FINC4101
Investment Analysis
Instructor: Dr. Leng Ling
Topic: Interest Rate Risk
1
Learning objectives
1. Define
interest rate risk for bonds.
2. Recognize maturity as a major determinant of
interest rate risk.
3. Compute Macaulay’s duration and modified
duration.
4. Identify the determinants of Macaulay’s duration.
2
Concept Map
Foreign
Exchange
Portfolio
Theory
Asset
Pricing
FI400
Equity
Derivatives
Market
Efficiency
Fixed Income
3
Interest rate risk
Why are changes in interest rates a risk for bonds?
Changes
in interest
rates
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, interest rate risk and interest rate
sensitivity are synonymous.
4
Bond Pricing Relationships (1)
D
C
B
A
5
Bond Pricing Relationships (2)
For all 4 bonds, we observe that:
 Bond prices and yields are inversely related.



As YTM increases, bond price falls
As YTM decreases, bond price rises
Price-yield relationship is convex.


An increase in YTM results in a smaller price change
than a decrease in YTM of equal magnitude.
This property is called “convexity”.
6
Bond Pricing Relationships (3)
Comparing bonds A and B, we see that:
 Interest rate sensitivity increases with maturity.


Prices of long-maturity bonds tend to be more
sensitive to interest rate changes than prices of shortmaturity bonds.
Interest rate sensitivity increases at a decreasing
rate as maturity increases.
7
Bond Pricing Relationships (4)
Comparing bonds B and C, we see that,
 Interest rate sensitivity is inversely related to
coupon rate.

Prices of high-coupon bonds are less sensitive to
changes in interest rates than prices of low-coupon
bonds.
Comparing bonds C and D, we see that,
 Interest rate sensitivity is inversely related to the
YTM at which the bond currently is selling.
8
Maturity and interest rate risk
A
bond’s maturity is a major determinant
of interest rate risk.
 Maturity alone is not enough to measure
interest rate risk.
 Cash flows received before maturity
affects the relationship between maturity
and interest rate sensitivity.

Consider the following…
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Coupon bond vs. zero coupon bond

Prices of 8% annual coupon bonds
YTM
T = 10 years
T = 20 years
8%
1000
1000
9%
935.82
908.71
% price change
-6.42%
-9.13%

Prices of zero-coupon bonds
YTM
T = 10 years
T = 20 years
8%
463.19
214.55
9%
422.41
178.43
% price change
-8.80%
-16.84%
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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.
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Macaulay’s duration (1)

A measure of interest rate sensitivity.

Macaulay’s duration is the weighted average of
the times to each coupon or principal payment
made by the bond.

Often referred to as ‘duration’.

The weight for each payment time is the PV of
the payment divided by the bond price.
12
Macaulay’s duration (2)
D  w1  2w2  3w3  4w4  ...  TwT
T
  t  wt
Time until 4th
cash flow
Weight of 4th
cash flow
t 1
CFt
t
(1  YTM )
where wt 
Bond Pr ice
Cash flow paid at time t
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Examples: 3-year bond, 8%, annual
coupon payments, YTM = 10%
A
B
A. 8% coupon bond
C
Sum:
E
Time until
Payment
Column (B)
Payment
Discounted
x
(Years)
Payment
at 10%
Weight Column (E)
1
80
72.727
0.0765
0.0765
2
80
66.116
0.0696
0.1392
3
1080
811.420
0.8539
2.5617
950.263
1.0000
2.7774
Sum:
B. Zero-coupon bond
D
1
0
0.000
0.0000
0.0000
2
0
0.000
0.0000
0.0000
3
1000
751.315
1.0000
3.0000
751.315
1.0000
3.0000
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Macaulay’s duration measures
interest rate sensitivity
Absolute percentage change in price when
Bond
YTM increases from
10% to 11%
YTM decreases from
10% to 9%
8% coupon bond
(duration =2.7774)
2.48%
2.57%
Zero coupon bond
(duration =3)
2.68%
2.78%
Higher duration is associated with larger percentage price
changes, i.e., greater interest rate sensitivity.
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Macaulay’s duration for
semi-annual payment coupon bond

For a semi-annual coupon bond, use the same
formula as for an annual coupon bond, but:




Each period is a semi-annual period
CFt : cash flow for semi-annual period t
Yt : semi-annual YTM
Result: semi-annual Macaulay duration.
Annual Macaulay duration
= Semi-annual Macaulay duration/2
16
Example: 3-year bond, 8%,
semi-annual coupon payments, YTM = 10%
A
B
A. 8% coupon bond
Sum
C
D
E
Time until
Payment
Column (B)
Payment
Discounted
x
(6-months)
Payment
at 5%
Weight Column (E)
1
40
38.095
0.0401
0.0401
2
40
36.281
0.0382
0.0764
3
40
34.554
0.0364
0.1092
4
40
32.908
0.0347
0.1387
5
40
31.341
0.0330
0.1651
6
1040
776.064
0.8176
4.9054
949.243
Semi-annual duration
5.4349
Annualized duration
2.7174
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Modified duration (1)
 Turns
out that we can transform
Macaulay’s duration into a measure that
estimates percentage price change for a
given change in YTM.
 This measure is called Modified duration.

Directly gauges the impact of interest rate risk
because it approximates the change in price
when yield changes (increases or decreases).
18
Modified duration (2)
 Modified
duration, D*
Macaulay's durat ion
D =
1 + init ial YT M
*
YTM (in decimal)
19
Using Modified duration to estimate
percentage price change
Dollar change in bond price
Change in YTM in decimal
P
  D  y
P0
Initial price
Modified duration
y  newYTM  initialYTM
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Using Modified duration to
estimate dollar price change
P  P0  ( D  y)
a given change in yield, y , the
predicted price = P0  P
 For
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Interpretation
 As
Macaulay’s duration ↑, interest rate
sensitivity ↑ , vice versa.
 As
modified duration ↑, interest rate
sensitivity ↑ , vice versa.
22
Problems (1)
A
nine-year bond has a yield of 10% and a
(Macaulay) duration of 7.194 years. If the
bond’s yield increases by 50 basis points,
what is the percentage change in the
bond’s price.
 Note:
1 basis point = 0.0001 or 0.01%
 100 basis points = 1%

23
Problems (2)
A
30-year maturity bond making annual
coupon payments with a coupon a rate of
12% has a duration of 11.54 years. The
bond currently sells at a yield to maturity of
8%. If the yield to maturity falls to 7%,
compute:
1.
2.
3.
The percentage price change.
The estimated dollar price change.
The predicted price.
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Problems (3)

A six-year 6.1% semi-annual coupon bond has a yield to
maturity of 10% and a semi-annual Macaulay duration of
10.014.
1.
What is the annual Macaulay duration?
2.
What is the modified duration on a semi-annual basis?
3.
What is the modified duration on an annual basis?
4.
If the yield increases by 25 basis points, what is the
percentage price change using the annual modified
duration?
25
Determinants of duration (1)

Rule 1: The duration of a zero-coupon bond
equals its time to maturity.

Rule 2: Holding time to maturity and yield to
maturity constant, a bond’s duration and interest
rate sensitivity are higher when the coupon rate
is lower.
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Determinants of duration (2)

Rule 3: Holding the coupon rate constant, a
bond’s duration and interest rate sensitivity
generally increase with time to maturity. Duration
always increases with maturity for bonds selling
at par or at a premium to par.

Rule 4: Holding other factors constant, the
duration and interest rate sensitivity of a coupon
bond are higher when the bond’s yield to
maturity is lower.
27
Determinants of duration (3)
 Rule
5: The duration of a perpetuity is
1+ YTM
Durat ion of perpet uit y =
YTM
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Graphical summary
Rule 1
Rule 4
Rule 2
Rule 3
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Conceptual problems (1)

Rank the following bonds in order of descending duration.
Bond
A
B
C
D
E
Coupon
15%
15
0
8
15
Time to Maturity
20 years
15
20
20
15
Yield to Maturity
10%
10
10
10
15
30
Conceptual problems (2)

Which set of conditions will result in a bond with
the greatest price volatility?
A high coupon and a short maturity
B. A high coupon and a long maturity
C. A low coupon and a short maturity
D. A low coupon and a long maturity
A.
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Conceptual problems (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
High, short
High, long
Zero, long
B.
C.
D.
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Conceptual problems (4)

With a zero-coupon bond:
A.
Duration equals the weighted average term to
maturity.
Term to maturity equals duration.
Weighted average term to maturity equals the
term to maturity.
All of the above.
B.
C.
D.
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Summary
1.
Interest rate fluctuations are a key risk to bonds.
2.
Maturity is a major determinant of interest rate risk.
3.
Compute Macaulay’s duration and modified duration.
4.
Determinants of Macaulay’s duration.
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Practice 7
 Chapter
11:
4,8,9,12,14.
35
Homework 8
1. A 10% semi-annual coupon bond currently sold at $950 will mature after 20
years. Investors expect that the firm will be able to make good on the remaining
interest payments but that at the maturity date bondholders will receive full
$1000 par value with 0.5 probability and 70% of par value with 0.5 probability.
Compute the expected bond equivalent YTM.
2. Suppose you are a money manager who manages a portfolio of two bonds.
A bond is a 5% semi-annual coupon bond with 5 years to maturity. Its par value
is $1000. The YTM is 7%. There are 10,000 unit of A bonds hold in the
portfolio. B bond is a 3-year 6% semi-annual coupon bond of $500 par value
and its YTM is 8%. You hold 5,000 unit of B bond. (You must keep 5 decimal
places in all steps, eg. $5.67867, otherwise you get 0)
(1) What is the duration of A and B bond? (you can use excel)
(2) If the YTM for both bonds increase by 50 basis points, how much is the
dollar amount change in the portfolio value ? You MUST use modified duration
method to solve this question.
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