Expected Value

Expected value is the number you'd get if you could repeat a gamble forever and average all the results. It's a weighted average: take every outcome, multiply it by its probability, and add them up. That's it. Almost every quant interview question is, underneath, asking you to compute one.

If a random outcome $X$ can take values $x_1, x_2, \dots$ with probabilities $p_1, p_2, \dots$, its expected value is

$$E[X] = \sum_i p_i\, x_i .$$

The nicest way to picture it: imagine each outcome as a weight placed on a number line, heavier where the outcome is more likely. The expected value is the balance point, the spot where the line would sit level on a fulcrum.

Notice the last point: the expected value need not be a value you can ever observe. It's the center of gravity of the outcomes, not a prediction of any single result.

Worked example: is a bet worth it?

A game costs $10 to play. You roll one die and win $4 times the number you roll. Should you play?

Your winnings are $4 \times (\text{die face})$. Because expectation is linear, the expected winnings are just $4 times the expected face:

$$E[\text{win}] = 4 \times E[\text{face}] = 4 \times 3.5 = 14 .$$

You pay $10 and expect to win $14, so your expected profit is $14 - 10 = 4$ per play. Over many plays you come out ahead by about $4 each time, so yes, play it. If the cost were $15, the expected profit would be $14 - 15 = -1$ and you should decline, every play bleeds a dollar on average.

Worked example: the two-outcome bet

Most trading questions reduce to a bet with a few outcomes. Suppose a coin pays $3 on heads and costs you $1 on tails, and it lands heads 40% of the time. The expected payoff is

$$E[\text{payoff}] = (0.40)(3) + (0.60)(-1) = 1.20 - 0.60 = 0.60 .$$

Positive, so it's a good bet, you'd expect to make 60 cents each time you take it. This is exactly the calculation a market maker does before quoting: find the fair value (the expected payoff), then quote a bid below it and an ask above it so you profit on average whichever side trades.

The one rule that does all the work

Linearity of expectation: for any random quantities $X$ and $Y$,

$$E[X + Y] = E[X] + E[Y],$$

whether or not they are independent. This is what makes hard-looking problems collapse. Asked for the expected number of heads in 100 coin flips? Don't sum a messy distribution, write it as 100 separate flips, each worth $E = 0.5$, and add: $100 \times 0.5 = 50$. Asked for the expected number of people who get their own hat back when $n$ hats are shuffled? Each person has a $1/n$ chance, and there are $n$ people, so the answer is $n \times \tfrac{1}{n} = 1$, no matter how large $n$ is.

Where expected value misleads

Expected value is the right compass for a decision you'll repeat many times. It can be a terrible one for a decision you'll make once, because it says nothing about risk.

• It ignores the spread. A bet with a tiny chance of a huge payout and a large chance of ruin can have a great expected value and still be a bad idea if a single bad draw wipes you out. That's why Kelly sizing and variance exist: to weigh the swings, not just the average.
• The St. Petersburg trap. A game whose expected value is infinite (double your money each time a coin keeps landing heads) is still something no sane person pays much to play, because the outcome is dominated by outcomes you'll essentially never see.
• Averages hide fat tails. When outcomes are heavily skewed, the mean can sit far from anything typical. Pair the expected value with the variance and higher moments before trusting it.

The habit to build: for any uncertain situation, list the outcomes, attach a probability to each, and compute the weighted average. Then ask the second question, how much does it vary? Those two numbers, the mean and the spread, are the start of almost every quant answer.