# Using interest rate volatility to build binomial interest rate tree - CFAI example

Pg 335 Example 5 in Reading 44 from CFAI text.

This isn’t actually a question, but I wanted to confirm the different forward rates given in question myself, assuming the volatility of 15%, in order to practise doing so.

I’m getting a spot rate for Year 1 of 4.707%, which yields a forward rate of 5.0149%.

Where I’m confused is, I figured if I used the 0.15e volatility calculation (ie. not multpilying by 2, because I’m using the average forward rate to calculate up and down forward rates) which comes to 0.8607 (e-0.15) and multiplied by avg. forward rate I would get the down rate in text, but I get 4.3164% instead of the given 4.2729% in text. Can anyone tell me what I’m doing wrong here?

The way they calculated 4.2729 is 0.057678e(-2*0.15) = 4.2729%. Remember to use negative sign for lower node to get the rates.

Yep. Since I(up) = I(down)*e^(2*0.15), I(down) = I(up)/e^(2*0.15)

If you bring the denominator to the numerator, you should add the negative sign to the power of e. I(down) = I(up)*e^(-2*.015)

yeah I realize how to get the up or down, once u already have either one. my question is how did they get EITHER number from just the forward/spot rate for that period? that is where I’m struggling with regards to this question. if u know the forward rate, that is, the average 0 volatility forward rate, how do u get either of the ups or downs with the volatility being X. and in particular in this question, how was either of the up or down forward rates calculated from the par/spot/forward rates?

The way the curriculum skims over it and explains that the process is iterative, it looks like we wont need to calculate the first one without more info. Notice the related LOSes are ‘compare’ or ‘describe.’

R43 has been the worst reading for me by far - it seems to skip around and assume a lot of knowledge that I personally dont have.

^^ see a part of me thinks this too, that we won’t be asked to do this on exam, but the other part sees page 293, example 4, reading 43, and in that example I believe they calculate the 2 forward rates by using the average forward rate and adjusting the equation for volatility by not multiplying by 2 to find the up and down rates using the average forward rate. I could be misinterpreting something tho…

There seem to be multiple ways to calculate the interest rates on the binomial interest rate tree. i didn’t get the exact answer that the text has though but from Elan notes i caught sight of this

Example: going back to Exhibit 2,3,4 and 5 from Reading 43.

it has one year rate, one year from now (f1,1) = 1.4%

and they calculate i,1L at time 1 as 1.4% * (1+15%) = 1.61% where 15% is the implied volatility

and same case for i,1H at time 1 = 1.4% * (1 - 15%) = 1.19%

from CFAI text, they took i,1L as “supposedly” 1.194% then calculate i,1H as 1.194% *(e^2*0.15) = 1.612%

pretty close to the simplistic calculation Elan has provided. I reckon the more accurate “suppose” value comes from iterative process of finding out the interest rate.

More than happy to know how we can calculate this “supposed” value instead of using the approximation if anyone can help

Cheers

jono85 - Although I haven’t looked at the particular example in detail, except to note what you said in your original post, I would think like the other posters that the rates have been calibrated to fit some (real or theoretical) market data.

You shouldn’t forget that building an interest rate tree is not primarily for answering exam questions, but rather using it in practical applications where consistency with real data is paramount, so it would make sense that CFAI didnt mind that point being borne out.

There is a small chance it could simply be errata, in the sense of numbers that werent updated from a prior edition, but I think this is unlikely.

EDIT: I just saw this is from April !