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Bootstrapping Discount Factors to Calculate a Forward Swap Rate

Article Quant Q&A · Author: Dylan Zhu
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Summary

The document explains how to estimate a forward-starting swap rate from annual par swap rates. One response offers a rough linear interpolation-style calculation from the two-year and five-year par rates, while acknowledging that it is approximate. A more precise method first bootstraps discount factors year by year from the par rates, using the present value relationship for annual fixed-leg payments.

With discount factors in hand, the forward swap rate for the period beginning at year two and ending at year five is calculated as the difference between the discount factors at the start and end, divided by the sum of discount factors for the payment dates. The example reports a result of 6.581%, compared with the rough estimate of about 6.50%. The calculation assumes annual swaps and the stated curve inputs. Its accuracy depends on the conventions and curve construction; different payment frequencies, day-count rules, compounding conventions, or interpolation choices would require corresponding adjustments.

Key ideas

  • Bootstrap discount factors sequentially from annual par swap rates.
  • A forward swap rate is derived from the discount factor at the forward start and the later discount factors for fixed-leg payments.
  • A simple weighted average of spot par rates can provide only an approximate forward rate.
  • Swap frequency and market conventions affect the curve bootstrap and resulting rate.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# How to calculate this swap rate


# How to calculate this swap rate












What is the 2x5 swap rate? here 2x5 swap rate refers to the 3-year swap, 2 years forward.

## Answer by Bozothegrey (score 3)

https://quant.stackexchange.com/a/28348

Give or take, should be 6.50%. I got this result by applying (5 * 5y swap - 2 * 2y swap) / (5 - 2). Clearly this is not exact and I feel a bit ashamed to publish it on a quant forum.

## Answer by David Duarte (score 3)

https://quant.stackexchange.com/a/42796

Assuming these are par rates for annual swaps, if you bootstrap the curve with DF = (1 - par * sum of dfs from prev annual payments) / (1 + par), you get:

```
1y 0.952381
2y 0.902613
3y 0.851161
4y 0.798483
5y 0.745020
```

The 2y3y swap rate would be (df2y - df5y) / sum(df3y, df4y, df5y), i.e., 6.581%

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.