# FI Binomial Model Qbank Question

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

i1,lower = 12% (given) i1,upper = i1,lower * e^(2*rho) i1,upper = 12*e^(2*0.12) = 14.6568% V2uu = 100 coupon = 7% V2ud = V2du = 100 coupon = 7% V2dd = 100 coupon = 7% V1u = 1/2 *[112/1.146568 + 112/1.146568] = 97.6828 V1d = 1/2 *[112/1.12 + 112/1.12] = 100 D??? EDIT: Guys, even I had to look up that formula from the book…

dinesh beat me to it… you are given I 1L = .12 so you know V 1L = 100 and then the answer is B or D and since I 1U must be > .12 you know V 1U is < 100 so it can’t be 101.125 and so D is your answer… Though this is a q where one of those silly formulas would come in handy. I 1L = .12 and we need I 1U I 1U = I 1L * e^2s = .12 * e^2(0.10) = .1466 So V 1U = 112/1.1466 = 97.683 and V 1L = 112/1.12 = 100 D (I had to look that formula up it wasn’t really given and it wasn’t on my sheet. it is now.)

D is correct. I didn’t recall seeing that formula ever or this topic at all. I have to look at that section again. Thats a nice short cut way of looking at it slouiscar.

you are given I 1L = .12 so you know V 1L = 100 and then the answer is B or D and since I 1U must be > .12 you know V 1U is < 100 so it can’t be 101.125 and so D is your answer… … Why is this so? Could you explain this part if you don’t mind? tks

What happens to the bond price when interest rates increase from 12% to 14.65%??? … got the hint!!

No…I get that part…I should have been more specific… My question is how do you know that the I 1U must be over .12 without doing the math? Is it because the volatility lower bound is always lesser than the volatility upper bound?

We can’t get away from that weird formula, we have to do the math to get to that number of 14.65% and hence select D.

dinesh.sundrani Wrote: ------------------------------------------------------- > i1,lower = 12% (given) > i1,upper = i1,lower * e^(2*rho) > i1,upper = 12*e^(2*0.12) = 14.6568% > > V2uu = 100 > coupon = 7% > > V2ud = V2du = 100 > coupon = 7% > > V2dd = 100 > coupon = 7% > > V1u = 1/2 *[112/1.146568 + 112/1.146568] = > 97.6828 > V1d = 1/2 *[112/1.12 + 112/1.12] = 100 > > D??? > > EDIT: Guys, even I had to look up that formula > from the book… isn’t the coupon 12%? where did you get 7% also…where is this formula?/ and why have i never seen it before… sheesh…i dunno whether to love or hate this forum dinesh.sundrani Wrote: ------------------------------------------------------- > We can’t get away from that weird formula, we have > to do the math to get to that number of 14.65% and > hence select D. I disagree. I think there is a LOS on describing the creation of the interest rate tree. I am foggy but I kind of recall the steps include determining the L rate and estimating volatility. From there you then use that formula to generate the H rate. My point here is that since the std dev is positive the H rate will be > than the L rate, and therefore no formula is needed to answer this q… I actually think this q is similar to something you might see on an exam… the LOS didn’t say calc the rates it just said describe the process and if you can describe the process then you can answer this q without calc. Those tricky q writers!

i don’t remember seeing LOS on constructing binomial trees. I’m pretty CFAI materials don’t cover that.

slouiscar Wrote: ------------------------------------------------------- > dinesh.sundrani Wrote: > -------------------------------------------------- > ----- > > We can’t get away from that weird formula, we > have > > to do the math to get to that number of 14.65% > and > > hence select D. > > > I disagree. > > I think there is a LOS on describing the creation > of the interest rate tree. I am foggy but I kind > of recall the steps include determining the L rate > and estimating volatility. From there you then > use that formula to generate the H rate. > > My point here is that since the std dev is > positive the H rate will be > than the L rate, and > therefore no formula is needed to answer this > q… > > I actually think this q is similar to something > you might see on an exam… the LOS didn’t say > calc the rates it just said describe the process > and if you can describe the process then you can > answer this q without calc. Those tricky q > writers! i agree slousicar…i liked the way you approached the question…would’ve saved you a helluva lotta time and strain of calculations on exam day… hats off to you

mumukada Wrote: ------------------------------------------------------- > dinesh.sundrani Wrote: > -------------------------------------------------- > ----- > > i1,lower = 12% (given) > > i1,upper = i1,lower * e^(2*rho) > > i1,upper = 12*e^(2*0.12) = 14.6568% > > > > V2uu = 100 > > coupon = 7% > > > > V2ud = V2du = 100 > > coupon = 7% > > > > V2dd = 100 > > coupon = 7% > > > > V1u = 1/2 *[112/1.146568 + 112/1.146568] = > > 97.6828 > > V1d = 1/2 *[112/1.12 + 112/1.12] = 100 > > > > D??? > > > > EDIT: Guys, even I had to look up that formula > > from the book… > > > > isn’t the coupon 12%? where did you get 7% > > also…where is this formula?/ and why have i > never seen it before… > sheesh…i dunno whether to love or hate this > forum mumu, I remember telling you that it was a bad-office-day, recall? Hence all those typos… it should obviously mean 12% Coupon.

What section (los, page, etc…) did this formula come from: I 1U = I 1L * e^2s = .12 * e^2(0.10) = .1466 I dont recall this formula?

slouiscar Wrote: ------------------------------------------------------- > My point here is that since the std dev is > positive the H rate will be > than the L rate, and > therefore no formula is needed to answer this > q… wow slouiscar , I’ll need to do some number crunching to see if this actually works. But Anyways a brilliant idea, that would have surely not struck me in those 6 hrs of massacre. Just to reconfirm, do you think, if std dev is negative, the Rate(H) < Rate(L)??

I just checked: Reading 57 (valuing bonds with embedded options) doesn’t have construction of binomial trees in LOS.