Hi,
Why do we multiple forward rates when discounting cash flows to find the value of a straight bond, whereas we just use a single forward rate to discount for that year when valuing callable or putable bonds (and we don’t even square it for discounting back a the cash flow at T=2 or for cash flow at T=3)
Can you give a reference to an example in CFAI or Schweser?
CFAI:
Valuation of Default-free and option-free bonds Exhibit 1 calculation on pg 181.
Valuation of Callable Bond at Zero Volatility Exhibit 2 on pg 181
You use every forward rate in the tree to value a straight bond, every forward rate in the tree to value a callable bond, and every forward rate in the tree to value a putable bond. They’re done using exactly the same methodology.
What makes you think that they’re being handled differently?
Each value is discounted for one period, using 1-period forward rates. You would square the rate (or, more accurately, 1 + r) when you’re discounting for 2 periods.
Valuation of Default-free and option-free bonds Exhibit 1 calculation on pg 181:
The three-year 4.25% annual coupon bond can now be valued using the spot rates:4
4.25/(1.02500) + 4.25/(1.03008)2 + 104.25/(1.03524)3 =102.114
An equivalent way to value this bond is to discount its cash flows one year at a time using the one-year forward rates:
4.25/(1.02500) + 4.25/(1.02500)(1.03518) + 104.25/(1.02500)(1.03518)(1.04564) =102.114
Valuation of a Default-Free Three-Year 4.25% Annual Coupon Bond Callable at Par One Year and Two Years from Now at Zero Volatility
Today Year 1 Year 2 Year 3
Cash Flow 4.250 4.25010 4.250
Discount Rate 2.500% 3.518% 4.564%
Value of the Callable Bond (100+4.250)/1.02500 (99.700+4.250)/1.03518 104.250/1.04564
=101.707 =100.417 Called at 100 =99.700 Not called
You can discount the cash flows to today at the spot rates, or one period at a time at the 1-period forward rates. You need to do the latter when you have embedded options because you need to determine whether, at each node, the option will be exercised or not.
The methodology (discounting one period at a time) is identical for option-free bonds and bonds with options. I’m not sure what you’re seeing that makes you think that they’re different.
What I’m trying to say is that when valuing option-free bonds we discount 2nd year cash flow by multiplying the first year forward rate and the second year forward rate.
Whereas for callable or putable bond we only used the 2nd year forward rate for the 2nd year cash flow in order to discount it.
This isn’t true.
We use the 2nd year forward rate to discount it back to the first year (i.e., t = 1), then the 1st year forward rate to discount it back to today.
Algebraically, they’re identical.
Thanks
You’re welcome.