- CFA Exams
- CFA Level I Exam
- Study Session 14. Fixed Income (1)
- Reading 44. Introduction to Fixed-Income Valuation
- Subject 7. The Maturity Structure of Interest Rates

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**CFA Practice Question**

Ben Nagler wants to purchase a bond and has narrowed his selection down to two choices. He is indifferent about holding a bond for 13 years or 18 years. He knows the spot rates for each bond but wants to compute the f

_{13, 5}. What is the five-year forward rate 13 years from now, assuming a spot rate of 5.0127% for 13 years and another spot rate of 5.2350% for 18 years? Use semi-annual compounding.A. 5.8141%

B. 6.3955%

C. 5.0873%

**Explanation:**Forward rate calculations require candidates to assess the future interest rate for some period in the future. This question asks what the five-year forward rate is 13 years into the future.

Candidates should remember that spot rates are related to forward rates. In fact, the spot rates contain the forward rates implied by the market. This calculation is very complicated and provides numerous opportunities for error. Candidates should remember to always place the longer period spot rate in the numerator and the smaller period in the denominator. Moreover, it is important to take one half of the spot rate and add "1" to each spot rate before using the exponent function. An important distinction to note when computing a forward rate for multiple time periods is the exponent computation after dividing the longer spot rate by the shorter spot rate. This additional computation recognizes the geometric return earned on the bond during the time period.

Finally, it is important to remember to subtract "1" at the end of the calculation and double (for the annualized forward rate). Errors are often made through forgetting to double (or halve) certain calculations or by forgetting to add or subtract the number 1 when dealing with the exponential functions. The detailed answer is shown below.

((Spot rate for 18 years)

^{18}/(Spot rate for 13 years)

^{13})

^{(1/5)}

The spot rate for 18 years is 5.2350

The spot rate for 13 years is 5.0127

The numerator becomes

(1+ 0.0262)

^{18}or 1.5922

and the denominator becomes (1 + 0.0251)

^{13}or 1.3796

Next, divide 1.5922 by 1.3796 = 1.1541 then take the 1/5 root = 1.0291

subtract 1 = the semi-annual rate of 0.0291%

Doubling the semi-annual rate provides the annualized forward rate (the answer of 5.8141%).

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**User Contributed Comments**
13

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

labsbamb |
Fr(13to18)=((Sr18+1)exp18/((Sr13+1)EXP13)exp(1/(18-13))-1 |

aggabad |
times 2 |

winterSun |
Why do we have to use the spot/2? |

achu |
winterSun: By convention, these are all listed as annual rates but periods are semiannual. |

octavianus |
How does the 1/5 root come into the picture? |

lazio |
Go and re-check the general formula for the xFy forward rate.The 1/x root is used for the general formula, so too, 1/5 root is used here, in computing 5f13 |

Andrewua |
I don't see any reason, why should we devide by 2? It's said, that we have periods, not years... why shoul |cinsider periods as years. I arrived at this answer without making half periods, and it matched ok! |

dini85 |
Simple way for prox answer [(18x5.2350)-(13x5.0127)]/5 = 5.8130 |

neworizon |
right dini85! |

Skrills |
awesome dini85 |

ruzzpo |
thank dini85 |

Shaan23 |
Dont know why they explained it that way. Just do this (1 + z13) ^26 + (1 + 5F13)^10 = (1 + Z18) ^36 Solve for 5F13 ---- and we're using bonds so use semi annual rates for everything. |

thebkr7 |
Thanks @dini85!!! |