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Why a Lower Three-Month Spot Rate Can Raise the Forward Rate

Article Quant Q&A · Author: Thomas Boyd
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Summary

The document explains the key-rate duration behavior of a Eurodollar futures contract by relating its value to a forward three-month LIBOR rate. The question concerns a down shock to the three-month spot key rate that appears to increase the corresponding forward rate, even though the contract is exposed to rates at its future settlement date.

The answer gives a compact explanation: the six-month rate is held constant while the three-month spot rate falls. Since the six-month rate incorporates the short spot period and the following forward period, keeping the longer rate unchanged requires the forward rate to rise to offset the lower spot rate. This helps explain the observed direction of the partial duration response. The exchange provides no derivation, numerical example, or detailed specification of the rate curve and shock construction, so the result should be understood in the context of that fixed six-month-rate assumption.

Key ideas

  • A Eurodollar future can be viewed as exposed to a forward three-month LIBOR rate at settlement.
  • The explanation assumes the six-month rate remains unchanged during the spot shock.
  • If the three-month spot rate falls under that assumption, the following forward rate must rise to keep the six-month rate constant.

Tags

FuturesFixed incomeDerivatives pricingRisk management
Full text
# Eurodollar Future Key Rate Duration


# Eurodollar Future Key Rate Duration












I am having trouble understanding the Key Rate (partial) Duration profile of Eurodollar Future contracts. Using market rates and pricing date as of 11/14/2018 I have calculated the partial duration profile of the EDH9 (March '19) contract as such:

With an 11/14/2018 reference date, this contract has ~3mo to expiry at which point it will cash settle on the value of 3mo LIBOR. So it should follow that the value of this contract (ignoring any effects of convexity) derives its value from 3mo LIB struck 3mo forward.

Turning back to the partial duration profile above, this contract has (-) KRD DV01 3M, implying its value falls when the 3mo key rate is shocked down 1bps. Stated alternatively, in the KRD 3M DN1 scenario, 3mo x 3mo LIB has risen (long Euro$, rates rise, value falls). This development in forward space is observed:

My question is why does a 1bps down shock to LIB Spot produce an increase of 1bps to LIB FWD 3M?

I have spent some time surfing the net and cannot seem to find a concise explanation. Thanks for your help!!

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/42669

Because you are keeping the 6m rate constant. Therefore, if the spot 3m rate goes down, the forward must go up.

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