The process of bond valuation is based on the fundamental concept that the current price of a security can be determined by calculating the present value of the cash flows that the security will generate in the future. There is a consistent and predictable relationship between a bond's coupon rate, its par value, a bondholder's required return, and the bond's resulting intrinsic value. Trading at a discount, trading at a premium, and trading at par refer to particular relationships between a bond's intrinsic value and its par value. These result from the relationship between a bond's coupon rate and a bondholder's required rate of return.
Remember, a bond's coupon rate partially determines the interest-based return that a bond (might/will) pay, and a bondholder's required return reflects the return that a bondholder(would like/is obligated) to receive from a given investment.
The mathematics of bond valuation imply a predictable relationship between the bond's coupon rate, the bondholder's required return, the bond's par value, and its intrinsic value
These relationships can be summarized as follows:
When the bond's coupon rate is equal to the bondholder's required return, the bond's intrinsic value will equal its par value, and the bond will trade at par.
When the bond's coupon rate is greater to the bondholder's required return, the bond's intrinsic value will (be less than/exceed/equal) its par value, and the bond will trade at a premium.
When the bond's coupon rate is less than the bondholder's required return, the bond's intrinsic value will be less than its par value, and the bond will trade (at a premium/at par/at a discount).