Convexity

(a) If \(f^{\prime \prime}\) exists, use the definition 2.11 and

\[f^{\prime \prime}(x)=\lim _{h ightarrow 0} \frac{f(x+h)+f(x-b)-2 f(x)}{h^{2}}\]

to show that \(f^{\prime \prime}(x) \geq 0\) for a convex function ( \(f^{\prime \prime} \leq 0\) for a concave function Formula 2.11).

(b) For a bond with periodic coupon payments, the \(i\) th cash flow is discounted using \(1 /(1+y / m)^{i}\). Show that \(1 /(1+y / m)^{i}\) is a convex function of \(y\).

(c) Show that Formula 2.5, for the price of a coupon bond is a convex function of yield. [Hint: Use the result from part

(b) of this problem.]