FINM 3421
Class 4: Duration and Convexity
CRICOS code 00025B
Price-Yield Curve
• We want to know the effect of a change in the yield on the price of a bond.
• Change in bond price from a change in yield is called price volatility.
• Change in yield is treated as a driving force of bond price changes.
price
At low yields, prices rise at an increasing rate as yields fall
At higher yields, prices fall at a decreasing
rate as yields rise
yield
FINM3421
CRICOS code 00025B
2
Bond Price Volatility
• An increase in the required yield decreases the present value of the bond’s
expected cash flows and therefore decreases the bond’s price.
• A decrease in the required yield increases the present value of the bond’s
expected cash flows and therefore increases the bond’s price.
FINM3421
CRICOS code 00025B
3
Bond Price Volatility
• Why do we need to understand bond price volatility?
o Know how yield changes affect your portfolio
o Know how to take advantage of expected changes in interest rates
o Understand the risks you face
o Know how to hedge those risks
FINM3421
CRICOS code 00025B
4
Price Volatility Characteristics of Option-Free Bonds
1. Time-to-Maturity
• High time-to-maturity  Higher sensitivity
• Low time-to-maturity  Lower sensitivity
2. Coupon Rate
• High coupon rate  Lower sensitivity
• Low coupon rate  Higher sensitivity
3. Initial Yield-to-Maturity
• High initial YTM  Lower sensitivity
• Low initial YTM  Higher sensitivity
FINM3421
CRICOS code 00025B
5
Price Volatility Characteristics of Option-Free Bonds
• What is the common link between all three bond characteristic –
price volatility relations?
• The higher the proportion of the bond’s present value that comes
from cash flows in distant periods – the greater the bond’s price
sensitivity to changes in YTM.
FINM3421
CRICOS code 00025B
6
Measures of Bond Price Volatility
• Money managers, arbitrageurs, and traders need to measure a bond’s price
volatility to implement trading strategies
• Need a way to compare bonds that differ on multiple dimensions
• E.g., Which bond is more sensitive to changes in interest rates: a 20-year
bond with 8% coupon or a 15-year bond with 4% coupon?
• Three measures are commonly used:
1. Price value of a basis point
2. Yield value of a price change
3. Duration
FINM3421
CRICOS code 00025B
7
Measures of Bond Price Volatility: Price Value of a Basis Point
• The price value of a basis point is the (dollar) change in the price of the
bond if the required yield changes by 1 basis point.
• Typically, the price value of a basis point is expressed as the absolute
value of the change in price.
• Price volatility is approximately the same for an increase or a decrease
of 1 basis point in required yield.
• Because this measure of price volatility is in terms of dollar price
change, dividing the price value of a basis point by the initial price
gives the percentage price change for a 1-basis-point change in yield.
FINM3421
CRICOS code 00025B
8
Measures of Bond Price Volatility: Price Value of a Basis Point
• Suppose you own a semi-annual bond with $100 par value, a coupon rate of
8%, 12 years to maturity, and the current YTM is 6%. What is the price value
of a basis point?
1. Current Price: 𝑃0 = 4
1−1/(1.03)24
0.03
100
+
(1.03)24
= 116.94
2. Price if YTM increases one basis point:
1 − 1/(1.03005)24
100
𝑃0 = 4
+
= 116.84
24
0.03005
(1.03005)
Price value of a basis point: $0.10
% change of one basis point: 0.10/116.94 = 0.09%
FINM3421
CRICOS code 00025B
9
Measures of Bond Price Volatility: Yield Value of a Price Change
• Another measure of the price volatility of a bond is the change in the yield for
a specified price change.
• First, calculate the bond’s yield to maturity if the bond’s price is decreased by,
say, X dollars
• Second, compare the difference between the initial yield and the new yield.
This is the yield value of an X dollar price change.
• The smaller this value, the greater the dollar price volatility, because it would
take a smaller change in yield to produce a price change of X dollars.
FINM3421
CRICOS code 00025B
10
Measures of Bond Price Volatility: Yield Value of a Price Change
• Suppose you own a semi-annual bond with $100 par value, a coupon rate of
4%, 6 years to maturity, and a current YTM of 4.5%. What is the yield value
of a price change of $1?
1. Current Price: Price = 97.40, if YTM=4.5%
2. Implied YTM if price were $98.40: 4.3%
Then, the yield value of a price change of $1 is: 0.20% (|4.5%-4.3%|)
FINM3421
CRICOS code 00025B
11
Price Yield Relation: Approximation
Price
Actual Price
𝑃0
𝑃1
Tangent Line at 𝑦0
(estimated price)
𝑦0 + ∆𝑦
𝑦0
FINM3421
Yield
CRICOS code 00025B
12
Price Yield Relation: Approximation
The basic idea of duration and convexity is to approximate the nonlinear
price-yield relationship by a simple Taylor expansion:
1
𝑃1 = 𝑃0 +𝑓 ′ (𝑦0 )∆𝑦+ 𝑓 ′′ (𝑦0 )(∆𝑦)2 +⋯
2
where
𝑓′
∙ =
𝑑𝑃
𝑑𝑦
is the first derivative and
𝑓 ′′
∙ =
𝑑2 𝑃
𝑑𝑦 2
is the second
derivative.
First-order approximation for percentage changes in prices:
More accurate approximation:
FINM3421
∆𝑃
𝑃
=
𝑑𝑃
𝑑𝑦
𝑃
∆𝑦 +
𝑑2 𝑃
𝑑𝑦2
1
2 𝑃
∆𝑃
𝑃
=
𝑑𝑃
𝑑𝑦
𝑃
∆𝑦
(∆𝑦)2
CRICOS code 00025B
13
Measures of Bond Price Volatility: Duration
Recall that the price of a fixed-rate coupon bond is:
𝐶
𝐶
𝐶
𝑃=
+
+ ⋯+
2
1+𝑦
1+𝑦
1+𝑦
Then,
d𝑃
𝑑𝑦
P
=
1
−
1+y
𝑀
+
𝑛
1+𝑦
𝑛
𝐶
2𝐶
𝑛𝐶
𝑛𝑀
+
+⋯+
+
1+𝑦 𝑛 1+𝑦 𝑛
1+𝑦 1 1+𝑦 2
P
Duration (Macaulay Duration) is defined as:
Macaulay Duration =
FINM3421
𝐶
2𝐶
𝑛𝐶
𝑛𝑀
+
+⋯+
+
𝑛
1+𝑦
1+𝑦 𝑛
1+𝑦 1 1+𝑦 2
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)
CRICOS code 00025B
14
Measures of Bond Price Volatility: Duration
• We could rewrite the formula as:
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝐷 =
𝑛
𝑡=1 𝑡
×(
𝐶𝐹𝑡
1+𝑦 𝑡
𝐶𝐹𝑡
𝑛
𝑡=1 1+𝑦 𝑡
)
This suggests that for a regular coupon bond, the Macaulay duration
represents a measure of the weighted average life of the bond.
FINM3421
CRICOS code 00025B
15
Duration: What Does it Mean?
• Recall that the first-order approximation for percentage changes in prices is:
∆𝑃
𝑃
=
𝑑𝑃
𝑑𝑦
𝑃
∆𝑦 = −
𝐷
∆𝑦
1+𝑦
• Higher duration means greater price sensitivity to changes in interest rates.
• Duration simultaneously takes into account all of the different bond features
(coupon, time-to-maturity, and initial YTM), to provide a single measure of
price sensitivity.
FINM3421
CRICOS code 00025B
16
Duration: Modified Duration
• Investors refer to the ratio of Macaulay duration to 1 + y as the modified duration.
The equation is:
𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛
𝐷∗
=
𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛
𝑌𝑇𝑀
1+
𝑚
where m = number of coupon payments per year. Then, the first-order approximation
for percentage changes in prices is:
∆𝑷
𝑫
=−
∆𝒚 = −𝑫∗ ∆𝒚
𝑷
𝟏+𝒚
FINM3421
CRICOS code 00025B
17
Duration: Modified Duration
• Consider the earlier duration example. The modified duration for this bond is:
• Modified duration:
𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛
1.86
=
𝑌𝑇𝑀 = 1.06 = 1.75
1+ 𝑚
• Suppose the required YTM increased from 12% to 13%.
% Δ Price = - 1.75 × 0.01 = -0.0175
New Price = 96.535 × (1-0.0175) = 94.846, which is close to the new price
using a financial calculator: 94.861
FINM3421
CRICOS code 00025B
18
Properties of Duration
• The modified duration and Macaulay duration of a coupon bond are less than
the maturity.
• Ceteris paribus:
1. Lower coupon rates implies a greater duration.
2. Longer maturity implies a greater duration.
3. Higher YTM implies a lower duration.
FINM3421
CRICOS code 00025B
19
Dollar Duration
• Modified duration is a proxy for the percentage change in price.
• Can also find Dollar Duration: the dollar change in a bond’s price for a
specified change in YTM
• For example, the dollar duration for a 1% change in YTM is: (0.01)* Modified
Duration * Price
• E.g., for the bond used in the previous example the dollar duration for a 1%
change in YTM is:
𝐷𝑜𝑙𝑙𝑎𝑟 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 = 0.01 × 1.75 × 96.535 = 1.69
FINM3421
CRICOS code 00025B
20
Spread Duration
• Market participants also compute a measure called spread duration
• The YTM on non-treasury bonds can change for two reasons:
1. Treasury bond YTM change
2. The credit spread between the bond and the treasury bond changes
• Spread duration measures a bond’s sensitivity to changes in the second component
• Conceptually, in terms of calculation, it is no different from regular duration. It is the
approximate change in the price for a 100-basis point change in the spread.
• Suppose you expect the economy to improve and the credit spread to narrow. You
do not expect the YTM on treasury bonds to change. Do you want to own corporate
bonds with a high spread duration or a low spread duration?
• Also, what is the spread duration of a treasury bond?
FINM3421
CRICOS code 00025B
21
Portfolio Duration
• So far we have looked at the duration of individual bonds. Similar to equities,
bonds are typically held in a portfolio. Therefore, bond portfolio
managers need to measure the portfolio duration.
• There are two methods to calculate the duration of a bond portfolio:
• The weighted average of time to receipt of the future cash flows
• The weighted average of the individual bond durations comprising the
portfolio
• The first method is the “theoretically” correct approach, but practically hard to
implement because you will have to estimate the portfolio yield to determine
the weight of each future cash flow.
• The second method is commonly used by fixed-income portfolio managers.
The textbook simply “defines” the duration of a portfolio as the weighted
average duration of the bonds in the portfolio.
FINM3421
CRICOS code 00025B
22
Portfolio Duration
• Portfolio managers also look at their interest rate exposure to a particular
bond in terms of its contribution to portfolio duration
• This measure is found by multiplying the weight of the bond in the
portfolio by the duration of the individual bond given as:
contribution to portfolio duration
= weight of bond in portfolio × duration of bond.
FINM3421
CRICOS code 00025B
23
Portfolio Duration
• The duration of a portfolio is simply the weighted average duration of the bonds in
the portfolios.
• Example: A portfolio manager holds the following bonds:
Bond
market value duration
A
$10 million
4
B
$40 million
7
C
$30 million
6
D
$20 million
2
What is the portfolio’s duration?
Total value = 10+40+30+20=100
10
×
100
FINM3421
4+
40
×
100
7+
30
×
100
6+
20
×
100
2 = 5.4
CRICOS code 00025B
24
Portfolio Duration
• Portfolio managers generally look at portfolio duration based on sectors of
the bond market.
• Also, two durations can be computed:
• The first is the duration of the portfolio with respect to changes in the
level of Treasury rates.
• The second is the spread duration.
FINM3421
CRICOS code 00025B
25
25
Index Duration
Calculation of Duration and Spread Duration for Bond Index
Sector
Treasury
Corporate Bonds
Mortgage-Backed
Junk Bonds (Corporate)
Total
FINM3421
Weight in
Index
Sector
Duration
Contribution to
Index Duration
0.40
0.30
0.20
0.10
1.000
4.95
3.58
5.04
6.35
1.98
1.07
1.01
0.64
4.70
CRICOS code 00025B
Contribution to
Index Spread
Duration
1.07
1.01
0.64
2.72
26
Bond Portfolio Duration
Calculation of Duration and Spread Duration for Bond Portfolio Manager
Sector
Treasury
Corporate Bonds
Mortgage-Backed
Junk Bonds (Corporate)
Total
Weight in Port
Sector
Duration
Contribution to
Port Duration
0.00
0.50
0.20
0.30
1.000
4.95
3.58
5.04
6.35
0.00
1.79
1.01
1.91
4.70
Contribution to
Port Spread
Duration
1.79
1.01
1.91
4.70
The portfolio has the same duration as the index: 4.70
The portfolio has higher spread duration than the index: 4.70 versus 2.72
What does the portfolio manager expect to happen?
FINM3421
CRICOS code 00025B
27
Convexity
• Duration is an approximation. It is a linear approximation of a non-linear
Price-Yield relation. This works for small changes in YTM. But does not work
well for large changes in YTM.
• Recall that the first and second-order approximation for percentage changes
in bond prices:
∆𝑃
𝑃
=
𝑑𝑃
𝑑𝑦
𝑃
∆𝑦 +
𝑑2 𝑃
𝑑𝑦2
1
2 𝑃
(∆𝑦)2
• Convexity captures the non-linearity of the Price-Yield relation for a bond.
FINM3421
CRICOS code 00025B
28
Convexity
• The tangent line represents an estimate of the bond price using duration.
• The approximation will always understate the actual price.
• When yields decrease, the estimated price increase will be less than the
actual price increase, thereby underestimating the actual price.
• When yields increase, the estimated price drop will be greater than the actual
price drop, resulting in an underestimate of the actual price.
FINM3421
CRICOS code 00025B
29
Measuring Convexity
• Recall:
∆𝑃
𝑃
=
𝑑𝑃
𝑑𝑦
𝑃
∆𝑦 +
𝑑2 𝑃
𝑑𝑦2
1
2 𝑃
(∆𝑦)2
• Duration (modified or dollar) attempts to estimate a convex relationship with a straight line (the
tangent line)
• The dollar convexity measure of the bond:
𝐝𝟐 𝐏
𝐝𝐨𝐥𝐥𝐚𝐫 𝐜𝐨𝐧𝐯𝐞𝐱𝐢𝐭𝐲 𝐦𝐞𝐚𝐬𝐮𝐫𝐞 = 𝟐
𝐝𝐲
• The dollar change in price due to convexity is:
1
∆𝑃 = 𝑑𝑜𝑙𝑙𝑎𝑟 𝑐𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 ∆𝑦 2
2
• We divide the dollar convexity measure by price to get the convexity measure:
𝐝𝟐 𝐏 𝟏
𝐜𝐨𝐧𝐯𝐞𝐱𝐢𝐭𝐲 𝐦𝐞𝐚𝐬𝐮𝐫𝐞 = 𝟐
𝐝𝐲 𝐏
• The percentage price change due to convexity is:
1
2
%∆P
=
convexity
measure
∆y
CRICOS code 00025B
FINM3421
2
30
30
Measuring Convexity
• Recall that
d𝑃
𝑑𝑦
=−
𝐶
2𝐶
+
1+𝑦 2
1+𝑦 3
𝑛
=−
𝑡=1
𝑡𝐶
1+𝑦
+⋯+
𝑛𝐶
1+𝑦 𝑛+1
+
𝑛𝑀
1+𝑦 𝑛+1
𝑛𝑀
+
𝑡+1
1 + 𝑦 𝑛+1
Then, we can calculate the second derivative (i.e., the dollar convexity) as:
𝑑2 𝑃
𝑑𝑦 2
FINM3421
=
𝑡(𝑡+1)𝐶
𝑛
𝑡=1 1+𝑦 𝑡+2
+
𝑛(𝑛+1)𝑀
1+𝑦 𝑛+2
CRICOS code 00025B
31
Measuring Convexity: Example
• Let’s use the data from our earlier example
Time
CF
(1.06)t+2
t(t+1) * CF
Column 4 /
Column 3
1
5
1.191
10
8.396
2
5
1.262
30
23.763
3
5
1.338
60
44.838
4
105
1.419
2,100
1480.5
1,557.50
FINM3421
CRICOS code 00025B
32
Measuring Convexity: Example
• Convexity measure in half years:
 Second derivative divided by price
1,557.50
96.535
= 16.13
• Convexity measure in years
 Convexity measure in half years divided by the number of periods per
year squared
16.13401
22
FINM3421
= 4.03
CRICOS code 00025B
33
Measuring Convexity: Example
• How would we use this convexity measure? We can use it to undo the
bias that comes from using duration alone.
• Using the previous example, the percent price change due to convexity
with 1% change in yield:
0.5 × Convexity Measure × (Δ YTM)^2
= 0.5 × 4.03 × (0.01)^2
= 0.0002 = 0.02%
FINM3421
CRICOS code 00025B
34
Measuring Convexity: Example
• The percentage change in price from duration is –modified duration *(Δ YTM)
-1.7536 × 0.01 = -0.017533 = -1.7533%
• Suppose YTM changed from 12% to 13%
• The price change would then be
-1.7533% + 0.02% = -1.7333%
• If the price was originally $96.535, now it would be
$96.535 × (1 – 0.017333) = $94.861
FINM3421
CRICOS code 00025B
35
Value of Convexity
• Using convexity can help improve approximations of how a bond’s price will
change when YTM changes.
• Convexity has important investment implications.
• Investors are willing to pay a premium for convexity.
• If the market price of convexity is high, investors with expectations of low
interest rate volatility will probably want to “sell convexity”.
FINM3421
CRICOS code 00025B
36
Convexity Properties
• All option-free bonds have three convexity properties
• The intuition behind all three of the convexity properties is the same: Cash
flows further out in time have much more effect on convexity than cash flows
that occur soon.
1. As the required yield increases (decreases), the convexity of a bond
decreases (increases).
 As the yield increases, the distant cash flows become relatively less
important. Since the distant cash flows have more effect on convexity,
convexity decreases.
FINM3421
CRICOS code 00025B
37
Convexity Properties
2. For a given yield and maturity, lower coupon rates will have greater
convexity
 For a given yield and maturity, a bond with a lower coupon rate has
proportionally more of its present value occurring further in the future.
 Since cash flows further in the future have a stronger effect on convexity,
the lower coupon bond has higher convexity.
FINM3421
CRICOS code 00025B
38
Convexity Properties
3. For a given yield and modified duration, lower coupon rates will have smaller
convexity.
 If two bonds have different coupon rates, but the same yield and modified
duration, then the bond with the lower coupon rate must have a less time
to maturity (e.g., a 7-year, 4% coupon rate bond and a 5-year zerocoupon bond could both have modified duration of 4.6 years)
 The higher coupon bond’s cash flows occur further out in time, and these
cash flows make a large contribution to convexity. So the higher coupon
bond (with longer time to maturity) has higher convexity.
 The investment implication is that zero-coupon bonds have the lowest
convexity for a given yield and modified duration.
FINM3421
CRICOS code 00025B
39
Concerns when Using Duration
• Because of convexity, relying on duration as the sole measure of the price
volatility of a bond may mislead investors.
• There are two other concerns about using duration:
 First, duration implicitly assumes that all cash flows for the bond are
discounted at the same discount rate. This problem will be further
discussed in later classes.
 Second, there is misapplication of duration to bonds with embedded
options.
FINM3421
CRICOS code 00025B
40
Duration and Convexity for Callable Bonds
price
noncallable bond
yield
• Negative convexity under low interest rates!
FINM3421
CRICOS code 00025B
41
Approximating a Bond’s Duration and Convexity Measure
• Because duration and convexity are essentially the first and second
derivatives of bond price with respect to yield, we can always numerically
approximate them.
𝑃− −𝑃+
2∙𝑃0 ∙∆𝑦
𝑃− +𝑃+ −2∙𝑃0
𝑃0 ∙ ∆𝑦 2
Effective 𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 =
Effective 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 =
• Where P0 is the initial price, P- is the new price if y (annualized yield) goes
down by ∆𝑦, and P+ is the new price if Y (annualized yield) goes up by ∆𝑦.
FINM3421
CRICOS code 00025B
42
Approximating a Bond’s Duration and Convexity Measure
• Example: There is a 25-year 6% coupon bond trading at 9%. The current
price is 70.357. If the annualized yield goes up by 10 basis points, the new
price will be 69.6164. If the yield goes down by 10 basis points, the new price
will be 71.1105
71.1105 − 69.6164
𝐸𝑓𝑓. 𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑 𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 =
= 10.62
2 × 70.357 × 0.001
𝐸𝑓𝑓. 𝐶𝑜𝑛𝑣𝑒𝑥𝑖𝑡𝑦 =
FINM3421
71.1105+69.6164−2 70.3570
70.3570 0.001 2
= 183.35
CRICOS code 00025B
43
Next Class
• Next class we will discuss term structure of interest rates.
FINM3421
CRICOS code 00025B
44