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When a bond is issued, it has a face value (also known as par value), coupon rate, and a maturity date. The Yield-to-Maturity (YTM) is the return that a bondholder can expect to receive if the bond is held until its maturity date. The relationship between the price of a bond and its YTM is inversely proportional: as the YTM increases, the price of the bond decreases, and vice versa. This inverse relationship is because when interest rates in the market rise (which causes YTM to rise), the fixed coupon payment of the bond becomes less attractive relative to other investment opportunities. Investors would be willing to pay less for the bond, leading to a decrease in its price. Conversely, when interest rates in the market fall, the fixed coupon payment of the bond becomes more attractive, leading to an increase in its price.

Bonds sell at a premium when their coupon rate is higher than the prevailing market interest rates, which means the bond's yield-to-maturity is lower than its coupon rate. This happens because investors are willing to pay more for bonds that offer higher coupon payments compared to the market rates. On the other hand, bonds sell at a discount when their coupon rate is lower than the prevailing market interest rates, which means their YTM is higher than the coupon rate. This occurs because investors are not willing to pay as much for bonds that offer lower coupon payments compared to the market rates. When a bond sells at par value, the coupon rate is equal to the YTM. Par value indicates that the bond's price is equal to its face value, and there is no discount or premium applied.

Current yield is the bond's annual coupon payment divided by its current market price. For bonds selling at a premium, the current yield will be lower than the YTM; this is because the price of the bond is higher than its face value, which results in a lower yield based on the market price. The YTM calculation takes into account both the bond's coupon payments and the capital loss when it matures at its lower face value. For discount bonds, the current yield will be higher than the YTM; this is because the bond price is lower than its face value, which results in a higher yield based on the market price. In this case, the YTM calculation takes into account the bond's coupon payments and the capital gain when it matures at its higher face value. For bonds selling at par value, the current yield and YTM will be equal since the bond's market price is the same as its face value. In this case, there is no gain or loss in the bond's principal value at maturity, and the return is entirely based on its coupon payments.