Need urgent help regarding fixed income EOC question 15 reading 44

Hello everyone

Here I am again with another problematic question.

I cant seem to get my head around a FI question (Q15, R44) where they set the coupon in arrears and floor/cap it at a certain level. I can figure out the coupon payment for the nod at the top and the bottom but i can not understand the process for the node which lie in the middle.

To give you a little more insight, the question is of valuing a bond with embeded options with a life of 3 yrs. They have given IRs for yr1-yr3 (1 IR for yr 1, 2 for yr 2 and 3 for yr3) … I discounted the CFs of the last year by taking all the relevant cash flows and have the DR for each as well. For 1 year on i am left with 3 CFs but i’ve 2 discount rates. That is ok when i am discounting the cash flows but i dont know what to do with the coupon payments? Should i take an average?

I hope i was able to explain you the problem … if not then please look at the EOC question and help me … needless to say i’ll be grateful for the help!!!

I literally don’t know what a cap/floor is, havent done this yet and probably won’t get to it.

A capped or floored is a ‘floater’. So as long as the one year fwd rate is less than the ‘cap’, you will get the fwd rate. When it goes above the cap, then discount it (100+Cap rate/(1+whatever the fwd rate which will be higher than the cap rate)), so you will end up with a value less than 100. Lets say, the rate never goes above the ‘cap’ rate, the value will be 100 (since it is a a floater.). That is why, value of the capped floater will be less than a straight bond.

Similarly in a floor, if the one year fwd rate is less than the floor then discount it by (100+floor rate/(1+whatever the fwd rate ,which will be lower than the floor) and hence value will be greater than 100. So value of a floor will be straight bond plus value of floor.

Net net, the coupon at each node depends on the ‘cap’ and ‘floor’. If rate is > cap, then coupon is the cap, else the actual rate. If rate is < floor, then coupon is the floor, else the actual rate.

nsbharath … thanks alot buddy but my question still remains … i am clear with the concept but i just cant solve the question i mentioned … if you can be generous enough with your time to walk me through this i’ll be grateful … i will completely understand if u can not

By “node in the middle”, do you mean the center Year 2 S=ud node?

It’s the same calculation as the S=uu node, because the rates for both the S=ud and S=uu nodes are both above the 5% cap, so the coupon is capped at 5.

So node S=ud Value (middle node, Year 2, is [((100+5) / 1.050092) + ((100+5) / 1.050092)] / 2 OR just 105/1.050092 in this case since the values+coupon in S=udu and S=udd are the same.

Remember, with Floaters, the coupon is paid in arrears, so the coupon in a particular node is based on the rate in a previous year’s node.

How did you end up with 2 CFs in year 1, it should be just 2. See in Year 3, use (105/1.063679) which will give you a value of 98.71. Similarly 105/1.050092 would give you 99.99. Average these two and arrive at ‘uu’ value of 99.35. Now to arrive at value at ‘ud’ average 99.99 (105/1.050092) and 100 (103.9404/1.039404). So at ‘ud’ value is 99.99.

Since the interest rate at year 1 is 4.5027% and 3.5419% where both are less than 5%, you don’t have to bother with any calculations. The value now is just (99.99+99.35)/2 = 99.67. If you go to 2 more decimal places you will probably get a more precise answer.