# Discount Factors

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

***

## 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.

***

## 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

***

## 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

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

***

## 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.

***

## Using Discount Factors in BlueGamma

### API

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

**Response:**

```json
{
  "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")
```

***

## 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              |

***

## 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

***

## 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.

***

## Related Documentation

* [Interest Rate Curves](/documentation/methodology/how-to-bootstrap-the-yield-curve.md) — How curves are constructed
* [Zero Rates](/documentation/methodology/zero-rates.md) — Zero rates and discount factors are equivalent representations
* [Forward Rates](/documentation/methodology/forward-rates.md) — How forward rates are calculated
* [Government Bond Curves](/documentation/methodology/government-bond-curves.md) — Treasury yield curve construction
* [How to Calculate Discount Factors Using Government Bond Yields](/documentation/integrations/excel-add-in/how-to-guides/how-to-calculate-discount-factors-in-excel.md) — Excel guide for Treasury-based discounting
* [API Reference](/documentation/integrations/api/api-reference.md) — Complete API documentation


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