Chapter 4
Bond Price Volatility
Introduction

Bond volatility is a result of interest rate
volatility:


When interest rates go up bond prices go down and
vice versa.
Goals of the chapter:


To understand a bond’s price volatility
characteristics.
Quantify price volatility.
Price-Yield Relationship - Maturity

Consider two 9% coupon semiannual pay bonds:


Bond A: 5 years to maturity.
Bond B: 25 years to maturity.
Yield
5 Years
25 Years
6
1,127.95
1,385.95
7
1,083.17
1,234.56
8
1,040.55
1,107.41
9
1,000.00
1,000.00
10
961.39
908.72
11
924.62
830.68
12
889.60
763.57

The long-term bond price is more
sensitive to interest rate changes than
the short-term bond price.
Price-Yield Relationship - Coupon Rate

Consider three 25 year semiannual pay bonds:


9%, 6%, and 0% coupon bonds
Notice what happens as yields increase from 6% to 12%:
Yield
9%
6%
0%
9%
6%
0%
6%
1,127.95
1,000.00
228.11
0%
0%
0%
7%
1,083.17
882.72
179.05
-4% -12% -22%
8%
1,040.55
785.18
140.71
-8% -21% -38%
9%
1,000.00
703.57
110.71
-11% -30% -51%
10%
961.39
634.88
87.20
-15% -37% -62%
11%
924.62
576.71
68.77
-18% -42% -70%
12%
889.60
527.14
54.29
-21% -47% -76%
Bond Characteristics That Influence
Price Volatility

Maturity:



For a given coupon rate and yield, bonds with longer maturity
exhibit greater price volatility when interest rates change.
Why?
Coupon Rate:

For a given maturity and yield, bonds with lower coupon rates
exhibit greater price volatility when interest rates change.
Shape of the Price-Yield Curve
If we were to graph price-yield changes for bonds
we would get something like this:

Price

What do you notice about
this graph?


Yield
It isn’t linear…it is convex.
It looks like there is more
“upside” than “downside”
for a given change in yield.
Price Volatility Properties of Bonds

Properties of option-free bonds:



All bond prices move opposite direction of yields, but the
percentage price change is different for each bond,
depending on maturity and coupon
For very small changes in yield, the percentage price
change for a given bond is roughly the same whether
yields increase or decrease.
For large changes in yield, the percentage price increase
is greater than a price decrease, for a given yield change.
Price Volatility Properties of Bonds

Exhibit 4-3 from Fabozzi text, p. 61 (required yield is 9%):
Measures of Bond Price Volatility

Three measures are commonly used in practice:
1. Price value of a basis point
(also called dollar value of an 01)
2. Yield value of a price change
3. Duration
Price Value of a Basis Point


The change in the price of a bond if the required yield
changes by 1bp.
Recall that small changes in yield produce a similar
price change regardless of whether yields increase or
decrease.

Therefore, the Price Value of a Basis Point is the same for yield
increases and decreases.
Price Value of a Basis Point

We examine the price of six bonds assuming yields are 9%. We
then assume 1 bp increase in yields (to 9.01%)
Bond
Initial Price New Price
(at 9% yield) (at 9.01%)
Price Value
of a BP
5-year, 9% coupon
100.0000
99.9604
0.0396
25-year, 9% coupon
100.0000
99.9013
0.0987
5-year, 6% coupon
88.1309
88.0945
0.0364
25-year, 6% coupon
70.3570
70.2824
0.0746
5-year, 0% coupon
64.3928
64.362
0.0308
25-year, 0% coupon
11.0710
11.0445
0.0265
Yield Value of a Price Change

Procedure:




Calculate YTM.
Reduce the bond price by X dollars.
Calculate the new YTM.
The difference between the YTMnew and YTMold is the
yield value of an X dollar price change.
Duration
The concept of duration is based on the slope of the
price-yield relationship:
 What does slope of a
curve tell us?

Price




Yield
How much the y-axis changes
for a small change in the x-axis.
Slope = dP/dy
Duration—tells us how much
bond price changes for a given
change in yield.
Note: there are different types
of duration.
Two Types of Duration

Modified duration:


Dollar duration:


Tells us how much a bond’s price changes (in percent)
for a given change in yield.
Tells us how much a bond’s price changes (in dollars)
for a given change in yield.
We will start with modified duration.
Deriving Duration

The price of an option-free bond is:
P
C
C
C



2
3
(1  y ) (1  y )
(1  y )







C
M

(1  y ) n (1  y ) n
P = bond’s price
C = semiannual coupon payment
M = maturity value (Note: we will assume M = $100)
n = number of semiannual payments (#years  2).
y = one-half the required yield
How do we get dP/dy?
Duration, con’t

The first derivative of bond price (P) with respect to yield (y) is:
dP
1  C
2C




2
dy
1  y  (1  y) (1  y)



nC
nM 

(1  y)n (1  y) n 
This tells us the approximate dollar price change of the bond for a
small change in yield.
To determine the percentage price change in a bond for a given
change in yield we need:
dP 1
1 1  C
2C



 
2
dy P
1  y  P  (1  y ) (1  y )
nC
nM  



n
(1  y ) (1  y )n  
Macaulay Duration
Duration, con’t

Therefore we get:
Modified Duration 

Macaulay Duration
1 y
Modified duration gives us a bond’s approximate percentage
price change for a small (100bp) change in yield.

Duration is measured on a per period basis. For semi-annual cash flows, we
adjust the duration to an annual figure by dividing by 2:
Duration in years 
Duration over a single period
2
Calculating Duration

Recall that the price of a bond can be expressed as:
1

1  (1  y ) n
PC
y




M

n
 (1  y )
Taking the first derivative of P with respect to y and
multiplying by 1/P we get:
C
 2
y
Modified Duration  


1  n( M  C / y ) 
1  (1  y)n   (1  y)n1 




P
Example

Consider a 25-year 6% coupon bond selling at 70.357
(par value is $100) and priced to yield 9%.
C 
1  n( M  C / y ) 
1


 2

n 
n 1
y
(1

y
)
(1

y
)



Modified Duration  


P
 3

2
0.045
Modified Duration  


 50(100  3 / 0.045) 
1

1  (1.045)50  
51
(1.045)



70.357

Modified Duration  21.23508 (in number of semiannual periods)

To get modified duration in years we divide by 2:
Modified Duration  10.62
Duration


duration is less than (coupon bond) or equal to (zero
coupon bond) the term to maturity
all else equal,




the lower the coupon, the larger the duration
the longer the maturity, the larger the duration
the lower the yield, the larger the duration
the longer the duration, the greater the price volatility
Properties of Duration
Macaulay
Duration
Modified
Duration
5-year
4.13
3.96
9% 25-year
10.33
9.88
5-year
4.35
4.16
6% 25-year
11.10
10.62
5-year
5.00
4.78
0% 25-year
25.00
23.98
Bond
9%
6%
0%


Duration and Maturity:


Duration increases with maturity.
Duration and Coupon:

The lower the coupon the greater the
duration.
Earlier we showed that holding all else constant:


The longer the maturity the greater the bond’s price volatility (duration).
The lower the coupon the greater the bond’s price volatility (duration).
Properties of Duration, con’t

What is the relationship between duration and yield?
Yield Modified  The lower the yield the higher the
(%) Duration
duration.
7
11.21
 Therefore, the lower the yield the
8
10.53
higher the bond’s price volatility.
9
9.88
10
9.27
11
8.7
12
8.16
13
7.66
14
7.21
Approximating Dollar Price Changes

How do we measure dollar price changes for a given change in yield?
We use dollar duration: approximate price change for 100 bp change in yield.

Recall:

Solve for dP/dy:

dP 1
= -  modified duration 
dy P
dP
  (modified duration)P
dy
dollar _ duration

Solve for dP:
dP  (dollar duration)dy
Example of Dollar Duration

A 6% 25-year bond priced to yield 9% at 70.3570.


Dollar duration = 747.2009
What happens to bond price if yield increases by 100 bp?
dP  (dollar duration)dy
dP  (747.2009)  0.01
dP  $7.47

A 100 bp increase in yield reduces the bond’s price by
$7.47 dollars (per $100 in par value)

This is a symmetric measurement.
Example, con’t

Suppose yields increased by 300 bps:
dP  (dollar duration)dy
dP  (747.2009)  0.03
dP  $22.4161

A 300 bp increase in yield reduces the bond’s price by
$22.4161 dollars (per $100 in par value)


Again, this is symmetric.
How accurate is this approximation?

As with modified duration, the approximation is good for small yield
changes, but not good for large yield changes.
Portfolio Duration


The duration of a portfolio of bonds is the weighted average of
the durations of the bonds in the portfolio.
Example:
Bond Market Value Weight Duration


A
$10 million
0.10
4
B
$40 million
0.40
7
C
$30 million
0.30
6
D
$20 million
0.20
2
Portfolio duration is: 0.10  4  0.40  7  0.30  6  0.20  2  5.40
If all the yields affecting the four bonds change by 100 bps, the
value of the portfolio will change by about 5.4%.
Price-Yield Relationship
Accuracy of Duration

Why is duration more accurate for small changes in yield than for
large changes?
Because duration is a linear approximation of a curvilinear (or convex) relation:

Price
 Duration treats the price/yield
relationship as a linear.
 Error is small for small Dy.
 Error is large for large Dy.
 The error is larger for yield decreases.
 The error occurs because of convexity.
P3, Actual
Error
P3, Estimated
P0
P1
P2, Actual
Error
P2, Estimated
y3
y0 y1
y2
Yield
Convexity

Duration is a good approximation of the price
yield-relationship for small changes in y.

For large changes in y duration is a poor
approximation.

Why? Because the tangent line to the curve can’t capture
the appropriate price change when ∆y is large.
How Do We Measure Convexity?
1.
Recall a Taylor Series Expansion from Calculus:
dP
1 d 2P
2
dP 
dy 
(
dy
)
 error
2
dy
2 dy
2.
Divide both sides by P to get percentage price change:
dP dP 1
1 d 2P 1
error
2

dy 
(
dy
)

P dy P
2 dy 2 P
P

Note:



First term on RHS of (1) is the dollar price change for a given
change in yield based on dollar duration.
First term on RHS of (2) is the percentage price change for a given
change in yield based on modified duration.
The second term on RHS of (1) and (2) includes the second
derivative of the price-yield relationship (this measures convexity)
Measuring Convexity



The first derivative measures slope (duration).
The second derivatives measures the change in slope (convexity).
As with duration, there are two convexity measures:



Dollar convexity measure – Dollar price change of a bond due to convexity.
Convexity measure – Percentage price change of a bond due to convexity.
The dollar convexity measure of a bond is:
d 2P
dollar convexity measure  2
dy

The percentage convexity measure of a bond:
d 2P 1
convexity measure  2
dy P
Calculating Convexity


How do we actually get a convexity number?
Start with the simple bond price equation:
C
C
C
P



(1  y ) (1  y )2 (1  y )3

C
M


(1  y )n (1  y )n
Take the second derivative of P with respect to y:
d 2 P n t (t  1)C n(n  1)M


2
t 2
dy
(1  y)n 2
t 1 (1  y )

Or using the PV of an annuity equation, we get:
d 2 P 2C 
1 
2Cn
n(n  1)(M  C / y)

1



dy 2
y3  (1  y)n  y 2 (1  y)n1
(1  y)n2
Convexity Example

Consider a 25-year 6% coupon bond priced at 70.357
(per $100 of par value) to yield 9%. Find convexity.
d 2 P 2C 
1 
2Cn
n(n  1)(M  C / y)

1



dy 2
y3  (1  y)n  y 2 (1  y)n1
(1  y)n2

d 2P
2(3) 
1
2(3)(50)
50(51)(100  3/ 0.045)

1



dy 2 0.0453  (1.045)50  0.0452 (1.045)51
(1.045)52
d 2P
 51, 476.26
2
dy

Note: Convexity is measured in time units of the coupons.
To get convexity in years, divide by m2 (typically m = 2)
d 2 P 51, 476.26

 12,869.065
2
dy
4
Price Changes Using Both Duration
and Convexity

% price change due to duration:


% price change due to convexity:


= -(modified duration)(dy)
= ½(convexity measure)(dy)2
Therefore, the percentage price change due to
both duration and convexity is:
1
(modified duration)(dy)  (convexity measure)(dy ) 2
2
Example

A 25-year 6% bond is priced to yield 9%.


Modified duration = 10.62
Convexity measure = 182.92

Suppose the required yield increases by 200 bps (from 9% to
11%). What happens to the price of the bond?

Percentage price change due to duration and convexity
1
 (modified duration)(dy )  (convexity measure)(dy )2
2
1
 (10.62)(0.02)  (182.92)(0.02) 2
2
 21.24%  3.66%  17.58%
Important Question:
How Accurate is Our Measure?

If yields increase by 200 bps, how much will the bond’s
price actually change?
Measure of Percentage Percentage
Price Change
Price Change

Duration
-21.24
Duration & Convexity
-17.58
Actual
-18.03
Note: Duration & convexity provides a better
approximation than duration alone.

But duration & convexity together is still just an approximation.
Why Is It Still an Approximation?

Recall the “error” in the our Taylor Series expansion?
dP
1 d 2P
2
dP 
dy 
(
dy
)
 error
2
dy
2 dy

The “error” includes 3rd, 4th, and higher derivatives:


The more derivatives we include in our equation, the more accurate
our measure becomes.
Remember, duration is based on the first derivative and convexity is
based on the 2nd derivative.
Some Notes On Convexity

Convexity refers to the curvature of the price-yield
relationship.



The convexity measure is the quantification of this curvature
Duration is easy to interpret: it is the approximate %
change in bond price due to a change in yield.
But how do we interpret convexity?

It’s not straightforward like duration, since convexity is based on the
square of yield changes.
The Value of Convexity

Suppose we have two bonds with the same duration and the same
required yield:

Price





Notice bond B is more curved (i.e., convex) than bond A.
If yields rise, bond B will fall less than bond A.
If yields fall, bond B will rise more than bond A.
That is, if yields change from y0, bond B will always be
worth more than bond A!
Convexity has value!
Investors will pay for convexity (accept a lower yield) if
large interest rate changes are expected.
Bond B
Bond A
y0
Yield
Properties of Convexity


All option-free bonds have the following
properties with regard to convexity.
Property 1:


As bond yield increases, bond convexity decreases (and
vice versa). This is called positive convexity.
Property 2:

For a given yield and maturity, the lower the coupon the
greater a bond’s convexity.
Approximation Methods

We can approximate the duration and convexity for any
bond or more complex instrument using the following:
P  P
approximate duration =
2( P0 )(Dy)
P  P  2 P0
approximate convexity =
( P0 )(Dy)2

Where:




P– = price of bond after decreasing yield by a small number of bps.
P+ = price of bond after increasing yield by same small number of bps.
P0 = initial price of bond.
∆y = change in yield in decimal form.
Example of Approximation



Consider a 25-year 6% coupon bond priced at 70.357 to yield 9%.
Increase yield by 10 bps (from 9% to 9.1%): P+ = 69.6164
Decrease yield by 10 bps (from 9% to 8.9%): P- = 71.1105.
approximate duration =
approximate convexity =

69.6164+71.1105  2(70.357)
P  P  2 P0
=
 183.3
2
2
(70.357)(0.001)
( P0 )(Dy)
How accurate are these approximations?



71.1105  69.6164
P  P
=
 10.62
2(70.357)(0.001)
2( P0 )(Dy)
Actual duration = 10.62
Actual convexity = 182.92
These equations do a fine job approximating duration & convexity.