Chapter 24 Bond Price Volatility

1
Chapter 24Bond Price Volatility

• Fabozzi Investment Management Graphics by

2
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.

3
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 bonds
  convexity.

4
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 bonds price volatility are
• Duration
• Convexity

5
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 gt capital loss if the yield rises by
  same amount of basis points

6
Factors that affect a bonds price volatility

• Coupon
• Term to maturity
• Trading yield level

7
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.

8
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.

9
Measures of price volatility

• The two most popular measures of price volatility
  are
• Price value of a basis point
• Duration

10
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

11
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

12
Duration
With modified duration stated as
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.
13
Macaulay duration and modified duration an
example

• Insert Table 24-4

14
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.

15
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.

16
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.

17
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.
  

18
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

19
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.

20
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

Insert Table 24-6
21
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

22
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.