Bootstrapping Spot Rates Under Coupon and Compounding Conventions
Summary
The discussion uses two coupon bonds to illustrate bootstrapping a one-year spot rate from a known six-month rate. It explains that, under a semiannual coupon convention, the annual yield is divided into two coupon-period rates and cash flows are discounted over the corresponding number of periods. The example frames this as a market convention rather than a universally required mathematical rule.
The answer contrasts the U.S. convention of splitting an annual coupon rate evenly across semiannual payments with a Brazilian convention that derives the half-year rate through compounding. It emphasizes that conventions reflect market history and investor expectations, so an issuer using an unfamiliar convention may face a higher required yield. The explanation is brief and gives no full worked solution for the unknown spot rate; actual calculations depend on the bond's contractual and market conventions.
Key ideas
- Bootstrapping solves for later spot rates by discounting known earlier cash flows and isolating the remaining unknown rate.
- With semiannual compounding, yields and coupon amounts are applied per half-year period under the relevant convention.
- Coupon conventions vary by market and can use either equal rate division or an equivalent compounded rate.
- Investors' familiarity with conventions can affect the yield demanded on a bond.
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Full text
# Bootstrapping adjustment for coupon # Bootstrapping adjustment for coupon I struggling to get why in bootstrapping I need to divide the YTM by 2 (for semiannual coupons) and not adjust the power for the semiannual period. Please see below example. Consider two bonds with a face value of $ 100, with the yield to maturity equal to the coupon rate: Maturity 0.5 Year 1 Year Yield to Maturity 3.0% 3.50% Now, for a zero-coupon with a maturity of 6 months, it will receive a single coupon equivalent to the bond yield. Hence, the spot rate for the 6-month zero-coupon bond will be 3%. For a 1-year bond, there will be two cash flows, at 6 months and at 1 year. The cash flow at 6 months will be (3.5%/2 * 100 = $ 1.75) and cash flow at 1 year will be (100 + 1.75 = $ 101.75) From the 0.5-year maturity the spot rate or the discount rate is 3% and let us assume the discount rate for 1-year maturity be x%, then 100 = 1.75/(1+3%/2)^1 + 101.75/(1+x/2)^2 why we divide the coupon by 2 and don't adjust the power as follows? (assume semi annual coupon so 180 days for the 1st cashflow and 360 for the 2nd cashflow): 100 = 1.75/(1+3%)^(180/360) + 101.75/(1+x)^(360/360) https://www.wallstreetmojo.com/bootstrapping-yield-curve/ Can you please explain the logic behind it as you would do to an undergraduate student? ## Answer by Dimitri Vulis (score 2) https://quant.stackexchange.com/a/78214 Many texts and sites are U.S./Europe centric, and make it sound like there must be some logic behind U.S./Europe market conventions, but, in general, market conventions are driven not by logic, but more by convenience tradition. In the U.S., "6% a year coupon paid semi-annually" usually means "6%/2=3%" coupons, but, for example, in Brazil it means "(1+6%)^(1/2)-1=2.95%" coupons (see, https://sisweb.tesouro.gov.br/apex/f?p=2501:9::::9:P9_ID_PUBLICACAO:27710 , page 8). This is because when the newly independent Americans first started issuing fixed-coupon bonds in the U.S., in late 18th century, it was hard to solve for square roots, so they did what the Europeans did for centuries. But when Brazilians started issuing fixed-coupon bonds, they felt less bound by European traditions, so they did what seemed more logical. A few other things are done in Brazil in ways that make more sense than in the U.S. You're free issue a fixed-coupon bond in the U.S. and specify in the prospectus that the coupon amount will use square roots. But some potential bondholders will be spooked by this, and won't be willing to buy your bond, unless you pay higher yield.
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