Random variables and expectation

A random variable is a number whose value is not yet known but whose possible values and their probabilities are. A share price next year, a claim amount, a portfolio's return.

Expectation

For a discrete random variable, the expected value is the probability-weighted sum:

$E[X] = \sum_i p_i x_i$

For a continuous one, the sum becomes an integral against the density. The idea is identical: average the outcomes, weighting each by how likely it is.

Expected value is not the most likely value, and often not a possible value at all. The expected number on a die is 3.5.

The algebra that makes everything else work

Four properties, and nearly all of portfolio mathematics is these four applied:

$E[aX + b] = a\,E[X] + b$

$E[X + Y] = E[X] + E[Y] \quad \text{always, even if } X \text{ and } Y \text{ are dependent}$

$\text{Var}(aX + b) = a^2\,\text{Var}(X)$

$\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\,\text{Cov}(X, Y)$

The second and fourth are the important pair. Expectations always add. Variances only add when the covariance is zero.

That single asymmetry is the whole mathematical content of diversification: combining assets averages their expected returns but can more than average their risk, because the covariance term can be small or negative. Chapter 3 of the Portfolio theory subject is this line expanded.

The constant in the variance rule being squared is also doing real work: doubling a position doubles expected return and quadruples variance. Leverage is not a neutral multiplier.

Where expectation misleads

Expected value answers "what is the average over many repetitions". Two problems with that in practice.

You do not get many repetitions

A retirement is one draw, not a thousand. An expected annual return of 11% does not mean you will get 11% — it means that is the centre of a distribution you will sample from exactly once, over a path that matters. Chapter 8 of the Risk subject calls this sequence risk; PFRDA states the same thing plainly about the NPS, that there is no implicit or explicit assurance of benefit and investments are subject to market conditions.

A positive expectation can still ruin you

The bet in the problem above pays 10× with probability 0.15:

$E = 0.15 \times 10 + 0.85 \times 0 = 1.5$

Fifty paise of expected profit per rupee staked. An excellent bet — and staking everything on it repeatedly is near-certain ruin, because surviving twenty rounds requires winning every one:

$0.15^{20} \approx 3 \times 10^{-17}$

The expectation is computed over a world where you can keep playing. Losing everything removes you from that world. This is why the useful question is rarely "is the expected value positive" and almost always "what fraction of capital does this put at risk" — the subject of chapter 7 of the Risk subject.

The same arithmetic, less dramatically, is the geometric-versus-arithmetic mean gap from chapter 3. Multiplicative processes are not described by their arithmetic average.

Covariance and correlation

$\text{Cov}(X, Y) = E\big[(X - E[X])(Y - E[Y])\big]$

Covariance has awkward units — percent squared — so it is usually standardised into correlation:

$\rho = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} \in [-1, 1]$

Two cautions that matter more than the formula:

Correlation measures linear association only. A perfect non-linear relationship can have a correlation near zero. An option's payoff against the underlying is exactly such a relationship, which is why correlation is a poor description of a portfolio containing options.

Correlation is estimated from a sample and is not stable. The correlation that matters is the one during the fall, and it is reliably higher than the one measured in calm. Chapter 6 of the Risk subject has the consequence.

The point

Expected value is the probability-weighted average of outcomes, and expectations add unconditionally while variances only add when covariance is zero — which is diversification stated in one line. Expectation describes the average of many repetitions, and a retirement is one draw; a bet with positive expected value can still be near-certain ruin if it is sized to remove you from the game.