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21 - Lecture 21 Bonds Coupon Paying Bonds Let c by the...

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Lecture 21 Bonds Coupon Paying Bonds Let c by the yearly coupon payment and m the face value. Suppose the time to maturity is n years. Then the yield to maturity, y, solves p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n Note: This follows the standard way of computing an internal rate of return. y is therefore the bond’s internal rate of return. Coupon Paying Bonds Let c by the yearly coupon payment and m the face value. Suppose the time to maturity is n years. Then the yield to maturity, y, solves p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n We now simplify this expression: Coupon Paying Bonds p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n We now simplify this expression. Let S = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c 1 + y ( ) n Then S 1 + y = c 1 + y ( ) 2 + c 1 + y ( ) 3 + c 1 + y ( ) 4 + ! + c 1 + y ( ) n + 1 Subtract the second equation from the first..
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Coupon Paying Bonds p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n We now simplify this expression. Let S = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c 1 + y ( ) n Then S 1 + y = c 1 + y ( ) 2 + c 1 + y ( ) 3 + c 1 + y ( ) 4 + ! + c 1 + y ( ) n + 1 S ! S 1 + y = c 1 + y ! c 1 + y ( ) n + 1 Thus S = ??? Coupon Paying Bonds p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n We now simplify this expression. Let S = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c 1 + y ( ) n Then S 1 + y = c 1 + y ( ) 2 + c 1 + y ( ) 3 + c 1 + y ( ) 4 + ! + c 1 + y ( ) n + 1 S ! S 1 + y = c 1 + y ! c 1 + y ( ) n + 1 Thus S = c y 1 ! 1 1 + y ( ) n " # $ % & ' Coupon Paying Bonds p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n We now simplify this expression. Let S = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c 1 + y ( ) n Then p = c y 1 ! 1 1 + y ( ) n " # $ % & ' + m 1 + y ( ) n Thus S = c y 1 ! 1 1 + y ( ) n " # $ % & ' Coupon Paying Bonds p = c 1 + y + c 1 + y ( ) 2 + c 1 + y ( ) 3 + ! + c + m 1 + y ( ) n p = c y 1 ! 1 1 + y ( ) n " # $ % & ' + m 1 + y ( ) n With semiannual coupon payments we get: p = 0.5 c 1 + y ( ) 1/2 + 0.5 c 1 + y + 0.5 c 1 + y ( ) 3/2 + ! + 0.5 c + m 1 + y ( ) n p = 0.5 c 1 + y ( ) 0.5 ! 1 1 ! 1 1 + y ( ) n " # $ % & ' + m 1 + y ( ) n
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Yield to Maturity p = c y 1 ! 1 1 + y ( ) n " # $ % & ' + m 1 + y ( ) n Solve this equation for y (usually, one must do this numerically). Exception p=m, i.e., bond is “sold at par”, then y=m/c.
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