Duration of par bonds

Since the bond is trading at par, this implies that the yield is equal to the interest rate.

Let the Principle amount be $S$ . The price of the bond is also equal to $S$ .

We first calculate the Macaulay Duration of the bond, which is equal to

$\frac{ \frac{ 10 \times S } { (1.05)^{10} } + \sum_ {i=1}^{10} \frac{ i \times 0.05 \times S } { (1.05)^ i } } { S } = 8.11$

Next, we calculate the modified duration, which is

$\frac{ 8.11 } { 1 + \frac{ 0.05 } { 1 } } = 7.724.$