Chapter 4
Bond Price Volatility
2-1
Interest Rate Risk
Why are changes in interest rates a risk for
bonds?
Changes in
yield of
bond
Changes in
bond price
(capital
gain/loss)
Uncertainty
in return
from bond
investment
• Interest rate risk: Sensitivity of a bond’s price to
changes in market interest rate (i.e., yield to
maturity).
– In our discussion, price volatility and interest rate risk are
synonymous.
2
Coupon bond vs. zero coupon bond
• Prices of 8% annual coupon bonds
YTM
T = 10 years
T = 20 years
8%
1000
1000
9%
934.96
907.99
% price change
-6.50%
-9.20%
• Prices of zero-coupon bonds
YTM
T = 10 years
T = 20 years
8%
456.39
208.29
9%
414.64
171.93
% price change
-9.14%
-17.69%
3
Review of the Price-Yield Relationship
for Option-Free Bonds
• As illustrated in Exhibit 4-1:
• An increase in the required yield ________ the present
value of its expected cash flows and therefore _________
the bond’s price.
• A decrease in the required yield ___________ the present
value of its expected cash flows and therefore _______ the
bond’s price.
• As shown in Exhibit 4-2…
4-4
Exhibit 4-2
Shape of Price-Yield Relationship for an
Option-Free Bond
Price
Maximum
Price
Yield
4-5
Price Volatility Properties
• Property 1: Although the prices of all option-free bonds move in the
opposite direction from the change in yield required, the percentage price
change is not the same for all bonds.
• Property 2: For very small changes in the yield required, the percentage
price change for a given bond is roughly the same, whether the yield
required increases or decreases.
• Property 3: For large changes in the required yield, the percentage price
change is not the same for an increase in the required yield as it is for a
decrease in the required yield.
• Property 4: For a given large change in basis points, the percentage price
increase is greater than the percentage price decrease
• Make sure you know and understand these four properties
4-6
Exhibit 4-1 Price–Yield Relationship for Six Hypothetical Bonds
Required
Yield (%)
6.00
7.00
8.00
8.50
8.90
8.99
9.00
9.01
9.10
9.50
10.00
11.00
12.00
Price at Required Yield (coupon/maturity in years)
9% / 5 9% / 25 6% / 5 6% / 25 0% / 5
0% / 25
112.795 138.594 100.0000 100.000 74.4094 22.8107
108.316 123.455 95.8417 88.2722 70.8919 17.9053
104.055 110.741 91.8891 78.5178 67.5564 14.0713
102.002 105.148 89.9864 74.2587 65.9537 12.4795
100.396 100.996 88.4983 71.1105 64.7017 11.3391
100.039 100.098 88.1676 70.4318 64.4236 11.0975
100.000 100.000 88.1309 70.3570 64.3928 11.0710
99.9604 99.9013 88.0943 70.2824 64.3620 11.0445
99.6053 99.0199 87.7654 69.6164 64.0855 10.8093
98.0459 95.2539 86.3214 66.7773 62.8723
9.8242
96.1391 90.8720 84.5565 63.4881 61.3913
8.7204
92.4624 83.0685 81.1559 57.6712 58.5431
6.8767
88.9599 76.3572 77.9197 52.7144 55.8395
5.4288
4-7
EXHIBIT 4-3 Instantaneous Percentage Price Change for Six Hypothetical Bonds
Six hypothetical bonds, priced initially to yield 9%:
9% coupon, 5 years to maturity, price = 100.0000 6% coupon, 25 years to maturity, price = 70.3570
9% coupon, 25 years to maturity, price = 100.000 0% coupon, 5 years to maturity, price = 64.3928
6% coupon, 5 years to maturity, price = 88.1309 0% coupon, 25 years to maturity, price = 11.0710
Yield (%)
Change to:
Change in
Basis Points
6.00
7.00
8.00
8.50
8.90
8.99
9.01
9.10
9.50
10.00
11.00
12.00
-300
-200
-100
-50
-10
-1
1
10
50
100
200
300
Percentage Price Change (coupon/maturity in years)
9% / 5
12.80
8.32
4.06
2.00
0.40
0.04
-0.04
-0.39
-1.95
-3.86
-7.54
-11.04
9% / 25 6% / 5 6% / 25
38.59 13.47
42.13
23.46
8.75
25.46
10.74
4.26
11.60
5.15
2.11
5.55
1.00
0.42
1.07
0.10
0.04
0.11
-0.10 -0.04
-0.11
-0.98
-0.41
-1.05
-4.75
-2.05
-5.09
-9.13
-4.06
-9.76
-16.93 -7.91 -18.03
-23.64 -11.59 -25.08
0% / 5 0% / 25
15.56 106.04
10.09
61.73
4.91
27.10
2.42
12.72
0.48
2.42
0.05
0.24
-0.05
-0.24
-0.48
-2.36
-2.36 -11.26
-4.66 -21.23
-9.08 -37.89
-13.28 -50.96
4-8
Property 1
Property 1: Although the prices of all option-free bonds move in the opposite
direction from the change in yield required, the percentage price change is not
the same for all bonds.
4-9
Property 2
Property 2 (price-yield relationship): For very small changes in the yield
required, the percentage price change for a given bond is roughly the same,
whether the yield required increases or decreases.
4-10
Property 3
Property 3: For large changes in the required yield, the percentage price
change is not the same for an increase in the required yield as it is for a
decrease in the required yield.
4-11
Properties of Price-Yield Relationship
12
Property 4
Property 4: For a given large change in basis points, the percentage price
increase is greater than the percentage price decrease
4-13
Price Volatility Characteristics
of Option-Free Bonds
• Characteristics of a Bond that Affect its Price
Volatility
• There are two characteristics of an option-free bond that determine
its price volatility: coupon and term to maturity.
• For a given term to maturity and initial yield, the greater the price
volatility of a bond, the lower the coupon rate.
• For a given coupon rate and initial yield, the longer the term to
maturity, the greater the price volatility.
Make sure you know and understand these two characteristics!
4-14
Characteristic 1
Characteristic 1: For a given term to maturity and initial yield, the lower the
coupon rate, the greater the price volatility
4-15
Characteristic 2
Characteristic 2: For a given coupon rate and initial yield, the longer the
term to maturity, the greater the price volatility.
4-16
Price Volatility Characteristics of
Option-Free Bonds
• Effects of Yield to Maturity
• In the real world, two bonds that have the same coupon
and maturity may trade at different yields.
– Due to all of those different types of risk!
• Holding other factors constant, the higher the yield to
maturity at which a bond trades, the lower the price
volatility.
4-17
EXHIBIT 4-4
Price Change for a 100-Basis-Point Change in
Yield for a 9% 25-Year Bond Trading at Different
Yield Levels
Yield Level (%)
Initial Price
New Price a
Price Decline
7
$123.46
$110.74
$12.72
Percent
Decline
10.30
8
110.74
100.00
10.74
9.70
9
100.00
90.87
9.13
9.13
10
90.87
83.07
7.80
8.58
11
83.07
76.36
6.71
8.08
12
76.36
70.55
5.81
7.61
13
70.55
65.50
5.05
7.16
4.42
6.75
14
65.50
61.08
a As a result of a 100-basis-point increase in yield.
4-18
Measures of Bond Price Volatility
• Money managers, arbitrageurs, and traders need
to have a way to measure a bond’s price volatility
to implement hedging and trading strategies.
• Three measures that are commonly employed:
1) price value of a basis point
2) yield value of a price change
3) duration
4-19
Measures of Bond Price Volatility
• Price Value of a Basis Point
• The price value of a basis point, also referred to as the
dollar value of a 01, is the change in the price of the bond if
the required yield changes by 1 basis point.
• This measure of price volatility indicates DOLLAR price
volatility as opposed to percentage price volatility
• Typically expressed as the absolute value of the change in
price.
4-20
Price Value of a Basis Point
Bond
Price Value of a
Basis Point
Initial Price (9% yield)
Price at 9.01%
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
5-year zero-coupon
25-year zero-coupon
70.3570
64.3928
11.0710
70.2824
64.3620
11.0445
0.0746
0.0308
0.0265
4-21
Measures of Bond Price Volatility
• Yield Value of a Price Change
• If we have a specific price change in mind, we can
calculate the corresponding yield change.
Example: There is a bond that pays a 12% coupon, has a
YTM of 12%, and maturity of 25 years. (Because coupon =
YTM, price = par). How much would yields have to change to
make the bond worth 104.376? Assume a par value of 100.
4-22
Measuring Interest Rate Risk
• An acceptable measure of interest rate risk must
account for:
– Time to maturity.
– Cash flows which are paid out during the life of
the bond.
• Such a measure is Macaulay’s duration,
23
Measures of Bond Price Volatility
• Duration
• The Macaulay duration is one measure of the approximate
change in price for a small change in yield:
1C
Macaulay duration 
1 +
1  y 
2C
1  y 
2 + . . .+
nC
1  y 
n +
nM
1  y 
n
P
where P = price of the bond
C = semiannual coupon interest (in dollars)
y = one-half the yield to maturity or required yield
n = number of semiannual periods (number of years
times 2)
M = maturity value (in dollars)
4-24
Measures of Bond Price Volatility
• Duration
• Investors refer to the ratio of Macaulay duration to 1 + y as the
modified duration. The equation is:
M acaulay duration
modi fi ed dur ati on =
1+ y
where y = one-half the yield to maturity or required yield.
• The modified duration is related to the approximate percentage
change in price for a given change in yield as given by:
dP 1
 modified duration
dy P
where dP = change in price, dy = change in yield, P = price of the bond.
4-25
Duration (graphically)
• Green line: Price-yield relationship
• Red line: Tangent at initial yield (Duration)
• Approximates a convex curve by a straight line
Price
Actual Price
Slope of tangent=dP/dy
p*
Tangent Line at y*
y*
Yield
26
Measures of Bond Price Volatility
• Duration
• In general, if the cash flows occur m times per year, the durations are
adjusted by dividing by m, that is,
du ration in m periods per year
duration in years 
m
4-27
Duration Properties
• The Macaulay duration of a zero-coupon bond is
equal to its maturity
• The Macaulay duration of coupon bonds is less than
its maturity
28
Duration Properties
• Holding all other factors constant, a bond’s modified
duration is higher when the coupon rate is lower.
• Holding all other factors constant, a bond’s modified
duration generally increases with time to maturity.
• Holding other factors constant, the modified duration
of a coupon bond is higher when the bond’s yield to
maturity is lower.
29
EXHIBIT 4-5
Calculation of Macaulay Duration and Modified
Duration for 5-Year 9% Bond Selling to Yield 9%
Coupon rate: 9.00%
Term (years): 5
Initial yield: 9.00%
PV of CF
t × PVCF
0.956937
4.306220
4.30622
4.50
0.915729
4.120785
8.24156
3
4.50
0.876296
3.943335
11.83000
4
5
6
7
8
9
10
4.50
4.50
4.50
4.50
4.50
4.50
$104.50
0.838561
0.802451
0.767895
0.734828
0.703185
0.672904
0.643927
3.773526
3.611030
3.455531
3.306728
3.164333
3.028070
67.290443
100.00000
15.09410
18.05514
20.73318
23.14709
25.31466
27.25262
672.90442
826.87899
Period, t
Cash Flow
PV of $1 at 4.5%
1
$ 4.50
2
4-30
Duration Calculation
31
EXHIBIT 4-6
Calculation of Macaulay Duration and Modified
Duration for 5-Year 6% Bond Selling to Yield 9%
Coupon rate: 6.00% Term (years): 5 Initial yield: 9.00%
Period, t
1
2
3
4
5
6
7
8
9
10
Total
Cash Flow
$
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
103.00
PV of $1 at 4.5%
PV of CF
0.956937
0.915729
0.876296
0.838561
0.802451
0.767895
0.734828
0.703185
0.672904
0.643927
2.870813
2.747190
2.628890
2.515684
2.407353
2.303687
2.204485
2.109555
2.018713
66.324551
88.130923
t × PVCF
2.87081
5.49437
7.88666
10.06273
12.03676
13.82212
15.43139
16.87644
18.16841
663.24551
765.89520
4-32
Duration Calculation (4-6)
𝐶
𝑛(M − 𝑦
𝐶
1
1
−
+
1+𝑦 𝑛
𝑦2
1 + 𝑦 𝑛+1
𝑚𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 =
𝑃
Substitute:
33
Intuition
• Modified Duration can be interpreted as the approximate percent
change in price for a 100-basis-point (1%) change in yield
• Bond 2 Modified duration=4.16 years
• This means if market yields goes up by 1% the bond price will drop
by 4.16% (approximately)
• Direct measure of the extent of exposure to interest rate volatility
• Widely used by practitioners!
• As Macaulay’s duration _____, interest rate risk _____ , vice versa.
• As modified duration _____, interest rate risk _____ , vice versa.
4-34
Measures of Bond Price Volatility
• Approximating the Percentage Price Change
• For a given change in required yield, we can approximate
percentage price change by using the formula:
dP
 ( m o d i f i ed d u r a t i o n)  d y 
P
• where dP = change in price, P = price of the bond and dy = change
in yield.
• Suppose that the yield on any bond changes by 100 basis points.
Then, substituting 100 basis points (0.01) for dy into the above
equation, we get:
dP
 ( m o d ified d u r a tio n )  0 .0 1  (m o d ified d u r a tio n ) 1 % 
P
• Thus, modified duration can be interpreted as the approximate
percentage change in price for a 100-basis-point change in yield.
4-35
Measures of Bond Price Volatility
• Approximating the Dollar Price Change
• Modified duration is a proxy for the percentage change in
price. Investors also like to know the dollar price
volatility of a bond.
• For small changes in the required yield, the below
equation does a good job in estimating the change in price
yield:
dP = (dollar duration)(dy)
where dP = change in price and dy = change in yield.
4-36
Problem 1
• A nine-year bond has a yield of 10% and makes semiannual coupon payments. The bond has a Macaulay
duration of 14.388 half-years, i.e. based on semi-annual
compounding. If the bond’s yield increases by 50 basis
points, what is the % change in the bond’s price.
37
Problem 1 Solution
• Compute annual modified duration
• Percentage price change =
38
Problem 2
• A 30-year maturity bond making semi-annual coupon
payments with a coupon a rate of 12% has a
Macaulay duration of 23.08 half-years. The bond has
a par value of $1,000 and currently sells at a yield to
maturity of 8%. If the yield falls to 7%, compute:
1. The percentage price change.
2. The predicted dollar price change.
3. The predicted price (based on duration).
4. The actual price
39
Problem 2 Solution
40
Problem 2 Solution
• Dollar price change
• Predicted price of bond (using duration)
• Actual price at new yield
41
Problem 3
• A six-year 6.1% semi-annual coupon bond has a yield to
maturity of 10% and a Macaulay duration of 10.014 (in
half-years).
1. What is the modified duration in years?
2. If the yield increases by 25 basis points, what is the percentage
price change using the modified duration?
42
Problem 3 Solution
43
Price Value of a Basis Point
• PVBP: If yields change by 1 basis point, how
much will the prices change by?
• For example, consider a 6% 25-year bond trading at 70.357
with a modified duration of 10.62. A 1 bp decrease in yields
would approximately lead to an increase of price by $0.0747.
44
Duration (graphically)
• Green line: Price-yield relationship
• Red line: Tangent at initial yield (Duration)
Price
Actual Price
p*
Tangent Line at y*
(estimated price)
y*
Yield
45
Price Approximation Using Duration
Actual Price
Price
Error in Estimating Price
Based only on Duration
Error in Estimating Price
Based only on Duration
p*
Tangent Line at y*
(estimated price)
y1
y2 y* y3
y4
Yield
46
Price Approximation Using Duration
• When there is a small change in yield: from y* to y2 or y*
to y3:
– The actual price change (as shown in green line) and the price
change approximated by duration (tangent line) are virtually
identical
• When there is a large change in yield: from y* to y4 or y*
to y1:
– The actual price (green line) is significantly different from the
price change approximated by duration.
– For a yield decrease, price gain is understated by duration.
– For a yield increase, price loss is overstated by duration
47
Measures of Bond Price Volatility
Portfolio Duration
• The duration of a portfolio is simply the weighted average
duration of the bonds in the portfolios.
Bond Market Value Portfolio Weight Duration
A
$10 million
0.1
4
B
$40 million
0.4
7
C
$30 million
0.3
6
D
$20 million
0.2
2
$100 million
4-48
Measures of Bond Price Volatility
• Analytical Versus Empirical Duration
• Credit risk and interest rate risk are the two major risks
affecting a bond
• For corporate bonds that have a high credit rating (i.e.,
investment-grade bonds), the _________________ risk is the
dominant risk.
• Hence, the duration measure as calculated by formula may not
be a good measure of interest-rate risk for such bonds.
4-49
Convexity
• When using modified duration to approximate the percentage price
change, we are assuming that there is a linear relationship
between yield change and percentage price change.
– But this works pretty well for very small changes in yields
– Price yield curve is convex, so it works less well or larger changes
• Supplement duration with convexity of a bond to capture the
curvature. (second derivative of the pricing equation)
𝑑2𝑃
𝑑𝑦
2 =
2𝐶
1
2𝐶𝑛
1
−
−
+
𝑦3
1+𝑦 𝑛
𝑦 2 1 + 𝑦 𝑛+1
𝐶
𝑛(𝑛 + 1)(𝑀 − 𝑦
1 + 𝑦 𝑛+2
4-50
Convexity
• Measuring Convexity
• The convexity measure is in terms of periods squared.
• In general, if the cash flows occur m times per year,
convexity is adjusted to an annual figure as follows:
convexity measure in year 
convexity measure in m period per year
m
2
4-51
Exhibit 4-13 Price Approximation Using Duration
Actual Price
Price
Error in Estimating Price
Based only on Duration
Error in Estimating Price
Based only on Duration
p*
Tangent Line at y*
(estimated price)
y1
y2 y* y3
y4
Yield
4-52
Convexity Problem
• Calculate convexity of a five-year 6% bond with semi-annual payments
selling to yield 9% (same bond as slide 32)
2
𝑑 𝑃
2𝐶
1
2𝐶𝑛
= 3 1−
− 2
+
𝑑𝑦 2
𝑦
1+𝑦 𝑛
𝑦 1 + 𝑦 𝑛+1
𝐶
𝑛(𝑛 + 1)(𝑀 − 𝑦
1 + 𝑦 𝑛+2
53
Convexity
• Measuring Convexity
• The dollar convexity measure of the bond:
2
d P
dollar convexity measure 
2
dy
• The approximate change in price due to convexity is:
dP   dollar convexity measure  dy 
2
• The percentage change in the price of the bond due to convexity:
2
d P 1
convexity measure 
2
dy P
» The percentage price change due to convexity is:
dP 1
2
  convexity measure  dy 
P 2
4-54
Using modified duration & convexity to
estimate percentage price change
• Percentage change in price due to convexity
• From previous discussion: percentage change in price
due to duration
• Percentage change in price due to duration &
convexity
𝑑𝑃
1
= −𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 ∗ 𝑑𝑦 + 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑑𝑦 2
𝑃
2
dy: Change in yield
55
Example
• Consider the 5-year, 6% bond selling to yield 9%.
• We have shown modified duration to be 4.16 and
convexity to be 20.84 (both annualized). Find the percent
change in price for a 3% rise in yield?
56
Duration & Convexity 1
• A six-year 6.1% semi-annual coupon bond has a yield to
maturity of 10%, a Macaulay duration of 10.014 (half years)
and a convexity of 110.88 in half-years.
1. If the yield increases by 200 basis points, what is the
percentage price change using the modified duration and
convexity?
2. The dollar price change based on duration and convexity
3. The predicted price based on duration and convexity
57
Duration & Convexity (1) Solution
58
Duration & Convexity (1) Solution
59
Exhibit 4-16
Comparison of Convexity of Two Bonds
Price
Bond B Has Greater
Convexity Than Bond A
Bond A
Bond B
Bond B
Bond A
Yield
4-60
Example: Value of Convexity
• Investors of a bond fund require the manager to keep duration at 4. The
manager foresees significant interest rate volatility, i.e. upward or
downward moves in interest rates. She has to pick between two bonds;
A with a convexity of 100 and B with a convexity of 50. Which one
should she pick?
% Price Change
Int. rates go up 3%
Int. rates go down 3%
Bond A
-4*0.03+0.5*100*(0.03)^2
=-0.075 (-7.5%)
-4*-0.03+0.5*100*(-0.03)^2
=0.165(16.5%)
Bond B
-4*0.03+0.5*50*(0.03)^2
=-0.0975 (-9.75%)
-4*-0.03+0.5*50*(-0.03)^2
=0.1425 (14.25%)
Pick Bond A:
lower
downside,
higher upside
61
Concerns with Duration and Convexity
• Term structure is flat: All cash flows are discounted at the
same discount rate. This assumption is questionable. See
chapter 5!
• Shifts in yield curve are parallel: When yields change, the
basis point change in yield is same for all maturities.
• Bonds have no embedded options: Effective duration
62
Bonds with Embedded Options (1)
• Bonds with call options
a1
Option-free bond
Price
b
b1
Area of negative
convexity
a
y*
Yield
63
Bonds with Embedded Options (2)
• Bonds with put options
–
64
Conceptual Problem (1)
• Rank the following bonds in order of descending
duration.
Bond Coupon Time to Maturity Yield to Maturity
A
15%
20 years
10%
B
15
15
10
C
0
20
10
D
8
20
10
E
15
15
15
65
Conceptual Problem (2)
Which set of conditions will result in a bond with the
greatest price volatility?
A. high coupon and a short maturity
B. high coupon and a long maturity
C. low coupon and a short maturity
D. low coupon and a long maturity
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Conceptual Problem (3)
An investor who expects declining interest rates would
be likely to purchase a bond that has a ________
coupon and a ________ term to maturity.
A. Low, long
B. High, short
C. High, long
D. Zero, long
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Conceptual Problem (4)
With a zero-coupon bond:
A. Macaulay duration equals the weighted average term
to maturity.
B. Term to maturity equals Macaulay duration.
C. Weighted average term to maturity equals the term
to maturity.
D. All of the above.
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