Derive the probability distribution of the $1-$year HPR on a $30-$year U$.$S$.$ Treasury bond with an $4\%$ coupon if it is currently selling at par and the probability distribution of its yield to maturity a year from now is as follows: $($Assume the entire $4\%$ coupon is paid at the end of the year rather than every $6$ months. Assume a par value of $$100.)($Leave no cells blank $-$ be certain to enter $"$O$"$ wherever required. Enter negative values with a minus sign. Do not round intermediate calculations. Round your answers to $2$ decimal places. Omit the $"$$$"$ & $"\%"$ signs in your response.$)$

$\backslash$table$[[$State of the Economy,Probability,YTM$,$Price,Capital Gain,Coupon Interest,HPR$],[$Boom$,10,6\%,$$$,72\mathrm{.}49,$$$,-23\mathrm{.}51,$$$,4,,1\%$