Binomial Interest Rate Tree

CFA Level II Fixed Income
#1

This question comes from Example 4 in Reading 36 on Interest Rate Trees and Arbitrage-Free Valuation

I understand the formula to calculate the price for a two-year zero(Time 1), but I do not understand how the three-year zero price(Time 2) is calculated from the middle, high, and low forward rates.

the spot rates are S1 = 2.0%, S2 = 3.015% and S3 = 4.055

Time 1

“Beginning at F1,1d = (4.040%)(e–0.15) = 3.477% and F1,1u = (4.040%)(e0.15) = 4.694% gives a price for the two-year zero of [(0.5)(1/1.03477) + (0.5)(1/1.04694)]/1.02 = 0.9419.***”

Then,

“We will begin with the average forward rate for Time 2, F2,1 = (1.040553/1.030152) – 1= 6.167% as the middle value with (6.167%)(e–0.3) = 4.569% and (6.167%)(e0.3) = 8.325% as the lower and upper values. Those values give a price for a three-year zero-coupon bond of 0.8866***”

Is the three-year zero coupon calculated by the following?:

[(.33)(1/(1+“UpperValue”)) + (.33)(1/(1+“MiddleValue”)) + (.33)(1/(1+“LowerValue”))]/(1.03015)^2

Thanks!

***Institute, CFA. 2017 CFA Level II Volume 5 Fixed Income and Derivatives. CFA Institute, 07/2016. VitalBook file.

#2

Nope.

First you have to calibrate the binomial tree from Exhibit 16 to Exhibit 18. Next you work backward to calculate tree values in Exhibit 19.

To answer your question:

The value of bond is calculated as 0.5(0.8923/1.02) + 0.5(0.9184/1.02) = 0.8876 (lets take this value instead of 0.8866, for accuracy sake). There is no “middle value” used.

0.8923 (upper value of Time 1 Exhibit 19) = 0.5(0.9245/1.04646) + 0.5(0.9430/1.04646)

Similarly, 0.9184 = 0.5(0.9430/1.03442) + 0.5(0.9571/1.03442)

1.02 is derived from Time 0 Exhibit 18

P.S. The exact values and close values from this example may create some confusion here.

P.S.2 My 2016-edition textbook copy contained some errata, ensure that your 2017-edition is free of errata to avoid further confusion.

#3

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  • Creating a tree
  • Using a tree to calculate OAS
  • Using a tree to value bonds with embedded options
  • Using a tree to value floating-rate bonds

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#4

Thanks for the response. I think I may be over thinking this.

My issue stems from the the portion of the example where you’re calculating the Time 2 values in exhibit 18. You can’t work backwards to fill the tree values in exhibit 19 without first correctly calculating the values for exhibit 18, right? I’m confused about which formula to use with Excel’s Solver to find the three correct one-year forwards (4.482%, 6.051%, and 8.%) based on the relationship between the average forward rates for Time 2 (6.167% as the middle value, 4.569% lower value, and 8.325% as the upper value) and the .8876 Bond Price. I’m trying to figure this out working from left to right…

#5

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