5.4.1 Risk Parity & ERC from Zero

Basic Prerequisites: none - this page needs no prior finance theory. Linear algebra and basic calculus help later.

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1. Intuition & Practical Objective

This page builds the why of risk parity with no prior knowledge of portfolio theory needed. The objective is one idea: diversification that counts dollars is not diversification that counts risk, and the fix is to allocate risk, not capital.

Start with the dumbest question: is a 60/40 stock/bond portfolio “balanced”? A typical equity index has an annualized volatility near 15%; a long government-bond index near 5%. One dollar of stock produces about three times the wiggle of one dollar of bonds. So in a portfolio that is 60% stocks and 40% bonds, the stocks supply roughly $0.60\times 3=1.8$ units of wiggle and the bonds $0.40\times 1=0.6$, meaning stocks dominate the portfolio’s ups and downs. When a big loss hits, ~9 in 10 dollars of that loss look just like a stock-market loss. Qian (2005) states it as a table of eggs: with $15\%$ stock vol and $5\%$ bond vol, a “60/40” portfolio holds the equivalent of 58 eggs, of which 54 (≈93%) come from the stock basket - at most six of the “diversification” eggs are actually doing any diversification.

Three steps:

1. Risk is what actually hurts you, dollars are not. A portfolio can look diversified by capital but be a single bet by risk. The amount of risk an asset contributes is the product of how much you hold and how much that holding moves the whole portfolio - not how volatile the asset is in isolation.
2. Equal capital ≠ equal risk; equal risk ≠ equal capital. Because most investors are more familiar with equal dollars ($1/N$, or 60/40), the discipline of risk allocation feels inverted: the safest asset class usually ends up with a larger capital weight, not a smaller one.
3. Risk-balancing is not return-free. To match a target return you get from a stock-heavy portfolio, a risk-balanced portfolio must leverage its low-risk legs. That single sentence carries both the promise and the danger of the whole strategy - the danger lives on 05 · Failure Modes.

A clean numerical example to carry through the folder. Take just two assets - say a broad equity index with ${\sigma }_E=15.1\%$ and a bond index with ${\sigma }_B=4.6\%$, correlation $\rho =0.2$ (exactly Qian’s illustrative numbers). A “balanced” 60/40 portfolio gives over 92% of its risk to the equities. To instead split the risk 50/50, the weights must be ≈23% equities / ≈77% bonds - genuinely “inverted” relative to the 60/40 intuition. (Both numbers are reproduced by the code below and again in 02 · Risk Contributions.)

The one-sentence essence. “Risk parity answers the question a 60/40 investor never asked: two thousand eggs are not diversified if 1,800 of them sit in one basket - split the risk, then decide how much risk you want with leverage.”

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2. Mathematical Ground Truth & Derivations

The only math this page needs is the definition that makes “contribution to risk” precise. Later pages derive it carefully; here is the shape.

Let $w=(w_1,\dots ,w_N)$ be the weights (fractions of capital, summing to 1) and $\Sigma$ the covariance matrix of the assets’ returns (${\Sigma }_{ii}={\sigma }_i^2$, ${\Sigma }_{ij}={\rho }_{ij}{\sigma }_i{\sigma }_j$). The portfolio’s volatility is

$$\sigma (w)=\sqrt{w^{\top }\Sigma w}.$$

Because $\sigma$ is homogeneous of degree 1 - scaling every weight by $\lambda$ scales the volatility by $\lambda$: $\sigma (\lambda w)=\lambda \sigma (w)$ - Euler’s homogeneous-function theorem guarantees the total risk splits exactly into $N$ per-asset pieces:

$$\sigma (w)=\sum _{i=1}^N\underset{=:R{C}_i}{\underbrace{w_i\frac{(\Sigma w{)}_i}{\sigma (w)}}}.$$

Each $R{C}_i$ is the risk contribution of asset $i$ - the slice of total volatility attributable to that position. This decomposition is not an approximation and does not require any assumption about returns: it is a property of the quadratic form. Risk parity / ERC simply asks for the weights that make all the $R{C}_i$ equal:

$$R{C}_1=R{C}_2=\cdots =R{C}_N=\frac{\sigma (w)}{N}.$$

Everything else in the folder - the closed forms, the convex programs, the failure modes - hangs off these two sentences.

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3. Computational Implementation - the two-asset picture, verified

The 60/40 → ~90% equity risk, and the ≈23/77 parity portfolio, computed in stdlib. Note how the equal-risk weights are inverted (more bonds than stocks).

The lesson is on the left column of the weights: 60/40 gives the equity leg ~93% of the portfolio’s risk, and to give it its fair 50% share of risk you must flip the dollars to ~23/77. Same risk appetite, radically different portfolio - one of the two numbers (92.7%) is what a “diversified” investor actually holds today; the other (50/50 risk) is the risk-parity promise.

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4. Failure Modes & First-Principles Breakdowns

1. The “costless leverage” assumption (from the start). The claim that you can fix risk balance and simply “lever the safe leg back up” assumes borrowing at or near the risk-free rate is freely available. It is not always: margin constraints, funding spreads and forced de-leveraging events change the answer (the core of 06 · Leverage Aversion).
2. Equal risk is not automatically better. Asness et al. (2012) are blunt: equalizing risk is only optimal if you believe you are not paid enough in expected return to accept the concentrated risk. Parity is a statement about expected returns, not just about diversification - a point the marketing of risk-parity funds often omits.
3. The 60/40 math depends on the correlation. The “90% equity risk” figure assumes low stock–bond correlation. If equities and bonds start to move together, the same 60/40 portfolio becomes even more equity-like in a crash - and the parity allocation’s leverage becomes dangerous (see 05 · Failure Modes).

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5. References

• Qian, Edward: Risk Parity Portfolios: Efficient Portfolios Through True Diversification, PanAgora (2005)
• Maillard, Roncalli & Teïletche: The Properties of Equally Weighted Risk Contribution Portfolios, JPM 36(4) (2010)
• Asness, Frazzini & Pedersen: Leverage Aversion and Risk Parity, FAJ 68(1) (2012)

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6. Connected Graph Bridges

• Base: Linear Algebra · Calculus
• Continue: 02 · Risk Contributions · Index Hub
• Sibling: Mean–Variance & Markowitz