Bootstrapping an OIS Discount Curve from Par Swap Rates

Summary

The document asks why an overnight indexed swap (OIS) par rate cannot simply be converted into a discount factor using a single-period formula. The response explains that discount factors at longer maturities must remain consistent with discount factors already derived for earlier cash flows. For example, a one-year factor can be decomposed into the product of a factor to an intermediate date and a forward factor from that date to one year.

In this view, the quoted one-year par rate helps determine the remaining discount factor after previously bootstrapped information is accounted for; it does not independently define the full maturity discount factor as though the whole period were a single deposit. The goal is a coherent curve for valuing future payments in one currency, assuming instruments have consistent credit quality. The answer is conceptual and omits coupon schedules, compounding conventions, and a numerical bootstrap, so implementation requires additional instrument details.

Key ideas

• A par swap rate is not generally a standalone single-period rate for converting directly to a maturity discount factor.
• Longer-maturity discount factors must be consistent with factors derived for earlier dates.
• Bootstrapping uses known curve information to infer the remaining factor over a later interval.
• A coherent discount curve can value future payments in a currency when input instruments have consistent credit quality.
• Actual curve construction also depends on instrument schedules and conventions not covered here.

Tags

# Bootstrapping OIS discount curve from OIS Swaps


# Bootstrapping OIS discount curve from OIS Swaps












I‘m trying to understand why we need Bootstrapping to determine discount factors when we already have the fix rates of OIS Par Swaps. So lets assume that we have the following (annualized) rates: 1 Day - 3,8% 1 Week - 3,79% 2 Weeks - 3,789% … 1 Year - 3,5% Why can‘t we for example use the one year rate directly to determine the 1y discount factor by $\frac{1}{1+3,5%}$ ?

## Answer by user68819 (score 2)

https://quant.stackexchange.com/a/81388

D(0,1y) = D(0,6m) x D(6m,12m). I'm arbitrarily using 6m here. But it shows you that to be consistent with other discount factors of differing tenors the 1y df must also incorporate preceding information, where available. This is why the procedure is commonly referred to as bootstrapping.

From this angle, really the 1y rate is telling you, to be consistent with whatever you have computed up till then, what the d(6m,1y) should be. All the preceding information is known.

This way, in an ideal world, for a single currency (implied from instruments of consistent credit quality) you have a single discount curve which can be used to compute the present value of 1 unit of that currency paid at some time in the future.

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.