Problem $5-8$

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.$)$

Answer is complete but not entirely correct.

$\backslash$table$[[$Economy$,$Probability,YTM$,,,$rice,$\backslash$table$[[$Capital$],[$Gain$]],\backslash$table$[[$Coupon$],[$Interest$]],$HPR$],[$Boom$,0\mathrm{.}10,6\mathrm{.}0,\%,$$$,98\mathrm{.}11\backslash$times $$,(1\mathrm{.}89)\backslash$times g$,4\mathrm{.}00$ V$,2\mathrm{.}11\backslash$times $],[\backslash$table$[[$Normal Growth$]],0\mathrm{.}60,5\mathrm{.}0,\%,,99\mathrm{.}05\backslash$times $,(0\mathrm{.}95)$x$,4\mathrm{.}00,3\mathrm{.}05$x$],[$Recession$,0\mathrm{.}30,4\mathrm{.}0,\%,,100\mathrm{.}00,0\mathrm{.}00,4\mathrm{.}00$ V$,4\mathrm{.}00?]]$