The finance director of AQR Co has heard that the market value of the company will increase if the weighted average cost of capital of the company is decreased. The company, which is listed on a stock exchange, has $100$ million shares in issue and the current ex div ordinary share price is $$2\mathrm{.}50$ per share. AQR Co also has in issue bonds with a book value of $$60$m and their current ex interest market price is $$104$ per $$100$ bond. The current after$-$tax cost of debt of AQR Co is $7\%$ and the tax rate is $30\%.$

The recent dividends per share of the company are as follows.

Year $20$X$020$X$120$X$220$X$320$X$4$

Dividend per share $($cents$)19\mathrm{.}3820\mathrm{.}2020\mathrm{.}4121\mathrm{.}0221\mathrm{.}80$

The finance director proposes to decrease the weighted average cost of capital of AQR Co$,$ and hence increase its market value, by issuing $$40$m of bonds at their nominal value of $$100$ per bond. These bonds would pay annual interest of $8\%$ before tax and would be redeemed at a $5\%$ premium to nominal value after $10$ years.

Required;

$($a$)$ Calculate the market value after$-$tax weighted average cost of capital of AQR Co in the following circumstances:

$($i$)$ Before the new issue of bonds takes place.

$($ii$)$ After the new issue of bonds takes place.

Comment on your findings. $(12$ marks$)$

$($b$)$ Discuss the director's view that issuing traded bonds will decrease the weighted average cost of capital of AQR Co and thereby increase the market value of the company.