# Backward induction with Caps or Floors

Hi guys,

this is the only area of backward induction that throws me. Can somone please explain the process here:

Example: Y2 7.02%

Y1 3.71%

Y0 2% Y2 5.20%

Y1 2.75%

Y2 3.85%

What is the value of a bond with a 5% cap and floating rate libor coupon.

Step 1) Calculate whether the final nodes are capped or not? So 7.02% capped at 5, 5.20% capped at 5, 3.85% not capped.

Step 2) 105/1.072 = 98.11 (what coupon do we add here? solution suggets no coupon added)

Step 3) Repear process back to T0 with usual process of averaging the nodes and dividing by previous node etc.

I am confused with which coupons we add back though.

Can anyone work through the example above and let me know please?

In the solution there is no coupon added at the T2 period but at T1 we are adding the variable rates (3.71 and 2.75 as coupons before inducting again)

Thanks

Remember that with floaters, the coupon is determined at the beginning of the period, but paid at the end of the period (a coupon t=1 will actually be paid at t=2. So the coupon when calculating the value at a particular node is the rate listed in that node, subject to the cap or floor.

Y2HH = ( ((100+5) + (100+5)) / 2 ) / 1.072 = 97.9478

Y2HL = ( ((100+5) + (100+5)) / 2 ) / 1.052 = 99.8099

Y2LL = ( ((100+3.85) + (100+3.85)) / 2 ) / 1.0385 = 100

Y1H = ( ((97.9478 + 3.71) + (99.8099 + 3.71)) / 2 ) / 1.0371 = 98.9190

Y1L = ( ((99.8099 + 2.75) + (100 + 2.75)) / 2 ) / 1.0275 = 99.9075

Y0 = ( ((98.9190 + 2) + (99.9075 + 2)) / 2 ) / 1.02 = 99.4235

Hopefully that’s the correct answer.

Thanks Jaywill you are clearing it up! Superstar. Not got the answers on me at the moment but i’ll let you know

I am trying to use your exact methodology for another example but not working (from the Wiley Mock 1 AM) - Question 16.

Annual coupon based on 12 month libor 320bps.

Cap = 5.40% 2.6865%

2.0908% 2.2893%

0.5430% 1.7817% 1.9508%

Step 1: Add 320bps to each yield I presume?

Step 2: Using the new rates with 320bps added on check whats ITM and calculate prices

(105.4 + 105.4)/2 / 1.058865 (2.6865+320bps) = 99.5405 (CORRECT)

(105.4 +105.4)/2 / 1.054893 (2.2893+320bps) = 99.9153 (CORRECT)

(105.1508 + 105.1508)/2 / 1.01508 = 100.

Up to here is fine.

Now following your methodology:

99.5405 + 5.2908 (Node 2 +320 bps) =104.8308

99.153+ 5.2908 = 104.4438

104.8308 + 104.4438/2 / 1.052908 = 99.37

Answer key is suggesting that is 99.7416?

Any thoughts?

your calculation of 104.8308 is wrong, should be 104.8313. Also, in your first calculation where you got 100, I believe that’s wrong too…believe it should be \$103.1388

Is anyone able to show me the full solution to the second question with 320bps added on? I cannot seem to get the right answer. I would be greatful a million times over.

Thanks

99.5405 + 99.9153 + 10.581 / 2 = 105.0187

10.581 = (2.0908 + 3.20) x 2

105.0187 / (1.020908 + .0320) = 99.7416

Your math wasn’t adding up when you got 104.8308 & 104.4438, that is why you were stuck.