P1.T4. EOC 12.6 and 12.9

[AUola2165]

Question 12.6:
Suppose that the duration of a bond position is zero and convexity is positive. Does the value of the bond position increase or decrease when there is a small parallel shift in rates?

Answer:
It increases. Duration alone predicts no change, but a positive convexity leads to an increase in value when there is a parallel shift.

I am a bit confused about this question. I thought that a bond with positive convexity will have larger price increase from 1% decline in yields than a price decline from 1% increase in yields. Could someone explain how a small parallel shift lead to increase in bond value when the duration is 0.


Question 12.9:
Why is yield-based convexity likely to be greater than yield-based duration for a ten-year bond (assume that rates are expressed with continuous compounding)?

Answer:
Yield-based convexity is calculated by squaring each cash flow’s time to maturity and then taking a weighted aver-age with weights proportional to the present values of the cash flows. Yield-based duration is a weighted average of the time to maturity of cash flows with the same weights. For a ten-year bond, the former is clearly greater than the later because, for nearly all the cash flows, the square of the time to payment is greater than the time to payment.

How can we compare duration to convexity? They are different units. I am no means a quant but comparing the size of duration to convexity or delta to gamma seems pointless... or am I missing something?

 

[David Harper CFA FRM]

Hi @AUola2165 You aren't confused, you are observant and I agree with you. How GARP can get the first wrong is beyond me. Briefly:

• The zero duration position implies approximately zero value change for a small parallel shift per the definition of duration. The correct answer to 12.6 is: Neither, because a small parallel shift is the scenario where we can rely on duration and exclude convexity.
  
  They apparently meant to suggest (but it is hazardous to attempt inferences when questions are lazy) that a large change in yield (for the zero duration portfolio) will increase value because they imagine a local minimum where the price-yield curve is flat. Put another way, apparently they are thinking of ΔP/P = -D*Δy + 0.5*C*(Δy)^2 where D = 0 such that it must be the case that ΔP/P > 0. However, they've neglected that the zero duration must be achieved with (also) short positions (aka, -P) such that this pathological instance is non-trivial. It's moot, however, because "small parallel shift" is a robust instruction (see Tuckman) to ignore the convexity term, hence the correct answer is approximately zero.
  
  Indeed, this question would be more interesting if it asked about a "large change in the yield." But the nuances here are beyond their capabilities and this pathological case (IMO) needs a better setup by more careful writers; but it's just not worth it, especially here.
  
  
• Yes, we must agree with you here, too. We know what they mean, right? A 10-year zero-coupon bond (under CC) has a 10 year duration and a 100 years-squared (i.e., the units of convexity are maturity-squares) such that the numeric value indeed is likely to be greater. But this is similar to saying that the length of a 5-feet long brick wall (itself without width) is less than its surface area of 5*height square-feet (feet^2). In my opinion, they are not directly comparable. Thank you for the observations!