The following information relates to Questions $3$ to $5$

An investor plans to retire in $10$ years. As part of the retirement portfolio, the investor buys a newly issued, $12-$year, $8\%$ annual coupon payment bond. The bond is purchased at par value, so its yield$-$to$-$maturity is $8\mathrm{.}00\%$ stated as an effective annual rate.

$1$

$3.$ Calculate the approximate Macaulay duration for the bond, using a $1bp(0\mathrm{.}01\%)$ increase and decrease in the yield$-$to$-$maturity and calculating the new prices per $100$ of par value to six decimal places.

$4.$ Calculate the duration gap at the time of purchase.

$($Hint: An investor plans to retire in $10$ years. So$,$ this investor's investment horizon is $10$ years.$)$

$5.$ Does this bond at purchase entail the risk of higher or lower interest rates?