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If V (t) is the value of an investment product at time t, where t is measured in...

Nov 27, 2024

If V (t) is the value of an investment product at time t, where t is measured in years, the instantaneous log-return at time t is defined as p(t)=d/dt(In(V(t)) The company which is selling the investment product would like to give investors an incentive to hold on to the product. For this reason, they ensure that the log-return is a strictly increasing function of time. Question 1 A twice differentiable function f(t) is said to be convex if f"(t) > 0 for all t. If p'(t) > 0 it is clear that In(V) is a convex function of t. Does this also imply that V is a convex function of t?

Solution by Steps

step 1

The instantaneous log-return is defined as

p(t) = \frac{d}{dt}(\ln(V(t)))

. Since p'(t) > 0 , it indicates that

p(t)

is increasing

step 2

If p'(t) > 0 , then

p(t)

is a strictly increasing function, which implies that

\ln(V(t))

is convex

step 3

The second derivative test for convexity states that if f''(t) > 0 , then

f(t)

is convex. Here, p'(t) > 0 implies that

V(t)

is also increasing

step 4

However, to determine if

V(t)

is convex, we need to check if V''(t) > 0 . Since

p(t)

being increasing does not guarantee

V(t)

is convex, we cannot conclude this without further information

Answer

No, p'(t) > 0 does not necessarily imply that

V(t)

is convex without additional information about the second derivative of

V(t)

Key Concept

Convexity of functions and the relationship between derivatives.

Explanation

While p'(t) > 0 indicates that the log-return is increasing, it does not directly imply that the original function

V(t)

is convex without examining its second derivative.