 # Positive and negative convexity

For a security that exhibits positive convexity, the duration changes in the desired direction; for a security that exhibits negative convexity, there is an adverse change in the duration. If interest rates rise, the duration of the MBS increases while the duration of the Treasury decreases. Why does duration of the MBS increase, but duration of Treasury decreases?

because of the negative convexity which is a result prepayment risk

• basically the slope of the tangent line to the MBS curve (the duration) decreases in absolute value (but increases not in absolute value)

Not to put too fine a point on it, but the slope of the price-yield curve is not the (negative of) the duration; it’s the (negative of) the _ dollar _ duration.

Thanks for the graph very useful. But where on the graph can we see duration increasing as interest rates increase? Aren’t the axis yield and price?

so draw a tangent line to two different yield points along the negative convex portion of the price-yield curve and compare their slopes…you will see

assume yield = interest rate for the sake of illustration

thanks S2000, you always got my back!

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so draw a tangent line to two different yield points along the negative convex portion of the price-yield curve and compare their slopes…you will see

assume yield = interest rate for the sake of illustration

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But where is duration on the price-yield curve? Duration is a function of payments and time, so not sure how that is on the graph?

Thats true, it is afunction of payments and time, BUT it can be used to approximate the % change in the bond price given a 100 bps change in the yield, hence, it can be graphed in the yield-price space

Thats true, it is afunction of payments and time, BUT it can be used to approximate the % change in the bond price given a 100 bps change in the yield, hence, it can be graphed in the yield-price space

I see. So a flattening slope as yield increases means duration decreases?