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For a discount bond, the current yield considers the gain that occurs as the discount disappears over the lifetime of the bond.

If a bond pays $90 interest annually, matures in 10-years and has a price of $1,100, its current yield is 9.0%.

The yield to maturity considers not only the coupon income from a bond but also any change in the value of the bond when its held to maturity.

If interest rates in general were to fall, the prices of existing bonds would fall as well.