A portfolio manager is assessing the interest rate risk of three bonds as she considers making an investment of USD$50$ million. All three bonds are issued on $1$ June $2026$ and mature on $1$ June $2030,$ and they have the following characteristics:

Characteristic Bond One Bond Two Bond Three

Coupon $($semiannual$)7\%3\%5\%$

Yield$-$to$-$maturity $3\%7\%5\%$

Question

Q$.$ The portfolio manager is interested in comparing the interest rate risk of Bond Three to that of Bond Four, a floating$-$rate note that resets every six months. On $1$ June $2026,$ both bonds were priced to yield $5\%.$ If the yield changes from $5\%$ to $5\mathrm{.}25\%$ halfway through the first coupon period, which bond has the greater Macaulay duration?

Bond Three

Bond Four

Neither: The Macaulay duration is the same for both bonds.

Please explain in detail how $0\mathrm{.}5$ for Bond $4$ can be calculated easily