An investor purchases an annual coupon bond with a $6\%$ coupon rate and exactly $20$ years remaining until maturity at a price equal to par value. The investor's investment horizon is eight years. The approximate modified duration of the bond is $11\mathrm{.}470$ years.

What is the duration gap at the time of purchase?

$($Hint: use approximate Macaulay duration to calculate the duration gap$)$