Zero Rates

How BlueGamma derives zero-coupon rates from bootstrapped discount factors, with the day count and compounding convention for each index and a worked SOFR example.

BlueGamma's zero rate for a date is derived from the bootstrapped discount factor at that date, r = (1/DF)^(1/t) − 1, and quoted with simple compounding on the index's day count (Actual/360 for SOFR and EURIBOR, Actual/365 for SONIA, semi-annual Actual/Actual for government bonds). Zero rates are not quoted in the market; BlueGamma bootstraps them from observable par swap 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.

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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.

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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

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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.

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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

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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

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Using Zero Rates in BlueGamma

API

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

Response:

{
  "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")

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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

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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

Note: The compounding convention affects how you convert between zero rates and discount factors. Always check the convention when comparing rates from different sources.

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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

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Related Documentation

• Interest Rate Curves — How curves are constructed
• Discount Factors — Converting zero rates to discount factors
• Forward Rates — Deriving forward rates from zero rates
• API Reference — Complete API documentation