copyright 2014 by diane s. docking1 duration & convexity
TRANSCRIPT
Copyright 2014 by Diane S. Docking 1
Duration & Convexity
Learning Objectives
Know how to the calculate duration of a security.
Know how to calculate the convexity of a security.
Understand the economic meaning of duration.
Copyright 2014 by Diane S. Docking 2
Duration
Duration allows for the comparison of securities of different coupons, maturities, etc.
Duration measures the weighted average life of an instrument. It equals the average time necessary to recover the initial cost. E.g..: A bond with 4 years until final maturity with a duration of 3.5 years indicates that
an investor would recover the initial cost of the bond in 3.5 years, on average, regardless of intervening interest rate changes.
Duration measures of the price sensitivity of a financial asset with fixed cash flows to interest rate changes.
3Copyright 2014 by Diane S.
Docking
Macaulay DurationUsing annual compounding, Macaulay duration (D) is:
where:CF = the interest and/or principal payment that occurs in period t,t = the time period in which the coupon and/or principal payment
occurs,i = the current market rate or current market yield on the security
N
tt
t
N
ttt
i
CFi
CFt
D
1
1
1
1
Numerator is:PV of Future CFs weighted by period of
receipt
Denominator is:PV of Future
CFs = Current Price
Duration is in
years or fraction of years
4Copyright 2014 by Diane S.
Docking
Copyright 2014 by Diane S. Docking 5
Duration For interest
rate increases, Duration overestimates the price decrease.
For interest rate decreases, Duration underestimates the price increase
Change in price predicted by durationyield
Price
Actual change in price
i0 i1
Copyright 2014 by Diane S. Docking 6
Example 1:Calculating Duration: 10-yr, 10% Coupon Bond; rm =10%
(1) (2) (3) (4) (5)
Year CF ($) PVCF @ rm% t x PVCF Duration
1 $100 $90.91 $90.91
2 $100 82.64 165.293 $100 75.13 225.394 $100 68.30 273.215 $100 62.09 310.466 $100 56.45 338.687 $100 51.32 359.218 $100 46.65 373.219 $100 42.41 381.6910 $100 38.55 385.5410 $1,000 385.54 3,855.43
Total 1,000.00$ 6,759.02$ 6.7590 yrs
< 10 years
This is current Price = P0
6,759.021,000.00
Copyright 2014 by Diane S. Docking 7
Example 2:Calculating Duration: 10-yr, 10% Coupon Bond; rm =20%
(1) (2) (3) (4) (5)Year CF ($) PVCF @ rm% t x PVCF Duration
1 $100 $83.33 $83.332 $100 $69.44 138.893 $100 $57.87 173.614 $100 $48.23 192.905 $100 $40.19 200.946 $100 $33.49 200.947 $100 $27.91 195.368 $100 $23.26 186.059 $100 $19.38 174.4310 $100 $16.15 161.5110 $1,000 $161.51 1,615.06
Total 580.75$ 3,323.01$ 5.7219 yrs
< 10 years
This is current Price = P0
Verify Price:FV = 1,000n = 10 yrs.Pmt = $100i = 20% PV = 580.75
3,323.01580.75
Copyright 2014 by Diane S. Docking 8
Example 3:Calculating Duration: 10-yr, 20% Coupon Bond; rm =10%
(1) (2) (3) (4) (5)Year CF ($) PVCF @ rm% t x PVCF Duration
1 $200 $181.82 $181.822 $200 $165.29 330.583 $200 $150.26 450.794 $200 $136.60 546.415 $200 $124.18 620.926 $200 $112.89 677.377 $200 $102.63 718.428 $200 $93.30 746.419 $200 $84.82 763.3810 $200 $77.11 771.0910 $1,000 $385.54 3,855.43
Total 1,614.46$ 9,662.61$ 5.9851 yrs
< 10 years
This is current Price = P0
Verify Price:FV = 1,000n = 10 yrs.Pmt = $200i = 10% PV = 1,614.46
9,662.611,614.46
Copyright 2014 by Diane S. Docking 9
Example 4:Calculating Duration: 5-yr, 10% Coupon Bond; rm =10%
(1) (2) (3) (4) (5)Year CF ($) PVCF @ rm% t x PVCF Duration
1 $100 $90.91 $90.912 $100 $82.64 165.293 $100 $75.13 225.394 $100 $68.30 273.215 $100 $62.09 310.465 $1,000 $620.92 3,104.61
Total 1,000.00$ 4,169.87$ 4.1699 yrs
< 5 years
This is current Price = P0
4,169.871,000.00
Copyright 2014 by Diane S. Docking 10
Example 5:Calculating Duration: 5-yr, Zero-Coupon Bond; rm =10%
(1) (2) (3) (4) (5)
Year CF ($) PVCF @ rm% t x PVCF Duration
1 $0 $0.00 $0.002 $0 $0.00 0.003 $0 $0.00 0.004 $0 $0.00 0.005 $0 $0.00 0.005 $1,000 $620.92 3,104.61
Total 620.92$ 3,104.61$ 5.0000 yrs
This is current Price = P0 D=maturity
Verify Price:FV = 1,000n = 5yrs.Pmt = 0i = 10% PV = 620.92
3,104.61620.92
Copyright 2014 by Diane S. Docking 11
Example 6:Calculating Duration: 10-yr, Zero-Coupon Bond; rm =20%
This is current Price = P0
(1) (2) (3) (4) (5)Year CF ($) PVCF @ rm% t x PVCF Duration
1 $0 $0.00 $0.002 $0 $0.00 0.003 $0 $0.00 0.004 $0 $0.00 0.005 $0 $0.00 0.006 $0 $0.00 0.007 $0 $0.00 0.008 $0 $0.00 0.009 $0 $0.00 0.0010 $0 $0.00 0.0010 $1,000 $161.51 1,615.06
Total $161.51 1,615.06$ 10.0000 yrs
1615.06161.506
Verify Price:FV = 1,000n = 10 yrs.Pmt = 0i = 20% PV = 161.51
D=maturity
Copyright 2014 by Diane S. Docking 12
Asset Properties and Duration
Maturity Coupon Rm Duration
Ex. 1 10 yrs. 10% 10% 6.759 yrs.
Ex. 2 10 yrs. 10% 20% 5.722 yrs.
Ex. 3 10 yrs. 20% 10% 5.985 yrs.
Ex. 4 5 yrs. 10% 10% 4.170 yrs.
Ex. 5 5 yrs. 0% 10% 5.000 yrs.
Ex. 6 10 yrs. 0% 20% 10.000 yrs.
Copyright 2014 by Diane S. Docking 13
Asset Properties and Duration
1. For bonds with the same coupon rate and the same yield, the bond with the longer maturity will have the________ duration. (Ex.1 vs. Ex.4)
2. For bonds with the same maturity and the same yield, the bond with the lower coupon rate will have the ________ duration. (Ex. 1 vs. Ex. 3; Ex. 2 & 6; Ex. 4 vs. Ex. 5)
3. When interest rates rise, the duration of a coupon bond ________. (Ex. 1 vs. Ex. 2)
4. The lower the initial yield, the _______ the duration for a given bond. (Ex. 1 vs. Ex. 2)
Copyright 2014 by Diane S. Docking 14
Key facts about duration
1. The longer a bond’s duration, the its sensitivity to interest rate changes
2. The duration of a______________ bond = bond’s term to maturity
3. The Macaulay duration of any coupon bond is always ______ than the bond’s term to maturity
4. Duration is the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each.
5. The more frequently a security pays interest or principal, the its duration.
Duration
Copyright 2014 by Diane S. Docking 15
Macaulay Duration Duration – annual interest pmts:
Duration – semi-annual interest pmts:
2
;
21
21
1
1yrsdouble
yrsN
tt
t
N
ttt
yrsdouble
DD
i
CF
i
CFt
D
N
tt
t
N
ttt
yrs
i
CFi
CFt
D
1
1
1
1
Copyright 2014 by Diane S. Docking 16
Modified Duration
Modified duration – annual interest pmts:
Modified duration – semi-annual interest pmts:
212
1
21yrs.)(in mod i
D
i
DD yrsyrsdouble
iD
D yrs
1yrs.)(in mod
Copyright 2014 by Diane S. Docking 17
Duration and Price Sensitivity So, an estimate of the percentage change in the
price of a financial asset is:
So, an estimate of $ price change in the price of a financial asset is:
or
00 1 i
iD
P
P
001P
i
iDP
Copyright 2014 by Diane S. Docking 18
Duration estimates of price change
If annual interest payments:
If semi-annual interest payments:
001P
i
iDP yrs
00
00
21or
212P
ii
DPiiD
P yrsyrsdouble
Copyright 2014 by Diane S. Docking 19
Duration For interest rate
increases, Duration overestimates the price decrease.
For interest rate decreases, Duration underestimates the price increase
Change in price predicted by durationyield
Price
Actual change in price
Copyright 2014 by Diane S. Docking 20
Problem 1: Duration – Annual Payments
Assume today is September 1, 2XX1. Midwest Bank owns the following security:G.E. Corporate bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid annually.
1. What is the security’s YTM?2. If market interest rates increase 1%, what will be the security’s
price?3. What is the security’s Macaulay Duration? Modified Duration?4. What is the dollar change in price for a 1% increase in market rates
as estimated by Duration?5. Compare the actual price change to the estimated price change.
Problem 1 Solution: Duration - Annual Payments
1. YTM: PV = 124 3/32 = 124.09375 = $1,240.94
FV = 1,000
Pmt = 107.50
n = 5 yrs.
therefore Rm = YTM =___________
2. Rm increases 1% to 6.1603%:
FV = 1,000
Rm = 6.1603%
Pmt = 107.50
n = 5 yrs.
therefore PV = ___________
Copyright 2014 by Diane S. Docking 21
Makes sense: rates increase, price decreases
Copyright 2014 by Diane S. Docking 22
Problem 1 Solution: Duration - Annual Payments
Calculating Duration on a Bond - Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.1603%
(1) (2) (3) (4)Year CF ($) PVCF @ rm% t x PVCF Duration
1 $107.50 $102.22 $102.222 $107.50 $97.21 $194.423 $107.50 $92.44 $277.324 $107.50 $87.90 $351.615 $107.50 $83.59 $417.955 $1,000 $777.57 $3,887.86
Total $1,240.94 $5,231.38
Macaulay 4.2157 years
Modified 4.0088 years
5,231.381,240.94 4.2157
1.051603
Problem 1 Solution: Duration - Annual Payments (cont.)
4. From Duration:
Macaulay
or
Modified
Copyright 2014 by Diane S. Docking 23
$49.751,240.941.051603
.012157.4ΔP duration todue
$49.751,240.9401.0088.4ΔP duration todue
Problem 1 Solution: Duration - Annual Payments (cont.)
5. Original Price when (Rm = 5.1603%) $1,240.94
Duration est. of price change < 49.75 >
Duration est. of new Price $1,191.19
vs.
$1,192.49
Copyright 2014 by Diane S. Docking 24
Diff = <$1.30>
Duration overestimated
price decrease.
Copyright 2014 by Diane S. Docking 25
Problem 2: Duration – Semi-annual payments
Assume today is September 1, 2XX1. Midwest Bank owns the following security:U.S. Treasury bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid semi-annually.
1. What is the security’s YTM?2. If market interest rates increase 1%, what will be the security’s
price?3. What is the security’s Macaulay Duration? Modified Duration?4. What is the dollar change in price for a 1% increase in market rates
as estimated by Duration?5. Compare the actual price change to the estimated price change.
Problem 2 Solution: Duration – Semi-annual Payments
1. YTM: PV = 124 3/32 = 124.09375 = $1,240.94
FV = 1,000
Pmt = 107.50/2 = 53.75
n = 5 yrs. x 2 = 10
therefore Rm = YTM = 2.6069% semi-annual; 5.2137% annual
2. Rm increases 1% to 6.2137% annual/ 2 = 3.1069%
FV = 1,000
Rm = 3.1069%
Pmt = 53.75
n = 10
therefore PV = $1,192.42
Copyright 2014 by Diane S. Docking 26
Copyright 2014 by Diane S. Docking 27
Problem 2 Solution: Duration – Semi-annual Payments (cont.)
Calculating Duration on a Bond - Semi-Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.2137% 0.026069
(1) (2) (3) (4)
Year CF ($) PVCF @ rm%/2 t x PVCF Duration
1 $53.75 $52.38 $52.382 $53.75 $51.05 $102.113 $53.75 $49.76 $149.274 $53.75 $48.49 $193.975 $53.75 $47.26 $236.306 $53.75 $46.06 $276.367 $53.75 $44.89 $314.238 $53.75 $43.75 $349.999 $53.75 $42.64 $383.7410 $53.75 $41.55 $415.5410 $1,000 $773.10 $7,731.01
Total $1,240.94 $10,204.90
8.2235 dbl-yrs
Macaulay 4.1118 yrs.
Modified 4.0073 yrs.
10,204.901240.94
8.22352
4.1118(1.026069)
Problem 2 Solution: Duration – Semi-annual Payments (cont.)
4. From Duration (in years):
Macaulay
or
Modified
Copyright 2014 by Diane S. Docking 28
$49.731,240.941.026069
.011118.4ΔP duration todue
$49.731,240.9401.0073.4ΔP duration todue
Problem 2 Solution: Duration – Semi-annual Payments (cont.)
5. Original Price when (Rm = 5.21%) $1,240.94
Duration est. of price change < 49.73 >
Duration est. of new Price $1,191.21
vs.
$1,192.42
Copyright 2014 by Diane S. Docking 29
Diff = <$1.21>
Duration overestimated
price decrease.
Copyright 2014 by Diane S. Docking 30
Duration
For interest rate increases, Duration overestimates the price decrease.
For interest rate decreases, Duration underestimates the price increase
Change in price predicted by durationyield
Price
Actual change in price
ConvexityTaking convexity
into account: For interest rate
increases, the actual reduction in price will be less than that predicted by duration
For interest rate decreases, the actual increase in price will be more than that predicted by duration
Change in price predicted by durationyield
Price
Actual change in price
Convexity adds amount back
31Copyright 2014 by Diane S. Docking
Copyright 2014 by Diane S. Docking 33
Convexity Convexity – annual interest pmts:
Convexity – semi-annual interest pmts:
2
22
2 2)1(
'
2
iyrsdouble
yrs
DCX
CX
2
'
1 i
DCX yrs
Copyright 2014 by Diane S. Docking 34
Example 1A:Calculating Macaulay Convexity: 10-yr 10% Coupon Bond; rm =10%Calculating Macaulay Convexity on a Bond - Annual Payments
Face = $1,000Maturity = 10 yearsCoupon Rate = 10.00%Market Rate = 10.00%
(1) (2) (3) (5) (6) (7)
Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity1 $100 $90.91 2 $181.822 $100 $82.64 6 $495.873 $100 $75.13 12 $901.584 $100 $68.30 20 $1,366.035 $100 $62.09 30 $1,862.766 $100 $56.45 42 $2,370.797 $100 $51.32 56 $2,873.698 $100 $46.65 72 $3,358.859 $100 $42.41 90 $3,816.8810 $100 $38.55 110 $4,240.9810 $1,000 $385.54 110 $42,409.76
Total 1,000.00$ $63,879.00
63.8790 52.7926 "yrs."
63.8790(1.10)2
Copyright 2014 by Diane S. Docking 35
Example 2A:Calculating Macaulay Convexity: 10-yr 10% Coupon Bond; rm =20%
(1) (2) (3) (5) (6) (7)
Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity
1 $100 $83.33 2 $166.672 $100 $69.44 6 $416.673 $100 $57.87 12 $694.444 $100 $48.23 20 $964.515 $100 $40.19 30 $1,205.636 $100 $33.49 42 $1,406.577 $100 $27.91 56 $1,562.868 $100 $23.26 72 $1,674.499 $100 $19.38 90 $1,744.2610 $100 $16.15 110 $1,776.5610 $1,000 $161.51 110 $17,765.61
Total 580.75$ $29,378.27
50.5865 35.1295 "yrs."
50.5865(1.20)2
Copyright 2014 by Diane S. Docking 36
Example 3A:Calculating Macaulay Convexity: 10-yr 20% Coupon Bond; rm =10%
(1) (2) (3) (5) (6) (7)
Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity
1 $200 $181.82 2 $363.642 $200 $165.29 6 $991.743 $200 $150.26 12 $1,803.164 $200 $136.60 20 $2,732.055 $200 $124.18 30 $3,725.536 $200 $112.89 42 $4,741.587 $200 $102.63 56 $5,747.378 $200 $93.30 72 $6,717.719 $200 $84.82 90 $7,633.7610 $200 $77.11 110 $8,481.9510 $1,000 $385.54 110 $42,409.76
Total 1,614.46$ $85,348.24
52.8650 43.6901 "yrs."
52.8650(1.10)2
Copyright 2014 by Diane S. Docking 37
Example 4A:Calculating Macaulay Convexity: 5-yr 10% Coupon Bond; rm =10%
(1) (2) (3) (5) (6) (7)
Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity
1 $100 $90.91 2 $181.822 $100 $82.64 6 $495.873 $100 $75.13 12 $901.584 $100 $68.30 20 $1,366.035 $100 $62.09 30 $1,862.765 $1,000 $620.92 30 $18,627.64
Total 1,000.00$ $23,435.69
23.4357 19.3683 "yrs."
23.4357(1.10)2
Copyright 2014 by Diane S. Docking 38
Example 5A:Calculating Macaulay Convexity: 5-yr Zero-Coupon Bond; rm =10%(assume annual payments)
(1) (2) (3) (5) (6) (7)
Year CF ($) PVCF @ rm% (t2+t) (t2+t)PVCF D' Convexity
1 $0 $0.00 2 $0.002 $0 $0.00 6 $0.003 $0 $0.00 12 $0.004 $0 $0.00 20 $0.005 $0 $0.00 30 $0.005 $1,000 $620.92 30 $18,627.64
Total 620.92$ $18,627.64
30.0000 24.7934 "yrs."
18,627.64620.92
30.0000(1.10)2
Copyright 2014 by Diane S. Docking 39
Asset Properties and Convexity
1. For bonds with the same coupon rate and the same yield,
the the bond with the longer maturity will have the greater convexity. (Ex.1A vs. Ex.4A)
2. For bonds with the same maturity and the same yield,the the bond with the lower coupon rate will have the greater convexity. (Ex. 1A vs. Ex. 3A)
3. When interest rates rise,the convexity of a coupon bond falls. (Ex. 1A vs. Ex. 2A)
4. The greater the initial yield,the less the convexity for a given bond. (Ex. 1A vs. Ex. 2A)
Copyright 2014 by Diane S. Docking 40
Key facts about convexity
1. Convexity increases with bond maturity
2. Given the same maturity, coupon bonds are _____convex than zero-coupon bonds.
Convexity
Copyright 2014 by Diane S. Docking 41
Price change explained by Convexity: If annual interest payments:
If semi-annual interest payments:
02
21 PiCXP yrsconvexitytodue
0
2
02
221or2
1 PiCXPiCXP yrsdoubleyrsCXtodue
Copyright 2014 by Diane S. Docking 42
Problem 1A: Duration and Convexity – Annual Payments
Assume today is September 1, 2XX1. Midwest Bank owns the following security:G.E. Corporate bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid annually.
1. What is the security’s YTM?2. If market interest rates increase 1%, what will be the security’s
price?3. What is the security’s Macaulay Duration? Modified Duration?4. What is the security’s Convexity?5. What is the dollar change in price for a 1% increase in
market rates:a) From Duration?b) From Convexity?
6. Compare the actual price change to the estimated price change.
Copyright 2014 by Diane S. Docking 43
Problem 1A Solution: Duration & Convexity Annual Payments
3. 4.
Calculating Macaulay Duration & Convexity on a Bond - Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.1603%
(1) (2) (3) (4) (5) (6) (7)
Year CF ($) PVCF @ rm% t x PVCF (t2+t) (t2+t)PVCF D' Duration Convexity
1 $107.50 $102.22 $102.22 2 $204.452 $107.50 $97.21 $194.42 6 $583.253 $107.50 $92.44 $277.32 12 $1,109.264 $107.50 $87.90 $351.61 20 $1,758.055 $107.50 $83.59 $417.95 30 $2,507.675 $1,000 $777.57 $3,887.86 30 $23,327.18
Total $1,240.94 $5,231.38 $29,489.86
23.7642 4.2157 years 21.4892 "years"
5,231.381,240.94
29,489.861,240.94
23.7642(1.051603)2
Problem 1A Solution: Duration & Convexity - Annual Payments (cont.)
5. a) From Macaulay Duration:
6. b) From Convexity:
Copyright 2014 by Diane S. Docking 44
33.1$1,240.9401.4892.2121ΔP 2
convexity todue
$49.751,240.941.051603
.012157.4ΔP duration todue
Problem 1A Solution: Duration & Convexity - Annual Payments (cont.)
6. Original Price when (Rm = 5.1603%) $1,240.94
Duration est. of price change < 49.75 >
Duration est. of new Price $1,191.19 vs. $1,192.49
Convexity est. of price change
not explained by duration + 1.33
Duration + Convexity est. of new Price $1,192.52
vs.
$1,192.49
Copyright 2014 by Diane S. Docking 45
Diff = <$1.30>
Duration overestimated
price decrease.
Diff = <$.03>
Copyright 2014 by Diane S. Docking 46
Problem 2A: Duration and Convexity – Semi-annual payments
Assume today is September 1, 2XX1. Midwest Bank owns the following security:U.S. Treasury bond, 10 ¾%, September 2XX6. Today’s closing price is 124 3/32. Assume interest is paid semi-annually.
1. What is the security’s YTM?2. If market interest rates increase 1%, what will be the security’s price?3. What is the security’s Duration?4. What is the security’s Convexity?5. What is the dollar change in price for a 1% increase in market rates:
a) From Duration?b) From Convexity?
6. Compare the actual price change to the estimated price change.
Copyright 2014 by Diane S. Docking 47
Problem 2A Solution: Duration & Convexity Semi-annual Payments
3. 4.
Calculating Macaulay Duration & Convexity on a Bond - Semi-Annual Payments Face = $1,000Maturity = 5 yearsCoupon Rate = 10.75%Market Rate = 5.2137%
(1) (2) (3) (4) (5) (6) (7)
Year CF ($) PVCF @ rm%/2 t x PVCF (t2+t) (t2+t)PVCF D' Duration Convexity1 $53.75 $52.38 $52.38 2 $104.772 $53.75 $51.05 $102.11 6 $306.323 $53.75 $49.76 $149.27 12 $597.084 $53.75 $48.49 $193.97 20 $969.855 $53.75 $47.26 $236.30 30 $1,417.816 $53.75 $46.06 $276.36 42 $1,934.507 $53.75 $44.89 $314.23 56 $2,513.818 $53.75 $43.75 $349.99 72 $3,149.929 $53.75 $42.64 $383.74 90 $3,837.3710 $53.75 $41.55 $415.54 110 $4,570.9610 $1,000 $773.10 $7,731.01 110 $85,041.15
Total $1,240.94 $10,204.90 $104,443.54
84.1650 8.2235 dbl-yrs 79.9427 "dbl-yrs"
4.1118 yrs. 19.9857 "yrs"
10,204.901240.94
104,443.541,240.94
84.1650
(1.026069)2
8.22352
79.9427
22
Problem 2A Solution: Duration & Convexity – Semi-annual Payments (cont.)
5. a) From Macaulay Duration:
6. b) From Convexity (in years):
Copyright 2014 by Diane S. Docking 48
24.1$1,240.9401.9857.1921ΔP 2
convexity todue
$49.731,240.941.026069
.011118.4ΔP duration todue
Problem 2A Solution: Duration & Convexity – Semi-annual Payments (cont.)
6. Original Price when (Rm = 5.21%) $1,240.94
Duration est. of price change < 49.73 >
Duration est. of new Price $1,191.21 vs. $1,192.42
Convexity est. of price change
not explained by duration + 1.24
Duration + Convexity est. of new Price $1,192.45
vs.
$1,192.42
Copyright 2014 by Diane S. Docking 49
Diff = <$1.21>
Duration overestimated
price decrease.
Diff = <$.03>