How to calculate the number of independent paths in a path-wise valuation method?

Schweser notes say 2^{(n-1)}, n being the number of periods.

However, at a number of places I don’t get the right answer using this formula? Help please!

How to calculate the number of independent paths in a path-wise valuation method?

Schweser notes say 2^{(n-1)}, n being the number of periods.

However, at a number of places I don’t get the right answer using this formula? Help please!

It’s 2_^{n}_ _ **2 ^{n}** __

*With 1 period, there’s 2 ^{1−1} =2^{0} = 1 path (i.e., discount at today’s spot rate).*

With 1 _ **2** _ period_ **s** _, there are _ **2 ^{2−1} =** _ 2

- Up
- Down

With 2 _ **3** _ periods, there are _ **2 ^{3−1} =** _ 2

- Up, up
- Up, down
- Down, up
- Down, down

With 3 _ **4** _ periods, there are _ **2 ^{4−1} =** _ 2

- Up, up, up
- Up, up, down
- Up, down, up
- Up, down, down
- Down, up, up
- Down, up, down
- Down, down, up
- Down, down, down

And so on.

A 30-year, monthly binomial tree (for valuing a 30-year, monthly-pay mortgage) will have _ **2 ^{360−1} = 2^{359}** _ paths, or slightly more than 2,348,542,582,773,830,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

_ **1,174,271,291,386,915,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000** _.

I’ll leave it to you to enumerate them all, but I’ll get you started:

- Up, up, (356 _
**355**_ more ups), up, up

this is super helpful…thanks S2000magician…you are a saviour!

My pleasure.

i am confused now. in the Practise Problems in CFAI website. under Ahti Maalouf Case Scenario, the four-year bond requires 2^{3} = 8 paths. Why is is not 2^{4} = 16 paths

The interest rate in each node represents a one-period forward rate.

For a four-year bond, the final (4th year) cash flow (principal + coupon) will be discounted using the 1-year forward rate in Year 3, which means you only need the interest rate tree up to Year 3 only, hence there is only 2^{(4-1)} = 8 paths.

As you should be, because I was mistaken. I was thinking of a tree for stock prices, not an interest rate tree.

Indeed, for a binomial interest rate tree, it should be 2^{n−1}. I’ve fixed my earlier post.

Thanks for your confirmation.!

My pleasure.