The instrument you wish to price has the following characteristics: floating rate bond issued on January $31,2020,$ with maturity on January $31,2024,$ semiannual coupons pegged to the $6-$months treasury rates plus $40$ bps $(0\mathrm{.}4\%).$ At the last reset date, July $31,2022,$ the next coupon payment $($due on January $1,2023)$ was set to the $6-$months

treasury rate at that time, $3\mathrm{.}2\%,$ plus the spread, $0\mathrm{.}4\%.[$Attention: Both rates are annualized, so the coupon payment for January $2023$ is really $1\mathrm{.}6\%+0\mathrm{.}2\%]$

$1-$ What is the price on October $31,2022,$ of the pure floating part of this bond?

$2-$ What is the price on October $31,2022,$ of the spread part of this bond? $3-$ What is the price on October $31,2022,$ of the entire floating rate bond with

spread?

$4-$ What are the Macaulay duration and the modified duration with respect to continuously compounding rates of the pure floating part of the floating rate bond? $[$i$.$e$.,$ ignore the spread$]$