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?
You know the forward rates TODAY with some certainty based on a broker quote. However you do not know with nearly as much certainty what those forward rates are going to be one , two , three periods from today. Its all guesswork assuming some rate structure for the future You use a binomial tree because you can set some bounds at each stage on high and low rates and compute a tree to get to your final known future price and then discount it all the way to the present price By adjusting this tree to meet market conditions you can study other characteristics such as duration
It’s useful for bonds with embedded options.
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?
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).
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
No, 3 year forward rate 1 year from now (3f1) = (1+4year spot)/(1+1yr spot rate).
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
the spot rate is a given rate that exists in the market, a forward rate is the spot rate starting some point in the future. The forward rate is implied by the spot rates. So your 2 year and 1 year spot rate will imply what your 1 year spot rate will be starting in one year (i.e. earning the 2 year spot rate for two years should be equal to earning the 1 year spot rate for one year and then the one year spot rate starting in one year, for one year).
You have $100 to invest for 3 years. You ask what is the spot rate (i.e., the rate quoted by the market right now), and you are told it is 5%. So, $100 grows at 5% compounded annually. If you decide to go with a 2-year investment then get your money at end of 2nd year and invest it in a 1-year bill, then you might find that the 2-year spot rate is 4.75%, as an example. This means your $100 will grow by 4.75% for 2 years, but then what happens for the final year? It’s anyone’s guess, but we should be able to figure out what the implied 1-year spot rate for that final leg. Call that 1f2, meaning the 1-year rate, two years from now, that’s what you are looking for. The quickest way to find that 1-year rate is to sum up the two number you have in 1f2, which equals 3, and that should go to the numerator (hang on you’ll see where). Then, take the number on the right (the 2), which is the “years from now” number, and use it as the denominator in the following: 1f2 = (1+z3)^3/(1+z2)^2. z means spot rate. In our example, the the 1-year rate, two years from now, 1f2 = (1.05^3)/(1.0475)^2 = 5.5%. What’s 3f5? … 8 goes up in the numerator, and 5 goes down. What’s 1f0?
is 2 yr ytm= 2 yr spot rate ?
Ignore this please