Can you please explain why key rate duration may be negative?

Fixed Income - Key Rate Duration

There is no quick explanation! This is a very technical, academic aspect.

It all depends on the notion of the par bond (on the Cognition slides, this is a 5% coupon bond and we have a flat yield curve of 5%).  If we hold this bond constant and therefore also hold the yield curve flat at 5%, then if one rate rises (e.g. 5y) then for the yield curve to still be flat at 5%, the other rates beyond year 5 through to maturity (year 15) must fall since we need the average to remain 5%.  For example if we assume 5y yields rise by 1% then maybe the remaining 10 y yields all fall by 10bp.

Now lets apply this to the cash flows.  Any cash flow at year 5 will have a lower PV due to the 1% increase in 5y yields and all of the CFs from years 6-15 will have slightly higher PVs (due to fall in rates beyond Yr 5).  With the par bond, these two effects cancel out however, if the coupon is very small then the 5 y CF will fall by a very small amount and the other CFs (year 6 to 15) will increase in value by more, resulting in an overall increase. Thus if 5 y yields rise, the value of the bond rises.