- CFA Exams
- CFA Level I Exam
- Topic 6. Fixed Income
- Learning Module 44. Introduction to Fixed-Income Valuation
- Subject 7. The Maturity Structure of Interest Rates

###
**CFA Practice Question**

Daniel Altman is undecided about purchasing a three period bond or a series of one period bonds. Unfortunately, the table that provides the forward rates and spot rates for the next few periods is torn and therefore incomplete. Daniel believes that he has ample information to calculate the three period spot rate even though that data is missing. Given the forward rates and spot rates provided below, what would Daniel earn on his three period spot rate bond?

Spot Rates and Six Month Forward Rates (Annualized Rates on a Bond Equivalent Basis)

_{j}f_{k}: forward rate from period k, for j period(s)A. 4.4977

B. 5.1723

C. 4.0479

**Explanation:**The spot rate is nothing more than all of the forward rates multiplied together taken to the nth root. While this calculation appears more difficult than it actually is, there are a number of areas that can potentially cause problems. For example, each spot rate needs to be annualized and needs to be divided by two in order to get the effective period rate. Also, after all of the forward rates have been multiplied by each other (being sure to add "1" to each rate before multiplying) it is important to take the nth root (depending on the number of periods) and to subtract "1" afterwards. The detailed calculation is shown below:

(((((1+0.040728/2)

^{2}x (1+0.0535/2))

^{(1/3)})-1) x 2 = 0.044977.

Note: In order to see the number in "percentage" form it is necessary to multiply by 200 rather than by 2.

###
**User Contributed Comments**
6

User |
Comment |
---|---|

eddeb |
Why don't we use 5.02%, instead of 5.35% to compute this rate? |

Shelton |
(1.0313 x 1.0502 x 1.0535)^(1/3)=4.4954% |

aggabad |
[(1+S3rd/2)^3]/[(1+S2nd/2)^2]=[F3rd/2+1]^(3-2) |

Xocrevilo |
And for those who prefer English (from the LOS): "The relationship between short-term forward rates and spot rates: The spot rate for a given period is related to the forward rates; specifically, the spot rate is a geometric average of the current six-month spot rate and the subsequent 6-month forward rates..." |

lazio |
Why could we not simply use the formula for 1f2 to solve for Z3 and then annualize? |

janglejuic |
Why couldn't they just say "Geometric Average" in the answer description? stupid as f |