Why Z-Spreads and Asset-Swap Spreads Converge Near Par
Summary
The document explains why a bond’s z-spread and asset-swap spread are often similar when its cash price is near par. The two measures reach a target price in different ways: an asset-swap spread adjusts cash flows, while a z-spread adjusts the discount rate applied to the bond’s cash flows. Similarity depends on the bond’s sensitivity to coupon changes matching its sensitivity to yield changes.
At par, coupon and yield begin equal, and those sensitivities are equal to first order. The answer illustrates this with a small coupon increase and an equivalent small yield decrease, using the present value of an annuity to explain the relationship. This is an intuition for a local approximation, not an exact equivalence: the annuity value itself shifts slightly as the discount rate changes. The explanation is conceptual and does not address conventions or implementation details for particular swap and curve setups.
Key ideas
- An asset-swap spread changes bond-related cash flows, while a z-spread changes discounting.
- The two spread measures are similar when coupon and yield sensitivities are similar.
- For a bond at par, those sensitivities match to first order.
- The relationship is approximate because annuity value changes with the discount rate.
- The explanation does not specify market conventions or implementation details.
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# why is z spread and asw spread similar when cash price is around par? # why is z spread and asw spread similar when cash price is around par? I break down my questions to below two sub questions - based on my understanding, asw spread is the spread added to the floating leg in the interest rate swap so that the present value of the floating leg is the same as the fixed leg (the bond cashflows). in this interest rate swap, are cashflows discounted using swap zero curve? - based on my understand, z spread is the spread added to the swap zero curve which forms the new discounting curve to discount bond cashflows. how do I understand that the results from these two sets of calculations (being z spread and asw spread respectively) are usually similar only when cash price is around par? is there an intuitive way to understand why are they similar in the first place? many thanks! ## Answer by dm63 (score 1) https://quant.stackexchange.com/a/81107 Method 1 is equivalent to changing the cashflows on the bond until you hit the right price. Method 2 is equivalent to changing the discount rate until you hit the right price. These will be similar if the bond sensitivity to a change in coupon equals its sensitivity to a change in yield. Turns out they are equal (to first order) for bonds at par. You can do the full bond math proof. My intuition for it is as follows: let $P(c,y)$ be the price of a bond with coupon $c$ and yield $y$. Since the bond is at par, $y=c$ initially. For example $P(5.00,5.00)=100$. Now I claim that $P(5.01,5.00)=P(5.00,4.99)$ because changing the coupon is the same as reducing the discount rate: $$P(5.01,5.00) =A(0.01,5.00) + P(5.00,5.00)= A(0.01,5.00) + P(4.99,4.99)= P(5.00,4.99)$$ as required,where A is the PV of an annuity. The first step is from additivity of cash flows, and the last step is correct to first order because the value of the annuity is slightly different in the two discount rates.
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