When discouting, what we learned at first is to discount each period cash flow with one single rate. But then now, when I’m reading the binomial interest rate tree, it says that each period cash flow is discounted using underlying spot rate. What’s the difference? Isn’t the single rate that we use to discount the cash flow the spot rate?
Spot rate is for a specific maturity. a zero-coupon bond with a 1 year maturity will earn a different spot rate than one with a 20-year maturity. But, if it’s a 20-year bond with 20 annual coupons, the 1st coupon is paid on the same date as that 1 year bond above matured. so, the 1st year coupon shouldn’t be discounted at the rate of the 20 year bond. that would be crazy. in fact, that specific 1-year coupon should be discounted at the same rate as that 1-year bond’s principal.
YTM is a bit more artificial. it’s the entire return of a bond, expressed as a single %. It’s all of the idiosyncracy of the 20 spot rates you’d imagine above, turned into the single interest rate that tells you the internal rate of return you receive if you buy the bond and reinvest the coupons, etc.
Though YTM is a single rate, the spot is a extra level of reality.
To discount bond cash flows using YTM you are implicitly assuming a flat yield curve, i.e. every year the interest rate is the same so the coupon that you receive you can reinvest back at that same market rate. That was Level 1 and it was overly simplistic.
Now that we know yield curves (term structure) are far from flat we try to assign spot rates to each cash flow separately.
Spot rates are calculated by the method of bootstrapping and from the spot rates we can then calculate forward rates.
