# Zero Rates

Zero-coupon rates (or "zero rates") represent the yield on a hypothetical bond that pays no coupons — just a single payment at maturity. They are the foundation for all other rate calculations.

***

## What Is a Zero Rate?

A zero rate is the interest rate earned on an investment that:

* Has no intermediate cashflows (no coupons)
* Pays a single lump sum at maturity
* Is held from today until the maturity date

Zero rates are also called **spot rates** because they represent the rate available "on the spot" for a given maturity.

***

## Why Zero Rates Matter

Zero rates are fundamental because:

1. **Discount factors are derived from them** — DF = 1/(1+r)^t
2. **Forward rates are calculated from them** — Using no-arbitrage relationships
3. **They enable apples-to-apples comparison** — Unlike par rates, zero rates can be directly compared across tenors
4. **They're used for valuation** — PV calculations require zero rates, not par swap rates

***

## Zero Rates vs Par Swap Rates

|                          | Zero Rate                    | Par Swap Rate                      |
| ------------------------ | ---------------------------- | ---------------------------------- |
| **Definition**           | Yield on a zero-coupon bond  | Fixed rate that makes swap NPV = 0 |
| **Coupons**              | None                         | Periodic payments                  |
| **Directly observable?** | No (derived)                 | Yes (quoted in market)             |
| **Use case**             | Discounting, PV calculations | Swap pricing, hedging              |

BlueGamma bootstraps zero rates from observable par swap rates.

***

## The Formula

The zero rate for maturity t is related to the discount factor by:

$$
r\_t = \left( \frac{1}{DF\_t} \right)^{\frac{1}{t}} - 1
$$

Or equivalently:

$$
DF\_t = \frac{1}{(1 + r\_t)^t}
$$

Where:

* **r** = Zero rate for maturity t
* **DF** = Discount factor for maturity t
* **t** = Time in years

***

## Example: SOFR Zero Curve

Here's the current SOFR zero curve (December 2024):

| Maturity        | Zero Rate | Discount Factor |
| --------------- | --------- | --------------- |
| 6M (Jun 2026)   | 3.63%     | 0.9821          |
| 1.5Y (Jun 2027) | 3.40%     | 0.9345          |
| 2.5Y (Jun 2028) | 3.41%     | 0.9036          |
| 3.5Y (Jun 2029) | 3.51%     | 0.8723          |
| 4.5Y (Jun 2030) | 3.63%     | 0.8407          |
| 6.5Y (Jun 2032) | 3.93%     | 0.7769          |
| 9.5Y (Jun 2035) | 4.45%     | 0.6833          |

**Observations:**

* The curve is slightly inverted at the short end (6M > 1.5Y)
* Rates rise steadily from 2Y onwards
* This shape reflects market expectations of near-term rate cuts followed by normalisation

***

## Using Zero Rates in BlueGamma

### API

```bash
curl "https://api.bluegamma.io/v1/zero_rate?index=SOFR&date=2030-06-15" \
  -H "x-api-key: your_api_key"
```

**Response:**

```json
{
  "index": "SOFR",
  "date": "2030-06-15",
  "zero_rate": 3.63,
  "day_count": "Actual360",
  "compounding": "Simple"
}
```

### Excel Add-in

```
=BlueGamma.ZERO_RATE("SOFR", "2030-06-15")
```

***

## Zero Curve vs Discount Curve

The zero curve and discount curve contain the same information in different forms:

| Maturity | Zero Rate | Discount Factor |
| -------- | --------- | --------------- |
| 1Y       | 3.47%     | 0.9647          |
| 2Y       | 3.34%     | 0.9345          |
| 3Y       | 3.34%     | 0.9036          |
| 4Y       | 3.39%     | 0.8723          |
| 5Y       | 3.45%     | 0.8407          |
| 6Y       | 3.51%     | 0.8088          |
| 7Y       | 3.58%     | 0.7769          |
| 8Y       | 3.65%     | 0.7453          |
| 9Y       | 3.71%     | 0.7141          |
| 10Y      | 3.77%     | 0.6833          |

You can convert between them:

* **Zero → DF:** `DF = 1 / (1 + r)^t`
* **DF → Zero:** `r = (1/DF)^(1/t) - 1`

***

## Compounding Conventions

BlueGamma zero rates use the following conventions:

| Index Type       | Day Count     | Compounding |
| ---------------- | ------------- | ----------- |
| SOFR             | Actual/360    | Simple      |
| SONIA            | Actual/365    | Simple      |
| EURIBOR          | Actual/360    | Simple      |
| Government Bonds | Actual/Actual | Semi-annual |

{% hint style="info" %}
**Note:** The compounding convention affects how you convert between zero rates and discount factors. Always check the convention when comparing rates from different sources.
{% endhint %}

***

## Common Use Cases

| Use Case                  | Description                                       |
| ------------------------- | ------------------------------------------------- |
| **Discounting cashflows** | Convert future values to present values           |
| **Comparing maturities**  | Zero rates allow direct comparison across tenors  |
| **Curve analysis**        | Identify inversions, steepening, or flattening    |
| **Model inputs**          | Many pricing models require zero curves as inputs |
| **Spread calculations**   | Calculate spreads between different curves        |

***

## Related Documentation

* [Interest Rate Curves](/documentation/methodology/how-to-bootstrap-the-yield-curve.md) — How curves are constructed
* [Discount Factors](/documentation/methodology/discount-factors.md) — Converting zero rates to discount factors
* [Forward Rates](/documentation/methodology/forward-rates.md) — Deriving forward rates from zero rates
* [API Reference](/documentation/integrations/api/api-reference.md) — Complete API documentation


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