 # Yield curve versus spot rate curve

I am having a bit of difficult differentiating between the terms yield curve and spot rate curve as used in reading 57 of the CFAI readings. Specifically on the bottom of page 101 they mention that nominal spreads are based off of the Treasury yield curve whereas the Z-spread and OAS use the Treasury spot rate curve. Does anyone have this concept down and can explain in a simple, straightforward way with an example? I’d really appreciate the help. Thanks, -J

You get the yield curve (sometimes called the “coupon” curve) if you plot yield-to-maturity of (say) Treasury bonds against their maturity. You get the spot curve (also called the “zero” curve) if you perform bootstrapping on the yield curve. As the result of bootstrapping, the zero curve now plots YTM of zero coupon bonds against their respective maturities.

One thing confuses me about the spot rate is: (1) spot rate is supposed to the the YTM for a zero-coupon bond (2) so if a one-year zero-coupon bond has a YTM of 8%, the spot rate is 8%, the price (PV)should be calculated using the discount rate of 8% If it’s a one-year coupon bond, I know that we should discount the first coupon payment by 4%, which is half of 8%, but I don’t understand why we should discount the last payment (2nd coupon plus principal by (1+4%)^2 rather than 8%.

Coupon payments are semi-annual. First coupon is for 1 period @ 4%. 2nd coupon is for 2 periods @ 4%. Both these coupons occur in the first year. so by the method of boot-strapping to determine the price of the bond CPN / 1.04 + (CPN + Face)/1.04^2 = Price of the bond. Does that answer the question? CP

Portfolio Wrote: ------------------------------------------------------- > One thing confuses me about the spot rate is: > > (1) spot rate is supposed to the the YTM for a > zero-coupon bond > (2) so if a one-year zero-coupon bond has a YTM of > 8%, the spot rate is 8%, the price (PV)should be > calculated using the discount rate of 8% > > If it’s a one-year coupon bond, I know that we > should discount the first coupon payment by 4%, > which is half of 8%, > > but I don’t understand why we should discount the > last payment (2nd coupon plus principal by > (1+4%)^2 rather than 8%. Also, if the 6 month spot rate is 4% in the example above, then 8% is the BOND EQUIV. YIELD. The actual yield, as you can see from (1.04)*(1.04)= 8.16% due to interest compounding on a semi-annual basis. ie. 100/1.08 = 92.59 but 100/(1.04)^2 = 92.45 If 8% was the actual annual yield on the bond, the you would discount each coupon payment at a 6 month rate of (1.08^1/2) -1 = 3.92%

Thanks a lot. So the spot rate is actually a BEY, we should not use it to discount the cash flows, rather, to get the discount rate, we should divide spot rate by 2 and elevate it to power of 2*n n is the year I was confused because I thought the spot rate is the result, rather than a BEY

Just to clarify: The BEY is simply 2 times the 6 MO SPOT rate. To get the discount rate for each semiannual cash flow (6 mo spot rate) you want to divide the BEY by 2. The 1 yr spot rate can then be found by: (1 + 6mo Spot)^2 If the 6 month SPOT rate was 4%, the BEY would be 8% and the 1 yr spot rate would be 8.16%

Actually, disregard that last post. I completely mixed up those terms and that is not correct… walked away and now I can’t edit however.

To use your example: If the 6 month SPOT rate was 4%, the BEY would be 8% and the 1 yr spot rate would be 8% as well However, to discount the cash flow one year from now, we should use (1+4%)^2 which is 1.0816. Is that correct? If so, what’s 1.0816 called? Effective annual rate?

McLeod81 Wrote: ------------------------------------------------------- > Actually, disregard that last post. I completely > mixed up those terms and that is not correct… > walked away and now I can’t edit however. Well that is just GREAT!!!

Portfolio Wrote: ------------------------------------------------------- > To use your example: > > If the 6 month SPOT rate was 4%, the BEY would be > 8% and the 1 yr spot rate would be 8% as well > > However, to discount the cash flow one year from > now, we should use (1+4%)^2 which is 1.0816. > > Is that correct? If so, what’s 1.0816 called? > Effective annual rate? Ok, given a semi-annual yield of 4%, doubling that will give you a 8% yield on a Bond Equivalent basis. Compounding (1.04)^2 will give you an 8.16% Effective Annual rate. Sorry for the confusion. If the 6 mo Spot rate was 4% and the 1 yr spot rate was 8%, that would actually mean that the 6 month forward rate 6 months from now was 3.846%: (1.04)*(1.03846) = (1.08), I doubt that’s what was being implied in the original question. The rates used and the conversions applied really depend upon which rates are given in the problem. ie If the BEY is given (usually the stated YTM on bonds in US) convert to the EAY and discount at that rate. If given the 6 mo and 1 yr spot rates, discount at those rates.

keep these formula in mind bond equivalent yield = 2 × semi-annual yield bond equivalent yield = 2 × [(1 + yield on annual pay) ^0.5 – 1] bond equivalent yield = 2 * [(1 + monthly cash flow yield) ^ 6 - 1]

In calculating a yield curve or a spot rate rate curve, the thing I didn’t get in level one is that you are calculating a benchmark against which you are going to value a bond. A benchmark is used so that you can tell if the current price of a bond you are considering is under or overvalued. If there is a difference between the two, there is a opportunity for arbitrage. A yield curve (the benchmark) is used to determine if the bond you are considerings yield is higher or lower A spot rate curve (benchmark again) is used to determine if the price of the bond is under or overvalued. I’ve just lost my train of thought and its not as clear to me as it was when I began typing… but the point of bootstrapping is to value the bond you are considering at the spot rates to determine if the bond you are considering is under or overvalued relative to the benchmark. You’ll use the spot rates implied by your benchmark to detrmine value… something like that…

or not…

The yield curve is the set of interest rates (spot rates) that the MARKET is valuing various coupon and principle payments at in the future. There is a separate yield curve for every grade of fixed income securities (ie treasury yc, AA muni yc, corp BBB yc). Market prices for bonds are determined by discounting each individual payment of interest or principle at the appropriate spot rate as determined by the yield curve for that particular grade of debt. If one feels that the market is incorrectly estimating the yield curve for a particular fixed income security, or incorrectly classifying the credit quality of a particular bond, this would lead one to think that a bond was over / undervalued. The yield curve itself, however, is not a static benchmark but rather a dynamic indication of the market’s opinion for future interest rates and fixed income valuation.

Thanks