1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Help with a proof of the yield curve.

  1. Mar 24, 2012 #1
    1. The problem statement, all variables and given/known data

    I need to show that the yield curve defined by

    r(t) = 1/t integral r(s) ds from 0 to t is a nondecreasing function iff:

    P(αt) ≥ (P(t))^α, for all 0<=α<=1 , t>= 0

    and P(t) is defined as:

    P(t) = exp{-integral r(s) ds from 0 to t}

    and r(s) is the spot rate function. So basically the yield curve is the average of all the spot rates until time t.

    2. Relevant equations



    3. The attempt at a solution

    So the definition of a nondecreasing function is that if f(b) > f(a) for all b>a.
    Now
    P(t) = exp{-t r(t)}
    so P(αt) = exp{-αt r(αt)}

    Now I have to show that P(αt) ≥ (P(t))^α results in r(t) being nondecreasing.
    Doing some simplification I came up with the following inequality

    ∫r(s) ds from 0 to αt ≥ α∫r(s) ds from 0 to t
    = r(αt) ≥ r(t)

    so that will be nondecreasing as long as that is satisfied for all αt > t

    But that means that α >= 1 which is not the case?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Help with a proof of the yield curve.
  1. Help with a Proof (Replies: 1)

  2. Proof help! (Replies: 3)

Loading...