Bond Duration

1
Bond Duration

• Linear measure of the sensitivity of a bond's
  price to fluctuations in interest rates.
• Measured in units of time always less-than-equal
  to the bonds maturity because the value of more
  distant cash flows is more sensitive to the
  interest rate.
• Duration" generally means Macaulay duration.

2
Macaulay Duration

• For small interest rate changes, duration is the
  approximate percentage change in the value of the
  bond for a 1 increase in market interest rates.
• The time-weighted average present value term to
  payment of the cash flows on a bond.

3
Macaulay Duration

• The proportional change in a bonds price is
  proportional to duration through the
  yield-to-maturity

4
Macaulay Duration

• A 10-year bond with a duration of 7 would fall
  approximately 7 in value if interests rates
  increased by 1.
• The higher the coupon rate of a bond, the shorter
  the duration.
• Duration is always less than or equal to the
  overall life (to maturity) of the bond.
• A zero coupon bond will have duration equal to
  the maturity.

5
Dollar Duration

• Duration x Bond Price the change in price in
  dollars, not in percentage, and has units of
  Dollar-Years (Dollars times Years).
• The dollar variation in a bond's price for small
  variations in the yield.
• For small interest rate changes, duration is the
  approximate percentage change in the value of the
  bond for a 1 increase in market interest rates.

6
Macaulay-Weil duration

• Uses zero-coupon bond prices as discount factors
• Uses a sloping yield curve, in contrast to the
  algebra based on a constant value of r - a flat
  yield.
• Macaulay duration is still widely used.
• In case of continuously compounded yield the
  Macaulay duration coincides with the opposite of
  the partial derivative of the price of the bond
  with respect to the yield.

7
Modified Duration

• Modified Duration where ncash flows per year.

and
8
Modified Duration
What will happen to the price of a 30 year 8
bond priced to yield 9 (i.e. 897.27) with D of
11.37 - if interest rates increase to 9.1?
9
Duration Characteristics

• Rule 1 the duration of a zero coupon bond is
  equal to its time-to-maturity.
• Rule 2 holding time-to-maturity and YTM
  constant, duration is higher when the coupon rate
  is lower.
• Rule 3 holding coupon constant, duration
  increases with time-to-maturity. Duration always
  increases with maturity for bonds selling at par
  or at a premium.
• Rule 4 cateris parabus, the duration of coupon
  bonds are higher when its YTM is lower.
• Rule 5 duration of a perpetuity is (1r)/r.

10
Bond Convexity

• Bond prices do not change linearly, rather the
  relationship between bond prices and interest
  rates is convex.
• Convexity is a measure of the curvature of the
  price change w.r.t. interest rate changes, or the
  second derivative of the price function w.r.t.
  relevant interest rates.
• Convexity is also a measure of the spread of
  future cash flows.
• Duration gives the discounted mean term
  convexity is used to calculate the discounted
  standard deviation of return.

11
Prices and Coupon Rates
Duration versus Convexity
Price
Yield