Bond Price Volatility
Outline
Price/Yield Relationship for Option Free
Bonds
Bond Price Theorems
Price Volatility of Option Free Bonds
Measures of Bond Price Theorems
– Bond Duration
– Bond Convexity
Price/Yield Relationship
A fundamental property of an option free
bond is that the price of the bond changes
in the opposite direction of the change in
the required yield for the bond
By how much bond price will change for a
given change in yield will depend on the
time to maturity, coupon, and interest rates
Bond Price Theorems
 Bond prices move inverse to the change in
interest rates
 If all other factors are held constant, a bond’s
interest rate risk rises with the length of time
remaining until it matures
• Bond price volatility and time to maturity are directly related
 A bond’s interest rate risk rises at a diminishing
rate as the time remaining until its maturity
increases
 The price change that results from an equal sized
increase/decrease in a bond’s YTM is
asymmetrical.
 A bond’s interest rate risk is inversely related to
the coupon
 High volatility
• Low coupon and
• High maturity
 Low volatility
• High coupon and
• Low maturity
 How do we measure bond’s volatility?
Bond Duration
Macaulay’s duration
Duration is defined as a weighted average
time to recovery of all interest payments
plus principal
Number of years needed to fully recover
the purchase price of a bond, given present
value of its cash flows
Examples
Modified Duration
An adjusted measure of Macaulay’s
duration is called modified duration.
Modified duration can be used to
approximate bond price volatility
Modified duration equals Macaulay’s
duration divided by one plus the current
YTM.
Examples
Approximating the percentage price
change using modified duration
Features of Bond Duration
 Duration of a bond with coupon payments will
always be less than maturity of the bond
 Inverse relationship between coupon and duration
 Positive relationship generally holds between
term to maturity and duration
• Duration increases at a decreasing rate with maturity
• The relationship between duration and maturity is not direct
• Shape of the duration/maturity curve depends on the coupon
and the yield to maturity
All else being the same, there is an inverse
relationship between YTM and duration
• More distant cash flows with smaller present value
will receive less weight, because they are being
discounted at a higher YTM
Sinking funds and call provisions can
accelerate the total cash flows for a bond,
and, therefore, significantly reduce the
bond duration
 Modified duration helps in approximating the
bond price change due to a small change in the
required yield
 Modified duration is a linear approximation of a
curvilinear relationship
 Graphical depiction of duration and price/yield
relationship
 What is yield changes are large?
• Does duration still provide a good approximation of bond
price change due to change in required yield?
Bond Convexity
 For large changes in bond yields, duration can be
supplemented with an additional measure to capture the
curvature or convexity of a bond
 Convexity is a measure of the curvature of the price/yield
curve
 Mathematically, convexity is the second derivative of
price with respect to yield divided by the price
 Convexity is a measure of how much a bond’s price-yield
curve deviates from the linear approximation of that curve
 For noncallable bonds, convexity is always positive
Measuring Convexity
Duration attempts to estimate a convex
relationship with a straight line
If we add convexity to duration, we get a
better approximation to the price of the
bond due to change in required yield