Chapter 24
Bond Price Volatility
Fabozzi: Investment Management
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Learning Objectives
• You will understand the factors that affect the price
volatility of a bond when yields change.
• You will be able to describe the price volatility
properties of an option-free bond.
• You will discover how to calculate the price value of
a basis point.
• You will learn how to calculate and explain what is
meant by Macaulay duration, modified duration, and
dollar duration.
Learning Objectives
• You will explore why duration is a measure of the
price sensitivity of a bond to yield changes.
• You will study the limitations of using duration as
a measure of price volatility.
• You will understand how price change estimated
by duration can be adjusted for the bond’s
convexity.
Introduction
Recall that the price of a bond is inversely related to the
required yield for the bond. Money managers need to
be able to quantify this relationship in order to predict
how bond prices can change. The two methods used to
measure a option-free bond’s price volatility are:
Duration
Convexity
Price volatility properties of
option-free bonds
1.For very small changes in the required yield, the percentage
price change for a given bond is about the same, whether the
required yield increases or decreases.
2.For large changes in the required yield, the percentage price
change is different for an increase in the required yield than for
a decrease.
3.For a large change in basis points, the percentage price
increase is greater than the percentage price decrease.
Price appreciation realized if required yield decreases > capital
loss if the yield rises by same amount of basis points
Factors that affect a bond’s price
volatility
 Coupon
 Term to maturity
 Trading yield level
The effect of the coupon rate and
maturity
Coupon rate effect
A low coupon rate increases the price volatility of a
bond.
Maturity effect
The longer the maturity, the greater the price volatility
of a bond.
Effects of yield to maturity on
price volatility
The higher the level of yields, the lower the price
volatility
Insert Figure 24-1
At the lower yield level, price changes are significant; at
higher yield level, these changes are much less.
Measures of price volatility
The two most popular measures of price
volatility are:
•Price value of a basis point
•Duration
Price value of a basis point
Measures the change in the price of the bond if the
required yield changes by one basis point
This is measured in terms of dollar value of each
basis point (01).
Insert Table 24-3
Duration
By taking the first derivative of a mathematical function,
we can use duration as a measure of bond price volatility.
If we take the first derivative of our bond price equation
in Chapter 23, we find the Macaulay duration:
Given:
P= price (in $)
n= number of periods (number of years x 2)
C= semiannual coupon payment (in $)
r= periodic interest rate (required annual yield  2)
M= maturity value
t= time period when the payment is to be receiv
Duration
With modified duration stated as
1(2)C 2(3)C
n(n  1)C N (n  1) M

 ...

1
2
n
(1  y ) (1  y )
(1  y )
(1  Y ) n
Convexity 
(1  y ) 2 P
Macaulay duration
(1  y)
And doing some substitution, we find, Approximate
percentage price change = - modified duration
The negative sign derives the inverse relationship between
bond prices and interest rates.
Macaulay duration and modified
duration: an example
Insert Table 24-4
Properties of duration
When computed, both types of duration are less than the
maturity. However, with a zero-coupon bond the Macaulay
duration is equal to maturity and the modified duration is
less.
Insert Table 24-5
The lower the coupon, the greater the modified duration.
The longer the maturity, the greater the price volatility.
At higher yields, modified duration decreases.
Approximating the percentage price
change
Approximate percentage price change = - modified duration x yield
change (decimal)
Example:
6%, 25 year bond selling at 70.357 to yield 9%
modified duration = 10.62
Yields increase to 9.1% (change of 10 basis points or +0.0010), the
approximate percentage change in price is:
-10.62 (+0.0010) = -0.0106 = -1.06%
Actual percentage price change from table 24-2 is +1.07%.
Note that with the small change in the required yield, modified
duration is a close figure.
Approximating the percentage
price change: a rule
Given: that the yield on any bond changes by 100 basis
points (0.01),
modified duration x (0.01) = modified duration %
We can say then that
Modified duration can be interpreted as the
approximate percentage change in price for a
100-basis-point change in yield.
Approximating the dollar price change
To measure the dollar price volatility of a bond we use the
following formula:
Approximate dollar price change = - modified duration x initial
price x yield change (decimal)
Dollar duration = modified duration x initial price
These equations work well for small changes in price, but when
the yield movement is large, dollar duration, like modified
duration, will not approximate the price reaction with any
accuracy.
Concerns with using duration
•Is only an approximation of price sensitivity
•Is not very useful for large changes in yield
•Assumes all cash flows are discounted at the same rate
•Misapplication of duration to bonds with embedded
options
Convexity
Insert Figure 24-2
The slope of the tangent line is related to dollar duration
and therefore the duration of the bond.
Steep tangent = longer duration
Flatter tangent = shorter duration
Duration decreases (increases) as yield increases
(decreases)
The price approximation will always be under the actual
price. Again, with small changes in yield, convexity
gives a good approximation; larger changes result in poor
approximations.
Adjusting duration for convexity
Both types of duration attempt to estimate a convex relationship
with the tangent line. An adjustment to the percentage change
estimated using duration is
Convexity adjustment = 0.5(convexity)(yield change in basis points)2
Using both convexity and duration provides a good approximation
of the actual price change for large movements
1(2)C 2(3)C
n(n  1)C N (n  1) M

 ...

1
2
n
(1  y ) (1  y )
(1  y )
(1  Y ) n
Convexity 
imation of the
2
(1  y ) P
actual price change for
Insert Table 24-6
Positive convexity
Positive convexity - As the required yield increases
(decreases), the convexity of the bond decreases
(increases).
Explains how if market yield rise, bond prices
fall. The decline is slowed by a decline the
duration as market yields rise.
Insert Figure 24-4
The value of convexity
Insert Figure 24-5
Given two bonds with the same duration and yield, there
can be two different convexities. In the above figure,
what is the effect of greater convexity on bond B? This
bond will have a higher price whether the market yield
rises or falls. For investors, there is an advantage in
owning B if they expect much volatility in market yields
and therefore, they will be willing to pay for the greater
convexity of B.