Portfolio Theory: Diversification as Arithmetic

"Diversification is the only free lunch in finance" is a slogan; this article is the arithmetic behind it. The entire subject reduces to one formula applied repeatedly.

The one formula

For a portfolio $P = \sum_i w_i X_i$ of assets with weights $w_i$:

$$\operatorname{Var}(P) = \sum_i w_i^2 \sigma_i^2 + \sum_{i \ne j} w_i w_j\, \rho_{ij}\, \sigma_i \sigma_j$$

Expected returns combine linearly ($E[P] = \sum w_i \mu_i$), but risk does not — it combines through the covariance structure. Everything in portfolio theory is a consequence of that mismatch.

The free lunch, quantified

Take $n$ assets, each with variance $\sigma^2$, equally weighted ($w_i = 1/n$), pairwise correlation $\rho$:

$$\operatorname{Var}(P) = \frac{\sigma^2}{n} + \left(1 - \frac{1}{n}\right)\rho\,\sigma^2 \;\xrightarrow{n \to \infty}\; \rho\,\sigma^2$$

Read the two terms. The first — idiosyncratic risk — dies at rate $1/n$: individually risky things become collectively safe if independent. The second — systematic risk — never diversifies away. With $\rho = 0$, portfolio vol falls like $1/\sqrt{n}$ forever; with $\rho = 0.2$, it floors at $\sqrt{0.2}\,\sigma \approx 0.45\sigma$ no matter how many assets you add.

Two interview-ready consequences:

• The number that matters about an asset is not its volatility but its correlation to what you already hold. A high-vol, zero-correlation asset can reduce portfolio risk.
• Correlation determines the value of adding bets. A strategy with Sharpe 1 run over $n$ independent markets has portfolio Sharpe $\sqrt{n}$; over perfectly correlated markets, still 1. This is why "how correlated are your signals?" is the first question any allocator asks.

Minimum variance and the efficient frontier

With two assets, minimizing $\operatorname{Var}(P)$ over the weight $w$ gives (for $\rho = 0$)

$$w_1^* = \frac{\sigma_2^2}{\sigma_1^2 + \sigma_2^2},$$

inverse-variance weighting — the low-risk asset gets weight proportional to $1/\sigma^2$, not $1/\sigma$. Tracing expected return against minimized risk across all target returns draws the efficient frontier: the upper-left boundary of what's achievable. Its punchline (Tobin's separation): once a risk-free asset exists, every efficient portfolio is a mix of cash and one single tangency portfolio — the one maximizing the Sharpe ratio.

The Sharpe ratio, honestly

$$\text{Sharpe} = \frac{E[R] - r_f}{\sigma_R}$$

It measures compensation per unit of total risk, and it's the right objective when the portfolio in question is your whole book (or will be levered to a target risk — leverage moves you along the line without changing Sharpe, which is exactly why it's the quantity to maximize).

Its honest limitations, which interviewers like to hear volunteered: it treats upside and downside vol identically; it's badly gamed by strategies that sell tail risk (steady small gains, rare catastrophic losses look high-Sharpe until they don't); and estimated Sharpes on short histories are extremely noisy — the standard error of an annual Sharpe estimated over $T$ years is roughly $\sqrt{(1 + \text{Sharpe}^2/2)/T}$, so distinguishing Sharpe 1 from Sharpe 0.5 takes on the order of a decade.

The interview version

"Two strategies, each 10% vol and Sharpe 1, correlation 0. What's the combined portfolio?" Equal-weight them: return stacks, vol is $\sqrt{0.05^2 + 0.05^2} \approx 7.1\%$ against a 10% blended return → Sharpe $\approx 1.41 = \sqrt{2}$. Then the follow-up — "and at correlation 0.5?" — uses the one formula again: vol $\sqrt{3}/2 \times 10\% \approx 8.7\%$, Sharpe $\approx 1.15$. If you can run that arithmetic aloud in thirty seconds, this topic is done.