## FRM 2- MR11

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### FRM 2- MR11

43.2 Assume a binomial interest rate tree with six steps. At the initial node (T0), the rate is
5.0%. The probability of an up-jump to the up-state is 60% (p = 0.6). The magnitude of an
up-jump is +50 basis points; the magnitude of a down-jump is -30 basis points (u = +0.5%, d
= -0.3%). In regard to the distribution at the end of the tree (T6), what is the MODE and
what is the probability that the final rate will be the mode?
a) 5.3% mode with 20.74% probability
b) 5.6% mode with 20.74% probability
c) 5.3% mode with 27.65% probability
d) 5.6% mode with 27.65% probability

source: web

edupristine
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### FRM 2- MR11

Hi,
For more clarity regarding the distribution and the respective mode, request you to kindly provide me with the exact weblink from where the question has been taken.
Thanks

anbu.edu
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### FRM 2- MR11

i got this question from BT
This was the answe they have given but i cound't understand the working
43.2 D. 5.6% mode with 27.65% probability
The final distribution is a binomial distribution with p=0.6 and n=6. BINOM.DIST(success =
3, n=6, p=0.6) = 27.65%.
Or, 27.65% = [6 choose 3]*60%^3*40%^3
This mode is reached by any sequence (since this tree recombines) of three ups and three
downs: +50*3 + -30*3 = +60 bps.
So the final mode rate is 5% + 60 bps = 5.6%
How did they concluded 6C3?

edupristine
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Hi,
First of all, we need to understand that the distribution involved is a binomial distribution. This is because a binomial interest rate tree is used where the interests can assume only one of the two values. It can either go up or down.
For a Binomial Probability Distribution,

P(x)= nCx ( px (1-p)n-x ) )
Where n = total no of binomial trials. In this case, n=6
x= the total number of success in a binomial trial where success is defined as the interest rates going up.
P= probability of success on a given binomial trial, ie here p would be the probability that the interest rate will go up. This is equal to 60% or 0.6
Now though there are a number of ways to reach at T6 node from T0 node, but an average case would be to assume that to reach at node T6 , we need to travel up and down the tree three times each.
So, we can that the total no of successes ( where success is equivalent to going up the tree) while reaching node T6 are equal to 3
Hence, x = 3.
So, substituting the values in the formula above, we get:
P(x) = 6C3 (0.63 * 0.4 6-3 ) = 27.65
Now, since T6 node can be reached by three ups and three downs, so the interest rate at T6 can be calculated as : (Int rate at T0 i.e 5%) +50*3 + -30*3 = 5.6 %