# Coupon Rate from Binomial Interest Rate Tree

1. Is it possible to find the coupon rate by hand (as opposed to using Excel) on a callable/putable bond using the market price of the bond and the forward rates used in the binomial interest rate tree? I was able to use Goal Seek on Excel to find the coupon rate but had trouble finding the coupon rate by hand.

2. Is it possible to move forward in a binomial interest rate tree? That is, is it possible to compound the market price to each node without knowing the coupon rate, but knowing the forward rates used in the tree?

What’s the problem with figuring out the coupon rate? If it’s fixed, it’s given. If it’s floating, then it will be a function of the forward rate at a given node (e.g., LIBOR +100 bp).

Perhaps I’m missing something here.

The problem gives me everything other than the coupon rate (in this case, it’s fixed). The question is asking for the coupon rate; the information given includes the forward rates in the tree, and the market price today. I’m not sure how to solve for the coupon rate by hand.

Is it multiple choice?

If so, try the middle coupon rate. If you get the correct price, that’s the correct answer. If you get too high a price, the coupon rate’s too high, so the lower rate is the correct answer. I’ll let you work out what to do if you get too low a price.

You can solve for the coupon rate algebraically, but it’s tedious. And silly.

Thank you! Yes it is multiple choice, but I didn’t know if there was a way to find the coupon rate quickly by hand instead of trial and error.

Not quickly.

It turns out to be a linear equation, so it’s easy to solve, but you have to figure out all of the discount factors and all of the weights multiplied by those discount factors and tot them all up to get the coefficient on the coupon and the constant. Here’s an example:

r0 = 2.000%

r1,H = 2.280%, r1,L = 1.900%

r2,HH = 2.592%, r2,HL = 2.160%, r2,LL = 1.800%

P0 = \$1,003.24

Working backward through the tree:

P2,HH = (\$1,000 + C) / 1.02592 + C

P2,HL = (\$1,000 + C) / 1.02160 + C

P2,LL = (\$1,000 + C) / 1.01800 + C

P1,H = [(\$1,000 + C) / 1.02592 + C + (\$1,000 + C) / 1.02160 + C] / 2 / 1.02280 + C

P1,L = [(\$1,000 + C) / 1.02160 + C + (\$1,000 + C) / 1.01800 + C] / 2 / 1.01900 + C

P0 = {[(\$1,000 + C) / 1.02592 + C + (\$1,000 + C) / 1.02160 + C] / 2 / 1.02280 + C

• [(\$1,000 + C) / 1.02160 + C + (\$1,000 + C) / 1.01800 + C] / 2 / 1.01900 + C} / 2 / 1.02000

\$1,003.24 = {[(\$1,000 + C) / 1.02592 + C + (\$1,000 + C) / 1.02160 + C] / 2 / 1.02280 + C

• [(\$1,000 + C) / 1.02160 + C + (\$1,000 + C) / 1.01800 + C] / 2 / 1.01900 + C} / 2 / 1.02000

I’ll let you take it from there. The answer is C = \$22.00; i.e., the coupon rate is 2.2%.

WOW! Thank you for taking time to explain it in such detail. You’re truly a magician!