How are liquidity horizons assigned to risk factors under FRTB, and why do they matter for capital calculations?

Question

AcadiFi · Accepted Answer

Under FRTB, the **liquidity horizon** represents the time needed to close out or hedge a risk factor position during a period of financial stress. Illiquid risk factors are assigned longer horizons, which increases their contribution to capital.

**Liquidity Horizon Buckets**

| Bucket | Horizon | Risk Factor Examples |
|---|---|---|
| 1 | 10 days | Large-cap equity prices, G10 FX rates, government bond yields |
| 2 | 20 days | Small-cap equity prices, corporate IG bond yields, commodity futures |
| 3 | 40 days | IG credit spreads, equity volatility (at-the-money) |
| 4 | 60 days | HY credit spreads, exotic FX, small-cap equity volatility |
| 5 | 120 days | Correlation parameters, structured credit, illiquid exotic rates |

**Composite ES Calculation**

The ES is not simply computed at the longest horizon. Instead, it uses a **nested square root of sums of squares** approach that reflects incremental risk at each horizon:

**ES = sqrt(ES_{10}^2 + (ES_{20}^2 - ES_{10,subset}^2) + (ES_{40}^2 - ES_{20,subset}^2) + ...)**

where each term adds the incremental ES from extending from one liquidity horizon to the next, applied only to the risk factors that need the longer horizon.

**Simplified Intuition**

```mermaid
flowchart LR
    A["10-day ES: All risk factors"] --> B["Add 20-day incremental: equity vol, corp bonds"]
    B --> C["Add 40-day incremental: credit spreads"]
    C --> D["Add 60-day incremental: HY spreads"]
    D --> E["Add 120-day incremental: correlations"]
    E --> F["Composite ES"]
```

**Worked Example**

Kelford Securities' interest rate desk has the following risk factors:

| Risk Factor | Liquidity Horizon | Standalone ES (10-day equivalent) |
|---|---|---|
| G7 government yields | 10 days | $12M |
| Swap spreads | 20 days | $5M |
| Corporate IG spreads | 40 days | $8M |

**Step 1:** Base 10-day ES for all factors = $15M (diversified)

**Step 2:** Extend swap spread factors to 20 days. The incremental ES from extending = sqrt(2) x the 10-day ES contribution of swap spreads minus the already-captured 10-day piece.

If the 20-day ES for swap-related factors = $5M x sqrt(20/10) = $7.07M, the increment = sqrt($7.07M^2 - $5M^2) = $5M.

**Step 3:** Similarly extend IG credit spreads to 40 days.

The final composite ES will be higher than a simple 10-day ES by the square-root-of-time scaled contributions of the longer-horizon factors.

**Why This Matters**

Under the old framework, all market risk used a uniform 10-day horizon. This dramatically underestimated capital for:
- Credit correlation books (120-day horizon vs. 10-day)
- High-yield bonds that cannot be liquidated in 10 days during stress
- Exotic options with illiquid vol surfaces

A desk heavy in 120-day risk factors might see its capital charge increase 3-4x compared to the old 10-day VaR framework.

For more on FRTB mechanics, check our FRM Part II materials.

How are liquidity horizons assigned to risk factors under FRTB, and why do they matter for capital calculations?

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