An investor purchases a portfolio consisting of three bonds. Bond A has annual coupons of $6\%$ and matures for its face amount of

$1,000$ in ten years. It is purchased for $1,000.$ Bonds B and C are zero$-$coupon bonds, maturing for $1,000$ each in five and ten years,

respectively. All three bonds have the same yield rate.

Calculate the Macaulay duration in years at the time of purchase of the portfolio with respect to the common yield rate.

$7\mathrm{.}23$

$7\mathrm{.}43$

$7\mathrm{.}60$

$8\mathrm{.}33$

$8\mathrm{.}38$