$4.$ You want to secure a payment of $$10$ million in $8$ years. You find a $12-$year, $10\mathrm{.}25\%$ coupon semi$-$annual bond with a face value of $$1000$ with a YTM $=6\mathrm{.}50\%.$ It has a duration of $8\mathrm{.}0.$ How many of these bonds will you purchase today to generate the appropriate payout in $8$ years? Now, show that if$,$ after you purchase the bond rates suddenly change so that the new YTM on the bond is $7\mathrm{.}5\%,$ you will still generate the appropriate $$10$ million payout in year $8.$