Derive the probability distribution of the $1-$year HPR on a $30-$year U$.$S$.$ Treasury bond with a $4\mathrm{.}0\%$ coupon 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.)$

Note: Leave no cells blank $-$ be certain to enter $"0"$ wherever required. Negative values should be indicated by a minus sign. Do

not round intermediate calculations. Round your answers to $2$ decimal places.