Schweser book 5 (p113) states: … If the spot yield curve is upward sloping, the Z-spread is larger than the nominal spread. … Why is this true? Thanks very much.
I don’t have a simple explanation for this except that I can prove it by looking at the function f(s) = Discounted cash flows with (ytm + s) - discounted cash flows at spot rates + s. Do Maclaurin series of the function and you get a proof pretty quickly. That’s not an intuitive explanation though…
This is going to be a long post, but hang on, it should make sense to you. The nominal spread is the difference between the YTM if a corporate bond and a similar treasury bond. 1. Say that the YTM of company XYZ 10 yr bond = 8% while that of the 10 yr treasury is 7%, so the price of the corporate bond is lower than the treasuries. Ignore embedded options and calls and puts so as not to complicate things… 2. Then the nominal spread is 100 basis points (BP), 8% - 7%. 3. Assume the spot rate yield curve is upward sloping, so each spot rate is higher than the previous rate. 4. The Z-spread would be some x number of basis points on top of each spot rate in the curve that would make the corporate bond’s price equal to its current market price. 5. Since the corporate bond is yielding 8% (higher than the treasury), we clearly need to discount all future values we get from this bond by a “higher” discount rate than those spot rates. Otherwise, its market price won’t be lower than the comparable treasury. * By trial and error, we can find such spread, called the Z-spread, lets say it is 140 bp. Your question is then, why should this Z-spread be larger than the nominal spread (100 bp in our example)? 1. First notice that if the yield curve is flat, the two spreads (z-spread and nominal) will be the same, i.e., 100 bp. This is because the difference in yields between the two bonds is same as the difference between their YTM’s. The nominal spread only looks at a single yield, the YTM, when disounting the cash flows, but the Z-spread looks at each spot rate. Thus if all spot rates are the same (flat yield curve) then both Z-spread and nominal spread produce same answer. 2. If the yield curve is upward sloping, the nominal spread still looks at the YTM (which is a single rate), but the Z-spread will be considering spot rates that are rising as we go up the maturity schedule. This makes the Z-spread larger than the “simple” nominal spread. More or less this is it, but I’m sure it could be explained better.
hmmm…I don’t think #2 is all that convincing…
You’re right, it’s not convincing, but I’m still struggling with some intuitive explanation.
thanks for every one here for replying. I think the using the discount cash flow approach by JoeyDVivre makes sense, although the result is not that much intuitive. To get a quick look, we can assume only 2 equal payments and look at the relationship; this will simplify the problem and will not lose generality. the main result is that in 2 period model, the YTM of treasury is more closer to Y(t2) than to Y(t1) when yield curve is not flat…