Transcript Document

Bond Duration
 Linear measure of the sensitivity of a bond's
price to fluctuations in interest rates.
 Measured in units of time; always less-thanequal to the bond’s maturity because the value of
more distant cash flows is more sensitive to the
interest rate.
 “Duration" generally means Macaulay duration.
1
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.
T
PV (CFt )t
D
Vb
1
 The time-weighted average present value term to
payment of the cash flows on a bond.
2
Macaulay Duration
 The proportional change in a bond’s price is
proportional to duration through the yield-tomaturity
 (1  r ) 
V
 D 

V
 (1  r ) 
3
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.
4
Dollar Duration
 Duration x Bond Price: the change in price in
dollars, not in percentage, and has units of DollarYears (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.
5
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.
6
Modified Duration
 Modified Duration – where n=cash flows per year.
Macaulay Duration
D* 
1 r
and
V
  D * r
V
7
Modified Duration
V
  D * r
V
V   D * r  V
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%?
11.37

0.001 $897.26  $9.36
1.09
8
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 [(1+r)/r].
9
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.
10
Duration
Convexity
Prices andversus
Coupon
Rates
Price
Yield
11