 # FI - Reading #50 - Binomial Trees

Are we expected to derive rates on the binomial trees? I’m pulling out my hair trying to figure out how these rates are derived.

As far as I know, there is no need to figure out the rates, the LOS just states to calculate the value of a bond from an interest rate tree (meaning rates are given). I’ve done the problems in the EOC CFAI curriculum and not one asks to calculate the rates. However, you can use bootstrapping method to calculate rates which is mentioned in the curriculum.

Interest rates will be provided, like swt326 said value of the bond needs to be calculated. Using binomial tree for bonds with embedded option is tricky and likely to be tested through this topic not figuring out the rates.

you may be asked to calculate the lower or upper rate given interest rate volitility.

Upper rate = Lower rate x e^(2σ)

Yes I agree. If interest rate volatility is given and upper rate or lower rate is given then using this formula you may be asked to calculate the missing rate.

Side question: how does one go about determining the rates for non treasury bonds? That is, how exactly does one determine the appropriate spread to use over treasuries when building a tree in the first place ?

I understand how you bootstrap treasury spot rates, but what about nontreasuries?

I dont exactly know about it but what if we add a premium over treasury spot rates to come up with an interest rate used in the tree like we do getting a yield of a corporate bond by adding risk premiums to risk free rate? Whats your take onto it?

I think i agree with what you’re saying, but what i don’t understand is how you come up with that initial premium (over the treasury) in the first place.

I think it would be helpful to hear from any practicing fixed income analysts…how do you truly value non-treasuries in the “real world” ? I always understood the non-theorectically correct way of using current YTM (or a benchmark YTM assuming the bonds you’re valuing aren’t listed) to revalue a bond, but the theroretically correct way, per CFAI, suggests using spot rates. I understand why this is the more correct method, but how do you do it in practice, for non-treasuries that is ??

I agree with you. A practicing fixed income analyst can give a better idea…

can someone help me understand these 2 questions?

Which of the following most accurately explains how the effective convexity is computed using the binomial model. In order to compute the effective convexity the:

A) binomial tree has to be shifted upward and downward by the same amount for all nodes. B) volatility has to be shifted upward and downward and the binomial tree recalculated each time. C) yield curve has to be shifted upward and downward in a parallel manner and the binomial tree recalculated each time.

A bond with a 12% annual coupon will mature in two years at par value. The current one-year spot rate is 14%. For the second year, the yield volatility model forecasts a lower bound of 12% for the one-year rate and a standard deviation of 10%. In a binomial interest rate tree describing this situation, what are the forecasted values for the bond in the first nodal period?

V1,U: upper rate value V1,L: lower rate value

A) 97.680 101.125 B) 94.676 97.664 C) 97.683 100.000

for the first one i would say C. this just describes the process for working out effective duration, it’s in the books. you bump the yield curve up a little then recalculate the bond price using the binomial tree, then do the same thing after reducing the yield curve. then plug that into your duration formula.

for the second one I would say C because if the bond’s coupon rate and the rate in your tree are the same, the bond should be valued at par.