For each of the two bonds described below, please answer:

a$.$ Calculate the modified duration of the two bonds.

b$.$ According to modified duration, which of the two bonds carries more risk, the $5-$ year U$.$S$.$ Treasury note or the $7-$year AT&T corporate bond?

Corporate Bond:

The pandemic years have been a very active time for Corporate Bonds, as many companies attempted to raise cash at record$-$low rates. For example, AT&T raised cash by selling a corporate bond in May $2020,$ with a time to maturity of $7$ years. They raised $2\mathrm{.}495$ billion USD, promising a face value of $2\mathrm{.}500$ billion USD, and a coupon rate of $2\mathrm{.}3\%$ per year, paid semiannually.

U$.$S$.$ Treasury Security:

Yields implicit in U$.$S$.$ Treasury securities are benchmark rates throughout the U$.$S$.$ economy, as well as in international capital markets. The $5-$year yield to maturity was $4\mathrm{.}10\%$ on the day our course started. What is the price of a $5-$year Treasury note with a coupon rate of $5\%$ per year, paid semiannually. Feel free to assume a face value of $100.$