Valuing bonds with embedded options - CFAI Text Reading 43 - pls help

Why are we adding the annual coupon in the following equation on Page 289 of CFA Fixed Income Text

0.5*((105/1.08 + 105/1.08)) _ + 5 _ = 102.2222 ought it not have been already captured in the first part of the equation?

please advice

In the first part of the equation you are discounting the coupon from the righthand (the future) node. After you get your PV of your future cash flow you are adding coupon to be recevied at current node.

It looks like CF is received in the beginning of the period. Is that correct?

Phantom, the formula you gave is for valuing option-free bonds.

0.5*((105/1.08 + 105/1.08)) _ + 5 _ = 102.2222

0.5*((Upper Bond Price/1+r + Lower Bond Price/1+r) + Coupon

However, your subject line says valuing bonds with embedded options, which is slightly different…

0.5*((Upper Bond Price+ Coupon)/ 1+r + (Lower Bond Price + Coupon)/ 1+r)

Maybe that’s what’s tripping you up? Its 2 different formulas depending on wether its an option free bond or a callable/putable bond.

If this is the in text example it may be because they decided to switch to valuing the bond as it were paying coupons in the beginning of the year, out of the blue, and didn’t mention it at all. I could be referring to something else, but I don’t have the time to check the text. It drove me insane when I saw it. s2000magician also tipped off a few people on what I’m talking about, so try digging back a bit on here.

Shoot, hope I’m not wrong here, but the part about coupons being paid in the beginning of the year (I thought) referred to floating caps/floor bonds, in which case the coupon rates are set by the previous year’s node’s rate. They do explicitly say “these coupon payments pays one year LIbor, set in arrears”…

I don’t recall the material writing something like that for regular embedded options, but they also don’t give a good explanation as to why there are 2 formulae. Hopefully someone good at FI can chime in…

i do not understand this either. hoping someone can chime in and clear this up.

I’ll give this one a shot, hopefully this helps us all learn and I don’t make error here, so don’t take this at face value, necessarily. Wow, that’s actually a decent pun. n1ksking, to me those formulas are the same, you just have the coupon outside in the first one to see what’s going on with the face value better. And that’s how coupon bonds trade when you look at their dirty prices. Kind of in a jigsaw pattern as they accrue interest. Phantom, the first formula separates out the clean bond and then plugs the coupon back in. The second is doing the same thing but just shows it with the coupon in the mix, because only then we can see if the bond is trading over $1xx.xx, indicating that interest rates are low, it’s at a premium, and it’s called. Remember, you start at the right, with the final payment, or FV, which here looks like 100 and the last coupon of 5. so FV 108, and work right to left solving the good old PV = FV / (1+r) and luckily it’s always annual r’s here to make it easier. Your PV at any node is just average of the next period’s FVs, i.e., two face values and two coupons because there’s a branch with two nodes, with a 50-50 chance each. Whats 105/1.08? 97.22. Now add it twice, like it is in parentheses and multiple by .5 That’s the same exact this as saying 97.22 *2 *1/2 or even 97.22*1. It’s just 97.22. Which is the present value of what we are going to get, 100+5. Except today it’s just worth 97.22 The coupon that Mr. Phantom put in bold is added into the mess because yes, it’s the PV at a node, but really it’s there because it’s also the FV of the node to the left of it. This is using a binomial tree, aka backward induction. The model would have already been calibrated to the spot rates, etc.

I think it’s safe to say that the real concept of valuation/analysis comes into play here is the actual calibration of the binomial model / interest rate tree’s volatility assumption. The size of the distance between the tree branches has a huge impact on your first node, especially if you’ve got a put you can use. (I’m assuming in reality the analyst has some sort of threshold for when they would put the bond at par/100/etc. on the tree, not just the second it went to PV=99, but who knows).

Hi Phantom9, a friend helped me figure this out today and I thought I’d share. I hope you see this in time. They are adding the 5 at the end of the formula you posted instead of adding it to the node at time = 1. Then, at t = 1, they add another 5 and discount that back to t=0. But notice in the last step, they do not add any coupons when discounting by 2%. This example is different from all the others I have seen (where they add the coupons at each node when they figure out the value of the bond at that node).

I hope this somewhat makes sense. Apologies if it’s a sloppy explanation