FRM 2- MR11

anbu.edu
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FRM 2- MR11

Postby anbu.edu » Mon Jun 16, 2014 7:12 pm

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

I dont kno how to proceed this question. please help me

source: web

edupristine
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FRM 2- MR11

Postby edupristine » Thu Jun 19, 2014 11:35 am

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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Posts: 205
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FRM 2- MR11

Postby anbu.edu » Thu Jun 26, 2014 7:36 pm

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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Postby edupristine » Sat Jun 28, 2014 11:48 am

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 % 


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