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Why Money-Market Cash Rates Differ from Continuously Compounded Zero Rates

Article Quant Q&A · Author: Tian
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Summary

The document explains why a quoted three-month cash rate can differ from a three-month zero rate when the rates use different day-count conventions and compounding definitions. Its example starts with a market rate quoted on an ACT/360 basis over 92 days, converts that quote into a discount factor, and then expresses the discount factor as a continuously compounded rate on an ACT/365 basis. A second answer describes an equivalent route through an effective rate before converting to the continuous convention.

The two calculations produce the same zero rate in the example, showing that the apparent discrepancy can arise from quote conventions rather than inconsistent pricing. The note is useful when reading or reconstructing a bootstrapped yield curve, where cash-market quotes and reported zero rates may be expressed on different bases. It only covers the stated short-dated example; it does not explain Bloomberg’s full curve-building process, instrument selection, or treatment of broader curve conventions.

Key ideas

  • A cash quote and a zero rate may use different day-count bases and compounding conventions.
  • The example converts the cash quote to a discount factor before deriving a continuous zero rate.
  • An effective-rate conversion provides another route to the same continuous rate in the example.
  • The calculation explains a rate-definition difference but does not document Bloomberg’s full bootstrapping method.

Tags

Fixed incomeStatistics
Full text
# Why the 3M Zero Rate is not equal to the 3M Cash Rate? On Bloomberg yield curve bootstrapping


# Why the 3M Zero Rate is not equal to the 3M Cash Rate? On Bloomberg yield curve bootstrapping












Can someone explain to me why the 3M Zero Rate is not equal to the 3M Cash Rate? Thanks.

## Answer by emot (score 10)

https://quant.stackexchange.com/a/71275

Market Rate, for this particular case, is a rate quoted on ACT/360 basis, start date = 21/06/2022, end date = 21/09/2022 on ACT/360 basis means year fraction 92/360. Discount Factor is $1/(1+92/360*2.02957\%)=0.99484008$. Zero rate is continous rate on ACT/365 basis, therefore $ZR=-ln(0.99484008)/(92/365)=2.05244\%$

## Answer by Daniele Penza (score 0)

https://quant.stackexchange.com/a/83721

you can also calculate the effective rate = (1+2.02957%/(92/360))^(92/360)=2.044953%. Then you take the continuous equivalent scaled on 365. ln(1+2.044953%)*365/360=2.05244%

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