Bond price

A bond paying coupons $C$ at times $t_1,\ldots,t_m$ and face $F$ at maturity $T$:

\[P = \sum_{i=1}^m \frac{C}{(1+y)^{t_i}} + \frac{F}{(1+y)^T}\]

where $y$ is the yield to maturity — the single discount rate equating present value to price.

Duration

Macaulay duration: the weighted average time to receive cash flows:

\[D_{Mac} = \frac{\sum_i t_i \cdot PV(C_i)}{P}\]

Modified duration measures price sensitivity to yield:

\[D_{mod} = -\frac{1}{P}\frac{dP}{dy} \approx \frac{D_{Mac}}{1+y}\]

A 1 basis-point (0.01%) rise in yield changes price by approximately $-D_{mod} \times P \times 0.0001$.

Convexity

Duration is a linear approximation. The second-order correction:

\[\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \frac{1}{2}\text{Conv}(\Delta y)^2\]

Convexity $= \frac{1}{P}\frac{d^2P}{dy^2}$. Positive convexity (all standard bonds) means price falls less than duration predicts when yields rise, and rises more when yields fall — beneficial asymmetry.