What is risk contribution, and how do you decompose a portfolio's total risk into individual obligor contributions?

Question

AcadiFi · Accepted Answer

**Risk contribution** measures how much each individual position contributes to the total portfolio risk. It decomposes aggregate risk into additive components that sum to the portfolio total, enabling fair capital allocation and informed concentration management.

**Key Concepts**

| Term | Definition |
|---|---|
| **Standalone risk** | UL of the exposure in isolation |
| **Marginal risk (incremental)** | Change in portfolio UL from adding or removing the exposure |
| **Risk contribution (Euler)** | The exposure's proportional share of portfolio UL, using partial derivatives |

**Euler Decomposition**

For a portfolio of n exposures, the risk contribution of exposure i satisfies:

**RC_i = w_i x (partial UL_p / partial w_i)**

where **SUM(RC_i) = UL_portfolio** (they add up exactly to total portfolio risk).

**Worked Example**

Dalton Credit Partners manages a $500M loan portfolio with three sectors:

| Sector | Exposure | Standalone UL | Correlation to Portfolio |
|---|---|---|---|
| Technology | $200M | $12.4M | 0.35 |
| Healthcare | $180M | $8.7M | 0.25 |
| Energy | $120M | $14.2M | 0.55 |

Sum of standalone ULs = $35.3M, but the diversified portfolio UL is only $22.8M (diversification benefit = $12.5M).

Risk contributions (from Euler decomposition):

| Sector | Risk Contribution | % of Portfolio UL |
|---|---|---|
| Technology | $7.9M | 34.6% |
| Healthcare | $4.6M | 20.2% |
| Energy | $10.3M | 45.2% |
| **Total** | **$22.8M** | **100%** |

```mermaid
flowchart TD
    A["Portfolio UL: $22.8M"] --> B["Technology RC: $7.9M (34.6%)"]
    A --> C["Healthcare RC: $4.6M (20.2%)"]
    A --> D["Energy RC: $10.3M (45.2%)"]
    E["Energy: 24% of capital but 45.2% of risk"] --> F["Concentration concern!"]
```

**Why Energy Dominates Despite Smaller Exposure**

Energy has the highest standalone UL ($14.2M) AND the highest correlation to the rest of the portfolio (0.55). High correlation means energy losses tend to coincide with losses in other sectors, reducing diversification benefit. This makes its risk contribution disproportionate.

**Business Applications**

1. **Capital allocation** — charge each business unit for its risk contribution, not standalone risk
2. **Concentration limits** — flag sectors whose risk contribution exceeds a threshold (e.g., 30%)
3. **Pricing** — add the marginal capital cost of a new loan to the lending spread
4. **Portfolio optimization** — reduce positions with high risk contribution per unit of return

**Marginal vs. Incremental**

- **Marginal risk contribution** = sensitivity (partial derivative) — used for Euler decomposition
- **Incremental risk** = portfolio UL with exposure minus portfolio UL without exposure — used for go/no-go decisions on new exposures

The difference matters: marginal is linear and additive; incremental captures nonlinear effects of large position changes.

For more portfolio credit risk concepts, visit our FRM Part II course.

What is risk contribution, and how do you decompose a portfolio's total risk into individual obligor contributions?

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