# 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 , then the **Z-spread** is the value such that

where

- denotes the
*coupon rate* - denotes the
*face-value*(i.e. what we paid for the bond upon initial purchase) - denotes the
*spot rate*of maturity for a*risk-free*bond (not annualized) - 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 in the such that in the formula above. Hence, it's simply the "average" percentage more earnings compared to the risk-free bond.