May i ask why negative duration means increasing in value when interest rate “increases” but not “decrease”?

with positive duration, an increase in interest rate decreases bond value, so i guess it goes the other way when duration is negative.

but i’m curious how can negative duration happen?

First, note that you’re talking about effective duration; neither Macaulay duration nor modified duration can be negative.

Second, recall the formula for effective duration from Level I:

Dur_{eff} = (P_{−} − P_{+}) / (2P_{0}Δ_y_)

If effective duration is less than zero, then,

(P_{−} − P_{+}) / (2P_{0}Δ_y_) < 0

If rates increase (i.e., Δ_y_ is positive), then, multiplying both sides by 2P_{0}Δ_y_ gives:

P_{−} − P_{+} < 0

or,

P_{−} < P_{+}

Thus, when rates increase, the price increases, and, consequently, when rates decrease, the price decreases.

Can someone give us insight as to when this would be possible?

Not insight, but understanding:

Interest only (I/O) strips at low YTMs can have negative effective duration because the increased prepayments means that the bondholders ultimately get less money.