Risk Parity
Key idea
Risk Parity allocates capital so that each asset contributes equal risk rather than equal capital.
Traditional portfolios allocate dollars.
Risk parity allocates volatility.
Core question:
How should capital be distributed so no asset dominates portfolio risk?
Definition
Portfolio volatility:
\[\sigma_p = \sqrt{w^\top\Sigma w}\]Risk contribution of asset $i$:
\[RC_i = w_i \frac{(\Sigma w)_i} {\sigma_p}\]Risk parity requires:
\[RC_1 = RC_2 = \cdots = RC_N\]Each asset contributes equally.
Equal Volatility Approximation
If correlations are ignored:
\[w_i \propto \frac1{\sigma_i}\]Interpretation:
- lower volatility → larger weight
- higher volatility → smaller weight
This approximation is commonly deployed.
Example
Two assets:
| Asset | Volatility |
|---|---|
| Equity | 20% |
| Bond | 10% |
Risk parity gives:
\[w \propto (5,10)\]Normalized:
\[(33%,67%)\]Capital is tilted toward lower-risk assets.
Leverage
Risk parity often requires leverage.
Example:
- bond allocation increases
- expected return decreases
Leverage restores target return.
This differs from traditional 60/40 portfolios.
Relationship to Mean–Variance
Risk parity ignores expected return.
Mean–variance includes:
\[\max w^\top\mu - \lambda w^\top\Sigma w\]Comparison:
| Method | Objective |
|---|---|
| Equal Weight | equal capital |
| Mean–Variance | utility |
| Risk Parity | equal risk |
Usage in Quantitative Equity
Risk parity ideas appear in:
- multi-asset portfolios
- factor allocation
- risk budgeting
- strategy blending
- volatility management
Practical implementations usually include:
- covariance shrinkage
- turnover constraints
- exposure controls
- leverage limits
Pure equal-risk solutions are rarely used directly.