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# Duration will understate the position

#### rahul.goyl

##### Member
Hi David,

I am not sure if am getting this correct,
In the below post, Duration will overstate for Negative Yield shock (- 25 Bps) instead Understate & (Vica-Versa) Understate
for Positive Yield Shock (+25 Bps).

>> http://www.bionicturtle.com/wiki/FRM2009.L1.12/

12c. If the bond has no embedded options, given the yield shock of +25 bps, is the actual bond position likely to be greater than, equal to, or less than the anticipated by the formula that uses duration and convexity?
Greater than; i.e., the loss in value will be less than predicted so the position value will be greater than predicted. Duration (first order linear approximation) will understate the position; convexity improves with a second-order measure but, if the bond is plain-vanilla, there will still be a gap between the estimated position and the actual position.

Regards,
Rahul

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
I made a mistake here, in regard to convexity (not in regard to duration).

In regard to duration, for vanilla bond as convexity adj is always positive, duration does give an estimate of bond price that underestimates the actual price.

the + yield shock (+ 25bps) gives the linear estimate of change. So, for example, perhaps duration alone gives an estimate of -2% change in bond price given a + 25 bps yield shock. As the price/yield curve has convexity, the actual bond price (at initial yield + 25 bps shock) is higher than predicted by the line. So, for + 25bps, loss in price is overstated and actual bond position is greater than predicted by duration only.

However, I goofed in regard to this question, which is about the error that convexity implies: as the third term Taylor series = 1/6*f'''(initial yield)*yield shock cubed (where f''' is third derivative, or first derivative of convexity), the subsequent adjustment, if we extended the taylor series would be:
duration + convexity +1/6*(a negative third derivative)*(yield shock)^3
i.e., if positive yield shock, a MINUS adjustment and if negative yield shock, a PLUS adjustment

... so, in regard to duration + convexity, a +25 bps understates the actual loss (i.e., b/c further taylor expansion would introduce a tiny MINUS) and so the actual is LESS THAN
... a -25 bps understates the gain, since (-)*(-) = (+), such that actual is greater than for a negative shock

so, thanks for pointing this out. Sorry if the language confuses: duration's error is correct, it always gives an estimated bond price that is below the actual price (i.e., underestimates the gain and overestimates the loss; as the line is always below the P/Y curve) but I was too simple re convexity's error: it is different in each direction.

Thanks for pointing this out! David