Class #6, Chap 9
1

Purpose: to understand what duration is, how to
calculate it and how to use it.

Toolbox: Bond Pricing Review

Duration




Concept
Interpretation
Calculation
Examples
2

Bond Pricing Review

Zero coupon bond with the YTM

Coupon bond with the YTM

Coupon bond with Yield Curve

Yield Curve and YTM
3
Price a zero coupon bond with 10 years left to maturity, face value of $1000
and YTM of 5%
Step 1: Find coupon payments & draw cash flows
Coupon Payments = $1,000*0 = $0
1
2
3
4
1000
5
6
7
8
9
10
Step 2: discount cash flows
1000
P
 $613.91
10
(1.05)
4
Price a 4 year coupon bond with face value of 1000 and an annual coupon of
7% if the yield to maturity is 13%
Step 1: Find coupon payments & draw cash flows
Coupon Payments = $1,000*.07 = $70
1000
70
70
70
1
2
3
70
4
Step 2: discount cash flows
70
70
70
1070
P



 $821.53
1
2
3
4
(1.13) (1.13) (1.13) (1.13)
5
• Yield curve gives the market rate for a pure discount bond at each maturity
• The market price of a coupon bond incorporates several different rates for
different time horizons (1 year money, 2 year money …. )
• Every bond has its own yield curve – treasury bonds, Ford bonds, GM bonds
• The yield curve will change every day
Current Yield Curve
1000
4.3%
70
70
1
2
70
3.1%
3
1.1%
1
2
3
6
Example: Find the price of a three year bond that pays an annual coupon of 8%. The following rates are
taken from the current yield curve. Face value is $1,000
Term
Rate
6 months
1.2%
1 year
2.1%
2 years
2.5%
3 years
2.7%
7
We used two different methods what is the difference?
Method 1: we pulled rates off the yield curve
P
80
80
1080


 1151.54
2
3
1.021 (1.025)
(1.027)
Method 2: we used one constant rate – lets value the bond above with a
constant rate of 2.676% (Yield to Maturity)
P
80
80
1080


 1151.54
2
3
1.02676 (1.02676) (1.02676)
What does this yield curve look like?
8

The yield to maturity (YTM) is basically a weighted
average of rates off the yield curve

It is the constant rate, over the full maturity, that
gives you the market price of the bond

When you use the YTM you assume that the yield
curve is flat!
9
DURATION



Concept
Calculation
Using duration
10
Concept of Duration
11

What does duration do?
 It measures the sensitivity of the asset price to changes in interests rates

How is that going to help us?
 We have been trying to measure interest rate risk
▪ The movement in asset prices in response to a change in interest rates
 Repricing gap gave us a rough measure but had several problems
 Duration improves upon some of these short falls

Advantages of Duration
 It is a market based measure so it takes into account current values
 It considers the current time to maturity rather than the defined term

Disadvantages of Duration
 It requires more information to calculate
12

Definition of Duration:

What the #@&! ?
This is what we are after. I want you to see
tells us:average
on average
 Duration is the present value that
cashduration
flow weighted
timewhen
to do
we receive the value of our bond/loan
maturity of a loan/bond
 Duration tells us, in terms of present value, the average timing of cash flows
from a loan or bond. (i.e. on average, when do we receive the value of our
bond)

Not much help? Lets split this discussion into two parts:
1.
2.
What is duration – work on explaining the definition
Why is it important – how does it help measure interest rate sensitivity
13
Duration is the present value cash flow weighted average time to maturity
of a loan/bond

First thing we want to realize is that any bond can be thought of as a
portfolio of zero coupon bonds

Consider a 6 year coupon bond that pays an annual coupon of 4% and
has a $1,000 face value
14
Duration is the present value cash flow weighted average time to maturity
of a loan/bond
1040
We can think of the coupon
bond as a portfolio
of 6 zero coupon bonds. The average maturity
40
40
40
40
40
Average TTM =
0
1
2
3
4
5
1 2  3  4  5  6
 3 .5
6
6
15
Duration is the present value cash flow weighted average time to maturity
of a loan/bond
1040
0
40
40
40
40
40
1
2
3
4
5
6
Average TTM = 3.5 yrs

Do you think that this tells you when, on average, you receive the value
of you payments?
Before or after 3.5 years?
Why?

Duration is going to depend on two things:
1. The timing of payments
2. The amount of payments – in terms of present value
Take away: We can calculate the average time to maturity of a bond but that
does not always tell us when, on average, we receive the full value of payments
16
Duration is the present value cash flow weighted average time to maturity
1040
of a loan/bond
 587.05
PV = 587.05
(1.10) 6
Which pmts is most/
40
least valuable?
 36
.36
PV =
36.36
1.10
40
0
1
1040
40
40
40
40
 30.05PV = 27.32
33.06 PV = 330.05

27
.
32

24
.
84
PV 2=33.06
PV = 24.84
(1.10)
(1.10)
(1.10) 4
(1.10) 5
40
40
40
40
2
3
4
5
6
Somewhere
around year 5 or
6 is a good guess
For this part, let’s just start with how much of the bond value we receive
at each point in time. Assume YTM = 10%
 On average, when do you think we receive the full bond value?
 How can we adjust the average to account for this? weighted average
 What do we use for weights? Present value of cash flows

Take away: On average we receive the full bond value close to
the largest PV(payment). So we need to weight by PV(CFs)
17
Duration is the present value cash flow weighted average time to maturity
of a loan/bond
1040
0
40
40
40
40
40
1
2
3
4
5
6
Duration = 5.35 yrs

On average we will receive the full value of our payments 5.35 years
from today.
18
1.
What is Duration – Definition
2.
Why is it important – how does it measure
interest rate sensitivity
19
Duration is important because it tells us the interest rate sensitivity
of a bond!!! But how?

It turns out that the weighted average time to maturity (duration) gives us the
maturity of the equivalent zero coupon bond.

This “equivalent” zero coupon bond will have the same interest rate
sensitivity as the coupon bond

Example: consider two bonds:
1040
40
40
40
40
40
Duration = 5.35
Bond 1:
0
1
2
3
4
5
6
1000
Bond 2:
0
5.35
20
Example: Price both bonds with YTM = 10% then again with YTM =
10.5% and compare the price sensitivity.
1040
40
40
40
40
40
Duration = 5.35
Bond 1:
0
V (10%) 
1
2
3
4
5
6
Sensitivity
40
40
40
40
40
1040





 738 .68
1.10 1.10 2 1.10 3 1.10 4 1.10 5 1.10 6
V (10.5%) 
40
40
40
40
40
1040





 721 .01
2
3
4
5
1.105 1.105 1.105 1.105 1.105 1.105 6
1000
721 .01  738 .68
 0.02393
738 .68
The two bonds have the
same interest rate sensitivity
Bond 2:
0
1000
V (10%) 
 600 .82
1.10 5.35
V (10.5%) 
1000
 586 .45
1.105 5.35
5.35
Sensitivity
586 .45  600 .82
 0.02395
600 .82
21
What does this get us? …. Lets see
What if we change the
principal amount?
1000
What is the duration
of this bond?
Bond 1:
0
5
20,728,012
1000
What is the duration
of this bond?
Bond 2:
0
10
Which bond is more interest rate
sensitive?sensitive?
Is it harder to see?
22
What does this get us? …. Lets see
1000
$40
Bond 1:
0
1
2
3
4
5
1000
$40
Bond 2:
0
1
2
3
4
5 6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
Which bond is more interest rate
sensitive?
23
What does this get us? …. Lets see
1000
$90
Bond 1:
0
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
Duration = 5.96
500
$40
Bond 2:
0
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
Duration = 6.95
Which bond is more interest rate
sensitive?
24
Bond 1
Face value
Time to Maturity
Coupon Rate
Compounding
Bond 2
100
100
100
200
0.50
0.05
8
2
Bond 1 Interest Rate Sensitivity
26261.5 - 28885.81
28885.81
= -0.0909
Bond 2 Interest Rate Sensitivity
Beginning of Period YTM
0.09
End of Period YTM
0.099
252.53 - 277.78
277.78
= -0.0909
Calculate
25
Conclusion:

Duration tells us:
 With respect to interest rate sensitivity, this bond will behave like a zero
coupon bond with D years to maturity (where D is the bond
duration/maturity)

We know that the zero coupon bond with longer maturity is
more interest rate sensitive

Therefore, we also know that a bond with longer duration is
more interest rate sensitive
26
Calculating Duration
27
Step 1: draw out the cash flows
Step 2: take the present value of all the cash flows
Step 3: calculate weights
Step 4: calculate the weighted average time to
maturity (duration)
28
Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM,
and $1,000 face value
Step #1: draw out the cash flows
1000
40
40
40
0.5
1
1.5
40
2
29
Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM,
1000
and $1,000 face value
40
40
40
0.5
1
1.5
40
Step #2: Take the present value of cash flows
40
 37.74
(1  .12 / 2)
40
 35.60
(1  .12 / 2) 2
40
 33.58
(1  .12 / 2) 3
1040
 823.78
(1  .12 / 2) 4
2
30
Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM,
and $1,000 face value
1000
Step #3: Calculate weights
40
 37.74
(1  .12 / 2)
40
 35.60
(1  .12 / 2) 2
40
 33.58
(1  .12 / 2) 3
1040
 823.78
(1  .12 / 2) 4
40
40
40
0.5
1
1.5
40
2
First thing we need to do is
sum the present values
930 .70
What is this
number?
31
Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM,
1000
and $1,000 face value
40
40
40
0.5
1
1.5
40
Step #3: Calculate weights
40
 37.74
(1  .12 / 2)
40
 35.60
(1  .12 / 2) 2
40
 33.58
(1  .12 / 2) 3
1040
 823.78
(1  .12 / 2) 4
37 .74
w.5 
 .0405
930 .70
w1 
35.60
 .0383
930 .70
w1.5 
33.58
 .0361
930 .70
w2 
823 .78
 .8851
930 .70
930 .70
Why do we take the present value?
2
What do the weights mean?
They are the percentages of the
present value of all cash flows
that occur on that time period
Example: 3.61% of the present
value of all cash flows is received
at year 1.5
We are trying to compare the relative importance of different cash flows so we need to compare them at the
same point in time – which is more valuable to an investor $1 today or $1.10 in one year if the one year interest
rate is 10%?
32
Calculate the duration of a two year treasury bond with an 8% semiannual coupon, 12% YTM,
and $1,000 face value
Step #4: Calculate the duration (present value cash flow weighted average
time to maturity)
weights
37.74
 .0405
930.70
35.60
 .0383
930.70
33.58
 .0361
930.70
823.78
 .8851
930.70
D  w1t1  w2t2  w3t3  ... wntn
D  .0405t1  .0383t2  .0361t3  .8851t4
D  .0405(.5)  .0383(1)  .0361(1.5)  .8851(2)  1.8829Years
What are the ts?
What are these?
Macaulay Duration
33



To this point, we know that duration is a measure of interest rate sensitivity
It turns out that duration is also the interest rate elasticity of a security
(bond) price
What more does that tell us?


Because it is an elasticity, we can use it to determine how much the bond price will move
in response to a change in interest rates
Elasticity Equation
X
%changein X
X
e

Y
%changein Y
Y
P
P
D
[(1  Rt 1 )  (1  Rt )]

(1  Rt )
P
R
P
(1  R)
34
We can rewrite the duration equation:
P
D 
P
 R

 (1  R ) 
This gives us an equation for calculating the percent change in the bond price
(return) due to a change in the interest rate
 R 
P
  D

P
 (1  R) 
35

We have been working with a two year 8% coupon treasury bond with
12% YTM and a price of $930.70

Suppose the interest rate decreased to .115 what would you expect the
percent change in the price to be?
1.
Calculate the percent change in the bond price:
p
p
2.
40
40
40
1040



 930.70
(1  .12 / 2) (1  .12 / 2) 2 (1  .12 / 2)3 (1  .12 / 2) 4
939 .12  930 .70
 0.00893
930 .70
40
40
40
1040



 939.12
2
3
(1  .115/ 2) (1  .115/ 2) (1  .115/ 2) (1  .115/ 2) 4
With Duration:
 R 
  .005 
P
 D


1
.
882888
 0.0088815



P
 (1  R) 
 (1  .12 / 2) 
Not exact but
pretty close
36

Duration gives us a way to measure the sensitivity of an asset
price to changes in the interest rate

Duration also gives us a way to calculate the magnitude of the
percent change in price in response to a change in interest
rates

Alternative forms of duration
 Bond traders developed more convenient ways to write duration
 Modified duration (MoD)
 Dollar duration
37


Macaulay Duration: (D)
Modified Duration (MoD)
 R 
P
  D

P
 (1  R) 
MoD 
D
1 R
P
  MoD R 
P
MoD allows you to calculate the %change in the bond
price just by multiplying by the change in interest rate

Dollar Duration
$ D  MoD  P
P  $DR
$D dollar duration allows you to calculate the change in bond
price from the change in the interest rate
38
Example: calculate the duration, modified duration and dollar duration for a bond with: face
value = 1000; annual coupon; coupon rate = 3%; YTM = 9%; and four years to maturity
39
Examples: Suppose the YTM = 9%
i)
Find the percent change in the bond price if YTM increases from 9% to 14% the duration is 3.8 years
ii)
The percent change in the bond price if the YTM increases to 11% given MoD = 3.492
iii) The raw change in the bond price if the YTM decrease to 8.5% if $D = 2813.06
40

How would things change if the bond had semiannual coupons?

The bond pricing would change as we have already seen

Modified duration would also change:
MoD 
D
3.8

(1  R) (1.09)
For semiannual
coupons we have
MoD 
D
3.8

(1  R / 2) (1  .09 / 2)
41
Different durations:
 Macaulay Duration = D
 Modified duration(MoD)

D
(1  R / k )
 Dollar duration = (MoD)(bond price)
D = Macaulay duration
R = the yield to maturity
k = compounding periods
42

We learned the meaning of duration (concept)

How to calculate duration (D, MoD, $D)

How to use duration to calculate the expected
change (%change in price)
43