Converting Bond Yields Between Semiannual and Continuous Compounding
Summary
The answers clarify that a bond’s coupon frequency determines how coupon payments are made, while the quoted annual yield depends on the compounding convention. A semiannual coupon quoted at an annual rate is divided by two to obtain each half-year payment rate. Compounding that rate over two periods gives the effective annual yield; expressing the same growth continuously requires taking the natural logarithm of the compounded growth factor.
The example relates a semiannually compounded yield near 6.87% to a continuously compounded rate of 6.75%. These are alternate quotations for the same annual growth factor, not competing solutions to the bond-pricing equation. The discussion also distinguishes the coupon from the yield and shows why the payment frequency appears in the pricing equation. It is a compact convention-focused explanation, not a full treatment of bond valuation, day-count rules, or other coupon schedules.
Key ideas
- A semiannual coupon is paid in two installments over a year.
- The quoted annual coupon rate is divided by two to obtain the rate for each half-year period.
- Annual growth under semiannual compounding differs numerically from its continuous-compounding quotation.
- Equivalent yield quotations can be related by matching their annual growth factors.
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Full text
# Hull's book par yield example
# Hull's book par yield example
In Hull's book (9th edition), on page 83, there is a simple example of par yield:
I am a bit confused when it says "this has semiannual compounding because payments are assumed to be made every 6 months. With continuous compounding, the rate is 6.75% per annum." Isn't the rate of 6.87% already assuming continuous compounding and is obtained by solving the equation here? What does it mean with continuous compounding, the rate is 6.75% per annum? Where is this coming from? And when it says "this has semiannual compounding", what is it referring to?
## Answer by Aksakal almost surely binary (score 2, accepted)
https://quant.stackexchange.com/a/60546
The way I like to explain this is with a notion of quoting. It's a convention to quote the coupons annualized by multiplying them by frequency. Suppose, the coupon is semiannual and equal to 3.375% of the outstanding. This is how much interest is accrued during 6 months. However, it is the convention to quote it on annualized basis, i.e. multiplied by 2 since it's semiannual. So, the coupon is quoted as c=6.75%.
Compounded, this coupon will yield the following in one year: $$(1+c/2)^2=1+y$$ $$y=(1+c/2)^2-1\approx 6.86\%$$
Alternatively you can quote the same yield on continuous compounding base: $$(1+c/2)^2=e^y$$ $$y=2\ln (1+c/2)\approx 6.64\%$$
## Answer by dm63 (score 5)
https://quant.stackexchange.com/a/60545
c is the coupon of the bond, so it is paid semiannually. You can see this from the LHS of the first equation, which is the sum of present values of the coupons and principal. The 6.87 and the 6.75 are related by
$$ (1+6.87/200)^2 = e^{0.0675} $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.