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 $"0"$ wherever required. Negative values should be indicated by a minus sign. Do not round intermediate calculations. Round your answers to $2$ decimal places.$)$

$\backslash$table$[[$Economy$,$Probability,YTM$,$Price,Capital Gain,$\backslash$table$[[$Coupon$],[$Interest$]],$HPR$],[$Boom$,0\mathrm{.}10,,,,4\mathrm{.}00,\%],[$Normal Growth,$0\mathrm{.}60,,,,4\mathrm{.}00,\%],[$Recession$,0\mathrm{.}30,7\mathrm{.}0\%,,,4\mathrm{.}00,\%]]$