Z-spread

The Z-spread, ZSPRD, Zero-volatility spread or Yield curve spread on a simple mortgage-backed security (MBS) is the flat spread over the treasury yield curve required in discounting a pre-determined coupon schedule to arrive at its present market price.

That is, the MBS yield curve spread is based on a comparison of the market price to a model of the bond which includes no variability in interest rate or mortgage repayment rates.

Definition

For mortgage-backed securities, a model of typical repayment rates tends to be given; for example, the PSA formula for a particular Fannie Mae MBS might equate a particular group of mortgages to an 8 year amortizing bond with a 5% mortality per annum. This gives a single series of nominal cash flows (like a riskless bond). If these payments are discounted to net present value with a static treasury yield curve the sum of their values will tend to overestimate the market price of the MBS. The parallel shift, which, if applied to the yield curve makes the NPV of the anticipated receipts equal to the market price is the Yield curve spread.

The Z-spread of a bond is the number of basis points one needs to add to the Treasury spot rates yield curve, so that the NPV of the bond cash flows (using the adjusted yield curve) equals the market price of the bond (after accounting for accrued interest). The spread is calculated iteratively and improves the accuracy of the value calculation as it uses the entire yield curve to value the cash flows.

If you calculate the present value of all future cash flows for a bond using prevailing spot rates, you may discover that the price you calculate is greater than the price observed in the market. This difference arises because the market price incorporates additional factors such as liquidity and credit risk. The Z-spread quantifies the impact of these additional factors. It is the spread you need to add to the curve you are discounting with in order to generate a price that matches the market price.

Conventionally, the zero rates are determined from the Treasury curve, with semi-annual compounding.

The Problem with YTM spreads

Coupon Paying bonds are essentially portfolios of Zero Coupon Bond components and the Yield to Maturity of such instruments can be thought of as being a complex blend of the component Zero Coupon bond yields. It therefore can be observed that, for example, in a positively sloped Yield Curve environment and comparing two bonds with the same cash flow dates and maturity, a higher coupon bond will offer a lower YTM than a low coupon bond. (The low coupon bond has cash flows more heavily influenced by a proportionally greater long term component).

Thus two fairly and correctly priced corporate bonds from the same borrower and having the same cash flow dates and maturity may well have significantly different Yields to Maturity. It follows that when these Yields are compared to that of a same-maturity benchmark, the resulting spreads may be markedly different. It is the spread then and this type of methodology in finding it, that is at fault and not the bonds or the market.

This measures the spread the investor or trader would capture over the entire benchmark Zero curve if the bond was held to maturity. The Z-spread is calculated as the spread that will make the present value of cash flows from the non-benchmark security when they are discounted at the benchmark Zero rates (plus the Z-spread) equal to the non-benchmark security's price. This is done by trial and error. This is different than the nominal spread because the nominal spread just uses one point on the curve.

For example, take the spot curve and add 50 basis points to each rate on the curve. If the two year spot rate is 3%, the rate you would use to find the present value of that cash flow would be 3.50%. After you have calculated all of the present values for the cash flows, add them up and see whether they equal the bond's price. If they do, then you have found the Z-spread, if not, you have to go back to the drawing board and use a new spread until the present value of those cash flows equals the bond's price.

The Advantage of Z-Spread

As the Z-Spread is not dependent upon only one point on the Yield Curve and takes account of all of the relevant term-structure, the distortions of Yield-to-Maturity spreads outlined above are eliminated. It is then a measure of Credit Spread without the distortions of YTM. It is widely used in the CDS and other markets by non bond traders who want a fair reading of Credit Spread undistorted by the complexities of individual bonds.

Benchmark for CDS Basis

The Z Spread is widely used as the 'cash' benchmark for the calculation of the CDS Basis. The CDS basis is commonly the CDS Fee minus the Z Spread for a bond of the same maturity. For the example, 10 Year RBS CDS is currently 199.7 bps, the Z Spread for their 10 year Global bond is 286.8, giving a negative basis of -87.1bps.

Example

Assume that a bond has three cash flows: $5 on 1/1/2009; $5 on 1/1/2010; and $105 on 1/1/2011.

The corresponding zero rates (compounded semiannually) are 4.5% on 1/1/2009, 4.7% on 1/1/2010 and 5% on 1/1/2011.

Assuming that the accrued interest is 0, and the Z-spread is 50 basis points (bp), the price P of this bond on 1/1/2008 is given by:

${\begin{aligned}P&={\frac {5}{(1+{\frac {4.5\%+50\mathrm {bp} }{2}})^{(2\times 1)}}}+{\frac {5}{(1+{\frac {4.7\%+50\mathrm {bp} }{2}})^{(2\times 2)}}}+{\frac {105}{(1+{\frac {5.0\%+50\mathrm {bp} }{2}})^{(2\times 3)}}}\\&=98.49861\\\end{aligned}}$

See also

• Option-adjusted spread
• I-spread

References

• Frank J. Fabozzi, The Handbook of Fixed Income Securities, 7th edition; McGraw-Hill; 2005