# ZSpread in multiple curve framework

how do I calculate ZSpread for a govt. bond in a multiple curve framework? I have not come across the exact details anywhere so I want to verify if I'm right. Below is my understanding, please correct me if I'm wrong:

1. Specify a discounting curve and a forecasting curve.
2. Using the above two curves, calculate the Zero Coupon Swap Rate for several maturities.
3. Estimate the parallel shift required to the above Zero Coupon Swap Curve to match the bond price in the market.

This parallel shift is the bond's ZSpread with respect to the specified discounting and forecasting curves. Depending on the set of curves specified, each bond can have multiple ZSpreads.

## 1 Answer

The math is actually simpler than what you proposed. Z-Spread is always computed as the parallel shift in a zero curve required so as to reprice the cash flows to a bond's cash flows; i.e., you solve for the $s$ in $$P + AI = \sum_{i=1}^N c_i \cdot d(t_i) \cdot e^{-t_i \times s}$$

In the multi-curve world, you simply compute both the LIBOR OAS and OIS OAS separately. To compute the LIBOR OAS, you plug the pseudo-LIBOR discount factors into the $d(t_i)$'s above; and to compete OIS OAS, you plug the OIS discount factors into $d(t_i)$.

Since z-spreads are not tradable, they're used mostly as relative value metrics. The key is internal consistency.

• To get the discount factors, will I not need to zero coupon swap rates? Or do I simply use discount factors from the disc curve? In case 1, I need to know the forecasting curve as well as discounting curve. In case 2, I only need the discounting curve. – InnocentR Nov 19 '14 at 20:32
• @InnocentR: Short answer is that you only need the discount curve. If you expand the formula I wrote, you'll notice that the $s$ is the continuously compounded spread added to the zero curve: $c_i e^{-r_i t_i} e^{-t_i * s} = c_i e^{-(r_i + s) * t_i}$. This is exactly correct for the OIS zero curve. For LIBOR OAS, it's debatable and frankly it's a matter of definition. Remember that it's not the "Swap curve OAS," it is the "Libor" OAS, so I think it's perfectly ok to just use the libor discount curve and be done with it. It's also what's being done in practice. – Helin Nov 19 '14 at 20:43
• Forgot to ask you, so the name ZSpread is a misnomer? As we are not really calculating spread off the Zero Coupon Curve but instead using the discounting curve directly. – InnocentR Nov 20 '14 at 9:06
• @InnocentR It's not. If you assume volatility is zero & continuous compounding, then it can be shown that z-spread = OAS for bonds without embedded options. Hence the name "z-spread" which is short for "zv spread" which in turn is short for "zero-volatility spread". – Helin Nov 20 '14 at 15:02