with 91 days to maturity yielding 5.56% on a bank discount basis, the bond-equivalent yield is calculated as follows:   Bond-equivalentyield= 365(0.0556) - = 0.0572 = 5.72% --------------------------------------------- 360- 91(0.0556)     Note the formula for the bond-equivalent yield presented above assumes that the current maturity of the discount instrument in question is 182 days or less.     Discount Instruments with More Than 182 Days to Maturity When a discount instrument (e.g., a 52-week Fannie Mae Benchmark bill) has a current maturity of more than 182 days, converting a yield on a bank discount basis into a bond-equivalent yield is more involved. Specifically, the calculation must reflect the fact that a Benchmark bill is a discount instrument while a coupon Treasury delivers coupon pay- ments semiannually and the semiannual coupon payment can be rein- vested. In order to make this adjustment, we assume that interest is paid after six months at a rate equal to the discount instruments bond-equiv- alent yield (BEY) and that this interest is reinvested at this rate. To find a discount instruments bond-equivalent yield if its current maturity is greater than 182 days, we solve for the BEY using the fol- lowing formula:7     7We can derive this using the following notation: P = price of the discount instrument BEY= bond-equivalent yield t = number of days until the discount instruments maturity then, P[1 + (BEY/2)] = future value obtained by the investor if $P is invested for six months at one-half the BEY   (BEY/365)[t - (365/2)][1 + (BEY/2)]P = the amount earned by the investor on a sim- ple interest basis if the proceeds are reinvested at the BEY for the discount instru- ments remaining days to maturity   Assuming a face value for the discount instrument of $100, then   P[1 + (BEY/2)]+ (BEY/365)[t - (365/2)][1 + (BEY/2)]P = 100   This expression can be written more compactly as   P[1 + (BEY/2)][(1+(BEY/2))(2T/365 - 1)] = 100