5.1.1 Modern Portfolio Theory from Zero

Basic Prerequisites: Linear Algebra (vectors, dot products will be enough).

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1. Intuition & Practical Objective

This page builds the why of Modern Portfolio Theory with no prior portfolio knowledge needed. The objective is one idea: an asset should never be judged in isolation - its value to a portfolio is entirely determined by how it covaries with everything else you hold. That single shift (asset-by-asset → covariance-aware) is the entire conceptual revolution of Markowitz (1952).

Before Markowitz, investors picked the “best” asset: highest yield, fastest growth. Markowitz replaced picking with averaging, and proved a stunning fact: combining risky assets can produce a portfolio less risky than any of them individually, at the same or better expected return. The mechanism is the correlation between assets, and the measure of its payoff is the portfolio-variance formula. Three “aha”s:

1. A portfolio’s risk is not the weighted average of its parts. It is a quadratic blend that includes the cross-terms (covariances). If two assets don’t move together, each wobbles while the other stays put, so the portfolio wobbles less.
2. Diversification is not a strategy, it’s arithmetic. The variance formula below forces the reduction. The only “free lunch” in financial economics is the one the quadratic form hands to you.
3. The lower the correlation, the bigger the gift. At $\rho =-1$ perfect negative correlation, you can drive variance to zero; at $\rho =+1$ diversification gives nothing. The correlation, not the individual volatilities, decides.

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2. Mathematical Ground Truth & Derivations

Portfolio of two assets. Holding weight $w$ in asset 1 and $1-w$ in asset 2, with individual variances ${\sigma }_1^2,{\sigma }_2^2$ and correlation $\rho$:

$${\sigma }_p^2=w^2{\sigma }_1^2+(1-w{)}^2{\sigma }_2^2+2w(1-w)\rho {\sigma }_1{\sigma }_2,$$

so the portfolio volatility is ${\sigma }_p=\sqrt{{\sigma }_p^2}$. The cross-term $2w(1-w)\rho {\sigma }_1{\sigma }_2$ is the entire plot: if $\rho <1$, the sum of the two positive “own” terms is partially cancelled, and ${\sigma }_p<w{\sigma }_1+(1-w){\sigma }_2$ (the weighted-average volatility). Note the risk is quadratic in $w$; that convexity is what makes a minimization well-posed.

The minimum-variance weight. Setting $d{\sigma }_p^2/dw=0$ gives the weight that makes the portfolio as quiet as possible:

$$w^{\ast }=\frac{{\sigma }_2^2-\rho {\sigma }_1{\sigma }_2}{{\sigma }_1^2+{\sigma }_2^2-2\rho {\sigma }_1{\sigma }_2}.$$

When the two assets have equal volatility and $\rho \ne 1$, $w^{\ast }=1/2$ - equal weight is exactly the minimum-variance portfolio, purely by symmetry. When their volatilities differ, you tilt toward the quieter asset; and if $\rho$ is high enough the formula can exceed $1$ or go negative, meaning the minimum-variance portfolio shorts one asset (a corner you can’t reach long-only).

The separable, general case ($N$ assets). With weights $w\in {\mathbb{R}}^N$, expected returns $\mu$, and covariance $\Sigma$,

$$R_p=w^T\mu ,{\sigma }_p^2=w^T\Sigma w,$$

with the budget constraint $w^{T}1=1$. The two-asset insight generalizes: the cross-covariances in $\Sigma$ (off-diagonal entries) are what diversification harvests. The full machinery is built out on 02 · The Efficient Frontier.

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3. Computational Implementation - diversification in numbers

The two-asset variance formula, stdlib only. Watch the correlation slash the portfolio volatility, and watch the minimum-variance weight tilt as $\rho$ changes.

Read the rows: even at zero correlation, two assets of volatility 0.30 blend to 0.2121 - a 29% volatility cut with no expected-return cost at all. At $\rho =-0.7$ it falls to 0.1162. And in the unequal-vol case, when $\rho =0.8$ the minimum-variance weight comes out 1.28, i.e. the quiet portfolio would short the louder asset - the seed of why long-only constraints matter (04 · Constraints).

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4. Failure Modes & First-Principles Breakdowns

1. Correlation vs. causality confusion. Diversification harvests historical correlation; if correlations rise to 1 in a crisis (the “correlation breakdown”), the free lunch disappears exactly when you need it. The formula is an identity given $\rho$, not a guarantee about $\rho$.
2. The "$N\to \infty$" myth. More assets help only because they add independent variance sources; buying 100 clones of the same factor adds nothing. What matters is the number of independent risk factors, not the number of names.
3. Volatility is not the only risk. Markowitz captures variance; it says nothing about skew, jumps, or tail dependence. Diversification that reduces variance can still leave correlated tail risk in place (bridge to the extreme-value/fat-tails material in Quantitative Risk).
4. The minimum-variance weight is an optimizer too. As soon as you minimize, you inherit input sensitivity - $w^{\ast }$ depends on the estimated ${\sigma }_1,{\sigma }_2,\rho$, and near $\rho \to 1$ the denominator $\to 0$ and $w^{\ast }$ becomes violently unstable. The fragility of every “optimal” weight is the theme of 05 · Failure Modes.

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5. References

• Markowitz, Harry: Portfolio Selection, Journal of Finance 7(1):77–91 (1952)
• Elton, Gruber, Brown & Goetzmann: Modern Portfolio Theory and Investment Analysis, Wiley
• Hastie, Tibshirani & Friedman (ESL)

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6. Connected Graph Bridges

• Back: Index Hub
• Forward: 02 · The Efficient Frontier · 03 · Tangency & CAPM
• Base: Linear Algebra · Probability