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Why Three-Month FRAs Cannot Determine the Six-Month Rate Curve

Article Quant Q&A · Author: Math
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Summary

The document addresses whether observed three-month forward rate agreements can be used to bootstrap a six-month interbank rate curve. Its central point is that three-month instruments alone do not reveal the spread between three-month and six-month rates. The tenors can reflect different credit and liquidity costs, and the size of that basis cannot be inferred from the shorter-tenor rates alone, even if an overnight indexed swap curve is treated as the risk-free reference.

A second answer describes how spot-rate observations can help interpolate missing maturities: combine forward rates with suitable spot rates on either side of the desired maturity. The examples explain the anchoring idea, but do not give a full curve-building procedure, market conventions, or a numerical calibration. The practical distinction is that interpolation within a curve requires anchoring observations, while converting a three-month curve into a six-month curve also requires information about the tenor basis.

Key ideas

  • Three-month FRA quotes alone do not identify the three-month versus six-month basis.
  • Different tenors can embody different credit and liquidity premia.
  • Spot rates can anchor implied spot-rate estimates when combined with forwards.
  • Interpolating maturities does not supply missing information about the basis between tenors.

Tags

Fixed incomeDerivatives pricingStatistics
Full text
# 6 month curve from 3 month forward rate agreements


# 6 month curve from 3 month forward rate agreements












Is it possible to bootstrap at least an approximate 6 month LIBOR curve (actually NIBOR, for Norway, in my case) if rates for 3 month FRAs are known? For example, say we know the rates for 1x4, 7x10, and 10x13 FRAs. Can we deduce any points on the 6 month curve?

## Answer by Ami44 (score 4, accepted)

https://quant.stackexchange.com/a/31603

No, you can't. You can never deduce the 3M/6M basis spread from 3 month instruments alone.

If you consider the OIS curve riskless, you can interpret the 3 month curve as riskless rate + additional cost for things like credit risk, liquidity and so on. The 6 month rate contains even more of these credit risk and liquidity cost. How much exactly though is impossible to say from 3 month rates alone.

## Answer by Will Gu (score 0)

https://quant.stackexchange.com/a/31597

You would need spot rates to anchor on. For example, in addition to the forward rates, if you have one-month spot, then you can calculate four-month spot based on your forward. you can impute the spot between one and four months.

To your question specifically, if you have spot at one-month and seven-month, then you can impute the six-month spot. Other combination like four and seven would work, too. The idea to is have spots on both sides so that you can impute.

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.