Expected value and variance

Every trading decision is a bet with a distribution of outcomes. Before you can say anything about a strategy, you need two summaries of that distribution: where its centre is, and how spread out it is.

Expected value

For a discrete random variable $X$ that takes value $x_i$ with probability $p_i$, the expected value is the probability-weighted average:

$$\mathbb{E}[X] = \sum_i x_i \, p_i$$

For a continuous variable with density $f$, the sum becomes an integral, $\mathbb{E}[X] = \int x f(x)\,dx$.

Expectation is linear, which is what makes it so useful: $\mathbb{E}[aX + bY] = a\,\mathbb{E}[X] + b\,\mathbb{E}[Y]$, whether or not $X$ and $Y$ are independent. The expected P&L of a portfolio is the weighted sum of the expected P&Ls of its parts — always.

Variance and standard deviation

Variance measures the average squared distance from the mean:

$$\operatorname{Var}(X) = \mathbb{E}\big[(X - \mathbb{E}[X])^2\big] = \mathbb{E}[X^2] - \mathbb{E}[X]^2$$

The standard deviation $\sigma = \sqrt{\operatorname{Var}(X)}$ is in the same units as $X$, so it is the number you quote. In finance, the standard deviation of returns is called volatility.

For the example above, $\mathbb{E}[X^2] = 0.4(9) + 0.6(1) = 4.2$, so $\operatorname{Var}(X) = 4.2 - 0.36 = 3.84$ and $\sigma \approx 1.96R$.

Unlike expectation, variance is not linear:

$$\operatorname{Var}(aX + bY) = a^2\operatorname{Var}(X) + b^2\operatorname{Var}(Y) + 2ab\operatorname{Cov}(X, Y)$$

That covariance term is the whole reason diversification works, and it is the starting point of portfolio construction.

Why the square root of n matters

Take $n$ independent trades with the same mean $\mu$ and volatility $\sigma$. The total P&L has mean $n\mu$ and standard deviation $\sigma\sqrt{n}$. The ratio of signal to noise therefore grows like

$$\frac{n\mu}{\sigma\sqrt{n}} = \sqrt{n}\,\frac{\mu}{\sigma}$$

Expected value is not enough

Two bets can share an expected value and be completely different to hold. A strategy that earns $+1\%$ a month with certainty, and one that earns $+11\%$ or $-9\%$ on a coin flip, both have mean $1\%$. Only one will ever bankrupt you. The rest of this curriculum is largely about what you need beyond the mean: variance and risk-adjusted return, growth and sizing, and tail risk.

Check yourself

• A trade wins $+2R$ with probability $p$ and loses $-1R$ otherwise. What is the break-even win rate? (Answer: $2p - (1-p) = 0 \Rightarrow p = 1/3$.)
• If two strategies each have volatility $10\%$ and correlation $0.5$, what is the volatility of an equal-weight blend? (Answer: $\sqrt{0.25(0.01) + 0.25(0.01) + 2(0.25)(0.5)(0.01)} \approx 8.66\%$.)