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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?

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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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    $\begingroup$ +1. "Spooked by roots" - May I quote that? $\endgroup$ Commented Feb 6 at 10:10
  • $\begingroup$ of course! An amusing anecdote. I attend sometimes a very nice seminar on the history of mathematics, and a few years ago we had a talk on the kind of (applied) math that George Washington studied as a teenager. I.e. that was what many teenagers of his background would study, but his notes are well preserved. I was surprised by his fluency in using logs of trig functions that would petrify most people today. $\endgroup$ Commented Feb 6 at 12:09

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