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Expectation, Linearity, and Conditional Expectation

Expectation is the operator underneath nearly every valuation. Three properties carry most of the weight.

Linearity, and why it matters for portfolios

E[X + Y] = E[X] + E[Y], \qquad E[aX] = aE[X]

The first holds whether or not $X$ and $Y$ are independent, which is the crucial part.

The portfolio consequence: expected return is the weighted average of component expected returns, regardless of correlation.

E[r_p] = \sum_i w_i E[r_i]

Correlation does not enter. It enters the variance and not the mean, which is the mathematical statement of why diversification reduces risk without reducing expected return, and therefore why it is close to a free lunch.

Key takeaway

Expected return is linear in weights and ignores correlation. Portfolio variance is quadratic and depends on it entirely. That asymmetry is the whole of portfolio theory.

Conditional expectation

E[X \mid Y]

What you expect of $X$ given knowledge of $Y$ . In finance this is nearly always the relevant quantity, since you are never estimating in a vacuum.

Pricing is a conditional expectation of a payoff given today's information. Hedging rests on how the expectation of your position changes conditional on the underlying moving, which is what a delta is. Regression estimates a conditional expectation, which is why $E[Y \mid X] = \alpha + \beta X$ is the model behind every beta calculation.

That last identity is worth seeing with numbers, because it is the point where an abstract operator becomes a trade you can put on.

Worked example: a conditional expectation is a hedge

Regressing a stock's daily returns on an index gives $\alpha = 0.01\%$ and $\beta = 1.3$ , so the fitted conditional expectation is

E[r_{\text{stock}} \mid r_{\text{index}}] = 0.01\% + 1.3\,r_{\text{index}}

Read it as a forecast and it says: if the index falls 2%, expect the stock to fall about 2.59%.

Read it as a hedge and it says something more useful. Short 1.3 units of index per unit of stock and the conditional expectation of the combined position collapses to $\alpha$ , which no longer depends on the index at all. The slope was the hedge ratio the whole time.

What remains after that hedge is the residual, and its variance is the risk you actually chose to hold. Isolating it is the entire business of an equity long/short book.

The tower property

E\left[E[X \mid Y]\right] = E[X]

Averaging a conditional expectation over the conditioning variable recovers the unconditional one.

This is the law of total probability for expectations, and it licenses a valuable technique: break a hard expectation into cases, compute each conditionally, and average.

It is also the formal basis for backward induction in derivative pricing. The value today is the expectation of the value tomorrow, which is itself an expectation of the value the day after, and so on to the payoff. Every tree-based pricing method is repeated application of the tower property.

Tip

When an expectation is hard, condition on something that makes it easy, compute, then average back. This is the same strategic conditioning move that solves most interview probability problems.

Test your knowledge

A portfolio puts $ 60\% $ in an asset with expected return $ 8\% $ and $ 40\% $ in an asset with expected return $ 12\% $. The two are highly correlated, with $ \rho = 0.9 $. What is the portfolio's expected return, as a percentage? Round your answer to one decimal place.

Pricing a derivative on a tree works backwards: the value at each node is the expected value at the nodes that follow it, repeated until you reach today. Which property of expectation licenses that procedure?

Linearity, $ E[aX + b] = aE[X] + b $, applied separately at each node

Jensen's inequality, which bounds the expectation of a convex function of the payoff

The law of large numbers, since a tree with enough nodes converges to the true price

The tower property, $ E[E[X \mid Y]] = E[X] $, which lets expectations nest

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