_i \ge 0$ and $\sum p_i = 1$), its expected value (often written as EV) is: $EV = \sum*{i=1}^{n} p_i \cdot x_i$

In prediction markets, EV is usually expressed in dollars (or stablecoins) per contract.

#Why It Matters in Prediction Markets

In a Prediction Market, every price implicitly encodes a probability and a potential edge. Expected value is the tool traders use to decide whether a position is mathematically favorable or not.

On platforms like Polymarket and Kalshi, most contracts are structured so that a “Yes” share is worth $1 if the event happens and $0 if it does not. If you buy at $0.55, your EV depends on your estimate of the true probability. If your estimated probability is higher than the market-implied probability, the trade may be positive EV.

Expected value also underlies Arbitrage strategies, portfolio construction, and risk management. Over many trades, consistently taking positive-EV bets (and avoiding negative-EV bets) is what allows a trader to have a sustainable edge.

#How It Works

#EV Calculation Flow

#Basic EV formula for a simple bet

For a single bet with a finite number of outcomes:

• Outcomes: $x_1, x_2, \dots, x_n$
• Probabilities: $p_1, p_2, \dots, p_n$

The expected value is: $EV = p_1 x_1 + p_2 x_2 + \dots + p_n x_n$

In prediction markets, outcomes often reduce to:

• “Event resolves Yes”
• “Event resolves No”

So the formula becomes: $EV = p*{\text{Yes}} \cdot x*{\text{Yes}} + p*{\text{No}} \cdot x*{\text{No}}$

#EV for a $1 binary contract

Consider a standard $1 “Yes” contract:

• Pays $1 if the event happens (“Yes”)
• Pays $0 if it does not (“No”)
• You pay a price $c$ (in dollars) to buy the contract

Let:

• $p$ = your estimated probability that the event happens
• $1 - p$ = your estimated probability that it does not

EV of the contract’s payoff (before cost):

$EV\_{\text{payoff}} = p \cdot 1 + (1 - p) \cdot 0 = p$

EV of your trade (net of cost):

Your net profit in each outcome:

• If “Yes”: $1 - c$
• If “No”: $-c$

So: $EV\_{\text{trade}} = p \cdot (1 - c) + (1 - p) \cdot (-c)$

Algebraically, this simplifies to: $EV\_{\text{trade}} = p - c$

This is the key insight: for a $1 binary contract, the EV of buying is simply (your probability estimate) minus (the price).

• If $p > c$, EV is positive.
• If $p = c$, EV is zero (ignoring fees).
• If $p < c$, EV is negative.

#Market-implied probability vs your probability

For a $1 binary contract:

• Market price $\approx$ market-implied probability (after adjusting for fees and spreads).

If a contract trades at $0.63, the market is roughly implying a 63% chance of the event. If you believe the true probability is 70%, your estimated EV is:

$EV\_{\text{trade}} = 0.70 - 0.63 = +0.07$

So on average you expect to gain $0.07 for every $1 contract bought, over many similar bets.

#Adjusting for fees and costs

Real platforms often charge:

• Trading fees (per trade, per contract, or as a percentage of volume)
• Settlement fees
• Gas costs (for on-chain trading)

To compute a more realistic EV, include these costs in your payoff or cost:

• Increase the effective price $c$ to include per-contract fees.
• Reduce the payoff $x\_{\text{Yes}}$ if settlement cuts into the payout.

The more friction in the trade, the more optimistic your probability estimate must be for the EV to remain positive.

#Examples

#Example 1: Fair coin-flip style market

Suppose a market is equivalent to a fair coin flip:

• If “Heads” occurs, a $1 contract pays $1.
• Otherwise, it pays $0.
• You can buy “Heads” at $0.50.

If you know the coin is fair:

• $p = 0.50$
• $c = 0.50$

Then: $EV\_{\text{trade}} = p - c = 0.50 - 0.50 = 0$

This is a zero-EV trade before fees: on average you neither gain nor lose (but in practice, fees make it negative EV to trade).

If, for some reason, you know the coin is biased to land “Heads” 55% of the time:

• $p = 0.55$
• $c = 0.50$

Then: $EV\_{\text{trade}} = 0.55 - 0.50 = +0.05$

You now expect to earn $0.05 per contract on average, over very many flips.

#Example 2: On-chain election market (Polymarket-style)

Imagine an on-chain election market similar to Polymarket:

• A "Yes" token pays $1 worth of stablecoins (e.g., USDC) if Candidate A wins.
• The token currently trades at $0.42.
• You estimate Candidate A has a 50% chance to win.

Then:

• $p = 0.50$
• $c = 0.42$

$EV\_{\text{trade}} = 0.50 - 0.42 = +0.08$

You expect to make $0.08 per token on average. If trading fees and gas work out to $0.01 per contract, your net EV becomes:

$EV\_{\text{net}} = 0.08 - 0.01 = +0.07$

#Example 3: Macro data release on Kalshi

Consider a macroeconomic event on Kalshi, a regulated exchange overseen by the CFTC as a Designated Contract Market (DCM):

• Contract: “Will the unemployment rate be above 5.0% in December?”
• “Yes” trades at $0.30.
• You believe there is a 25% chance this will happen.

Then:

• $p = 0.25$
• $c = 0.30$

$EV\_{\text{trade}} = 0.25 - 0.30 = -0.05$

This is negative EV according to your beliefs. Even if the contract offers attractive leverage, your expected value is -\0.05$ per contract; you would avoid this trade or consider taking the other side.

#Risks, Pitfalls, and Misunderstandings

Common issues traders face when working with expected value include:

• Confusing single outcomes with EV: A positive-EV bet can still lose money in any single instance. EV only “shows up” over many independent or similar trades.
• Overconfidence in probabilities: Mis-estimated probabilities lead to wrong EV calculations. If your model or intuition is poor, the EV math will not save you.
• Ignoring rare but large outcomes: Tail risks (e.g., extreme policy moves, data revisions) can dominate EV while being easy to overlook.
• Forgetting fees and market frictions: Trading fees, settlement costs, and poor execution can turn a slightly positive EV into a negative one.
• Overbetting on small edges: Using something like the full Kelly Criterion with noisy probability estimates can produce large drawdowns and risk of ruin.
• Resolution and rules risk: If you misread the market’s rules or the resolution source, the actual payoff distribution may differ from what you assumed, breaking your EV calculation.

#Practical Tips for Traders

• Use EV as a filter: Before entering a trade, explicitly compute $EV = p - c$ for simple $1 binaries. Avoid trades that are clearly negative EV under any reasonable probability estimate.
• Stress-test your probabilities: Check EV under a range of plausible probabilities (e.g., 45–65%). If the trade is only slightly positive EV in a narrow band, your edge may be fragile.
• Account for costs and execution quality: Include trading fees and likely Slippage when computing EV. Check the Order Book and Liquidity to see whether you can actually fill size near the quoted price.
• Think in repeated-trade terms: EV matters most when you can take similar edges many times. Be cautious about extrapolating EV from one-off, high-stakes trades.
• Use conservative sizing: Even with positive EV, size positions modestly. Consider using a fraction of the Kelly sizing to reduce volatility and drawdown risk.
• Watch correlation across markets: Multiple positions on related events share risk. Evaluate EV at the portfolio level, not just position-by-position.

#From EV to Position Size: The Kelly Criterion

Knowing a bet has positive EV tells you to bet, but not how much. The Kelly Criterion uses your EV and edge to calculate the optimal bet size to maximize long-term growth.

$f^* = \frac{bp - q}{b}$

Where:

• $f^*$ is the fraction of your bankroll to bet.
• $b$ is the odds received (decimal odds - 1).
• $p$ is the probability of winning.
• $q$ is the probability of losing ($1-p$).

Warning: Full Kelly betting is extremely volatile. Most traders use "Fractional Kelly" (e.g., betting half of what the formula suggests) to reduce risk.

#Tools

• Simple EV Calculator: (Probability * Payout) - Cost
• Kelly Criterion Calculator: Many online tools exist (e.g., EatTheBlocks Kelly Calc) to help size bets based on your EV.

#Related Terms

• Prediction Market
• Binary Market
• Scalar Market
• Kelly Criterion
• Arbitrage
• Order Book
• Liquidity
• Slippage

#FAQ

#How do you calculate expected value for a $1 binary contract?

For a standard $1 “Yes” contract, let $p$ be your estimated probability that the event happens and $c$ the current market price. The expected value of buying is simply $EV = p - c$. If $p > c$, the trade is positive EV (ignoring fees); if $p < c$, it is negative EV.

#Is using expected value risky for small traders?

Expected value itself is just a calculation, but relying on it without considering variance and bankroll can be risky. Small traders may face large swings even on positive-EV bets, especially if they size positions too aggressively. Combining EV with prudent position sizing and diversification helps manage this risk.

#Is expected value enough to decide whether to trade in a prediction market?

Expected value is necessary but not sufficient for making good trading decisions. You also need to consider your confidence in the probability estimate, your risk tolerance, correlation with existing positions, and practical factors like fees and liquidity. A trade that is theoretically positive EV but hard to execute or extremely volatile may not be attractive.

#How does expected value relate to prediction market prices?

In a $1 binary prediction market, the price of a contract roughly equals the market’s consensus probability of the event. Expected value for you is the difference between your probability and the price. If you think the true chance is higher than the price suggests, buying has positive EV; if lower, selling or shorting may have positive EV.

#How is expected value different from the Kelly Criterion?

Expected value measures how favorable a bet is on a per-unit basis, while the Kelly Criterion uses EV and variance to suggest an optimal fraction of your bankroll to stake. You generally need a positive EV to justify a bet at all; Kelly then addresses how much to bet, given your edge and risk tolerance.