Bonds

Table of Contents

Definitions

Z-spread

The zero-volatility spread is a single value that needs to be added to every spot yield on the curve in order to make the present value of the risky bond equal to the present value of the risk-free bond (Treasury bond).

That is, if the present value of the risk bond is bonds_6142a120a9589f1aa3120b87b4f66c3b67dfb742.png, then the Z-spread bonds_9c15196dd07b1add486b8b54592e74bfe946ed95.png is the value such that

bonds_3dded562505bba5dd0ef77699c61e264deda7f94.png

where

  • bonds_e20dcda5f035650122343c61053a7c3ad6acacaa.png denotes the coupon rate
  • bonds_9f1ebb0bf85685e45d9db55407657645b96add5e.png denotes the face-value (i.e. what we paid for the bond upon initial purchase)
  • bonds_45961c5d6073ad971344e4f311998eefecb8ac3a.png denotes the spot rate of maturity bonds_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png for a risk-free bond (not annualized)
  • bonds_e10e2b430f95617381cdd6d6b52aed29fb971dff.png is the number of "periods"

Say in the case where we had a bond for 4 yrs before it matured, which had a market price of 975:

  1 2 3 4
Cash flow 120 120 120 1120
Treasury spot rates (risk-free) 0.05 0.06 0.065 0.07
Discounted Cash Flow 114.28571 106.79957 99.341891 854.44264
Value of bond (at maturity)       1174.8698
Market price       975

Then the Z-spread is the value of bonds_9c15196dd07b1add486b8b54592e74bfe946ed95.png in the such that bonds_9b0a5039ff6e754ae5d724866af762880b52c0a5.png in the formula above. Hence, it's simply the "average" percentage more earnings compared to the risk-free bond.