Sign up  |  Log in

Fixed Income- Pathwise valuation

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!

Master the Level II curriculum by creating custom quizzes in the SchweserPro™ QBank. Question difficulty automatically adapts to your ability level on a given topic, measuring your knowledge and keeping you motivated. Included in every Schweser Study Package.

It’s 2n 2n−1.

With 1 period, there’s 21−1 =20 = 1 path (i.e., discount at today’s spot rate).

With 1 2 periods, there are 22−1 = 21 = 2 paths:

  • Up
  • Down

With 2 3 periods, there are 23−1 = 22 = 4 paths:

  • Up, up
  • Up, down
  • Down, up
  • Down, down

With 3 4 periods, there are 24−1 = 23 = 8 paths:

  • 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 2360−1 = 2359 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

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

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

My pleasure.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

i am confused now. in the Practise Problems in CFAI website. under Ahti Maalouf Case Scenario, the four-year bond requires 23 = 8 paths. Why is is not 24 = 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.

——————————————
Find useful resources on the CFA exams at http://www.youtube.com/c/FabianMoa

playstan211 wrote:
i am confused now. in the Practise Problems in CFAI website. under Ahti Maalouf Case Scenario, the four-year bond requires 23 = 8 paths. Why is is not 24 = 16 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 2n−1.  I’ve fixed my earlier post.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/

Thanks for your confirmation.!

My pleasure.

Simplify the complicated side; don't complify the simplicated side.

Financial Exam Help 123: The place to get help for the CFA® exams
http://financialexamhelp123.com/