U.S. Treasury Bills are quoted based on an Act / 360 / 360 discount yield, P B

U.S. Treasury Bills are quoted based on an Act $/360$ discount yield, $P_B=$ $1-y_{B}N/360$, where $N$ is the number of days from settlement date to maturity date.

(a) Provide a formula for converting the Act/360 discount yield to an Act $/360$ simple (add-on) yield, $P_B\times (1+y_{s}N/360)=1$.

(b) The semiannual coupon equivalent yield (bond equivalent yield, $BEY$ ) of a T-Bill is defined as follows: let $w=N/365$ (366 if there
is a leap day between settlement and maturity dates). If $w\le 1/2$, the BEY $y$ solves

$$P_B\times (1+yw)=1$$

If $w>1/2,y$ is the positive solution to the following quadratic equation

$$P_B(1+y/2)(1+y(w-1/2\left)\right)=1$$

Provide a formula for $y$ in terms of $P_B$ and $w$ for each case $(w\le 1/2$, $w>1/2$ ).

(c) On Thursday, 12-Aug-2021, the U.S. Treasury issued a 52-week ("1-year") T-Bill with maturity date Monday 11-Aug-2022. If the price of this T-Bill for settlement date 8-Oct-2021 is $98\%$, compute its discount yield, simple (add-on) yield, and the bond-equivalent yield on the settlement date.