Bootstrapping Swap Discount Factors from Par Swap Rates

Summary

The document addresses how to derive discount factors, and hence zero rates, from EUR plain vanilla swap par rates. The response points out that the referenced EUR swaps versus six-month rates use annual payments in the curve setup described, despite the question’s assumption of twice-yearly payments. Given the first-year discount factor, it presents a sequential formula that solves each later discount factor from the corresponding par rate and the sum of earlier discount factors.

A year-two calculation illustrates the recurrence using a negative par rate and produces a discount factor above one. This is a compact example of bootstrapping rather than a complete curve construction. The formula relies on the stated annual-payment convention and on using the same curve for forwarding and discounting; different payment schedules, day-count rules, curve conventions, or separate forward and discount curves would require adjustments. The document does not detail interpolation or the conversion from discount factors to zero rates.

Key ideas

• Bootstrap later discount factors sequentially from par swap rates and previously solved factors.
• The answer uses annual swap payments for the cited EUR curve convention.
• The recurrence assumes one curve is used for both forwarding and discounting.
• A negative par rate can produce a discount factor above one in the example.

Tags

# Pricing IRS: bootstrapping zero rate (spot rate) from the swap curve


# Pricing IRS: bootstrapping zero rate (spot rate) from the swap curve












I would like to ask about swap zero curve calculation algorithm used by Bloomberg. Below is a plain vanilla EUR IRS. I want to calculate >= 2 year spot rates from the market rates. I don't know how to bootstrap them for the valuation date = 04/14/2019. The swap pays twice a year on Jan 19 and Jul 19. Below I have attached screenshots of the swap yield curve (with Bloomberg zero rates that I'm trying to replicate) and swap details.

## Answer by David Duarte (score 1, accepted)

https://quant.stackexchange.com/a/53202

It seems you are using the same curve for forward and discounting.

The EUR Vanilla Swaps vs 6M actually have yearly payments, so to obtain the discount factors, and after having the DF for year 1, you can sequentially solve for them just using the par swap Rates.

$$DF_n = \frac{1-par_n \times \sum^{n-1}_{i=1} DF_i}{1+par_n}$$

So the DF for year 2 would be:

$$DF_2 = \frac{1-(-0.0019925)\times 1.002337 }{ 1 + (-0.0019925) } = 1.003998$$

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.