Option adjusted duration Vs interest rate volatility

[Sumit.K]

Hi David,

Can you help me with this question.

When is the option-adjusted duration of a callable bond closer to the duration of a similar non-callable bond?

1) When bond has a high volatility or 2) when bond trades much lower than the call price

I think, effective duration of both callable and non-callable bonds will be close to each other only when the volatility is very low. In case of high volatility the effective duration of non-callable bond will always be higher than that of a callable bond. Considering this, I think option (2) is a definite true but what about (1)? Am I right about high volatility effect? I am looking at this question considering the yield curves for callable bond and non-callable bond in addition to the formula for effective duration. Am I missing something?

How does it actually work?

please suggest..

Thanks in advance..

 

[Aleksander Hansen]

Yes, you are correct wrt to both.
The non-callable bond will always have a higher duration than the callable bond; however, whereas the duration of the non-callable bond is monotonically decreasing wrt interest rates, the duration of the callable bond is decreasing for very low rates up to a certain inflection point, where the duration is sharply increasing as a function of interest rates. As interest rates increase more and more, the duration of the callable bond increases, but when it gets closer to the duration of the non-callable bond, it will converge asymptotically to the duration of the non-callable bond.

 

[Sumit.K]

Thanks a lot Aleksander.
I agree that as we move from left to right (increasing interest rates) the gap between the duration of callable and non-callable bond decreases. Considering your explanation the reason behind this is lower rate of price reduction (although higher price differences) wrg to non-callable bond as compared to that of callable bond for which the rate of price reductions will increase (though with comparatively lower price gaps) till the inflection point.

However, I understand that soon after the point of inflection(the pt. at which the callable bond curve changes from negative convexity to positive convexity and the point from where the two curves overlap each other) the price reductions for both, the callable as well as non-callable bond, will almost be similar resulting in equivalent duration for a give change in interest rates. Why do you say that at inflection point duration is sharply increasing as a function of interest rates?

 

[Aleksander Hansen]

Two points, 1)where rates are very low...move along the curve a bit, and it shoots up (as a function of interest rates) - it is convex;
2) second point is where it becomes concave.