Help with a proof of the yield curve.

[Kuma]

Homework Statement

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.

Homework Equations

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?

 

[mighty2000]

Your analysis is correct, but you have overlooked an important detail. The inequality P(αt) ≥ (P(t))^α must hold for all values of α from 0 to 1, not just for αt > t. This means that your inequality must hold for all values of t, not just for t > 0. Therefore, your conclusion that α ≥ 1 is incorrect. Instead, you should consider the case where t > αt and see if the inequality still holds.