DURATION, IMMUNIZATION & CONVEXITY

  1. Duration
    1. Definitions
    1. Duration:

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    2. Modified Duration:

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    3. Effective Duration:

      4.  

       

       

       

       

       

       

       

       

    4. Dollar Duration:

 

 

 

 

 

 

 

    1. Calculations

 

    1. Duration:

      2. D = (1/Bo) x S [t x Ct/(1 + i)t]

       

      Where:

      Bo =

      t =

       

      Ct =

       

      i =

       

      Note that: S Ct/(1 + i)t is

       

       

       

       

      e.g.

      You are preparing a report for a 3 year bond with a 12% coupon rate and 9% YTM.

      Find the bond’s duration and interpret its meaning for your report.

       

      (1) (2) (3) (4)

      t CF PV CF (1)x(3)

      1________________________________________

      2________________________________________

      3________________________________________________

      Totals S S

       

       

      D = S (4)/S (3) =

       

    2. Modified Duration:

      3. D* = -D/(1 + io)

      Find the modified duration for the above bond and explain it’s meaning.

       

       

    3. Effective Duration:

      4.  

      D = (B- -B+)/(2xBoxD i)

       

       

      Where all variables are as previously defined and:

      B- =

      B+ =

      D i =

      Assume the above bond is callable. Find it’s effective (or option adjusted) duration assuming a 200 basis point change (ignore the impact of the change in pre-payment on the price of the bond).

       

       

       

       

       

       

    4. Dollar Duration:

$D = -Bo x D*

 

What is the dollar duration of the non-callable bond? What does it mean?

 

 

 

 

 

 

 

 

    1. Properties of Duration

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4.

 

 

 

 

  1. Immunization

*Immunization Defined:

 

 

 

 

 

*How does immunization work?

 

 

 

  1. If yields fall to 6% immediately after purchase and the bond is held until maturity, what is the average annual yield?

    2. (1 + i) = (Bn/Bo)1/n

  2. If yields fall to 6% immediately after purchase and the bond is held until duration, what is the average annual yield?

 

(1 + i) = (BD/Bo)1/D

Where BD is defined as:

 

 

 

 

 

 

 

 

 

  1. Convexity

There are three measures of bond price volatility which are helpful only when yield changes are small: (1) Price Value of a Basis Point

(2) Yield Value of a price change:

(3) Duration

    1. Definitions
    1. Convexity:

      2.  

       

       

      Price

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      |_____________________Yield
    2. Effective Convexity:

      3.  

       

       

       

       

    3. Dollar Convexity:

      4.  

       

       

       

    4. Negative Convexity:

 

 

 

 

 

 

 

Price

|

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|_____________________Yield

 

 

 

 

 

 

 

 
    1. Calculations
    1. Convexity:

C = [{S ((t2 + t) x CFt)(1 + i)-(t+2)}]/(Bo x m2)

Where all variables are as previously defined.

e.g.

You are preparing a report for a 3 year bond with a 12% coupon rate and 9% YTM.

Find the bond’s convexity and interpret its meaning for your report.

 

(1) (2) (3) (4) (5)

t CF (1+i)-(t+2) (t2 + t)CF (3) x (4)

1 12 0.7722 24 18.53

2 12 0.7084 72 51

3 112 0.6499 1,344 873.4656

Totals ______S 943__

 

C = S (5)/[Bo x m2] = 943/107.59 = 8.765

Your supervisor asked you to show how much of the bond price change will be due to convexity and how much due to duration if yields change by 200 basis points. Also, show the new bond price assuming a 200 basis point increase and a 200 basis point decrease.

@7%: B= 113.12 Price D = 107.59 - 113.12 = +5.53

@11%: B= 102.44 Price D = 107.59 - 103.44 = -5.15

Note:

  1. The change in the bond price due to convexity: (convexity on bonds with no embedded options is always positive).

    2. D B = 0.5 x Bo x C x (D i)2 = .5x107.59x8.765x(.02)2 = 0.1886

  2. The change in the bond price due to duration: (duration can be positive or negative)

 

D B = -D* x Bo x D I = -2.48x107.59x.02 = -5.336

-2.48x107.59x-.02 = +5.336

Therefore, for the 7% bond: 5.336 + .1886 = 5.53

And for the 11% bond: -5.336 + .1886 = -5.15

 

 

 

 

 

 

    1. Effective Convexity:

      2. C = [B- + B+ - 2xBo]/[Bo(D i)2]

       

       

       

       

       

       

    2. Dollar Convexity:

$C= (C* x Bo)/2

Where C*= C x (D i)2

 

 

    1. Properties
  1. As yields increase (decrease) the convexity decreases (increases). This is positive convexity.
  2. The lower (higher) the coupon rate the greater (smaller) the convexity.
  3. As yields decrease (increase), the market price of bonds with negative convexity move in the opposite (same) direction as bonds with positive convexity.
  4. Neither convexity nor duration alone are good approximations of bond price changes when yield changes are large. Both must be used together.
  5. Two bonds that have equal duration but different convexity, the bond with the greater convexity will be more valuable because no matter whether yields increase or decrease the bond will have a higher price.