Yield Based DV01 and Duration - pg 222 Tuckman - GARP Handbook

[Dr. Jayanthi Sankaran]

Hi David,

I wanted to find out whether the FRM requires us to know equations of the type (12.34) and (12.35) for yield based DV01 and (12.36) and (12.37) for yield based modified or adjusted duration. I am not very good at differential calculus to arrive at these complicated equations. Will they provide us with these equations or should we just learn them using numerical examples?

Thanks a tonne!
Jayanthi

 

[Dr. Jayanthi Sankaran]

On closer examination, I do understand these equations on a very intuitive level. So long as I know the basic YTM equation, all I need to do is to make time-based adjustments and those that are needed for DV01 and Duration!

Thanks
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran No, you do not need to know (have memorized) those equations; they are equations 4.34 to 4.37 in my kindle version of Tuckman. As a practical matter, it's too tedious to compute the Macaulay duration of a coupon-bearing bond (but I suppose such a question is always possible if the bond only had a few cash flows remaining!), such that the exam has leaned on either:

• computation of effective duration; i.e., the approximation of an analytical modified duration via the simulated ΔP/(PΔy) which is relatively quick on the calculator, or
  
• simply using a zero-coupon bond to forgo calculations (where mac duration = maturity).

Then there is a heavy reliance on the relationship: DV01 = Mod duration*Price/10,000. In short, you don't need to worry about complicated duration formulas. Instead, I think it's good to understand the text that follows the equations as it's good to understand how Mac duration is the weighted average maturity of a bond:

"There is a certain structure to equations (4.34) and (4.36). Each term in the brackets is the present value of a bond payment multiplied by the time to receipt of that payment, . The contribution of a payment to the interest rate risk of a bond varies directly with its present value and with its time to receipt. In addition, duration can be viewed as a weighted-sum of times to receipt, with each weight equal to the corresponding present value divided by the total of the present values, i.e., the price. Viewed this way, duration is a weighted-sum of times to receipt of payments and can be said to be measured in years. Hence, practitioners often refer to a duration of six as six years." -- Tuckman, Bruce; Serrat, Angel (2011-10-11). Fixed Income Securities: Tools for Today's Markets (Wiley Finance) (Kindle Locations 3739-3745). Wiley. Kindle Edition.

 

[Dr. Jayanthi Sankaran]

Thanks David - I did do the numerical example on Table 12-6 page 223 of Tuckman to understand the computation of DV01 and Modified Duration. It is very intuitive. Also, the corresponding formulae of yield based DV01 and Modified Duration of zero-coupon bonds, par bonds and perpetual bonds are easy to understand. Just to get this straight:

Modified Duration = Macaulay Duration/(1 + y/2)

Am I right in stating the above definition for Modified Duration?

Also, isn't DV01 = [(1/10,000)*(1/1+y/2)]*sum of time-weighted PV?
i.e. DV01 = [Modified Duration*Sum of Time-weighted PV]/10,000?

Thanks!
Jayanthi

 

[Dr. Jayanthi Sankaran]

Thanks David - I did do the numerical example on Table 12-6 page 223 of Tuckman to understand the computation of DV01 and Modified Duration. It is very intuitive. Also, the corresponding formulae of yield based DV01 and Modified Duration of zero-coupon bonds, par bonds and perpetual bonds are easy to understand. Just to get this straight:

Modified Duration = Macaulay Duration/(1 + y/2)

Am I right in stating the above definition for Modified Duration?

Also, isn't DV01 = [(1/10,000)*(1/1+y/2)]*sum of time-weighted PV?
i.e. DV01 = [Modified Duration*Sum of Time-weighted PV]/10,000?

Thanks!
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, the formula of Modified duration=Macaulay Duration/(1+y/k). In this case the k=number of compounding per year.
Thank you

 

[David Harper CFA FRM]

Agreed! @Jayanthi Sankaran you have a slight typo in your parenthesis, should be: DV01 = (1/10,000)*[1/(1+y/2)]*sum of time-weighted PV. It's true because:
DV01 = Price*Mod duration/10,000 = price*[Mac duration/(1 + y/k)]/10,000 = (1/10,000)*[1/(1+y/k)]*Price*Mac_Duration = (1/10,000)*[1/(1+y/k)]*Sum of time-weighted PV;
i.e., (Sum of time-weighted PV)/Price = Mac Duration, because:
Mac Duration = 0.5*(PV of cash flow at 0.5)/Bond_price + 1.0*(PV of cash flow at 1.0)/Bond_price + ... + Final_term*(PV of cash flow at Final_term)/Bond_price
Mac Duration = 1/Bond_price * [0.5*(PV of cash flow at 0.5)+ 1.0*(PV of cash flow at 1.0) + ... + Final_term*(PV of cash flow at Final_term)]
Mac Duration = 1/Bond_price * [0.5*(PV of cash flow at 0.5)+ 1.0*(PV of cash flow at 1.0) + ... + Final_term*(PV of cash flow at Final_term)] = 1/Bond_price *(Sum of time-weighted PV)