Z-Spread: Definition and Calculation

For the purpose of calculation, we start with an assumption for the z-spread. One takes the Treasury spot rates for each maturity, adds the z-spread to it, and uses this new rate as a discount rate for each maturity to price the bond. The correct z-spread is the one that makes the present value of cash flows equal to the price of the bond.

$P = \frac{C_1}{(1+r_1+z)} + \frac{C_2}{(1+r_2+z)^2} + \frac{C_3}{(1+r_3+z)^3} + \cdots + \frac{T}{(1+r_n+z)^n}$

Where,

• P represents the clean price of the bond plus any accrued interest.
• C represents the coupon payments.
• T represents the total cash flow received at maturity
• r1,r2… represent the zero spot rates for each maturity
• z represents the z-spread.
• n is the number of periods

Note that the benchmark for calculating z-spread is the spot rate curve. Unlike nominal spread, the z-spread is spread over the entire Treasury spot rate curve.

The z-spread represents the additional risk the investor is taking in the form of credit risk, liquidity risk, and option risk.

In most cases, such as for the vanilla coupon paying bonds, the z-spread will only slightly diverge from the nominal spread. The difference mainly comes from the shape of the term structure and the bond characteristics. For example, the difference will be high for amortizing bonds for which the principal is repaid over time, bonds with high coupon, and when the yield curve is steep.

The Z-spread is particularly useful because:

• It helps compare bonds with different structures across the yield curve
• It provides a measure of the bond's credit/liquidity risk premium over the risk-free rate
• It's more accurate than a simple spread calculation since it considers the entire term structure of interest rates