Bootstrapping Discount Factors for Coupon-Bearing Swap Curve Instruments

Summary

The document asks how to reproduce discount rates for the long end of a zero-rate curve shown in a Bloomberg terminal. It distinguishes short maturities, described as zero-coupon swaps and calculated with a simple rate-and-period formula, from longer maturities, described as swaps with daily compounding and quarterly coupon payments. The author says the short-end calculation matches an observed market example, while recognizing that the same direct formula does not apply to coupon-bearing instruments.

The unresolved issue is how to derive long-end discount factors from the swap cash flows. In general, this requires valuing the fixed and floating legs consistently and solving for discount factors or zero rates across payment dates, using market conventions and previously bootstrapped curve points. The document itself provides no full instrument terms, formula, or replication evidence for that process, so details such as day-count basis, calendars, and compounding conventions remain unspecified.

Key ideas

• The document distinguishes zero-coupon short-end instruments from coupon-bearing long-end swaps.
• A direct rate-and-period formula is reported to work for the short end.
• Long-end discounting must account for the timing and amount of coupon cash flows.
• Reproducing a market curve depends on instrument and market conventions that the document does not fully specify.

Tags

# How Bloomberg calculates discount rates for zero rate curves?


# How Bloomberg calculates discount rates for zero rate curves?












I would like to ask about discount rates calculation algorithm by Bloomberg terminal.

In the image above is possible to notice the discount rate for each term. The short end, instruments from 1 DY up to 18 MO, is composed by zero coupon swaps. The long end, instruments from 2 YR up to 20 YR, is composed by swaps compounded daily with quartly coupon payments.

We can see the 2 YR instrument description below:

I was able to calculate the discount rate for the short end by using the following formula:

We can easily check that above formula works for every term of the short end, e. g., for 3 MO term (curve date 10mar22), we have that: market rate is 5.905 and period is 92 days (14jun22 minus 14mar22).

Since the long term instruments has coupon payments, the formula above is not apropriate.

1. How can I replicate the discount for long end terms?

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