Binomial Interest Rate Tree

I’m working through example 4 in Reading 36. I can’t figure out how excel solver is applied in this case/what formula to use. How do I find the correct one-year forwards in Time 2??? I don’t understand how to get the three values in time 2 in exhibit 18.

***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, which is close to the correct price of 0.8876. Using numerical methods (again, Excel’s Solver), we find that the three correct one-year forwards are 4.482%, 6.051%, and 8.167%.

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

As I said in an earlier thread, exact values and close values are creating the confusion here.

The binomial tree are based on exact values (not 5 decimal places), generated from Excel.

Exhibit 18 displays the exact values as:




The related close values are:




The calculation method for these numbers is displayed in your own above post. The only reason why this method is inaccurate is because only 5 decimal-place values are being considered in the calculations.

Gd luck!


There is no confusion between what are the “exact” values and what are the “close” values. I wholly understand there is a difference, but I want to know how to calculate those exact values that are displayed in Exhibit 18.



My bad. Good if you are not confused over this. Because back then it took me a lot of time before realising those exact and close values!

It seems from the post that by now you are sorted out on your question of how to find the rates mentioned in the curriculum. I am still confused. Are the rates derived using Excel sheet? If yes, doesn’t it mean we would be provided these rates?

The rates are typically derived numerically, using a computer in some capacity.

On the exam they’ll have to give you the rates in the tree.

(They could omit some rates, but they have to give you at least two rates at each time (or the volatility) so that you can calculate the others at that time.)

Thank you

My pleasure.