F520_Bond
1
Valuation of Debt Contracts
and Their Price Volatility Characteristics
I. Review of Bond Pricing Basics
Example of a basic bond:
What is the value of a 10% bond that pays interest annually
and has a yield to maturity of 7 percent? Assume the bond
has 3 years from today till maturity.
0
|
1
|
pmt
N
I
N
3
I
7%
2
|
pmt
PV
cpt
$1,078.73
3 years
|
pmt
fv
PV
PMT
PMT
100
FV
1000
FV
F520_Bond
2
What is the value of a 10% bond that pays interest semiannually and has a yield to maturity of 7 percent? Assume the
bond has 3 years from today till maturity.
0
|
|
p
1
|
p
N
|
p
2
|
p
I
N
6
I
3.5%
|
p
3 years
|
p
fv
PV
PV
cpt
$1,079.93
PMT
50
PMT
FV
1000
FV
F520_Bond
Remember: The yield-to-maturity is a nominal rate.
What is the effective annual rate (EAR)?
Based on semiannual compounding our EAR is
(1 + k/2)2 - 1
(1 + .07/2)2 - 1 = 7.1225%
Or using the “I Conv” function on your calculators:
NOM = 7
N=2
cpt EFF (= 7.1225%)
Coupon Bond (10% coupon 3 years)
N
I
PV
PMT
FV
6
3.5000%
cpt
50
1000 Quote Yield is 7%
$1,079.93
N
I
PV
PMT
FV
3
7.1225%
cpt
100
1000 Compounds to EAR of 7.1225%
$1,075.35
Semi-annual has greater value becase of earlier coupon payments.
Zero Coupon (0% coupon 3 years)
N
I
PV
PMT
FV
6
3.5000%
cpt
0
1000 Quote Yield is 7%
$813.50
N
I
PV
PMT
FV
3
7.1225%
cpt
0
1000 Compounds to EAR of 7.1225%
$813.50
3
F520_Bond
4
What drives bond prices?
7:15 p.m. EST 02/01/13Treasurys
Price
Chg
1-Month Bill*
0/32
3-Month Bill*
0/32
6-Month Bill*
0/32
1-Year Note*
0/32
2-Year Note*
0/32
3-Year Note*
0/32
5-Year Note*
0/32
7-Year Note*
0/32
10-Year Note*
0/32
30-Year Bond*
0/32
* at close
Yield
(%)
0.030
0.076
0.112
0.140
0.285
0.410
0.892
1.409
2.017
3.221
August
2009
August 2009
1-Month Bill
3-Month Bill
6-Month Bill
1-Year Note
2-Year Note
3-Year Note
5-Year Note
10-Year Note
30-Year Bond
February 2013
Price
Chg
0/32
-0/32
-0/32
1/32
-1/32
-2/32
-3/32
-4/32
6/32
Yield
(%)
0.114
0.155
0.254
0.451
1.071
1.566
2.459
3.453
4.213
August 2009
February 2013
F520_Bond
5
Understanding Bills: (quote February 1, 2013)
Treasury Bills – Issue maturity Less than 1 year
Maturity Days to Bid
Ask
Chg
Maturity
6/13/2013
132
0.080
2/1/2013
|
0.070
-0.005
Ask Yld
0.071
Maturity (6/13/2013)
|
fv
$1000.00
$999.7397
Using the ask yield:
1
PV = 1000 *
(1 + r) n
1
PV = 1000 *
.00071
)
360
PV = 1000 * .9997397  999.7397
(1 + 132 *
Treasury Bills do not have compounding interest and their interest is
determined on a 360 day year. This differs from Treaury notes and bonds,
which are based on semi-annual compounding and a 365 day year.
One could also start with price, calculate the Holding Period Yield (HPY)
and convert to the Bond Equivalent Yield (BEY).
PV = 1000 * 99 .97397% = 999.7397
(1  HPY)  1000 / 999.7397  1.00026
HPY  .00026  0.026%
BEY  HPY
360
360
 0.026%
 0.071%
132
132
F520_Bond
6
Understanding Bills: (quote February 1, 2013)
Treasury Bills – Issue maturity Less than 1 year
Days to
Maturity
Bid
Ask
Chg
Maturity
6/13/2013
Today
2/1/2013
0.08
0.08
0.07
Maturity
6/13/2013
999.73967 formula PRICEDISC()
Ask Yld
-0.005
Par
132 days
0.071
Yield
1000
0.0710%
F520_Bond
7
Notes and Bonds: quote 2/1/2013
Treasury Notes – Issue Maturity Less than 10 years
Maturity
Rate
Bid
Ask
Chg
5/15/2019
3.125
111.7266
111.8203
-0.1719
Treasury Bonds – Issue Maturity Greater than 10 years
Maturity
Rate
Bid
Ask
Chg
11/15/2042
Today is
2/1/2013
2.750
91.2031
91.2656
11/15/12 5/15/13 11/15/13
|
|
0
1
2
1.5625 1.5625
|
-0.6875
Ask Yld
1.166
Ask Yld
3.207
5/15/2019
|
|
|
|
|
|
13 payments
1.5625 Coupon payments (%), = 3.125%/2
100
Maturity Value (%)
Let’s compare our “dirty price” as it is called which is the
Ask Price plus Accrued Interest to the calculation of Ask
plus Accrued.
CPN % A
AI 
*
2
E
CPN = Annual coupon as a percent
A = Number of days from the beginning of the coupon
period to the settlement date (today)
E = Number of days in coupon period in which the
settlement date falls
F520_Bond
CPN
A=
E=
8
3.125%
11/15/2012
11/15/2012
2/1/2013
5/15/2013
78 days
181 days
=ACCRINT
AI =
3.125%
2
AI + Ask =
*
78
181
=
Price
Dirty Price
0.67334%
0.67334
111.82030%
112.49364%
Let’s see if this matches up with the time value of money concepts you learned.
(1)First you would need to find the value (price) on the date of the last coupon
payment (11/15/12) just after the coupon was paid using TVM techniques.
(2)Then you would have to find the value on 2/1/2013.
N
13
1.1660%
I
0.5830%
N
0.4309
I
0.5830%
PV
cpt
112.2286474
PV
112.2286474
3.125%
PMT
1.5625
PMT
0
Difference
Bid
111.7266
Ask
111.8203
FV
100
FV
cpt
$112.51
-0.01650
0.09370
Bid Ask Spread
This is the actual sale price of the bond 112.49% of par or
$1124.90. Note that our calculations do not equal the ask
price in the quote. The quote price is called the clean price,
it does not include accrued interest.
F520_Bond
9
Understanding a Stripped Bond: (Prices as of Feb 1, 2013) First, I
start with a Treasury Note and then compare this to a Treasury Strip
Treasury Notes – Issue Maturity Less than 10 years
Maturity Coupon Bid
Ask
Chg
Ask Yld
5/15/2014
Accrued interest is
4.750 105.7891 105.7969
4.750% 78
AI 
*
 1.02348%
2
181
-0.0391
0.200
Dirty Price = Ask + AI
= 105.7969% + 1.02348% = 106.82038% =$1068.20
fraction of coupon period from last coupon (11/15/12) to today (2/1/13) is 78
days in coupon period (11/15/12) to (5/15/13) is 181
Treasury Strips – Stripping of coupon and principal on
Bonds and Notes
2013 May 15
99.988
99.990 0.001
0.04 Coupon
2013 Nov 15
99.880
99.888 0.004
0.14 Coupon
2014 May 15
99.726
99.739 0.006
0.20 Coupon
2014 May 15
99.729
99.742 -0.007
0.20 Principle
0
1
2
3
|
|
|
|
pmt
pmt
pmt
$23.75
$23.75
$23.75
1000.00
23.7476
23.7234
23.6880
997.4200
$1068.5790
I use the ask price multiplied by the payment. You can use either
the ask price or the ask yield to discount. The slight difference in price is possible between these
two sets of securities, arbitrage transaction costs of .01% for the buy and .01% for the sell (round-trip
.02%) are about $0.20.
I could buy the four components of the bond in the strip market for $1068.5790 or I could buy the complete
bond in the bond market for $1068.20. I will choose to purchase in whichever market I can buy it at the
lowest price, so in this case the bond market for $1068.20.
F520_Bond
10
Understanding Inflation Indexed Securities (February 1, 2013)
Treasury Inflation Indexed Securities
Maturity Rate Bid
Ask
Chg
Ask Yld
Accrued
Principle
2019 Jul
15
1.875 121.27 121.31
-4
-1.372
1078
2042
Feb 15
0.750 104.29 105.10
- 42
0.552
1018
*-Yld. to maturity on accrued principal.
Compare the Inflation Indexed Securities Yield to the Treasury
Note yield, why such a large difference in Ask Yield?
Treasury Notes – Issue Maturity Less than 10 years
Maturity
Rate
Bid
Ask
Chg
5/15/2019
3.125
111.7266
111.8203
-0.1719
Treasury Bonds – Issue Maturity Greater than 10 years
Maturity
Rate
Bid
Ask
Chg
11/15/2042
2.750
91.2031
91.2656
-0.6875
Ask Yld
1.166
Ask Yld
3.207
F520_Bond
11
II. Controlling the level of interest rate risk:
Understanding where interest rate risk comes from and
what a good measure of interest rate risk should
incorporate.
Basic types of interest rate risk
 Reinvestment Risk – The risk that the future
interest rate at which the coupon can be
reinvested will be less than the yield to maturity
at the time the bond is purchased. [Risk of rates
falling.]
 Price Risk (interest rate risk)-- The risk that the
bond will have to be sold at a loss, if sold prior to
maturity. [Risk of rates rising.]
The measure we will use is called Macaulay Duration.
Macaulay Duration is the an indication of the
proportional sensitivity of the price of an
asset/liability to changes in the market rate of
interest.
F520_Bond
12
Bond Price Volatility
3%
4%
5%
6%
7%
8%
9%
10%
11%
12%
13%
14%
15%
16%
17%
10% coupon 10% coupon 10% coupon 5% coupon 5% coupon 10% coupon10% coupon
5 YrsTM
10 YrsTM
20 YrsTM
5 YrsTM
20 YrsTM
5 YrsTM 20 YrsTM
$1,320.58
$1,597.11
$2,041.42
$1,091.59
$1,297.55
32.1%
104.1%
1,267.11
1,486.65
1,815.42
1,044.52
1,135.90
26.7%
81.5%
1,216.47
1,386.09
1,623.11
1,000.00
1,000.00
21.6%
62.3%
1,168.49
1,294.40
1,458.80
957.88
885.30
16.8%
45.9%
1,123.01
1,210.71
1,317.82
918.00
788.12
12.3%
31.8%
1,079.85
1,134.20
1,196.36
880.22
705.46
8.0%
19.6%
1,038.90
1,064.18
1,091.29
844.41
634.86
3.9%
9.1%
$1,000.00
1,000.00
1,000.00
810.46
574.32
0.0%
0.0%
963.04
941.11
920.37
778.25
522.20
-3.7%
-8.0%
927.90
887.00
850.61
747.67
477.14
-7.2%
-14.9%
894.48
837.21
789.26
718.62
438.02
-10.6%
-21.1%
862.68
791.36
735.07
691.02
403.92
-13.7%
-26.5%
832.39
749.06
687.03
664.78
374.07
-16.8%
-31.3%
803.54
710.01
644.27
639.83
347.83
-19.6%
-35.6%
776.05
673.90
606.06
616.08
324.67
-22.4%
-39.4%
F520_Bond
13
III. What are the Characteristics of Bond Prices that a good
measure of interest rate risk will include:
1. Bond prices move inversely to bond yields
F520_Bond
2. Holding the coupon rate constant, for a given change in
market yields, percentage changes in bond prices are
greater the longer the term to maturity.
3. The percentage price changes described in Theorem 2
increase at a decreasing rate as N increases
14
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4. Holding N constant and starting from the same market
yield, equal yield changes up or down do not results in
equal percentage price changes. A decrease in yield
increases prices more than an equal increase in yield
decreases prices. In more formal terms, price changes
are asymmetric with respect to changes in yield.
15
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5. Holding N constant and starting from the same market
yield, the higher the coupon rate, the smaller the
percentage change in price for a given change in yield.
16
F520_Bond
17
Conclusions:
Long-term securities and securities with lower coupon
payments have higher price risk and lower
reinvestment risk than short term securities and
securities with higher coupon payments.
The best measure of interest rate risk also accounts for
convextity (the curvature in the bond pricing).
IV. Duration is our basic measure of interest rate risk.
Duration controls for
 Maturity  D 
 Level of coupon payments  D 
 Frequency of Coupon payments  D 
 Level of interest rate  D 
 Convexity of bond prices  D ?
(when using most precise measure)
F520_Bond
18
Macaulay Duration (D) is the weighted average term to
maturity of the components of a bond’s cash flow, in which
the time of receipt of each payment is weighted by the
present value of that payment.
T
D
 PV t * t
t 1
T
 PV
t 1
T

 PV
t 1
t
*t
P
t
Modified Duration (MD) is the an indication of the
proportional sensitivity of the price of an asset/liability to
changes in the market rate of interest.
Modified duration = Macaulay Duration / (1+y)
MD = D / (1+y)
Modified duration is the approximate percentage price
change for a 100 basis point change in yields.
Modified Duration measures the sensitivity of our
portfolio/balance sheet/assets to changes in market interest
rates.
http://faculty.darden.virginia.edu/conroyb/Valuation/Val20
02/F-1238.pdf
F520_Bond
V.
A.
19
Calculation of Modified Duration
You can calculate modified duration for any fixed
income security using this general formula:
Bond Pr ice 
C    C    C    ...  C     C  
1 y  1 y  1 y 
1 y  1 y 
T
1
3
2
1
2
n
3
t
n
t 1
1 P
*
P y
Use _ First _ Derivative _ Bond _ price
MD  
P

y
t * C  
1 y  1 y 
1
1
MD   * 
t* C  

P
1 y  1 y 
T
1
t
t
t 1
T
t
t
t 1
drop _ double _ negative _& _ rearrange
t * C  
1 y 
T
t
MD 
t
t 1
1
1 y 
P
t* C  

 PV * t
1 y 
D

P
 PV
T
T
t
t 1
T
t
t
t 1
MD 
t 1
t
1
1 y 
D
where
Ct = Cash flow received on the security at end of
period t
T = The last period in which the cash flow is
received
t
F520_Bond
20
Basic Duration Example (Annual payments):
What is the duration of a 10% bond that pays interest
annually and has a yield to maturity of 7 percent? Assume
that as of today the bond has three years left to maturity.
What is the current value of this bond?
Precise Method
PV(CF(t)) @
t
CF(t)
7%
t*PV
1
100
$93.46
93.46
2
100
$87.34
174.69
3
1100
$897.93
2,693.78
Totals $ 1,078.73
2,961.93
Macaulay Duration = 2,961.90 / 1,078.72 = 2.7458
Modified Duration = 2.7458/(1+.07) = 2.5662
Using Duration:
%  Price = dP   D *
P
r
  MD * r
1 r
Use the bond information above to determine the
percentage change in price if interest rates increase by
50 basis points? Also show the dollar value change in
price.
dP
0.0050
 2.7458 *
 2.5662 * 0.0050  0.012831  1.2831%
P
1  .07
-1.2831% * $1,078.72 = -$13.84
If rates rise by 50 basis points (.5%) the price will fall
by approximately $13.84.
F520_Bond
21
C. Calculating Duration using the approximation method
MD 
V V
2 *V * y


0
where
MD = Modified Duration = Macaulay Duration/(1+r)
y is the change in rate (on an annual basis) used
to calculate the new prices
V+ is the new price when the yield of the bond is
increased by y.
V- is the new price when the yield of the bond is
decreased by y.
V0 is the current price of the bond.
Approximation Method:
N
I
V0
3
7
V+
3
7.1
V-
3
6.9
MD 
PV
cpt=
1078.73
cpt=
1075.97
cpt=
1081.50
PMT
100
FV
1000
100
1000
100
1000
V V  1081.50  1075.97  2.56
2 *V * y
2 *1078.73 * .001


0
V is used to symbolize approximation formulas and P the
more precise methodology.
F520_Bond
B.
22
Economic Meaning of Duration
Duration is the Interest-Elasticity or sensitivity of a
security’s price to small changes in interest rates.

dP
r
 D *
 MD * r
P
1 r
= percentage change in price
 For small changes, bond prices move in an
inversely proportional fashion according to size of
D.
 If interest (coupon) payments are semi-annual,
rather than annual, then

dP
r
 D *
 MD * r
r
P
1
2
= percentage change in price
 If there are n payments per year, then

dP
r
 D *
 MD * r
r
P
1
n
= percentage change in price
Definitions:
D
= Macaulay Duration
MD = Modified Macaulay Duration
r
= discount rate (or YTM)
r
= change in discount rate (as a decimal)
F520_Bond
23
Example 2: (Semi-annual payments):
What is the duration of a 10% bond that pays interest semiannually and has a yield to maturity of 7 percent? Assume
that as of today the bond has three years left to maturity.
What is the current value of this bond?
Precise Method:
PV (CFt@7%
annual, 3.5 semi)
CFt
t*PV
50
48.31
48.31
50
46.68
93.35
50
45.10
135.29
50
43.57
174.29
50
42.10
210.49
1050
854.18
5,125.05
Totals
1079.94
5,786.78
Macaulay Duration = 5,786.78 / 1,079.94 = 5.3584
= 5.3584/2 = 2.68 years
Modified Duration = 2.68 / 1.035 = 2.59
t
1
2
3
4
5
6
Approximation Method:
N
I
PV
PMT
FV
V0
3*2=
7 / 2=
Cpt=
50
1000
6
3.5
1079.93
V+
3*2=
7.1 / 2= Cpt=
50
1000
6
3.55
1077.14
V3*2=
6.9 / 2= Cpt=
50
1000
6
3.45
1082.73

1082.73  1077.14
MD  V V 
 2.58 is the modified duration
2 *V * y
2 *1079.93 * .001


0
F520_Bond
24
Based on Macaulay Duration of 2.68, what is the
expected value of a bond if interest rates go up 50
basis points (.50%)? Remember, this bond has semiannual payments.
dP
r
 D *
  MD * r
r
P
1
2
dP
0.0050

 2.68 *
 .012947  1.2947%
.07
P
1
2
Price = Price* dP
= $1079.94 * -.012947
P
%  Price =
= $-13.98
New Price = P0 + Price =
$1079.94 + $-13.98 = $1065.96
F520_Bond
25
Calculating Exact Duration when the quote date is not on
the coupon date. Today is February 1, 2013
Maturity
11/15/2016
Today is
2/1/2013
Coupon
Bid
Ask
Chg
Ask Yld
7.500
125.8203
125.8516
-0.0156
0.570
11/15/12 5/15/13
11/15/2016
|
|
|
0
1
p
2
p
|
|
|
|
|
|
6 payments
p
Coupon % / 2
fv
Maturity Value (%)
F520_Bond
26
Wall Street Journal Quote 02/01/13: (Understanding Duration)
Maturity
Rate
Bid
Ask
Chg
Ask Yld
11/15/2016
7.5
125.8203
125.8516
-0.0156
0.57
Maurity
11/15/2016
Par 1000
Today
2/1/2013
Last Coupon Date
11/15/2012
78 days ago
Col. for
Calculate Exact Duration:
convexity only
#
t
CF
PV(CF)
PV*t
PV*t*(t+1)
11/15/2012
0
5/15/2013
1
37.5
$37.39
37.39
74.79
11/15/2013
2
37.5
$37.29
74.57
223.72
5/15/2014
3
37.5
$37.18
111.54
446.17
11/15/2014
4
37.5
$37.08
148.30
741.51
5/15/2015
5
37.5
$36.97
184.85
1,109.10
11/15/2015
6
37.5
$36.87
221.19
1,548.33
5/15/2016
7
37.5
$36.76
257.32
2,058.58
11/15/2016
8
1037.5 $1,014.15
8,113.16
73,018.48
$1,273.68
9,148.34
79,220.69
t
is semi-annual periods
# Used the 6 month rate (YTM/2)
7.18 Semi-annual periods
Duration as of last coupon date is
3.59 Years
Adjusting Duration for today.
Subtract last coupon from today
0.21 78/365 = (02/01/13 less 11/15/12)/365
Exact Duration (as of Trading Date)is
3.38 Years (Macaulay Duration)
Excel calculation
3.38 Years (Macaulay Duration)
Total Convexity Calculation
37.67 semi-annual periods
9.42 yearly convexity
F520_Bond
27
VI. Duration and Immunization
A.
Duration and Immunizing Future Payments
With immunization, the expected return on a
portfolio is protected from both reinvestment and
market value risk.
Instructions:
Use F9 to update this link.
Input Section:
Purchase an asset with duration = duration of liability
Annual Bond
Par
$
1,000
Today
9/16/2013 yrs
Maturity Date
9/16/2018 yrs
Coupon Rate
8.00%
Current Rate
8.00%
Interest Rate Change
8.00%
Define Liability
Target Payment
$
1,469
Our holding period is
9/16/2018
Preliminary Calculations:
Years in holding period
Years to maturity
Present Value of Bond
Duration of Bond
Number of Bonds Needed
to obtain target
$
Value of Bonds to Purchase
$
5.0
5.0
1,000
4.31 years
1.000 bonds
1,000 bonds
The model which assumes constant interst rates and is used for estimating the number of contracts needed is included
on this Excel worksheet just below the model which fully adjusts.
This model adjusts for any changes in interest rates as well as some minimal changes in the desired holding
period (<6), changes in the maturity date, and changes in the amount of funds to hedge.
|--------------- |--------------- |--------------- |--------------- |--------------- |--------------- |
Time Period
0
1
2
3
4
5
6
Cash Flows
Interest From Bond to be reinvested
$ 80.00 $ 80.00 $ 80.00 $ 80.00 $
80.00
Interest on Interest Reinvestment
6.40
13.31
20.78
28.84
Return of Principal on original bond
999.78
Total Cash Flows
If we desire to have
of duration matching.
$
1,469 at the end of 5 years, we can guarantee this amount through the use of
sum
$
$
$
$
400.00
69.33
999.78
1,469.10
F520_Bond
28
VII. Weaknesses of the Duration Model
 For large interest rate changes, the duration model
over-predicts the fall and underpredicts the rise in
prices; moreover, the price-yield relationship may
not be linear; but rather convex.
As convexity increases, the fall in values is smaller
as interest rates rise, and the gain in value is greater
as interest rates fall. This can be approximated
through the second derivative.
2
2
dP
1
1
  MD * R  * dP
* (R)
2
P
2 d R P
2
dP
1
  MD * R  CX (R)
P
2
Taylor Series Expansion
where
MD = Modified Duration =
D
1 R
2
CX = Total Convexity = dP * 1
d R P
Note, Bloomberg reports convexity as
CX = Total Convexity/100 = dP * 1 * 1
d R P 100
When reporting convexity, make sure you know the
notation requested.
2
2
2
F520_Bond
29
1* 2 * C
1 
2 *3*C
2  
3* 4 *C
3   ... 
1 y  1 y  1 y 
Total _ Convexity 
1 y 
2 
1* 2 * C

1 2 
1 y 
Total _ Convexity 
2 *3*C

2  2 
1 y 
 1  y  t
1  y 
Total _ Convexity 
2
Ci
t
t 1
2
t
3* 4 *C
 ... 
3  2 
1 y 

i
P
Approximation of Total Convexity:
CX  V 
 V   2 *V 0
V
0
*
y 
2
1 y 
n 

n * (n  1) * M
1 y 
n 
*P
P
n
1
n * (n  1) * C
n * (n  1) * C
1 y 
n  2 

n * (n  1) * M
1 y 
n  2 
F520_Bond
30
Example 2 continued: (see page 23 for details)
What is the duration of the semi-annual bond when
incorporating convexity?
Approximation Method:
N
I
PV
PMT
FV
V0
3*2=
7 / 2=
Cpt=
50
1000
6
3.5
1079.93
V+
3*2=
7.1 / 2= Cpt=
50
1000
6
3.55
1077.14
V3*2=
6.90/ 2= Cpt=
50
1000
6
3.45
1082.73
Duration with Total Convexity:

 2 *V
1082.73  1077.14  2 *1079.93
0.01
CX  V V


 9.2598
0
.
001080
1079
.
93
*
*
0.001
V y 


0
2
2
0
Rounding mistakes are common in estimating convexity. The greater the number of
decimal places used, the greater the accuracy of your results. While I used 2 for this
example, 6 decimal places would be more appropriate given that we are dividing by
(.001)2.
Expected Price Change assuming +50 basis points :
Using Total Convexity
dP
1
  MD * R  CX (R)
P
2
2
2
dP
0.0050 1
 2.68 *
 9.26 (.0050)
.07 2
P
1
2
dP
 .012947  .00011575  .012831  1.2831%
P
Price = Price* dP
= $1079.94 * -.012831
P
New Price = P0 + Price
= $1079.94 + $-13.86 = $1066.08
= $-13.86
F520_Bond
31
Convexity improves on our solution, especially for
larger price changes.
N
I
PV
PMT
FV
3*2
7/2
CPT=
.10*1000/2 1000
1079.93
=50
3*2
7.5/2
CPT=
.10*1000/2 1000
1066.06
=50
3*2
10/2
CPT=
.10*1000/2 1000
1000.00
=50
On page 24, we had predicted 1065.96 for 50 bp
increase(off 10 cents), with our new measure of convexity
we are only 2 cents off.
Expected Price Change assuming +300 bp:
dP
0.03
1
 2.68 *
 9.26 (.03)
.
07
P
2
2
1
2
dP
 .07768  .004167  .073513  7.3513%
P
w/ convexity $1079.94 * -.073513 = $-79.39
$1079.94 + $-79.39 = $1000.55
w/o convexity $1079.94 * -.07768 = $-83.90
$1079.94 + $-83.90 = $996.04 over-estimates decrease.
And this is with a very small convexity measure.
F520_Bond
32
 The duration measure, which we have been
calculating, assumes a flat yield curve.
The appropriate duration can be calculated by
discounting the coupons and principal value of the
bond by the discount rates or yields on appropriate
maturity zero coupon bonds. In an upward sloping
yield curve, the adjusted duration measure will be
smaller, since more discounted cash flows are
discounted at higher rates.
 Our models have assumed no default risk. An
adjustment can be made by calculating the expected
cash flow in the duration measure.
 The detailed duration calculation is only for fixed
rate securities, but can be adjusted. (The
approximation will work for other securities if
properly done.) Floating rate securities have a
duration equal to the time period between the now
and when the interest rate is readjusted.
Restructuring the balance sheet in order to reduce
duration gap is time consuming and can be
expensive.
 Immunization based on duration requires costly
continuous portfolio re-balancing to ensure that
the investment duration exactly matches the
investment horizon.
F520_Bond
5% coupon bond with 20 years to maturity (calculated in textbook reading).
33
F520_Bond
34
Cont (demonstration of textbook problem)
Approximation method:
Annual
SemiAnnual
Coupon
5%
2.50%
Yield
9%
4.50%
20
40
100
100
Maturity
Par
Value
V0 63.1968311594
V+ 62.5444842325
V- 63.8593439431
9.00%
9.10%
8.90%
10.40
annual
annual
annual
Modified Duration
160.8602 Approximate total CX
Without Convexity: 2% increase in rates
dP
  MD * R
P
dP
 10.4 * .02
P
dP
 .208  20.8%
P
With Total Convexity: 2% increase in rates
2
dP
1
  MD * R  CX (R)
P
2
2
dP
1
 10.4 * .0200  160.86 * (.0200)
P
2
dP
 .208  .0322  .1758  17.58%
P
F520_Bond
35
With Total Convexity: 2% decrease in rates
2
dP
1
  MD * R  CX (R)
P
2
2
dP
1
 10.4 * .0200  160.86 * (.0200)
P
2
dP
 .208  .0322  .2402  24.02%
P
Actual Change in Price
Annual
Coupon
Yield
Maturity
Par
Original Price at 9%
SemiAnnual
5%
2.50%
7.00%
3.50%
20
40
100
100
63.1968
Value
78.6449 Price with new yield
24.44% is the Percent Change in Price
new value
F520_Bond
36
Defining Table 18-3 convexity to be consistent with
the price adjustment example.
Approximation method:
Annual
SemiAnnual
Coupon
9%
4.50%
Yield
9%
4.50%
5
10
100
100
Maturity
Par
Value
V0
100.0000000000
V+
99.6053349239
V100.3966103380
9.00%
9.10%
8.90%
3.96
19.4526
annual
annual
annual
Modified Duration
Aproximate total convexity
F520_Bond
37
Without Convexity: 2% increase in rates
dP
  MD * R
P
dP
 3.96 * .02
P
dP
 .0792  7.92%
P
With Total Convexity: 2% increase in rates
2
dP
1
  MD * R  CX (R)
P
2
2
dP
1
 3.96 * .0200  19.4526 * (.0200)
P
2
dP
 .0792  .00389052  .07530948  7.53%
P
Actual Change in Price
Annual
Coupon
Yield
Maturity
Par
SemiAnnual
9%
4.50%
11.00%
5.50%
5
10
100
100
Value
92.4623741714 Price with new yield
-7.54% is the Percent Change in Price
With Total Convexity: 2% decrease in rates
2
dP
1
  MD * R  CX (R)
P
2
2
dP
1
 3.96 * .0200  19.4526 * (.0200)
P
2
dP
 .0792  .00389052  .08309052  8.31%
P
new value
F520_Bond
38
Actual Change in Price
Annual
Coupon
Yield
Maturity
Par
SemiAnnual
9%
4.50%
7.00%
3.50%
5
10
100
100
Value
108.3166053226 Price with new yield
8.32% is the Percent Change in Price
new value