I am really stuck trying to figure out how to calculate a coupon rate.
 Most of the equations and examples I have found do not have this as
an unknown.  Please point me in the right direction.  I know the sell
price, bond life, and Yield-to-maturity and have been asked to find
the coupon rate of the bonds.

Please show me an example.  You can make up your own numbers for the
sell price, bond life and YTM.  I don't want anyone to give me the
answer to my problem.  i just want to know how to do it.

Thanks!

Hi!!


Always take into account this:
"Market value of a bond = PV of future payments (coupons and principal)
discounted at cost of debt (the yield to maturity)".

And:
"PV of future payments (coupons and principal) discounted at cost of
debt (the yield to maturity) = PV coupons + PV of principal"


You know:
Market value of the bond (selling price), PB ;
bond life (the number of future payments), n ;
YTM;
Face value or principal (the last payment, usually if it is not
indicated in the problem statement it is equal to $1,000), Pr.

PV principal = Face value / (1+YTM)^n

PB = PV coupons + Face value/(1+YTM)^n

Then:
PV coupons = PVc = PB - Face value/(1+YTM)^n


Note that:
PVc = Coupon/YTM * [1 - (1 / (1+YTM)^n)] 

Then:
Coupon = (PVc * YTM) / [1 - (1 / (1+YTM)^n)] 

Knowing Coupon payments we have that:
Coupon rate = 100 * Coupon / Face value   [%]



For example if the selling price is $1086; YTM = 6.8% and the bond
life is 14 years we have (use a face value of $1000):

PV principal = Face value / (1+YTM)^n =
             = $1000 / (1.068)^14 =
             = $398.11

PVc = PB - Face value / (1+YTM)^n =
    = $1086 - $1000 / (1.068)^14 =
    = $1086 - $398.11 =
    = $687.89

Then:
Coupon = (PVc * YTM) / [1 - (1 / (1+YTM)^n)] =
       = (687.89 * 0.068) / [1-(1/1.068)^14] =
       = 46.77652 / 0.6019 =
       = $77.715

Finally:
Coupon rate = 100 * Coupon / Face value =
            = 100 * 77.715 / 1000 =
            = 7.7715 %
       

I hope this halps you. Feel free to request for a clarification if you need it.

Regards,
livioflores-ga

Thank you!