### The spot yield curve can be used to estimate the price or value of a bond

#### Example

A company wants to issue a bond that is redeemable in four years for its par value or face value of $100, and wants to pay an annual coupon of 5% on the par value.

Estimate the price at which the bond should be issued and the gross redemption yield.

The annual spot yield curve for a bond of this risk class is as follows:

Year | Rate |
---|---|

1 | 3.5% |

2 | 4.0% |

3 | 4.7% |

4 | 5.5% |

#### Solution

The market price of the bond should be the present value of the cash flows from the bond (interest and redemption value) using the relevant yearâ€™s yield curve spot rate as the discount factor.

Year | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Cash flows | 5 | 5 | 5 | 105 |

Df | 1.035^-1 | 1.04^-2 | 1.047^-3 | 1.055^-4 |

Present value | 4.83 | 4.62 | 4.36 | 84.76 |

The market price = $98.57 |

Given a market price of $98.57, the gross yield to maturity is calculated as follows:

Year | CF | DF10% | PV | DF5% | PV | |
---|---|---|---|---|---|---|

0 | MP | (98.57) | 1 | (98.57) | 1 | (98.57) |

1-4 | gross interest | 5 | 3.170 | 15.85 | 3.546 | 17.73 |

4 | Redemption value | 100 | 0.683 | 68.3 | 0.823 | 82.3 |

NPV | (14.42) | 1.46 |

IRR or to maturity = 5% + (1.46 / 1.46 + 14.42) X(10% - 5%) = 5.46%

Note that the yield to maturity of 5.46% is not the same as the four year spot yield curve rate of 5.5%.

#### The reasons for the difference are as follows:

The yield to maturity is a weighted average of the term structure of interest rates.

The returns from the bond come in earlier years, when the interest rates on the yield curve are lower, but the largest proportion comes in Year 4.