$>$ Ouestion $4($Callable Bond$)$

Canton Industries Limited $($CIL$)$ is considering raising funds over a shoyt period. In its last management meeting, the finance director, Ben Lo $($Ben$),$ suggested Q$1$L to issue perpetual callable bonds with a face value of $$1,000$ each with a coupon rate of $7\mathrm{.}037\%$ paid in annually. The bond can be called one year after issuance and the call premium will be two annual coupons. The current market interest rate is $7\%.$ Ben estimates a $0\mathrm{.}3$ probability that next year's interest rate will increase to $10\%,$ and a $0\mathrm{.}7$ probability that it will fall to $6\%.$

$($i$)$ Determine what the new coupon rate of the callable bonds should be if the bonds are issued at par. Assume that the bonds will be called if the interest rates fall and the call premium is equal to twice the annual coupon.

$($ii$)$ Evaluate the value of the call provision. $($Assume that the bonds will be called if the interest rates fall and the call premium is equal to twice the annual coupon.$)$

$($iii$)$ If one year later, the actual interest rate applicable to CIL fall to $6\mathrm{.}5\%$ instead of the predicted $6\%,$ will CIL call back the bond? Why?