Properties of Bond Duration

Consider a fixed-rate bond with semiannual coupon payments. Which of the following statements _best_ characterizes the relationship between coupon payment dates and their durations?

Correct. On the coupon payment date, the modified duration of a bond jumps upward. You may remember the formula for calculating the Macaulay duration: $$\displaystyle \text{MacDur} = \frac{1+r}{r} - \frac{1+r+[N \times (c-r)]}{c \times \left[ (1+r)^N - 1 \right] + r} - \frac{t}{T} $$ This jump is caused by the term $$\frac{t}{T}$$. As $$t$$ equals $$T$$, the coupon is paid. Immediately after the coupon, $$\frac{t}{T}$$ goes from 1 to 0 and there is a "saw-tooth" pattern. This pattern is also applicable to the modified duration.

Incorrect. As the bond approaches its coupon payment date, its Macaulay duration actually decreases.

Incorrect. The modified duration of a bond does not decrease smoothly; there is a jump in duration on the coupon payment date.

The modified duration of the bond increases significantly on the coupon payment date

The Macaulay duration of the bond increases as the bond approaches its coupon payment date

The modified duration of the bond decreases smoothly, regardless of the coupon payment date