Derive the probability distribution of the $1-$year HPR on a $30-$year U$.$S$.$ Treasury bond with a coupon of $4\mathrm{.}0\%$ if it is currently selling at

par and the probability distribution of its yield to maturity a year from now is as shown in the table below. $($Assume the entire $4\mathrm{.}0\%$

coupon is paid at the end of the year rather than every $6$ months. Assume a par value of $$100.)($Leave no cells blank $-$ be certain to

enter $"O"$ wherever required. Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round

your answers to $2$ decimal places.$)$