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Homework Help: 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.
    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
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