Bond Duration
 

 Bond Duration


Duration is interest rate sensitivity of the price of a bond. Technically, it is the weighted average of the maturity of the bond's cashflows.


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== Effective Duration ==

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 Pf - Pr

---------   = Effective Duration

2 * P * Y


Pf and Pr are the prices when the yields rise and fall, respectively. P is the original price and Y is the change in yield. 


For example a bond that trades at par and goes to 102 when rates fall 1% and 99 on rates rise 1%. The effective duration is:


(102 - 99) / (2*100*0.01) = 3 / 2


Why does that make sense? Break it down: 


Yield falls, the price rises from 100 to 102:


(102 - 100)    2

----------- = --- = 1

 100 * .01     1


When yield rises, the price drops from 100 to 99


(100 - 99)     1

----------- = --- = 1 [ignoring the sign here]

 100 * .01     1

 

The average change is then: 


  (102 - 100)   (100 - 99) 

  ----------- + ----------

   100 * .01     100 * .01

----------------------------

              2


This reduces to:



   (102-100) + (100-99)      102 - 99      3

   -------------------- = -------------- = - 

      2 * 100 * .01        2 * 100 * .01   2

 

Which is the value we we got from the effective duration formula above. Notice that by taking the average of the up and down price move with respect to the change in yield, we effectively recreated that formula.


Effective duration can be directly used to estiamte the price moves of a bond for a given yield change expressed in the raw percent: 


 (Change in bond's price) = (Effective Duration) * (Yield Change)


So for our bond above, we'd estimate: (3/2)% change in price for a 1% change in yield. Since the effective duration is computed directly off of the bond's prices, it captures the effects of embeded options.


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== Macaulay Duration ==

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Estimates when a bond's cashflows will arrive. For example, a zero-coupon bond gets all of its flows at maturity, so Macaulay Duration = Maturity. For a bond with coupons, Macaulay Duration is less than maturity since some of the flows arrive earlier. This measure of duration obviously does not cover bonds with embedded options, since their cashflows are not known with certainty.


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== Modified Duration ==

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Modified Duration is an improvement over Macaulay duration since it encorporates the current YTM into the equation. Since it's based on Macaulay, it doesn't work for bonds with options. Modified Duration and Effective Duration may be similar.


                          Macaulay Duration

Modified Duration =  ---------------------------

                     1 + "Periodic Market Yield"



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== Factors that affect Duration ==

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All other things being equal:


Higher Coupons - decrease duration. Think about Macaulay duration: the more payments upfront, the less relative weight is given to the final payment at maturity.


Longer Maturity - increases duration. Think back to Macaulay again, for the case of the zero-coupon: duration = maturity, so longer maturity means longer duration.


Higher Yield - decreases duration. Based on the formula for modified duration - the higher the denominator (which is based on yield) the lower the duration.



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== Price Value of a Basis Point (PVBP) ===

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Remember that duration is % change in price for a % change in yield. A basis point in this case is 1% of 1%. PVBP simply captures the dollar amount by which the price of the bond changes in response to a 1 basis point change in rates.


  (% change in price) = Duration * 0.0001

  

  ($ change in price) = (% change in price) * Price


For example, a bond with a value of $1,000 and duration of 9.42


  (% change in price) =  9.42 * .0001 = 0.000942

  

  ($ change in price) = 0.000942 * $1,000 = $0.942


So for each basis point change in rates, the bond gains or looses 94.2 cents.