Macaulay and Modified Duration of a Fixed-Coupon Bond

Scenario

A portfolio manager holds a fixed-rate bond with annual coupon payment $C$, face value $M$, maturity $n$ years, and yield to maturity $y$ (annual compounding).

Your Task

1. Define the Macaulay duration $D_{mac}$ as the weighted average maturity of the bond's cash flows.
2. Derive the Modified duration $D_{mod} = -\frac{1}{P} \frac{dP}{dy}$ and prove that $D_{mod} = \frac{D_{mac}}{1+y}$.
3. Calculate the bond's price, Macaulay duration, Modified duration, and estimated percentage price change for a $+50$ bps shift in yield given: C = \6.00$,$M = $100.00$,$n = 5$years, and$y = 5%$.

Constraints

• Annual coupon payments and flat yield curve.
• First-order linear approximation for yield shift.

Examples For a 5-year $6\%$ coupon bond yielding $5\%$, the price is $P > 100$ (trading at premium). Macaulay duration is between 4.2 and 4.5 years.