P1.T4.16 Tuckman :hedging given DV01 or effective duration

[BawBawBaw]

Hi David,
In Tuckman's question set (question 16.3 page 58) : "at a rate of 4% a bond has a price of $107.93. If the rate drops by one bp to 3.99%, the bond price increases to $108. What is the estimate of the bond's effective duration ?

To answer that question, I applied the effective duration's formula : Bond Price - Bond Price if increase / 2 x Original Price x change in interest rate
which gives (108-107.93) / 2 x 107.93 x 0.0001 = 3.2428
The answer is 6.486, which is twice the answer given by the formula above.

Do I have to multiply the duration given by 2, if yes on what basis ?
Many thanks for your help
PS : I get the right answer with the DV01 formula, but I find the effective duration formula more convenient.

 

[David Harper CFA FRM]

Hi @BawBawBaw

Yes, that's common mistake. The effective duration is (symbolically) = [dollar duration]/price = -1/Price * [slope of the tangent to the P/Y curve]; the "dollar duration" is the slope of the tangent line which approximates the change in price given a change in yield. The reason I am focusing on that is to convey how the effective duration is just -1/price multiplied by a slope; slope is rise/run where the key is to "match them up." The (2*) in the denominator assumes you are pricing up and down by the yield shock, so if the shock is one basis point (0.01%), then our slope is:

• [price(@ yield = 4.00% - .010%) - price(@ yield = 4.00% + .010%)] / [price(@ 4.0%)*2*(0.010%)]; i.e., the overall yield shock is 0.02% to account for the difference between 4.01% and 3.99%

But if the prices differ based only upon a 0.01% shock, then the denominator must be only 0.01%

In this way, there is no magic to the (2*) in the denominator! (And why we want to understand the formula...). At 4.0%, there are in fact an almost infinite variety of exact effective durations. Here is a valid effective duration at 4.0%:

• [price(@ yield = 4.00% - 0.030%) - price(@ yield = 4.00% + 0.050%)] / [price(@ 4%)*0.080%)]; i.e., as we are pricing the bond with an overall (asymmetric) shock of 8.0%, the denominator should match. It's just rise/run from a slope perspective.

I hope that explains why you don't need the (2) in the denominator; i.e., the prices are based on 3.99% and 4.00 not 4.01%. Thanks!

P.S. If you want to discuss further, the source Q&A is here @ https://forum.bionicturtle.com/threads/p1-t4-16-hedging-given-dv01-or-effective-duration.5515/

 

[Praveen_India]

Hi David,

Duration is % change in price of a bond for 1% (100bps) change in interest rates. Here the price is changing by .07 (from 107.93 to 8) for 1 bp (4% to 3.99%) change.
Hence would not duration be change in price for 100 bps change i.e 0.07*100 i.e 7 years?

Kindly advise.


Hi BawBawBaw,

David has given a very clear explanation. Just to add it

The general formula for duration is: Bond Price if decrease - Bond Price if increase / 2 x Original Price x change in interest rate. Denominator should be like the one i have mentioned (underlined part). If you want to use this formula then you need calculate the price by increasing the yield by 1 bps also. Original Price (107.93)and Bond Price if decrease (108) are available.

Thanks,
Praveen

 

[Praveen_India]

I think i got it, dollar 7 change is w.r.t 107.93, in terms of percentage it is 6.49%, which is the duration of the bond.

Thank you.

 

[BawBawBaw]

Many thanks for your answer David, all very clear now.

 

[Sahil1999]

Hi David (@David Harper CFA FRM),


I finding it a bit confusing to understand that why are we dividing the DV01 of existing position and hedging position by 100 here. From what I understood, it's done to calibrate the Face Value for both positions to compare equal dollar change when yield changes by 1 basis point. Couldn't understand how dividing DV01 by 100 could do that.


Regards,
Sahil