YTM, spot rates, forward rates, binomial tree, recap. par, different, corporate, discount

anshultongia

Some review... then some questions at the end. Spot me, no pun intended.

1. You are given the yield curve of some corporate issuer, all bonds are option-free. So, you know the YTM for different maturities (1-yr bond has YTM = 5%, 2-yr bond has YTM=5.5%, etc). Say, the 4-year par bond has 5% coupon.

2. You know that if you use the YTM of 5%, you can discount all coupons plus principle and get a value of $100.

3. You should know that you can also discount those same cash flows using spot rates instead of the YTM, and get the same value of $100. How do you get those spot rates? use bootstrapping, where you set 1-yr spot rate to 1-yr YTM. Then 2-year spot rate can be determined by solving a simple equation: Coupon of 2-year bond/1-year spot rate + coupon+principal of second year/1+2yr spot rate, set these two terms to $100. Use this obtained rate to solve for 3-yr spot rate, etc. Not hard.

4. The forward rates are future spot rates. So, the 2-year spot rate is the rate you get for two years, compounded for 2 years that is. The 1-year forward rate, 1 year from now, is the spot rate on a 1-year bond a year later ....(note the difference).

5. You will also get the same $100 bond value if you use the forward rates instead of YTM and spot rates...all three give the same value.

Now, what the heck is a binomial interest rate tree? I can do the math well, where you start at the far right...get the coupon value for the final year, add principal, then discount using the 1yr rate in previous node, take half of that value. Do the same using the previous lower node, and take half of that value, add them up, and that's the bond's value at that node...etc. etc.

Question, why do we do that? If I have all those rates (spot, forward, and YTM), what do I get by using a model that assumes certain interest rate volatility and goes through the tree trying out different rates to see what set of rates would set the value of the tree to par, or whatever the value is, based on coupon?

What do I use those rates I get from the model for? I can do this mechanically like a charm, but how does it help in valuing a bond, that I already have various rates for?

onelife1

The critical part is that the binomial model is used to value the embedded options, the actual rate doesn't matter so much as its effect on the price of the bond (i.e. in a low rate environment a bond's value may be capped at par if it's callable at par).

sameeragarwal

Just to check my understanding about the relation between spot and forward rates
If you know the 4 year future spot rate (from bootstrapping) then you can have the following interpretations for forward rates:
4 year spot rate =
3 year forward rate 1 year from now
2 year forward rate 2 years from now
1 year forward rate 3 years from now

Please confirm

Thanks

profil

No, 3 year forward rate 1 year from now (3f1) = (1+4year spot)/(1+1yr spot rate).

xilige

I am strugglling in understanding

4. The forward rates are future spot rates. So, the 2-year spot rate is the rate you get for two years, compounded for 2 years that is. The 1-year forward rate, 1 year from now, is the spot rate on a 1-year bond a year later ....(note the difference).

Could you please expand this ....(note the difference).


Thanks

mp3bu

It's useful for bonds with embedded options.

NO EXCUSES

jarobi04

That's a good point regarding how it's based on a *known* future price, principal and coupon known ahead, which then can be used for "guessing" what rates could achieve this price. Also, the fact that it's a good way for valuing bonds with embedded options. All fine and dandy, but may be the below statement is the real benefit:

> By adjusting this tree to meet market conditions you can study other characteristics such
> as duration

Anyone can cite some examples?