Discount Factors

How BlueGamma calculates discount factors from interest rate curves.

Discount factors convert future cashflows to their present value. BlueGamma derives discount factors directly from our constructed curves.

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What Is a Discount Factor?

A discount factor represents the present value of £1 (or $1) received at a future date. It answers: "What is a future cashflow worth today?"

$$\text{Present Value} = \text{Future Cashflow} \times \text{Discount Factor}$$

Example: If the 2-year discount factor is 0.9358, then £100 received in 2 years is worth £93.58 today.

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The Formula

Discount factors are derived from zero-coupon rates:

$$DF_t = \frac{1}{(1 + r_t)^t}$$

Where:

• DF = Discount factor for time t

• r = Zero-coupon rate for maturity t

• t = Time in years to the cashflow

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Example Calculation

Given: SOFR zero-coupon rates (as of December 2024)

Maturity	Date	Zero Rate	Discount Factor
6M	2025-06-15	3.63%	0.9821
2Y	2027-12-15	3.39%	0.9358

Verify the 2-year discount factor:

Time to maturity: 2 years

$$DF = \frac{1}{(1 + 0.0339)^{2}} = \frac{1}{1.0689} = 0.9356$$

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How BlueGamma Calculates Discount Factors

1. Bootstrapping

We start with market instruments (deposits, futures, swaps) and bootstrap a zero-coupon curve.

2. Compounding Forward Rates

The discount factor at time t is the product of all overnight discount factors:

$$DF_t = \prod_{i=0}^{t-1} \frac{1}{1 + r_{i,i+1} \times \delta_i}$$

Where:

• r_{i,i+1} = Forward rate between day i and i+1

• δᵢ = Day count fraction for that period

3. Interpolation

For dates between curve points, we interpolate on log discount factors to ensure smooth, no-arbitrage results.

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Using Discount Factors in BlueGamma

API

curl "https://api.bluegamma.io/v1/discount_factor?index=SOFR&date=2026-06-15" \
  -H "x-api-key: your_api_key"

Response:

{
  "index": "SOFR",
  "date": "2026-06-15",
  "discount_factor": 0.9821,
  "timestamp": "2025-12-16T12:50:53"
}

Excel Add-in

=BlueGamma.DISCOUNT_FACTOR("SOFR", "2026-06-15")

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Common Use Cases

Use Case	Description
Swap valuation	Discount fixed and floating leg cashflows to present value
Bond pricing	Calculate the PV of coupon and principal payments
Loan valuation	Value amortising debt schedules
DCF models	Discount projected cashflows in financial models
Mark-to-market	Calculate current value of existing positions

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Key Properties

• Always positive — Discount factors are always > 0

• Decrease with time — Longer maturities have lower discount factors (assuming positive rates)

• DF at t=0 is 1 — Today's value of £1 today is £1

• Related to zero rates — DF and zero rates are mathematically equivalent representations

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Discount Factors vs Zero Rates

Both represent the same information in different forms:

Representation	Formula	Use Case
Discount Factor	DF = 1/(1+r)^t	Direct PV calculations
Zero Rate	r = (1/DF)^(1/t) - 1	Comparing rates across tenors

BlueGamma provides both via the API and Excel Add-in.

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Related Documentation

• Interest Rate Curves — How curves are constructed

• Zero Rates — Zero rates and discount factors are equivalent representations

• Forward Rates — How forward rates are calculated

• Government Bond Curves — Treasury yield curve construction

• How to Calculate Discount Factors Using Government Bond Yields — Excel guide for Treasury-based discounting

• API Reference — Complete API documentation