# Compute coupon of treasuries

How do you compute coupon starting from yield and price? Since yield is `coupon / par`, which is `2 / 100` in this case, would it just be `yield * price`, `0.01906 * (102 + 1 / 32)` with the data below? However that doesn't add up to the 2.000% that WSJ reports for US30.

``````4:58 PM EDT 09/17/21

Yield
1.906% 0.021

Price
102 1/32

Coupon 2.000%
``````

https://www.wsj.com/market-data/quotes/bond/BX/TMUBMUSD30Y

yield is coupon / par

That is the formula for current yield, which is not the yield that the WSJ reports.

The yield they report is most likely the Yield to Maturity, which is the equivalent interest rate that would give you the same net return for a bond if you reinvested the coupons as they were received. In other words, at what interest rate could you invest the 102 1/32 and get the same effective return as buying the bond?

That yield calculation requires iterating on the interest rate (except for very simple bond schedules), but knowing the yield you can figure out the coupon with just a little algebra.

The formula for price given a coupon and yield would be:

``````P = C/(1+r) + C/(1+r)^2 + C/(1+r)^3... + (100+C)/(1+r)^n
``````

where `r` and `C` are the periodic (e.g. semi-annual) yield and coupon, and n is the number of periods until maturity. So you can solve the above equation for C without iteration.

An alternative bond pricing system does not account for reinvestment of coupons and therefor is an approximate system. But everything can be calculated without iteration:

x / 102.03125 = (1.906 / 2) * 60 / 100

x = 58.3415

and

58.3415 + 102.03125 = (coupon/2 * 60) + 100

coupon/2 = 1.0062

coupon = 2.012

http://www.kbhscape.com/bond.htm

Note that "yield" is yield-to-maturity and allows for the current bond price progressively reaching the redemption price.

• Or, 1.906 * 102.03125 / 100 = 1.9447 . Then 2.03125 / 30 = 0.0677. Finally, 1.9447 + 0.0677 = 2.012 coupon . That's without compounding but allowing for a return to redemption value. Sep 20 at 1:59