Bond Duration and Convexity Approximation

You hold a fixed-rate bond with face value $F$, annual coupon rate $c$, maturity $T$ years, and yield-to-maturity $y$ (annual compounding). 1. Write the bond price $P(y)$ as a discounted cash-flow sum. 2. Define Macaulay duration $D_{\text{Mac}}$, modified duration $D_{\text{mod}}$, and convexity $C$ in terms of the derivatives $\frac{dP}{dy}$ and $\frac{d^{2}P}{dy^{2}}$. 3. Derive the duration-convexity approximation for the relative price change $\frac{\Delta P}{P}$ under a small yield shock $\Delta y$. When do you expect this approximation to be accurate, and when does it break down or become biased?