Expected Value

Expected value (EV) is the single most important concept in quantitative finance and probability theory. It is the probability-weighted average of all possible outcomes — the number you would converge to if you could repeat the same random event an infinite number of times.

The concept is intuitive. If you flip a fair coin and win $10 on heads but lose $6 on tails, the expected value is:

EV = 0.5 × $10 + 0.5 × (-$6) = $5 - $3 = +$2

On average, you make $2 per flip. You won't make exactly $2 on any individual flip — you'll either gain $10 or lose $6 — but over hundreds of flips, your average gain will converge to $2. This convergence is guaranteed by the Law of Large Numbers.

In trading, every decision can be framed as an expected value calculation. Should you take a position? What is the expected profit of a trade after accounting for the probability of winning, the magnitude of wins and losses, and transaction costs? Quant traders think in terms of EV constantly — it is the lens through which every opportunity is evaluated.

For a discrete random variable X with possible outcomes x₁, x₂, ..., x_n and corresponding probabilities p₁, p₂, ..., p_n:

E[X] = ∑ p_i × x_i = p₁x₁ + p₂x₂ + ... + p_nx_n

For a continuous random variable with probability density function f(x):

E[X] = ∫ x · f(x) dx

Key properties of expected value:

• Linearity: E[aX + bY] = aE[X] + bE[Y]. This holds even if X and Y are dependent — an extremely useful property.
• Constants: E[c] = c. The expected value of a constant is itself.
• Indicator variables: E[1_A] = P(A). The expected value of an indicator equals the probability of the event.
• Products of independent variables: If X and Y are independent, E[XY] = E[X]E[Y].

Linearity of expectation is one of the most powerful tools in probability — it allows you to break complex problems into simpler pieces. For example, the expected number of fixed points in a random permutation of n items is simply n × (1/n) = 1, by linearity, regardless of the complex dependencies between positions.

Example 1 — Trading decision:

You're considering a trade with a 60% chance of making $500 and a 40% chance of losing $400. Transaction costs are $20.

EV = 0.60 × $500 + 0.40 × (-$400) - $20 = $300 - $160 - $20 = +$120

The trade has positive expected value, so it's worth taking (assuming you can handle the variance).

Example 2 — The St. Petersburg paradox:

A casino offers a game: flip a fair coin until it lands tails. If tails appears on flip n, you win $2ⁿ. So: heads-tails pays $4, heads-heads-tails pays $8, etc. How much would you pay to play?

EV = ∑_n=1^∞ (1/2ⁿ) × 2ⁿ = ∑ 1 = ∞

The expected value is infinite! Yet no rational person would pay $1,000 to play. This paradox illustrates that expected value alone is insufficient for decision-making — you must also consider variance and the diminishing utility of wealth. The Kelly criterion resolves this by maximizing the expected logarithm of wealth rather than expected wealth.

Example 3 — Interview-style question:

You roll two fair dice. What is the expected value of their sum?

By linearity: E[D₁ + D₂] = E[D₁] + E[D₂] = 3.5 + 3.5 = 7

Expected value is the foundation of every trading decision at a quant firm:

• Trade evaluation: Before entering any trade, a quant trader estimates the EV. If the expected profit minus all costs (spread, market impact, commissions) is positive, the trade is worth considering. If negative, it's not.
• Market making: A market maker who quotes a bid-ask spread implicitly calculates the EV of each quote. The spread revenue per trade must exceed the expected adverse selection cost.
• Interview questions: EV problems are among the most common in quant interviews. You might be asked: "I offer you a game where you flip a coin and I pay you $1 for each head in a row before the first tail. How much would you pay to play?" (Answer: EV = ∑ n × (1/2)ⁿ⁺¹ = 1).
• Strategy design: Systematic strategies are designed to have positive EV on each trade. The Law of Large Numbers ensures that over thousands of trades, the actual P&L converges to the expected P&L — which is why high-frequency strategies with tiny per-trade EV but enormous trade counts can be highly profitable.
• Position sizing: EV tells you whether to trade; the Kelly criterion tells you how much to trade based on both EV and variance.