The return a bond provides to an investor is measured by its yield, which is quoted as a percentage.
Current yield is a commonly quoted yield calculation, used to evaluate the return on a bond for a oneyear period. It only accounts for the interest, or coupon payments, that the bond returns to investors. This yield is calculated as the bond's coupon rate divided by its current market price, but it does nothing to account for any capital gains or losses when the bond is sold. If the bond is not sold within the year, this yield calculation will provide the bondholder with an accurate assessment of his or her return. The only way a bond's current yield could be negative, using this basic evaluation, is if the investor were receiving negative interest payments, or if the bond somehow had a market value below $0  both of which are very unlikely to occur.
Other yield calculations will take into account different factors and can be used to better evaluate the returns an investor may receive, given different events.
As its name suggests, the yield to maturity (YTM) calculates the return (expressed as an annual percentage) on a bond if the investor were to hold the bond until maturity. This formula takes into account all of the coupon payments and the face  or par  value on the bond (assuming no principal defaults), and it can be seen as a more complete evaluation than current yield. However, calculating a bond's YTM is complex and involves trial and error. It is usually done by using a programmable business calculator, but you can also get an approximate YTM by using a bond yield table.
Let's consider an example: say an investor pays $800 for a bond that has has exactly two years to maturity, a face value of $1,000 and interest payments of $8 per year. Using a bond table, we could determine that the bond will have a YTM of about 10.86%. If the bond holder paid $1,200 for the bond, the YTM would be about 9.41% (It is worth noting, however, that a bond will not necessarily have a negative actual yield just because the investor paid more than face value for it.) When using the YTM calculation, it is possible to have a negative yield on a bond  it depends largely on how much you initially pay for the bond and the time to maturity.
Yields can be calculated using different formulas  there are many more than the two mentioned here  and depending on the formula used, you can end up with drastically different yields. To learn more, see our Bond Basics Tutorial, Advanced Bond Concepts and The Basics Of The Bond Ladder.

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What is the relationship between current yield and yield to maturity (YTM)?
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What is the difference between yield to maturity and holding period return yield?
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Yield To Maturity (YTM)
The total return anticipated on a bond if the bond is held until ... 
Current Yield
Annual income (interest or dividends) divided by the current ... 
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The amount of return an investor will realize on a bond. Several ... 
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The return a bond must offer in order to be a worthwhile investment. ... 
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The percentage of par, or face value, at which a bond is quoted. ... 
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