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.
My attempt:
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?