2006 FRM Practice Exams #52 - Convexity

[dennis_cmpe]

I have attached an image pertaining to this question. I chose answer D for this question, but the correct answer was C. I know the graph displays a callable bond due to the negative convexity from points y1 to y2. But why would convexity be 0 at point y2?

52. What bond type does the following price-yield curve represent and at which yield level is convexity equal to zero?

a. Puttable bond with convexity close to zero at y2.
b. Puttable bond with convexity close to zero at y1 and y3.
c. Callable bond with convexity close to zero at y2.
d. Callable bond with convexity close to zero at y1 and y3.

ANSWER: C
Convexity measures how interest rate sensitivity (i.e., duration) changes with interest rates. Callable bonds exhibit negative convexity at certain yield combinations. Negative convexity means that as the market yield decreases
duration decreases as well.

The correct answer is ‘C’. The graph represents the price yield relation for callable bonds. Convexity is close to zero at y2.

 

[David Harper CFA FRM]

Dennis,

The first derivative is the slope of the tangent line (I added red tangent lines), the second derivative is the rate of change of the slope of that tangent. So, if you start from the left (y axis), moving from Y1 to Y2, the plot is negative convex (aka, concave down) because the 2nd derivative is negative; i.e., the slope of the tangent is decreasing. Then, at Y2, the slope of the tangent stops decelerating and starts to accelerate (e.g., maybe the slope of the tangent starts near 0 at the y-axis then drops to -1 at Y2, then starts to head back to zero as it flattens toward Y3). That's an inflection point, where the *rate of change* of the slope of the tangent line decelerates to 0 then accelerates again: the second derivative is 0 at the inflection point.

(Actually, to be precise, convexity is not the 2nd derivative. Rather convexity is the 2nd derivative * 1/P; both duration/convexity are "infected" by price. To be technically accurate, the slope of the line is (convexity*P), alhough it is commonly misunderstood to be the convexity per se. That's why we say duration is similar to delta, convexity is similar to gamma. Because option delta/gamma are pure derivatives, but duration/convexity are infected by price)

David